Parrino

19.3 A Better Financial Planning Model 617

EXHIBIT 19.3 Blackwell Sales: Current and Pro Forma Income Statements
($ thousands)

The pro forma income statement for Blackwell Sales assumes that the income statement

items vary directly with sales.

Current Pro Forma Assumptions

Net sales $2,000 $2,600 Sales increase: 30%
Costs 1,700 2,210 Total costs ϭ 85% of sales
Taxable income $ 300 $ 390
Taxes (34.1%) Dividend policy: 33.5% of net income
Net income 102 133
$ 198 $ 257
Dividends
Addition to retained earnings $ 66 $ 86
$ 132 $ 171

These amounts relate to two ratios we will use in this chapter: the dividend payout ratio dividend payout ratio
and the retention ratio, or plowback ratio. Their formulas and calculations for Blackwell are the proportion of net income
as follows: paid out (distributed) as
dividends
Dividend payout ratio ϭ Cash dividends (19.2)
Net income retention (plowback) ratio
the proportion of net income
ϭ $86,000 ϭ 0.335, or 33.5% retained in the firm
$257,000

Retention 1plowback2 ratio ϭ Addition to retained earnings (19.3)
Net income

ϭ $171,000 ϭ 0.665, or 66.5%
$257,000

The dividend payout ratio shows the percentage of the firm’s earnings paid out as
cash dividends to stockholders. Similarly, the retention ratio tells what percentage of the
firm’s earnings is retained in the firm. Generally speaking, smaller, fast-growing compa-
nies plow back all or most of their earnings into the business; whereas more established
firms with slower growth rates and larger cash flows distribute more of their profits to
stockholders. Notice that the retention ratio plus the payout ratio equals 1.000 (0.335 ϩ
0.665 ϭ 1.000). This must be true, because net income is paid out as a cash dividend and/
or retained in the firm.

The Balance Sheet

To generate a pro forma balance sheet, we start with the current balance sheet, as shown in
Exhibit 19.4. For each account that varies directly with sales, the exhibit gives the relation as a
percent of sales for the current year. Notice that these percentages differ among the accounts.
How do we determine which accounts vary with sales, and how do we know the relevant per-
centages? Fortunately, the process is straightforward.

Historical Trends

We begin by looking at balance sheet accounts that might vary with sales. To do this we
gather four or five years of historical accounting data and express those data as a percent of
sales. A trend may be self-evident, or some simple trend lines can be fit to the data to identify
trends. In either case, this process allows the financial manager to decide which financial
accounts can safely be estimated as a percent of sales and which must be forecast using other
information.

The following table shows several years of historical data from Blackwell’s balance sheet ac-

618 CHAPTER 19 I Financial Planning and Forecasting

EXHIBIT 19.4 Blackwell Sales: Current Balance Sheet ($ thousands)

In this balance sheet for Blackwell Sales, many accounts vary directly with sales. The projected percent of sales is shown for
each of these accounts. The accounts labeled “n/a” do not change proportionately with sales.

Assets Liabilities and Stockholders’ Equity

Current Projected Current Projected
% of sales % of sales

Current assets $ 100 5% Current liabilities $ 60 4%
Cash 120 6 Accounts payable 140 n/a
Accounts receivable 140 7 Notes payable n/a
Inventory 18% Total $ 200 n/a
Total $ 360 $ 200
32 Long-term debt n/a
Net fixed assets 640 50% Owner’s equity $ 10 n/a
Total assets $ 1,000 590 n/a
Common stock
Retained earnings $ 600 50%

Total equity $ 1,000
Total liabilities and
stockholders’ equity

for assigning a percent of sales figure to each balance sheet account. We look first at the working
capital accounts: cash, accounts receivable, inventory, and accounts payable.

Percent of Sales

Forecast

2008 2009 2010 2011 2012

Cash 5% 5% 4% 5% 5%
Accounts receivable 10 9 9 9 6
Inventory 7 8 76 7
Accounts payable 4 4 43 4
Net fixed assets 30 32 34 32 32

Working Capital Accounts

The key working capital accounts tend to vary directly with sales. Take inventory as an ex-
ample. As sales increase, the firm needs to increase the level of inventory proportionately to
support the higher sales level. The historical data in the table support this view. Inventory
levels have been a relatively constant percentage of sales, varying from 6 to 8 percent. In se-
lecting the appropriate percentage for the planning process, management must consider what
the firm’s optimal inventory ratio is. On the one hand, as discussed in Chapter 14, manage-
ment would like to minimize inventory levels, because inventory is expensive to finance. On
the other, if inventory levels become too low, the firm may lose sales because of stockouts,
which occur when an order comes in and there is no product to sell. Let’s assume that Black-
well’s management determines that 7 percent of projected sales is the right inventory-to-sales
ratio for the firm.

The ratio of accounts receivable to sales has been 9 percent for the last several years. How-
ever, firms with similar credit policies operate with a receivables-to-sales ratio of 6 percent. As
sales have increased, Blackwell has provided proportionately more credit to its customers. To
improve the firm’s performance to industry standards, management decides to collect receiv-
ables more aggressively and targets the ratio at 6 percent. The firm has targeted the cash ac-
counts at 5 percent of sales. Management believes that a 5 percent cash ratio provides adequate
liquidity to fund ongoing operations and for unexpected emergencies, yet does not tie up an
excessive amount of cash in low-yielding assets.

On the liability side, the firm’s historical data show that accounts payable vary with sales.

19.3 A Better Financial Planning Model 619

place with its suppliers. Management is satisfied with the firm’s vendor relationships and the
payment schedule for vendors. Hence, accounts payable are forecast to be 4 percent of sales.

Fixed Assets

We assume that the company’s net fixed assets vary with the level of sales. An examination of
historical data shown earlier confirms that this is a reasonable assumption. Blackwell’s man-
agement decides to use the firm’s four-year historical average—32 percent—for the projected
ratio of fixed assets to sales. Thus, for every $100 in sales, the firm needs $32 of fixed assets to
support the sales.

We should note that the relation of fixed assets to sales may not always hold. The reason is
that fixed assets may vary directly with sales only when a firm is operating at full capacity and
fixed assets can be added in small increments. For example, if a firm has a large amount of
unused capacity, its sales could increase by 20 percent without adding any new fixed assets. We
will come back to this issue in more detail later in the chapter. For Blackwell, the data support
the proportional fixed assets-to-sales ratio, so we can proceed on that basis.

As a final comment, notice in Exhibit 19.4 on the asset side of the balance sheet that the
total percent of sales for asset items adds up to 50 percent. This means that total assets are 50
percent of sales. The ratio of total assets to sales is called the capital intensity ratio and is calcu-
lated for Blackwell Sales as follows:

Capital intensity ratio ϭ Total assets (19.4)
Net sales

ϭ $1 million ϭ 0.5, or 50%
$2 million

The capital intensity ratio, which is the inverse of the total asset turnover ratio discussed in
Chapter 4 (Equation 4.7), tells us something about the amount of assets the firm needs to generate
$1 in sales. The higher the ratio, the more capital the firm needs to generate sales—that is, the more
capital intensive the firm. Firms that are highly capital intensive tend to be more risky than similar
firms that use less fixed assets. As discussed in Chapter 12, if there is a downturn in sales, profits
decrease sharply for firms with high fixed costs because fixed costs cannot be reduced in the short
term. High capital intensities are generally associated with high fixed assets and high fixed costs.
With a 50 percent capital intensity ratio, Blackwell Sales is not a highly capital-intensive firm. Ex-
amples of capital-intensive industries are the airline and the automobile industries; for example, both
United Airlines and Ford Motor Company have capital intensity ratios greater than 100 percent.

Liabilities and Equity

For most firms, the remaining liability accounts on the balance sheet do not vary with sales.
Their values typically change because of management decisions, such as the decision to pay off
a loan or issue debt. Thus, each liability and equity account must be evaluated separately.

Turning to individual accounts, notes payable typically represent short-term borrowing.
This account value will only change with some decision by Blackwell’s management, such as
making a payment on a note or borrowing more money from a bank. Thus, the account’s value
does not vary with sales, as indicated by the “n/a,” or “not applicable,” in Exhibit 19.4. Similarly,
the account value for long-term debt changes only when management decides to issue or retire
debt. The same argument holds for the common stock account, which changes only when
management decides to sell or retire common shares. The last account is retained earnings.
Retained earnings may or may not vary directly with sales. The reason for the ambiguity is that
the amount of funds in retained earnings depends not only on the firm’s earnings, but also on
the firm’s dividend policy, which is set by management. Thus, for now, both the common stock
and the retained earnings accounts are entered as n/a in Exhibit 19.4.

The Preliminary Pro Forma Balance Sheet

We are now in a position to construct a preliminary pro forma balance sheet, as shown in
Exhibit 19.5. The preliminary pro forma balance sheet is a first approximation in deciding
how the firm should fund the assets it needs to support an increase in sales of 30 percent.

620 CHAPTER 19 I Financial Planning and Forecasting

EXHIBIT 19.5 Blackwell Sales: Preliminary Pro Forma Balance Sheet ($ thousands)

This preliminary pro forma balance sheet for Blackwell Sales is a first approximation in deciding how to fund anticipated
growth. At this stage of the analysis, the balance sheet will not balance (A L ϩ OE), and the difference will be the plug value,
which is usually the amount of external financing the firm will need in order to fund investments and operations.

Assets Liabilities and Stockholders’ Equity

Projected Change Projected Change

Current assets $ 130 $ 30 Current liabilities $ 104 $ 44
Cash 156 36 Accounts payable 140 0
Accounts receivable 182 42 Notes payable
Inventory Total $ 244 $ 44
Total $ 468 $ 108 $ 200 $0
Long-term debt
Net fixed assets 832 192 Owner’s equity $ 10 $0
Total assets $ 1,300 $ 300 761 171
Common stock
Retained earnings $ 771 $ 171

Total equity $ 1,215 $ 215
Total liabilities and $ 85 $ 215
stockholders’ equity
External financing needed (EFN)

To construct the preliminary pro forma balance sheet, we follow these steps:

1. We first calculate the projected values for all the accounts that vary with sales, and we
enter these values into the preliminary pro forma balance sheet.

2. We then compute and enter the projected value of any other balance sheet accounts for
which an end-of-period value can be forecast or otherwise determined.

3. For all the accounts for which end-of-period values could not be forecast or otherwise
determined (the n/a accounts), we enter the current year’s value.

4. Typically, the balance sheet will not balance at this point. We thus compute the plug value,
which balances the balance sheet. The plug value will involve the accounts marked “n/a”
in the initial balance sheet (Exhibit 19.4). We must analyze these accounts in light of the
firm’s capital structure and dividend policies. The plug value is usually the amount of ex-
ternal financing needed, because we are usually adding new assets to the balance sheet to
support growth; thus, total assets exceed total liabilities plus equity.

Let’s work through each step using numbers from the Blackwell case.

Step One. We calculate the projected balance sheet values for the accounts that vary with
sales as follows (projected sales are $2.6 million):

• Cash is projected to be 5 percent of sales: $2.6 million ϫ 0.05 ϭ $130,000.
• Accounts receivable is projected to be 6 percent of sales: $2.6 million ϫ 0.06 ϭ

$156,000.

• Inventory is projected to be 7 percent of sales: $2.6 million ϫ 0.07 ϭ $182,000.
• Net fixed assets are projected to be 32 percent of sales: $2.6 million ϫ 0.32 ϭ

$832,000.

• Accounts payable is projected to be 4 percent of sales: $2.6 million ϫ 0.04 ϭ

$104,000.

These values, along with the differences between the current and forecast amounts, are shown
in Exhibit 19.5.

Step Two. We now consider the balance sheet accounts that do not vary with sales. We can
determine the value of retained earnings, since the firm has a dividend policy of paying
out one-third of earnings as dividends. Recall from our earlier discussion that pro-
jected net income is $257,000 and the proportion of that amount going to retained
earnings is $171,000 (0.665 ϫ $257,000 ϭ $171,000). Thus, the end-of-year account
balance is $761,000 ($590,000 ϩ $171,000 ϭ $761,000), where $590,000 is the current

19.3 A Better Financial Planning Model 621

Step Three. The remaining accounts that do not vary with sales represent sources of fi-
nancing for the firm: notes payable, long-term debt, and common stock. These ac-
counts are entered into the preliminary pro forma balance sheet at their current values,
as shown in Exhibit 19.5.

Step Four. As predicted, the preliminary pro forma balance sheet does not balance at this
point: projected assets total $1.3 million, and projected sources of funding (debt and
equity) total $1.215 million. The difference between these two values is our plug value.
The plug value represents external funding needed, which is $85,000 ($1.3 million Ϫ
$1.215 million ϭ $85,000). Since we are dealing with a financing decision, all accounts
with the n/a designation represent possible financing options. Management must use
financial judgment and its knowledge of Blackwell Sales to select the appropriate
financing for the firm.

What the Findings Mean

What does all the information in Exhibit 19.5 tell management? First, if sales increase as pro-
jected, the firm’s total assets will expand by $300,000. Of that $300,000 increase, $108,000 will
go to increase current assets and $192,000 will go to increase the firm’s fixed assets.

Second, the $300,000 in additional assets could be financed as follows: $171,000 from in-
ternally generated funds (the addition to retained earnings), $44,000 from expanded trade
credit (the increase in accounts payable), and $85,000 of external financing from the sale of
debt or equity or both.

Management’s Decision

How should Blackwell Sales fund the $300,000 to support the 30 percent increase in sales? The
firm could issue debt, equity, or reduce dividends. Alternatively, the firm could rethink its
strategy and scale back the 30 percent targeted growth figure. Suppose Blackwell’s manage-
ment team meets to discuss the findings from Exhibit 19.5. After much discussion, they reach
a consensus on the following points:

1. The firm has a unique opportunity to ride a strong market for oil and gas development and
wants to pursue the 30 percent sales growth targeted.

2. Management is concerned about issuing more debt because of the volatility of the oil and
gas exploration business.

3. Management prefers not to issue more common stock for fear of diluting earnings.
4. Management would like to pay an annual dividend but only when justified.

What does management do? In the end, management decides to pay no cash dividend to
stockholders for the coming year. Thus, the $300,000 increase in assets is funded entirely from
earnings. This decision is made to avoid the risks associated with additional debt and the dilu-
tion of earnings that would result from issuing additional common stock.

The Final Pro Forma Balance Sheet

Exhibit 19.6 shows the final pro forma balance sheet reflecting the decision to temporarily
suspend dividends and fund the expansion with internal funds (retained earnings). As you
recall, Blackwell’s net income is $257,000; and thus, the retained earnings account is increased
by $257,000, making the final balance $847,000 ($590,000 ϩ $257,000 ϭ $847,000). Since the
proposed dividend of $86,000 now goes entirely into retained earnings, a source of funds, and
the firm’s additional financing needs are $85,000, there is $1,000 ($86,000 Ϫ $85,000 ϭ $1,000)
available to reduce debt. The most likely course of action is to reduce notes payable by $1,000,
making notes payable $139,000 rather than $140,000.5

5Alternatively, we could have redone the preliminary pro forma balance sheet and found: Total Assets ϭ $1.3 million
and Total Liabilities and Stockholders’ Equity ϭ $1.301 million ($244,000 ϩ $200,000 ϩ $857,000 ϭ $1,301,000).
Since Liabilities and Stockholders’ Equity Ͼ Total Assets, we have more funds than we need. To make the balance sheet

622 CHAPTER 19 I Financial Planning and Forecasting

EXHIBIT 19.6 Blackwell Sales: Final Pro Forma Balance Sheet ($ thousands)

The final pro forma balance sheet reflects Blackwell management’s decision to temporarily suspend dividends and fund
its growth with internal funds (retained earnings). Although financial models can determine the amount of EFN needed,
management must make the final decision about how to fund the firm’s capital requirements.

Assets Liabilities and Stockholders’ Equity

Projected Change Projected Change

Current assets $ 130 $ 30 Current liabilities $ 104 $ 44
Cash 156 36 Accounts payable 139 Ϫ1
Accounts receivable 182 42 Notes payable 43
Inventory Total $ 243 0
Total $ 468 $ 108 $ 200
Long-term debt 0
Net fixed assets 832 192 Owner’s equity $ 10 257
Total assets $ 1,300 $ 300 847 257
Common stock
Retained earnings $ 857 $ 300
$ 300
Total equity $ 1,300
Total liabilities and $0

stockholders’ equity
External financing needed (EFN)

Finally, it is important to note that financial models do not make decisions; only the firm’s
management can do that. Financial models can only generate numbers given the inputs and
assumption made when constructing the model. Once constructed, financial models can help
management evaluate strategic alternatives, assess their financial impact on the firm, and
determine whether they are consistent with the firm’s financial policies. In the Blackwell case,
management suspended its dividend policy.

LEARNING Blackwell’s Alternative Plan
BY
DOING APPLICATION 19.2 PROBLEM: Let’s continue the Blackwell Sales case. Suppose that Blackwell’s manage-
ment now decides to pay a cash dividend but to reduce the payout to 10 percent of net
income. Reconcile Blackwell’s retained earnings account.

APPROACH: First, we must calculate the new dividend payout and the amount of
funds going into retained earnings. Since net income remains unchanged at $257,000,
we calculate the dividends and addition to retained earnings by multiplying the net in-
come by the payout and the retention percentages. Second, we must calculate the im-
pact of the new dividend policy on the retained earnings account. An easy way to do this
is to reconcile the retained earnings account.

SOLUTION: The calculations for the new dividend payout and the addition to retained
earnings are:

(1) Cash dividends ϭ 0.10 ϫ $257,000 ϭ $25,700.
(2) Addition to retained earnings ϭ 0.90 ϫ $257,000 ϭ $231,300.

The calculations to reconcile the retained earnings account are:

Beginning retained earnings balance $590,000
ϩNet income 257,000
ϪDividends 25,700
Ending retained earnings balance
$821,300

Thus, the new retained earnings balance is $821,300.

19.4 Beyond the Basic Planning Models 623

> BEFORE YOU GO ON
1 . How are historical financial data used to determine the forecast values of bal-

ance sheet accounts?
2 . Why might you expect accounts receivable to vary with sales?

19.4 BEYOND THE BASIC PLANNING MODELS

In this section, we tie up some important loose ends concerning financial planning models. We LEARNING OBJECTIVE 4
first consider some shortcomings of the simple models we have been discussing and describe
how more sophisticated models address those shortcomings. We then discuss additional ben-
efits of financial planning.

Improving Financial Planning Models lumpy assets
fixed assets added as large,
Much of the discussion concerning the planning models developed in this chapter focuses on discrete units; these assets
the underlying process for generating pro forma statements. We readily admit that our models may not be used to full
lack sophistication. However, our goal is to have you understand how planning models work capacity for some time,
so that when you move to more elegant computer-based models, you will be an informed user leaving the company with
capable of understanding the models’ limitations and strengths. We now discuss some of the
improvements you should expect to find incorporated in most computer-based models.

Interest Expense

One omission from the models presented in the chapter is that they fail to account for interest
expense in the financial statements. A problem we face in modeling is that interest expense
cannot be estimated accurately until the cost and amount of borrowing have been determined,
and the cost of borrowing depends in part on the amount of borrowing. Thus, we cannot ac-
curately estimate one without the other. More sophisticated financial models estimate the in-
terest payments and borrowings simultaneously.

Working Capital Accounts

Another weakness in our percent of sales model is the assumption that working capital in-
creases proportionally with sales. Seasoned financial managers know that increases in some
working capital accounts are not proportional to sales; this is particularly true for cash balances
and inventory. Exhibit 19.7, for example, shows the inventory-to-sales ratios for two situations:
one where inventory varies directly with sales and one where it does not. The black line illus-
trates the assumption that changes in inventory vary in proportion to changes in sales. Notice
that inventory gets very small as sales approach zero. When inventory varies in proportion to
sales, the inventory/sales ratio is 50 percent, regardless of the level of sales. The red line illus-
trates a different relation. Here, at sales of $300, the inventory/sales ratio is 70 percent ($210/$300
ϭ 0.70, or 70 percent), and at sales of $500 it declines to 50 percent ($250/$500 ϭ 0.50, or 50
percent). The important point here is not the ratio calculations but the fact that working capital
does not increase directly with sales. Instead, it increases at a decreasing rate as sales increase.
This is a common relation between inventory and sales and between cash and sales.

Fixed Assets

Another issue concerns the way we handled fixed assets. Specifically, we assumed that when sales
increase, fixed assets are added in small increments and that production facilities are always
operating near or at full capacity. This is not typically the case. In most instances, fixed assets
are added as large discrete units, and much of a firm’s capacity may not be utilized for some
period of time. These types of assets are often called lumpy assets. Let’s look at an example.

Suppose you and a group of investors decide to enter the market for frozen Mexican snack

624 CHAPTER 19 I Financial Planning and Forecasting

Base stock Inventory $400 The black line shows inventory in the
$350 same proportion as sales—the
$300 inventory/sales ratio is 50 percent,
$250 regardless of the sales level.
$200
$150 The red line shows a varying relation
$100 between inventory and sales. At sales
of $300, the inventory/sales ratio is 70
$50 percent ($210/$300), and at sales of
$0 $500, it declines to 50 percent
$0 ($250/$500).

$100 $200 $300 $400 $500 $600 $700
Sales

EXHIBIT 19.7
Relation Between Inventory Levels and Changes in Sales

This graph shows inventory-to-sales ratios for two situations: one in which inventory varies
directly with sales (black line) and one in which it does not (red line). Financial managers
know from experience that most working capital accounts, such as inventory, do not
increase directly with sales. Instead, they increase at a decreasing rate as sales increase.

can easily be converted to manufacture Mexican snack foods. Exhibit 19.8 illustrates your initial
situation. After you make the purchase, your sales are zero, and you have $100,000 in fixed assets,
which will support sales of up to $150,000. Thus, the facility has $150,000 in excess capacity.

Over time, sales expand to $75,000. At this level, no additional assets are needed (Point A
in the exhibit) because the firm still has excess capacity of $75,000 ($150,000 Ϫ $75,000 ϭ
$75,000). When the firm’s sales expand to $150,000 (Point B), however, the firm no longer has

$700 At Point A ($75,000 sales), excess manufacturing
$600 capacity is still available, but at Point B ($150,000
$500 sales), no excess capacity remains.

Fixed Assets $400
($ thousands)
$300
The addition of
$200 Point $200,000 of fixed
$100 A assets creates
sufficient capacity
Point B to support sales of
up to $400,000.

$0 $400 $600
$0 $75 $150
Sales
($ thousands)

EXHIBIT 19.8
Fixed Assets Are Usually Acquired in Large, Discrete Units

In real-world situations, fixed assets usually do not vary directly with sales, as we
assumed with our simplified financial models. Management often adds fixed assets
in very large increments in order to add capacity in the most economical way.

19.5 Managing and Financing Growth 625

idle capacity. Your production manager determines that a $200,000 addition to fixed assets is
the most economical way to gain additional capacity. If you make this investment, the firm will
have $300,000 ($100,000 ϩ $200,000 ϭ $300,000) in fixed assets, which will support sales up
to $400,000. Notice that when your firm is at Point B, the threshold point, even a small increase
in sales results in more than doubling the firm’s fixed assets.

In financial planning, management must account for the fact that investments in fixed assets
often come in very large increments, or “lumps.” Furthermore, a significant amount of lead time
is often required to bring them on line. Thus, as a firm nears full manufacturing capacity, man-
agement should begin planning to acquire additional fixed assets in the future. In contrast, if a
firm has considerable excess capacity, sales growth will not require additions to fixed assets.

> BEFORE YOU GO ON

1 . Why is it that some working capital accounts may not vary proportionately
with sales?

2 . What are lumpy assets, and how do these assets vary with sales?

19.5 MANAGING AND FINANCING GROWTH LEARNING OBJECTIVE 5

We close the chapter with a discussion of how a business can grow and the need to manage
growth. When companies add assets through acquisition or the capital budgeting process, they
grow in size. If the rate of growth is rapid, much of the asset expansion will likely require ex-
ternal financing. Rapid growth is often a goal of management because it helps a company gain
market share quickly and strengthen its competitive position in the marketplace. In addition,
management in companies with high growth rates often receives accolades and recognition
from investors and their peers for their business acumen. Overall, rapid growth is considered
a desirable achievement for the management of a firm.

Rapid growth can have a dark side, however. As a firm grows rapidly, management might
finance the growth with long-term debt in a way that increases the firm’s overall financial lever-
age. Higher financial leverage increases the probability that a firm will face bankruptcy if
business conditions deteriorate. If management is using a lot of debt financing and sales then
unexpectedly plunge, causing cash flows to decline, the firm may not have enough cash to pay
long-term debt holders and other creditors.

An example of a firm that used too much debt to finance growth is Boston Chicken, Inc., the
former operator and franchiser of Boston Market, a chain of fast-food restaurants offering reason-
ably priced home-style meals. The firm burst onto the national scene in 1993 as one of the hottest
initial public offerings (IPOs) of the year. The first day of trading, its stock price shot up 143 per-
cent! Early successes allowed the firm to expand rapidly from an initial 33 stores to over 1,200.

Beginning in 1996, management decided to go head-to-head with McDonalds and Burger
King in the highly competitive fast-food lunch market. Boston Chicken used a lot of debt to
finance this effort, but the market proved to be difficult. As sales began to slow, management
began to tinker with the menu. Management teams moved in and out, menus grew more com-
plex, service and quality suffered, and sales continued to drop. By October 1998, poor sales and
a debt load of over $900 million forced Boston Chicken into bankruptcy. In December 1999,
McDonalds bought the firm for a bargain-basement price of $173.5 million.

How does rapid growth cause businesses like Boston Chicken to fail? The classic formula
for business failure is rapid expansion, a lack of solid long-term planning, and an insufficient
equity base or, put another way, the use of too much financial leverage.

External Funding Needed

When a firm expands rapidly, its operations might not be able to generate sufficient cash flows
to meet all of its current financial obligations. If this happens, management must look for out-
side funding—debt or equity. We now explore the factors affecting management’s decision to
seek external financing. We do so by developing some relations between a firm’s growth rate and

626 CHAPTER 19 I Financial Planning and Forecasting

Growth and External Funding

The best way to understand the relation between growth and external funding is in the context
of a rapidly growing firm and its financial statements. The firm we use is called Empire Enter-
prises, which is a hypothetical real estate investment firm located in New York City that en-
gages in real estate development and property management. Empire is a public company whose
stock is listed on the NYSE.

Exhibit 19.9 shows the current income statement and balance sheet for Empire Enter-
prises. Last year Empire had total assets of $50 million, book equity of $30 million, and gener-
ated $10 million of earnings on $100 million in sales. Empire’s management team believes the
firm can increase sales by 20 percent for the coming year. All costs and assets are assumed to
grow at the same rate as sales, 60 percent of earnings are paid as cash dividends, and the board
of directors is reluctant to issue additional common stock.

Given this information, we can prepare the pro forma income statement and balance sheet for
Empire Enterprises, which appear in Exhibit 19.10. The income statement shows both sales and
costs increasing by 20 percent for the year: projected sales are $120 million ($100 million ϫ 1.20
ϭ $120 million), projected costs are $108 million ($90 ϫ 1.20 million ϭ $108 million), and thus,
the firm’s projected net income is $12 million ($120 million Ϫ $108 million ϭ $12 million).

Turning to the pro forma balance sheet, we see that the total assets for the firm are $60 million
($50 million ϫ 1.20 ϭ $60 million). For the moment, total debt remains constant at $20 million
because this account will be the plug value—the EFN to support the 20 percent increase in sales.
The firm’s payout policy calls for 40 percent of earnings to be retained in the firm, since 60 percent
will be paid to stockholders as a dividend. Thus, given net income of $12 million, the addition
to retained earnings is $4.8 million (0.40 ϫ $12 million ϭ $4.8 million). The equity account is
increased to $34.8 million ($30.0 million ϩ $4.8 million ϭ $34.8 million).

After these changes have been made, the pro forma balance sheet does not balance. Total
assets equal $60.0 million, and total liabilities and equity equal $54.8 million. The difference,
$5.2 million ($60.0 million Ϫ $54.8 million ϭ $5.2 million), is the EFN. The $10 million ($4.8
million ϩ $5.2 million ϭ $10 million) investment is being financed from two sources: $4.8
million from the addition to retained earnings and $5.2 million from external funding. The
EFN could be either debt or equity, but in Empire’s case it will be long-term debt, since Em-
pire’s board is reluctant to issue equity.

So far, we have calculated EFN exactly as we did in the previous sections. However, we are
now going to build a mathematical model to calculate EFN. The model will allow us to better
understand the relation between a firm’s growth ambitions and the amount of EFN.

EXHIBIT 19.9 Empire Enterprises: Income Statement and Balance Sheet ($ millions)
The exhibit shows the current income statement and balance sheet for Empire Enterprises. Management believes that the firm
can increase sales by 20 percent for the coming year. All costs and assets are assumed to grow at the same rate as sales,
60 percent of earnings are paid out as dividends, and the directors do not want to issue additional common stock.

Income Statement

Net sales $ 100.0
Costs 90.0
Net income
Dividends $ 10.0
Addition to retained earnings $ 6.0
$ 4.0

Balance Sheet

Assets Liabilities and Stockholders’ Equity

Percentage Percentage
of Sales of Sales

Assets $ 50.0 20.0% Total debt $ 20.0 n/a
Total assets $ 50.0 Equity 30.0 n/a

Total liabilities and stockholders’ equity $ 50.0

19.5 Managing and Financing Growth 627

EXHIBIT 19.10 Empire Enterprises: Pro Forma Income Statement and Balance Sheet ($ millions)
The pro forma balance sheet for Empire Enterprises does not balance, and the difference is the amount of EFN. Because the
company’s board does not wish to issue common stock, the funding will have to take the form of long-term debt.

Income Statement (Pro Forma)

Net sales $120.0
Costs 108.0
Net income $ 12.0
Dividends $ 7.2
Addition to retained earnings $ 4.8

Assets Change Balance Sheet (Pro Forma) Change
Projected $ 10.0 Liabilities and Stockholders’ Equity
$ 60.0 $ 10.0 Projected $ 0.0
4.8
Assets $ 60.0 Total debt $ 20.0
Total assets Equity 34.8 $ 4.8
$ 5.2
Total liabilities and stockholders’ equity $ 54.8
External financing needed (EFN) $ 5.2

A Mathematical Model

Looking at the pro forma balance sheet calculations for Empire Enterprises (Exhibit 19.10),
we can see that new investments are determined by the firm’s total assets and projected growth
in sales:

New investments ϭ Growth rate ϫ Initial assets

For Empire, the calculation is $10 million ϭ 0.20 ϫ $50 million. Note that to calculate new
investments, we multiply the firm’s initial total assets by the expected growth rate in sales fore-
casted by management. The new investments are the capital expenditures and the increase in
working capital necessary to sustain the increase in sales.

Conceptually, the new investments are funded first by internally generated funds, which
come from earnings retained in the firm. Once those funds are exhausted, the remainder of
new investments must be financed externally by the sale of debt or equity, or some combina-
tion of both. Thus, the amount of EFN can be expressed as:

EFN ϭ New investments Ϫ Addition to retained earnings (19.5)

Substituting Growth rate ϫ Initial assets for New investments in Equation 19.5 yields the following:

EFN ϭ 1Growth rate ϫ Initial assets2 Ϫ Addition to retained earnings (19.6)

Applying Equation 19.6 to our Empire Enterprise situation, we get the following results:

EFN ϭ 10.20 ϫ $50 million2 Ϫ $4.8 million ϭ $10 million Ϫ $4.8 million
ϭ $5.2 million

The result, $5.2 million, agrees with the financial planning model calculation for Empire En-
terprises presented earlier.

Equation 19.6 highlights two important points. First, holding dividend policy constant,
the amount of EFN depends on the firm’s projected growth rate. The faster management ex-
pects the firm to grow, the more the firm needs to invest in new assets, and the more capital it
has to raise. The potential sources of external capital are the sale of new stock and the sale of
long-term debt. Second, the firm’s payout policy also affects EFN. Holding the growth rate
constant, the higher the firm’s dividend payout ratio, the larger the amount of external debt or
equity financing needed. Also, since EFN is the net amount of external funding needed, the
more stock a firm repurchases, the more new capital it must raise to satisfy its EFN require-

628 CHAPTER 19 I Financial Planning and Forecasting

EXHIBIT 19.11 EFN = (Growth rate × Initial assets) – Addition to retained earnings
External Funding Needed
(EFN) and Growth for Empire External Funding Needed ($ millions) At some growth rate A positive EFN
Enterprises (9.6% for Empire means that the firm
Enterprises), EFN = 0. must find external
The exhibit graphically funding to finance
illustrates Equation 19.6, its growth.
showing the connection
between growth rate in sales $5.2 9.6% 20%
and EFN. The horizontal axis $0
plots the firm’s projected Projected Growth Rate
growth rate, and the vertical –$4.8
axis plots EFN. The upward
slope of the line illustrates
how external financing
increases with the growth rate,
assuming that the dividend
policy is held constant.

At low growth rates, EFN is
negative (EFN < 0), and the
firm has surplus funds.

nternal growth rate (IGR) A Graphical View of Growth
he maximum growth rate that
a firm can achieve without Exhibit 19.11 illustrates Equation 19.6—the relation between sales growth rate and EFN—for
external financing Empire Enterprises. The horizontal axis plots the firm’s projected growth rate, and the vertical
axis plots EFN. The slope of the line illustrates how EFN increases with the growth rate, as-
suming that dividend policy is held constant. As you can see, the line is upward sloping. This
means that as the growth rate increases, the amount of EFN increases.

As a reference point in the exhibit, we plotted Empire’s EFN value of $5.2 million when the
firm’s sales are growing at a 20 percent rate. If you want to generate the line yourself, all you
need to do is make another calculation of EFN at a different growth rate, plot the points, and
connect them with a straight line. However, the important point here is not the mechanics of
generating the graph in Exhibit 19.11 but the interpretation of the line.

Turning to the exhibit, you can see that at low growth rates Empire Enterprises will
generate more funds from earnings than it will spend on new investments. In these situa-
tions, the calculated value for EFN is negative (EFN Ͻ 0), and the firm has a surplus of
funds. In other words, the internally generated funds exceed the firm’s planned investments.
With the surplus funds, management may elect to retire some debt or repurchase some com-
mon stock. For example, at a 0 percent rate of growth, no funds are needed for expansion,
and all the retained earnings are surplus, as we can see by using Equation 19.6:

EFN ϭ 1Growth rate ϫ Initial assets2 Ϫ Addition to retained earnings
ϭ 10.0 ϫ $50 million2 Ϫ $4.8 million
ϭ Ϫ$4.8 million

With a higher growth rate, the surplus becomes smaller and smaller as more and more
funds are used to finance the new investments. At a growth rate of 9.6 percent, the surplus
equals zero, as does the calculated value of EFN. Next we explain how to calculate the growth
rate at which the surplus equals zero. The key point here is that the higher the rate at which a
firm grows, the more external funding it requires.

The Internal Growth Rate

Management often has an interest in knowing the rate at which the firm can grow using just
internally generated funds. This rate is called the internal growth rate (IGR). The IGR is the
maximum growth rate that a firm can achieve without external financing. To determine this
rate, we set Equation 19.6 equal to zero (EFN ϭ 0) and solve for the growth rate. Thus,

19.5 Managing and Financing Growth 629

Rearranging terms yields the internal growth rate:

Addition to retained earnings (19.7)
IGR ϭ

Initial assets

The managerial implications of the formula are straightforward. Firms that can generate a higher
volume of retained earnings and/or use fewer assets can sustain a higher growth rate without rais-
ing more capital. For the Empire Enterprises example, the internal growth rate is calculated as:

Addition to retained earnings
IGR ϭ

Initial assets
ϭ $4.8 million ϭ 0.096, or 9.6%

$50 million

To gain more insight into what factors determine a firm’s internal growth rate, we can
manipulate Equation 19.7 by multiplying both the numerator and the denominator by net in-
come and total equity, as follows:

IGR ϭ Addition to retained earnings ϫ Net income ϫ Total equity
Total assets Net income Total equity

If we then rearrange terms, we arrive at the following expression:

IGR Addition to retained earnings ϫ Net income Total equity
ϭ ϫ
Net Income Total equity Total assets

From the discussions in Chapter 4 and in this chapter, we know the following: (1) plowback

ratio ϭ addition to retained earnings/net income; (2) return on equity ϭ net income/equity; and
(3) equity multiplier ϭ total assets/total equity.6 This means that we can write the above equation as:

IGR ϭ Plowback ratio ϫ Return on equity ϫ Measure of leverage (19.8)

Equation 19.8 tells us that firms that achieve higher growth rates without seeking external
financing tend to have one or more of the following characteristics:

• They have dividend policies that retain a high proportion of earnings inside the firm—that

is, they have a high plowback ratio.

• They are able to generate a high net income with a smaller amount of equity than other

firms and hence have a high return on equity (ROE).

• They use low amounts of leverage; thus, their debt-to-equity ratios are low.

The Sustainable Growth Rate

Another growth rate helpful in long-term planning is the sustainable growth rate (SGR), sustainable growth rate
which is the rate of growth that the firm can sustain without selling additional equity while
maintaining the same capital structure. You may wonder why management is interested in the (SGR)
sustainable growth rate. The sustainable growth rate is important to managers of firms that are the rate of growth that a firm
likely to generate excess funds internally and who want to determine the payout ratio that en- can sustain without selling
ables them to fund their firms’ growth while maintaining their current capital structures. additional equity while
maintaining the same capital
The sustainable growth rate is the rate at which a firm can grow using only (1) internally structure
generated funds from earnings and (2) external funds from the sale of new debt while main-
taining a constant debt-equity ratio. As it turns out, SGR is a function of the firm’s plowback
ratio and the return on equity (ROE). SRG can be expressed as follows:

SGR ϭ Plowback ratio ϫ ROE (19.9)

For Empire Enterprises, the sustainable growth rate is:

SGR ϭ 0.4 ϫ $10 million
$30 million

ϭ 0.4 ϫ 0.333
ϭ 0.133, or 13.3%

6Note that the measure of leverage in Equation 19.8, total equity/total assets, is the inverse of the equity multiplier,

630 CHAPTER 19 I Financial Planning and Forecasting

The 13.3 percent rate is a fairly high SGR that is driven by the company’s rather hefty 33.3 per-
cent return on equity.

An analysis of a company’s SGR relative to the company’s actual growth rate can provide
management with some insights into problems the firm may face in the future. For example, if
a firm’s actual growth rate consistently exceeds its SGR, management knows that unless they
sell new equity, the firm will have a cash shortage problem in the future because of the need to
purchase new assets to generate the growth. The SGR model does not, however, tell manage-
ment how fast the firm should grow. That decision requires informed judgment about the at-
tractiveness of the investment opportunities available to the firm.

DECISION Empire’s Ambitious Growth Plan
MAKING
SITUATION: You are part of the Empire Enterprises finance team. The firm’s strategic
EXAMPLE 19.2 plan calls for revenues to grow at 20 percent next year. As mentioned, the board of direc-
tors is not interested in using any additional external equity financing. Some members of
the team question whether these goals are realistic.

You have just been asked to comment on the proposed growth plan at a meeting. You
have a little over an hour to prepare. During the time available, you completed the fol-
lowing calculations using data from the most recent and the pro forma income statements
and balance sheets (Exhibits 19.9 and 19.10):

• EFN ϭ (Growth rate ϫ Initial assets) Ϫ Addition to retained earnings ϭ (0.20 ϫ
$50 million) Ϫ $4.8 million ϭ $5.2 million

• IGR ϭ Addition to retained earnings/initial assets ϭ $4.8 million/$50 million ϭ
0.96, or 9.6%

• SGR ϭ Plowback ratio ϫ ROE ϭ 0.40 ϫ 0.333 ϭ 13.3%

Given the above information, what can you say about this ambitious growth plan?

DECISION: You begin by applauding the visionary nature of the strategic plan. Clearly,
you want to keep your job. You point out, however, that the firm is facing some chal-
lenges. First, Empire’s IGR is 9.6 percent, which is the maximum growth rate the firm can
achieve without any kind of external financing. This amount is substantially below the de-
sired growth rate of 20 percent. Second, you note that Empire’s EFN is $5.2 million. This
means that $5.2 million of external capital will have to be raised by selling equity, debt,
or some combination of the two. Finally, Empire’s SGR is 13.3 percent—also below the
20 percent growth target. Empire cannot grow more than 13.3 percent without selling
equity if management wants to keep the firm’s capital structure at its current level.

LEARNING APPLICATION 19.3 Sustainable Growth and Financial Statements
BY
DOING PROBLEM: Because of your presentation (see Decision-Making Example 19.2), Empire’s
top management team has had second thoughts about their goal of growing the firm
20 percent during the next year. As a result, they have asked that you prepare pro forma
financial statements at a sales growth rate equal to the firm’s SGR of 13.3 percent.

APPROACH: For the income statement, all costs grow at the same rate as revenues.
Thus, you can multiply the current period’s sales and costs by 1.133 to calculate the
projected values of sales and costs. To construct the balance sheet, you must first
compute the values of accounts that vary with sales. Since you have no information
about how much of Empire’s total debt is long-term debt, you should enter its total
debt value of $20 million, along with all the information you have on the balance sheet
accounts. Finally, to make the balance sheet balance, you should calculate the amount
of EFN.

19.5 Managing and Financing Growth 631

SOLUTION:

Sales ϭ $100 million ϫ 1.133 ϭ $113.30 million
Costs ϭ $90 million ϫ 1.133 ϭ $101.97 million

Income statement:

Empire Enterprises
Pro Forma Income Statement ($ millions)

Sales $113.30
Costs 101.97
Net income
$ 11.33

Dividend ϭ Net income ϫ Payout ratio ϭ $11.33 million ϫ 0.60 ϭ $6.80 million

Addition to retained earnings ϭ Net income ϫ Plowback ratio ϭ $11.33 million ϫ 0.4
ϭ $4.53 million

Forecast value of the assets: $50 million ϫ 1.133 ϭ $56.65 million

Value of the equity: $30 million ϩ $4.53 million ϭ $34.53 million, where $30 million is the
initial value and $4.53 million is the addition to retained earnings

Value of debt plus equity: $20 million ϩ 34.53 million ϭ $54.53 million

The balance sheet does not balance ($56.65 million assets Ͼ $54.53 million debt
plus equity), and the difference ($2.12 million) is the plug number, which is the EFN.
Thus, to achieve the 13.3 percent rate of growth, Empire will need to issue $2.12 million
in long-term debt, which will bring the debt account to $20 million ϩ $2.12 million ϭ
$22.12 million. The resulting balance sheet is as follows:

Assets Empire Enterprises
Pro Forma Balance Sheet ($ millions)

Liabilities and Stockholders’ Equity

Assets $ 56.65 Debt $ 22.12
Total assets $ 56.65 Equity 34.53
Total liabilities and stockholders’ equity
$ 56.65

Growth Rates and Profits

So far, we have focused on a firm’s rate of growth. In the final analysis, however, the critical
question in business is not how fast the firm can grow, but whether the firm can sustain rapid
growth and maintain a satisfactory level of profits. In reality, it is very difficult to achieve and
sustain rapid growth in a competitive market and remain profitable. The business arena is lit-
tered with failed growth firms like Boston Chicken.

To provide a reality check, only 7 percent of publicly traded U.S. companies increase both
revenues and operating profits by an average of 10 percent a year. Experts generally agree that
growth rates at or above 10 percent are very difficult to sustain for established companies.

Growth As a Planning Goal

The final question we address is whether growth by itself is an acceptable strategic goal.
We pose this question because it is common for top management to set growth rates as
goals for the firm or operating divisions. In fact, there is nothing a CEO likes to do better
at the annual meeting than point out that “last year, under my leadership, Sleepy Hollow
exceeded its goal of 10 percent growth,” followed by a hearty round of applause. Growth
rate goals are popular because they are easy to communicate and understand. But are they
appropriate goals for financial planning? The short answer is “no.” Let’s consider why this
is the case.

As we discussed in Chapter 1, an appropriate goal for management is maximizing the

632 CHAPTER 19 I Financial Planning and Forecasting

NPVs, finances them at the lowest possible cost, and skillfully manages these assets, the com-
pany should be profitable and grow in size. This growth results from making sound business
decisions and executing strategies that create sustainable competitive advantages over the long
term. Thus, growth is an acceptable goal as long as it is anchored to a sound business strategy
that will generate an increase in stockholder value.

> BEFORE YOU GO ON
1 . What two factors determine the amount of EFN?
2 . What is IGR, and why is it of interest to management?
3 . If a firm continually exceeds its SGR, what problems may it face in the future?

S um m a ry of Learning Objectives

1 Explain what a financial plan is and why financial plan- trend lines to the data to see what type of relation exists be-
ning is so important. tween that variable and sales. Many income statement and bal-
ance sheet items vary directly with sales, but others may vary in
A financial plan is a set of actionable goals derived from the firm’s a nonlinear manner. The analysis in the Blackwell Sales Com-
strategic plan and other planning documents, such as the invest- pany example illustrates how to analyze a strategic investment
ment and financing plans. The financial plan focuses on select- decision.
ing the best investment opportunities and determining how they
will be financed. The financial plan is a blueprint for the firm’s 4 Describe the conditions under which fixed assets vary
future. Financial planning is important to management because directly with sales, and discuss the impact of so-called
the plan communicates the firm’s strategic goals throughout the lumpy assets on this relation.
organization, builds support for the firm’s strategies, and helps
align operating units with the firm’s strategic goals. Fixed assets vary directly with sales only when assets can be
added in small increments and production facilities are oper-
2 Discuss how management uses financial planning mod- ating near full capacity. This is typically not the case. In most
els in the planning process, and explain the importance situations, fixed assets are added in large, discrete units, and
of sales forecasts in the construction of financial plan- as a result, much of the new capacity may go unused for a
ning models. period of time. These types of assets are often called lumpy
assets. After lumpy assets are added, sales can increase for a
Financial models are the analytical part of the financial planning period of time with no corresponding change in the level of
process. A planning model is simply a series of equations that fixed assets.
model a firm’s financial statements, such as the income statement
and balance sheet. Once the model is constructed, management 5 Explain what factors determine a firm’s sustainable
can generate projected (pro forma) financial statements to de- growth rate, discuss why it is of interest to management,
termine the financial impact of proposed strategic initiatives on and compute the sustainable growth rate for a firm.
the firm.
A firm’s sustainable growth rate (SGR) is the maximum rate at
For most financial planning models, a forecast of the firm’s which the firm can grow without external equity financing and
sales is the most important input variable. The sales forecast is with leverage held constant. The determinants of a firm’s SGR
the key driver in financial planning models because many items are: (1) profit margins (the greater a firm’s profit margin, the
on the income statement and balance sheet vary directly with greater the firm’s SGR); (2) asset utilization (the more efficiently
sales. Thus, once sales are forecasted, it is easy to generate pro- a firm uses its assets, the higher its SGR); (3) financial leverage
jected financial statements using the historical relation between (as a firm increases its use of leverage, its SGR increases); (4)
a particular account and sales. payout policy (as a firm decreases its payout ratio, its SGR in-
creases); and (5) economic conditions (the more favorable the
3 Discuss how the relation between projected sales and economic environment, the higher the firm’s SGR). Management
balance sheet accounts can be determined, and ana- may be interested in knowing the SGR for two reasons. First,
lyze a strategic investment decision using a percent of the SGR is the rate of growth at which a firm’s capital structure
sales model. (debt to equity) will remain constant without the firm selling or
repurchasing stock. Second, if a firm’s actual growth rate exceeds
Historical financial data can be examined to determine wheth- its SGR, the firm could face cash shortage problems in the future
er and how a variable changes with sales. One way to do this unless it can sell new equity. Learning by Doing Application 19.3
is to prepare a table that shows four or five years of historical

Self-Study Problems 633

S um m a ry of Key Equations

Equation Description Formula
19.1 %¢S ϭ 1Stϩ1 Ϫ St2
Percent change in sales
19.2 St
Percent of net income paid out Dividend payout ratio ϭ Cash dividends
19.3 as dividends
Net income
19.4 Percent of net income retained Addition to retained earnings
19.5 & (plowed back into the firm)
19.6 Retention 1plowback2 ratio ϭ
19.7 & Level of assets needed to generate Net income
19.8 $1 of sales
Capital intensity ratio ϭ Total assets
19.9 External funding needed to support Net sales
growth in sales
Internal growth rate (level of growth EFN ϭ New investments Ϫ Addition to retained earnings
that can be supported without raising ϭ (Growth rate ϫ Initial assets) Ϫ Addition to retained earnings
external funds)
IGR ϭ Addition to retained earnings
Sustainable growth rate (level of
growth that can be supported without Initial assets
raising external equity or increasing
current leverage) ϭ Plowback ratio ϫ Return on equity ϫ Measure of leverage

SGR ϭ Plowback ratio ϫ ROE

Self-Study Problems

19.1 The Starlight, Inc., financial statements for the fiscal year ended June 30, 2011, are presented below.
The firm’s sales are projected to grow at a rate of 20 percent next year, and all financial statement
accounts will vary directly with sales. Based on that projection, develop a pro forma balance sheet
and income statement for the fiscal year ending June 30, 2012.

Starlight, Inc. Balance Sheet as of June 30, 2011

Assets: Liabilities and Stockholders’ Equity:
Cash
Accounts receivables $ 25,135 Accounts payables $ 67,855
Inventories 43,758 Notes payables 36,454
Total current assets 167,112
Net fixed assets Total current liabilities $ 104,309
Other assets $ 236,005 Long-term debt 223,125
325,422 Common stock 150,000
Total assets 13,125 Retained earnings 97,118

$ 574,552 Total liabilities and equity $ 574,552

Starlight, Inc. Income Statement for
the Fiscal Year Ended June 30, 2011

Net sales $ 1,450,000
Costs 812,500
EBITDA
Depreciation $ 637,500
EBIT 175,000
Interest
EBT $ 462,500
Taxes (35%) 89,575
Net income
$ 372,925
130,524

$ 242,401

634 CHAPTER 19 I Financial Planning and Forecasting

19.2 Use the financial information for Starlight from Problem 19.1. Assume now that equity accounts
do not vary directly with sales, but change when retained earnings change or new equity is issued.
The company pays 45 percent of its income as dividends every year. In addition, the company
plans to expand production capacity by building a new facility that will cost $225,000. The firm
has no plans to issue new equity this year. Any funds that need to be raised will be raised through
the sale of long-term debt. Prepare a pro forma balance sheet using this information.

19.3 Use the financial statements from Problem 19.1 and the information from Problem 19.2 to calcu-
late the company’s retention (plowback) ratio, external funds needed (EFN), internal growth rate
(IGR), and sustainable growth rate (SGR).

19.4 Northwood Corp. has a dividend payout ratio of 60 percent, return on equity of 14.5 percent, total
assets of $11,500,450, and equity of $4,652,125. Calculate the firm’s internal rate of growth (IGR).

19.5 Renewal Company has net income of $1.25 million and a dividend payout ratio of 35 percent. It
currently has equity of $2,875,223. What is the firm’s sustainable growth rate?

Solutions to Self-Study Problems

19.1 The pro forma statements for Starlight are as follows:

Starlight, Inc. Balance Sheet as of June 30, 2012

Assets: Liabilities and Stockholders’ Equity:
Cash
Accounts receivables $ 30,162 Accounts payables $ 81,426
Inventories 52,510 Notes payables 43,745
Total current assets 200,534
Net fixed assets Total current liabilities $ 125,171
Other assets $ 283,206
390,506 Long-term debt 267,750
Total assets 15,750 Common stock 180,000
Retained earnings 116,542
$ 689,462 $ 689,462
Total liabilities and equity

Starlight, Inc. Income Statement for
the Fiscal Year Ended June 30, 2012

Net sales $ 1,740,000
Costs 975,000
EBITDA
Depreciation $ 765,000
EBIT 210,000
Interest
EBT $ 555,000
Taxes (35%) 107,490
Net income
$ 447,510
156,629

$ 290,882

19.2 The pro forma income statement is the same as that shown in the solution to Problem 19.1. We
now have to account for the payment of dividends. Since the company pays 45 percent of its net
income as dividends, the retained earnings for 2012 is calculated as follows:
Retained earnings from 2012 income ϭ $290,882 ϫ (1 Ϫ 0.45) ϭ $159,985.

• This is the amount by which retained earnings will increase in 2012, from $97,118 to $257,103.
• No new equity is added.
• The increase in assets is financed externally through the sale of long-term debt.

The pro forma balance sheet is as follows:

Starlight, Inc. Balance Sheet as of June 30, 2012

Assets: Liabilities and Stockholders’ Equity:

Cash $ 30,162 Accounts payables $ 81,426
Accounts receivables 52,510 Notes payables 43,745
Inventories 200,534
Total current assets Total current liabilities $ 125,171
$ 283,206
Net fixed assets Long-term debt 382,188
Addition to fixed assets 390,506 Common stock 150,000
Other assets 225,000 Retained earnings 257,103
15,750

Critical Thinking Questions 635

19.3 The retention (plowback) ratio, external funds needed, internal growth rate, and sustainable
growth rate are calculated as follows:

Addition to retained earnings
Retention 1plowback2 ratio ϭ

Net income
ϭ $159,985

$290,882
ϭ 0.55, or 55%

EFN ϭ 1Growth rate ϫ Initial assets2 Ϫ Addition to retained earnings
ϭ 10.20 ϫ $574,5522 Ϫ $159,985
ϭ Ϫ$45,075

Thus, without considering the investment of $225,000 for the new facility, the firm will not need
any external financing. However, if you add the investment, then,

EFN ϭ New investments Ϫ Addition to retained earnings
ϭ 10.20 ϫ $574,5522 ϩ $225,000 Ϫ $159,985
ϭ $179,925

IGR ϭ Addition to retained earnings

Initial assets

ϭ $159,985
$574,552

ϭ 0.278, or 27.8%

SGR ϭ Plowback ratio ϫ ROE

ϭ Addition to retained earnings ϫ Net income
Net income Total equity

ϭ 0.55 ϫ 0.715

ϭ 0.393, or 39.3%

19.4 We calculate Northwood’s internal growth rate as follows:

IGR ϭ Plowback ratio ϫ ROE ϫ Measure of leverage

ϭ 0.40 ϫ 0.145 ϫ $4,652,125
$11,500,450

ϭ 0.0235, or 2.35%

19.5 Renewal’s sustainable growth rate is:

SGR ϭ Plowback ratio ϫ ROE

ϭ 0.65 ϫ $1,250,000
$2,875,223

ϭ 0.283 ϭ 28.3%

Critical Thinking Questions

19.1 What is financial planning? What four types of plans are involved in financial planning?
19.2 Why is the capital budget an important part of a firm’s financial planning?
19.3 Why do financing and investment decisions have to be made concurrently?
19.4 Explain how sales can be used to develop pro forma financial statements.
19.5 Why is sales not always a good measure to use in forecasting fixed assets?
19.6 List all the accounts that can be affected by the “plug” value. How does this value help managers?
19.7 Explain why the fixed asset account may or may not vary with sales.
19.8 How does the dividend payout ratio affect the amount of funds needed to finance growth?
19.9 Define internal growth rate (IGR). Identify the characteristics of a high-growth firm that has no

636 CHAPTER 19 I Financial Planning and Forecasting

Questions and Problems

BASIC > 19.1 Strategic plan: Explain the importance of the strategic plan.

19.2 Capital budget: What are the various steps in preparing a capital budget?

19.3 Financing plan: What are the elements of a financing plan?

19.4 Financial planning: Identify the steps in the financial planning process.

19.5 Financial modeling: List the various elements of financial modeling.

19.6 Payout ratio: Define the retention (plowback) ratio and the dividend payout ratio.

19.7 Addition to retained earnings: Northwood, Inc., has revenue of $455,316, costs of $316,487,
and pays taxes at a rate of 31 percent. If the firm pays out 45 percent of its earnings as dividends
every year, how much earnings are retained and what is the firm’s retention ratio?

19.8 Payout and retention ratio: Goodwin Corp. has revenues of $12,112,659, costs of $9,080,545,
interest payments of $412,375, and a tax rate of 34 percent. It paid dividends of $1,025,000 to its
stockholders. Find the firm’s dividend payout ratio and retention ratio.

19.9 Percent of sales: Cattail Corporation’s financial statements for the fiscal year just ended are
shown below:

Cattail Corporation
Financial Statements for Fiscal Year Just Ended ($ thousands)

Income Statement Balance Sheet

Net sales $1,500 Assets $700 Debt $600
Costs 350 Total Equity $100
Net income $700
$1,150 $700 Total

Cattail management expects sales to increase by 14 percent next year. Assume that the financial
statement accounts vary directly with changes in sales and that management has no financing
plan at this time. Given this information, develop a pro forma income statement for Cattail for
the next fiscal year.

19.10 Percent of sales: Given the data for Cattail Corporation in Problem 19.9, if you assume that all
balance sheet items also vary with the change in sales, develop a pro forma balance sheet for Cattail
for the next fiscal year. Assuming that the firm did not sell or repurchase stock, what is the cash
dividend implied by the pro forma income statement and balance sheet?

19.11 Capital intensity ratio: Define capital intensity ratio, and explain its significance.

19.12 Capital intensity ratio: Tantrix Confectioners has total assets of $3,257,845 and net sales of
$5,123,951. What is the firm’s capital intensity ratio?

19.13 Capital intensity ratio: McDonald Metal Works has been able to generate net sales of
$13,445,196 on assets of $9,145,633. What is the firm’s capital intensity ratio?

19.14 Capital intensity ratio: For McDonald Metal Works in Problem 19.13, how much must net
sales grow if the capital intensity ratio has to drop to 60 percent? State your answer as both a
percent of sales and a dollar sales increase.

19.15 Internal growth rate: Swan Supply Company has net income of $1,212,335, assets of $12,522,788,
and retains 70 percent of its income every year. What is the company’s internal growth rate?

19.16 Sustainable growth rate: If Newell Corp. has a ROE of 13.7 percent and a dividend payout
ratio of 32 percent, what is its sustainable growth rate?

19.17 EFN and growth: Refer to Exhibits 19.10 and 19.11 in the text. The EFNs for several growth
rates for Empire Enterprises are as follows:

Growth Rate (%) EFN ($ millions)

0% Ϫ$4.8
5 Ϫ2.3
9.6
10 0.0
15 0.2
20 2.7
5.2

Questions and Problems 637

19.18 Retention ratio: Refer to Problem 19.7. Northwood expects to increase its sales by 15 percent < I N T E R M E D I AT E
next year. All costs vary directly with sales. If Northwood wants to retain $65,000 of earnings
next year, will it have to change its dividend payout ratio? If so, what will be the new dividend
payout and retention ratios for the firm?

19.19 Capital intensity: Identify two industries (other than airlines) that are capital intensive. Using
online or other data sources, compute the capital intensity ratio for the largest firm in each of the
chosen industries.

19.20 Percent of sales: Tomey Supply Company’s financial statements for the most recent fiscal year
are shown below. The company projects that sales will increase by 20 percent next year. Assume
that all costs and assets increase directly with sales. The company has a constant 33 percent
dividend payout ratio and has no plans to issue new equity. Any financing needed will be raised
through the sale of long-term debt. Prepare pro forma financial statements for the coming year
based on this information, and calculate the EFN for Tomey.

Tomey Supply Company Income Statement and Balance Sheet

Income Statement Balance Sheet

Net sales $ 1,768,121 Assets: $ 280,754
Costs 1,116,487 Current assets 713,655
EBT 651,634 Net fixed assets
Taxes (35%) 228,072 $ 994,409
Net income Total assets
$ 423,562 $ 167,326
Liabilities and Equity: 319,456
Currrent liabilities 200,000
Long-term debt 307,627
Common stock
Retained earnings $ 994,409

Total liabilities and equity

19.21 Internal growth rate: Using the pro forma financial statements for Tomey Supply Company
developed in Problem 19.20, find the internal growth rate for Tomey.

19.22 Sustainable growth rate: Use the following pro forma information for Tomey Supply Com-
pany for next year: net income ϭ $508,275; addition to retained earnings ϭ $340,544; common
equity ϭ $848,171; net sales ϭ $2,121,745. Assume that management does not want the ratio of
long-term debt to equity to exceed the current long-term debt-to-equity ratio of 63 percent and
also does not want to issue new equity. What level of sales growth can Tomey Supply Company
sustain? Calculate the new sales level.

19.23 Sustainable growth rate: Rowan Company has a net profit margin of 8.3 percent, debt ratio
of 45 percent, total assets of $4,157,550, and sales of $6,852,654. If the company has a dividend
payout ratio of 67 percent, what is its sustainable growth rate?

19.24 Sustainable growth rate: Refer to the information for Rowan Company in Problem 19.23.
The firm’s management desires a sustainable growth rate (SGR) of 10 percent but does not wish
to change the company’s level of debt or its payout ratio. What will the firm’s new net profit mar-
gin have to be in order to achieve the desired growth rate?

19.25 Sustainable growth rate: Rocky Sales, Inc., has current sales of $1,215,326 and net income
of $211,253. It also has a debt ratio of 25 percent and a dividend payout ratio of 75 percent. The
company’s total assets are $712,455. What is its sustainable growth rate?

19.26 Sustainable growth rate: Ellicott Textile Mills management has reported the following finan-
cial information for the year ended September 30, 2011. The company generated a net income
of $915,366 on a net profit margin of 6.4 percent. It has a dividend payout ratio of 50 percent, a
capital intensity ratio of 62 percent, and a debt ratio of 45 percent. What is the company’s sustain-
able growth rate?

19.27 Internal growth rate: Given the information in Problem 19.26, what is the internal growth
rate of Ellicott Textile Mills?

19.28 Internal growth rate: Fantasy Travel Company has a return on equity of 17.5 percent, a total
equity/total assets ratio of 65 percent, and a dividend payout ratio of 75 percent. What is the
company’s internal growth rate?

19.29 EFN: Maryland Micro Brewers generated revenues of $12,125,800 with a 72 percent capital
intensity ratio during the year ended September 30, 2011. Its net income was $873,058. With the

638 CHAPTER 19 I Financial Planning and Forecasting

next year. Assume that all costs vary directly with sales and that the firm maintains a dividend
payout ratio of 70 percent. What will be the EFN needed by this firm? If the company wants to
raise no more than $750,000 externally and is not averse to adjusting its payout policy, what will
be the new dividend payout ratio?

19.30 EFN: Ritchie Marble Company has total assets of $12,899,450, sales of $18,174,652, and net
income of $4,589,774. Management expects sales to grow by 25 percent next year. All assets and
costs (including taxes) vary directly with sales, and management expects to maintain a payout
ratio of 65 percent. Calculate Ritchie’s EFN.

19.31 EFN: Norton Group, Inc., expects to add $1,213,777 to retained earnings and currently has
total assets of $23,159,852. If the company has the ability to borrow up to $1 million, how much
growth can the firm support if it is willing to borrow to its maximum capacity?

19.32 EFN: Capstone Marketing Group has total assets of $5,568,000, sales of $3,008,725, and net in-
come of $822,000. The company expects its sales to grow by 12 percent next year. All assets and
costs (including taxes) vary directly with sales, and the firm expects to maintain a payout ratio of
55 percent. Calculate Capstone’s EFN.

19.33 Maximum sales growth: Given the data for Capstone Marketing Group in Problem 19.32,
what would Capstone’s payout ratio have to be in order for the firm’s EFN to be zero?

19.34 Maximum sales Growth: Rockville Consulting Group expects to add $271,898 to retained
earnings this year. The company has total assets of $3,425,693 and wishes to add no new ex-
ternal funds for the coming year. If assets and costs vary directly with sales, how much sales
growth can the company support while retaining an EFN of zero? What is the firm’s internal
growth rate?

ADVANCED > 19.35 The financial statements for the year ended June 30, 2011, are given below for Morgan Construc-

tion Company. The firm’s sales are projected to grow at a rate of 25 percent next year, and all
financial statement accounts will vary directly with sales. Based on that projection, develop a pro
forma balance sheet and an income statement for the 2012 fiscal year.

Morgan Construction Company Balance Sheet as of
June 30, 2011

Assets: Liabilities and Stockholders’ Equity:

Cash $ 3,349,239 Accounts payables $ 9,041,679
Accounts receivables 5,830,754 Notes payables 4,857,496
Inventories 22,267,674
Total current assets Total current liabilities $ 13,899,175
Net fixed assets $ 31,447,667 Long-term debt 29,731,406
Other assets 43,362,482 Common stock 19,987,500
1,748,906 Retained earnings 12,940,974

Total assets $ 76,559,055 Total liabilities and equity $ 76,559,055

Morgan Construction Company
Income Statement for the Fiscal Year

Ended June 30, 2011

Net sales $ 193,212,500
Costs 45,265,625
EBITDA
Depreciation $ 47,946,875
EBIT 23,318,750
Interest
EBT $ 24,628,125
Taxes (35%) 11,935,869
Net income
$ 12,692,256
4,442,290

$ 8,249,966

19.36 Use the financial information for Morgan Construction Company from Problem 19.35. As-
sume now that equity accounts do not vary directly with sales but change when retained
earnings change or new equity is issued. The company pays 75 percent of its income as

Questions and Problems 639

expanding the current facility and acquiring additional equipment. This will cost the firm
$10 million. The firm has no plans to issue new equity this year. Prepare a pro forma bal-
ance sheet using this information. Any funds that need to be raised (in addition to changes
in current liabilities) will be in the form of long-term debt. What is the external financing
needed in this case?

19.37 Using the information for Morgan Construction Company in the preceding problem, calculate
the firm’s internal growth rate and sustainable growth rate.

19.38 Use the information for Morgan Construction Company from Problems 19.35 and 19.36.
Assume that equity accounts do not vary directly with sales, but change when retained
earnings change or new equity is issued. The company’s long-term debt-to-equity ratio is
approximately 90 percent, and its equity-to-total assets ratio is about 43 percent. The com-
pany management wishes to increase its equity-to-total assets ratio to at least 50 percent.
Management is willing to reduce the company’s payout ratio, but will retain no more than
40 percent of earnings. The company will raise any additional funds needed, including
funds for expansion, by selling new equity. No new long-term debt will be issued. Prepare
pro forma statements to reflect this new scenario.
a. What is the external financing needed to accommodate the expected growth?
b. What is the firm’s internal growth rate?
c. What is the firm’s sustainable growth rate?
d. How much new equity will the firm have to issue?
e. What is the firm’s new equity ratio and debt-to-equity ratio?

19.39 Munson Communications Company has just reported earnings for the year ended June 30, 2011.
Below are the firm’s income statement and balance sheet. The company had a 55 percent divi-
dend payout ratio for the last 10 years and does not plan to change this policy. Based on internal
forecasts, the company expects the demand for its products to grow at a rate of 20 percent for
the next year and has projected the sales growth for 2012 to be 20 percent. Assume that equity
accounts and long-term debt do not vary directly with sales, but change when retained earnings
change or additional capital is issued.

Munson Communications Company Balance Sheet as of
June 30, 2011

Assets: Liabilities and Stockholders’ Equity:

Cash $ 1,728,639 Accounts payables $ 4,666,673
Accounts receivables 3,009,421 Notes payables 2,507,094
Inventories 11,492,993
Total current assets Total current liabilities $ 7,173,767
Net fixed assets $ 16,231,054 Long-term debt
13,345,242 22,380,636
Other assets Common stock 10,165,235
1,748,906

Retained earnings 9,676,351

Total assets $ 40,360,595 Total liabilities and equity $ 40,360,595

Munson Communications Company
Income Statement for the Fiscal Year

Ended June 30, 2011

Net sales $ 79,722,581
Costs 59,358,499
EBITDA
Depreciation $ 20,364,082
EBIT 7,318,750
Interest
EBT $ 13,045,332
Taxes (35%) 3,658,477
Net income
$ 9,386,855
3,285,399

$ 6,101,456

a. What is the firm’s internal growth rate (IGR)?
b. What is the firm’s sustainable growth rate (SGR)?
c. What is the external financing needed (EFN) to accommodate the expected growth?
d. Construct the firm’s 2012 pro forma financial statements under the assumption that all external

640 CHAPTER 19 I Financial Planning and Forecasting

Sample Test Problems

19.1 Mercury Corp. has annual sales of $2,512,654, costs of $1,080,227, interest payments of $132,375,
and a tax rate of 34 percent. It pays annual dividends of $525,000 to its stockholders. Calculate the
firm’s dividend payout ratio and retention ratio.

19.2 Assume that Rex Corp. is operating at a capital intensity ratio of 63.5 percent and is able to gener-
ate net sales of $3,123,443. What is the book value of the firm’s assets?

19.3 Centennial Beverages currently has sales of $1,415,326, net income of $411,253, a debt ratio
of 25 percent, and a dividend payout ratio of 70 percent. The company also has total assets of
$1,850,325. What is its sustainable growth rate?

19.4 Given the information in Sample Test Problem 19.3, what is the internal growth rate of Centen-
nial Beverages?

19.5 Mirabelle Company has total assets of $3,267,450, sales of $5,174,652, and net income of $1,789,774.
The company’s management expects sales to grow by 20 percent next year. All assets and costs
(including taxes) vary directly with sales, and management expects to maintain a payout ratio of
75 percent. Calculate the external financing needed (EFN).

Options and Corporate

20Finance

Learning Objectives

1 Define a call option and a put option, and
describe the payoff function for each of
these options.

2 List and describe the variables that affect CHAPTER TWENTY
the value of an option. Calculate the value of
a call option and of a put option.

3 Name some of the real options that occur

in business and explain why traditional NPV

analysis does not accurately incorporate

Ethan Miller/Getty Images, Inc.; Getty Images, Inc.; Barry Sweet/ZUMA their values.
Press/©Corbis 4 Describe how the agency costs of debt and

S equity are related to options.
ometimes even the most carefully laid business plans must be 5 Explain how options can be used to manage
changed. This fact was especially apparent in Las Vegas, Nevada a firm’s exposure to risk.

after the U.S. economy went into a recession in December 2007. A

comparison of the decisions and outcomes at three casino/hotel/

condo projects illustrates how different management teams adjusted their plans to accommo-

date a changing economic environment.

Construction of Echelon Place, Fontainebleu, and The Cosmopolitan, all $3 to $4 billion

projects, were in relatively early stages when gaming revenues began to go into a freefall in late

2007. How severe the recession would be and how long it would last was the subject of much

debate in the business community throughout the following year. This uncertainty forced the

owners of the three Las Vegas projects to reexamine their timelines and operating plans.

The developer of Echelon Place quickly decided that the cash flow forecasts associated with

its project were unlikely to be realized in the deteriorating economic climate and announced in

August 2008 that construction would be suspended for three or four quarters. As conditions

642 CHAPTER 20 I Options and Corporate Finance

worsened over the following year, the suspension was extended to three to five years. Today, in
early 2011, the steel frame for the Echelon project stands near the north end of the Las Vegas
strip waiting for construction to resume.

Rather than suspending construction, the developer of Fontainebleu pushed ahead
with the project, apparently expecting the economy to turn around relatively quickly.
However, things only worsened. Fontainebleu was forced to file for Chapter 11 bankruptcy
protection and suspend construction in June 2009. A few months later the entire project,
on which $2 billion had been spent, was purchased by Carl Icahn in an auction for only
$156 million. Mr Icahn’s plans for the property were unknown, and the construction site
remained idle in July 2011.

The developers of The Cosmopolitan also decided to move forward with their project,
and, like the Fontainebleu project, The Cosmopolitan project was forced into bankruptcy.
However, in this case, the major lender, Deutche Bank, took ownership and decided to push
ahead with the project. The first stage of the Cosmopolitan opened on December 15, 2010,
but not without changes in its original plans. In an effort to increase long-term cash flows,
changes were made in the casino and a number of planned residential units were converted
into hotel rooms.

In 2007 the developers of these three projects all had a number of options with regards to
how they might react to the economic downturn. They could suspend construction, they could
slow construction, they could press ahead as planned, or they could sell their project. Further-
more, they could adapt to changes in the market by altering the design of their facilities and
how they would be used in an effort to increase forecasted cash flows. All of these options are
what we refer to as real options. This chapter discusses various ways in which options enter
into corporate financial decision making, and how they affect the value of a business.

CHAPTER PREVIEW Since financial options are commonly traded, we know a lot
about how they are valued.
Options and option-like payoffs complicate the analytical
frameworks that we have discussed in this book. Financial op- We then turn to real options, which affect the value of corpo-
tions, such as the right to buy or sell the shares of a company rate investments. As illustrated in the chapter opener, manag-
at a prespecified price, are often found in financial securities ers often have options to delay investing in a project, expand
that firms issue and therefore must be considered in the valua- a project, abandon a project, change the technology em-
tion of those securities. Real options, such as those discussed ployed in a project, and so on. You will see that the value of
in the chapter opener about the three Las Vegas casino/hotel/ these options is not adequately reflected in an NPV analysis.
condo projects, make calculation of the true NPV of a project
more complex. In order to fully understand the implications of We next revisit the agency costs of debt that we discussed in
these complications for financial analyses, it is important that Chapter 16. In particular, we show how option-like payoffs
you understand what options are and the types of options that contribute to the dividend payout, asset substitution, and
are available to managers or that they must contend with. underinvestment conflicts. We follow this discussion with a
related discussion of how option-like payoffs contribute to
We begin with a discussion of financial options and how they conflicts between stockholders and the managers who work
are valued because financial options are, in many ways, sim- for them. We conclude the chapter with a discussion of the
pler than real options to illustrate and value. Many financial ways in which managers use financial options to alter their
options are traded independently in the financial markets companies’ exposures to various types of risks.
while others are bundled with the financial instruments that
managers issue and that also trade in the financial markets.

20.1 Financial Options 643

20.1 FINANCIAL OPTIONS

A financial option is a derivative security in that its value is derived from the value of another LEARNING OBJECTIVE 1
asset. The owner of a financial option has the right, but not the obligation, to buy or sell an as-
set on or before a specified date for a specified price. The asset that the owner has a right to buy financial option
or sell is known as the underlying asset. The last date on which an option can be exercised is the right to buy or sell a financial
called the exercise date or expiration date, and the price at which the option holder can buy security, such as a share of stock,
or sell the asset is called the exercise price or strike price. on or before a specified date for
a specified price
Call Options
derivative security
Let’s consider how the value of an option is derived from the value of an underlying asset. Suppose a security that derives its value
you own an option to buy one share of IBM stock for $150 per share and today is the exercise from the value of another
date—if you don’t exercise the option today, it will expire and become worthless. If the price of IBM’s asset; an option is an example
stock is less than $150 per share, it does not make sense to exercise your option, because if you did, of a derivative security
you would be paying $150 for something you could buy for less than $150 in the open market.
Similarly, if the stock price is $150, there is no benefit to be had from exercising your option. If, underlying asset
however, the price is above $150, then you will benefit from exercising the option. Even if you do not the asset from which the value
want to own IBM stock, you can buy it for $150 and immediately turn around and sell it for a profit. of an option is derived
The value of the option to you is the difference between the market price of IBM stock and the exer-
cise price of the option. For example, if the IBM stock is trading for $160 per share in the market, exercise (expiration) date
then the option is worth $10 ($160 stock price Ϫ $150 exercise price ϭ $10) to you. If the stock is the last date on which an
trading for $170 per share, then the value of the option is $20 ($170 Ϫ $150 ϭ $20), and so on. option can be exercised

The relation between the value of an option and the price (value) of the underlying exercise (strike) price
asset—such as the IBM stock—is known as the option payoff function. Figure A in Ex- the price at which the owner of
hibit 20.1 illustrates the payoff function at expiration (actually, the instant before the op- an option has the right to buy
tion expires) for the owner of an option that is like the IBM stock option we just discussed. or sell the underlying asset
This option is known as a call option because it gives the owner the right to buy, or “call,”
the underlying asset. option payoff function
the function that shows how
With an exercise price of $150, the value of the IBM call option equals $0 if the price of the the value of an option varies
underlying stock is $150 or less. As we noted earlier, it would not make sense to exercise the option with the value of the
underlying asset

call option
an option to buy the
underlying asset

Value of CallFigure A. Owner (buyer) of a call option EXHIBIT 20.1
Option at Expiration Payoff Functions for a Call
The value of a call option increases Option at Expiration
dollar for dollar with an increase in theValue of Seller's
value of the underlying asset when thePosition at ExpirationAt the instant before it
value of that asset is above the exercise expires, the value of a call
price.of Call Option option to the owner equals
$0 either (1) $0, if the value of
Exercise the underlying asset is less
Price than or equal to the exercise
Value (Price) of Underlying Asset price, or (2) the value of the
underlying asset minus the
Figure B. Seller of call option exercise price, if the value of
the underlying asset is greater.
$0 The value of the seller's position
decreases dollar for dollar with an The value of the seller’s
increase in the value of the underlying position equals either (1) $0 if
asset when the value of that asset is the value of the underlying
above the exercise price of a call option. asset is less than or equal to
the exercise price or (2) the
Exercise exercise price minus the value
Price of the underlying asset if the
value of the underlying asset
Value (Price) of Underlying Asset is greater.

644 CHAPTER 20 I Options and Corporate Finance

all premium if the price of the stock is not greater than $150. Since an option is the right to buy or sell an under-
he price that the buyer of a lying asset, rather than an obligation to buy or sell, the owner of the option can simply let it expire
all option pays the seller for if it does not make sense to exercise it. This limits the downside for the owner of the option to $0.
hat option
If the underlying asset price is above the exercise price, the value of the call option at ex-
ercise increases dollar for dollar with the price of the underlying asset. You can see this relation
in Figure A of the exhibit. For every dollar that the asset price exceeds the exercise price, the
value of the call option increases by one dollar. In other words, the slope of the payoff function
equals one when the underlying asset price is above the exercise price.

Figure B of Exhibit 20.1 illustrates the payoff function for a person who sells a call option.
Notice that the payoff function for the seller is the mirror image of that for the owner (buyer)
of the call option. This makes sense, since any gain for the owner is a loss for the seller. To see
why this is true, let’s return to the IBM option example. Recall that if the stock is trading at
$160 when the option expires, the call option is worth $10 to the owner, who can purchase the
stock for $150 and then immediately sell it on the market for $160. The seller of the call option,
though, must sell a share of stock that is worth $160 for $150—resulting in a $10 loss.

Figure B of Exhibit 20.1 shows that the payoff to the seller of the call option is never posi-
tive. It is negative when the price of the underlying asset is greater than the exercise price, and
it equals zero when the price of the underlying asset is equal to or less than the exercise price.
You may be wondering why anyone would ever sell a call option if the return is never positive.
The reason is simply that the buyer pays the seller a fee to purchase the option. This fee, which
is known as the call premium, makes the total return to the seller positive when the price of
the underlying asset is near or below the exercise price.

A call premium is just like the premium you pay when you purchase insurance for your
car. In return for the insurance premium, the insurance company agrees to pay you if certain
events occur, such as if you collide with another car or if a hailstorm damages the car. The seller
of a call option is simply selling insurance to the buyer that pays the buyer when the value of
the underlying asset is above the exercise price.

put option Put Options
an option to sell the
underlying asset While the owner of a call option has the right to buy the underlying asset at a pre-specified price
on or before the expiration date, the owner of a put option has the right to sell the underlying
put premium asset at a pre-specified price. The payoff function for the owner of a put option is similar to that
he price that the buyer of a for a call option, but it is the reverse in the sense that the owner of a put option profits if the price
put option pays the seller of of the underlying asset is below the exercise price. This is illustrated in Exhibit 20.2.
hat option
Figure A of the exhibit shows that the owner of a put option will not want to exercise that
option if the price of the underlying asset is above the exercise price. Obviously, it does not
make sense to sell an asset for less than you can get on the open market. When the value of the
underlying asset is below the exercise price, however, the owner of the put option will find it
profitable to exercise the option. For example, suppose that you own a put option that is expir-
ing today and that entitles you to sell a share of IBM stock for $150. If the current price of IBM
stock in the market is $145, the put option is worth $5 because exercising the option will enable
you to buy a share of stock for $145 and then turn around and sell it for $150. Similarly, if the
current price of IBM stock is $130, the put option is worth $20 because you can buy the stock
for $130 and sell it for $150.

Figure B of Exhibit 20.2 shows that the payoff for the seller of the put option is negative
when the price of the underlying asset is below the exercise price. This is because the seller of
the put option is obligated to purchase the asset at a price that is higher than its market price.
For instance, in the IBM put option example, if the exercise price is $150 and the current mar-
ket price is $130, the seller of the put option must buy the stock for $150 but can only sell it for
$130. This results in a $20 loss.

As with a call option, the payoff for the seller of a put option, which is illustrated in
Figure B of Exhibit 20.2, is never positive. The seller of a put option hopes to profit from
the fee, or put premium, that he or she receives from the buyer of the put option.

American, European, and Bermudan Options

At the beginning of this section, we said that the owner of a financial option has the right to

Figure A. Owner (buyer) of a put optionValue of Put 20.1 Financial Options 645
Option at Expiration
The value of an expiring put option EXHIBIT 20.2
increases dollar for dollar with a decreaseValue of Seller'sPayoff Functions for Put
in the value of the underlying asset whenPosition at ExpirationOption at Expiration
the value of that asset is below the
exercise price.of Put Option At the instant before it
$0 expires, the value of a put
Exercise option to the owner equals
Price either (1) $0, if the value of
Value (Price) of Underlying Asset the underlying asset is greater
Figure B. Seller of a put option than or equal to the exercise
price, or (2) the exercise price
$0 The value of the seller's position minus the value of the
decreases dollar for dollar with a underlying asset, if the value
decrease in the value of the underlying of the underlying asset is less.
asset, when the value of that asset is
below the exercise price of a put option. The value to the seller of a put
option equals either (1) $0, if
Exercise the value of the underlying
Price asset is greater than or equal
to the exercise price, or (2) the
Value (Price) of Underlying Asset value of the underlying asset
minus the exercise price, if the
value of the underlying asset
is smaller.

there are actually several different arrangements concerning when an option can be exercised. You can learn more
Some options can only be exercised on the expiration date. These are known as European op- about call options and
tions. Other options, known as American options, can be exercised at any point in time on or put options on the
before the expiration date. There are also exotic options, such as so-called Bermudan options, Wikipedia Web site at
which can be exercised only on specific dates during the life of the option. Most exchange- http://en.wikipedia.org/
traded options (even in Europe) are American options. wiki/Call_option and
http://en.wikipedia.org/
More on the Shapes of Option Payoff Functions wiki/Put_option.

It is important to note that the payoff functions in Exhibits 20.1 and 20.2 illustrate the values
of options to owners and sellers at the instant before they expire. These payoff functions have
similar, but somewhat different, shapes at earlier points in time. We discuss why this is the case
in the next section.

When It Makes Sense to Exercise an Option DECISION
MAKING
SITUATION: You own a call option and a put option on a share of Ford Motor Com-
pany stock. The exercise price for both of these options is $18 per share, and both op- EXAMPLE 20.1
tions expire today. If the current price of Ford stock is $17, would you exercise either of
these options? If so, which one?

DECISION: You should exercise the put option. It allows you to sell a share of Ford
stock for $18 that would cost you only $17 to buy. It does not make sense to exercise the
call option because the exercise price is greater than the market price of Ford stock.

646 CHAPTER 20 I Options and Corporate Finance It is also important to recognize that the payoff
functions in Exhibits 20.1 and 20.2 are not straight
PAYOFF FUNCTIONS FOR OPTIONS ARE lines for all possible values of the underlying asset.
Each payoff function has a “kink” at the exercise
BUILDING NOT LINEAR price. This kink exists because the owner of the op-
INTUITION Payoff functions for options are not straight tion has a right, not an obligation, to buy or sell the
underlying asset. If it is not in the owner’s interest
lines. This is because the owners of options have to exercise the option, he or she can simply let it
the right, rather than the obligation, to buy or expire. Later, we will discuss how this feature of op-
sell the underlying assets. If it is not in the owner’s best interest to tions causes agency problems and how it can be
exercise an option, he or she can simply let it expire without exer- useful in managing the risks faced by a firm.
cising it. This limits the owner’s potential loss to the value of the
premium he or she paid for the option.

> BEFORE YOU GO ON

1 . What is a call option, and what do the payoff functions for the owner and
seller of a call option look like?

2 . What is a put option, and what do the payoff functions for the owner and
seller of a put option look like?

3 . Why does the payoff function for an option have a kink in it?

20.2 OPTION VALUATION

LEARNING OBJECTIVE We saw in the last section that determining the value of a call or a put option at the instant
before it expires is relatively simple. For a call option, if the value of the underlying asset is less
than or equal to the exercise price, the value of the option to the owner is $0. If the value of the
underlying asset is greater than the exercise price, the value to the owner is simply the value of
the underlying asset minus the exercise price. For a put option, if the value of the underlying
asset is greater than or equal to the exercise price, the value of the option is $0 to the owner. If
the value of the underlying asset is less than the exercise price, the value to the owner is the
exercise price minus the value of the underlying asset.

It is more complicated to determine the value of an option at a point in time before its
expiration date. We don’t know exactly how the value of the underlying asset will change over
time, and therefore we don’t know what value we will ultimately receive from the option. In
this section, we discuss the key variables that affect the value of an option prior to expiration
and describe one method that is commonly used to value options. Our objective is not to make
you an expert in option valuation but rather to help you develop some intuition about what
makes an option more or less valuable. This intuition will help you better understand how op-
tions affect corporate finance decisions.

Limits on Option Values

Let’s begin by using some common sense to put limits on what the value of a call option can
possibly be prior to its expiration date. We focus on call options here because, as you will see,
there is a simple relation that enables us to calculate the value of a put option once we know the
value of a call option with the same exercise price.

We already know that the value of a call option can never be less than zero, since the
owner of the option can always decide not to exercise it if doing so is not beneficial. A second
limit on the value of a call option is that it can never be greater than the value of the underlying
asset. It would not make sense to pay more for the right to buy an asset than you would pay for
the asset itself. These two limits suggest that the value of a call option prior to expiration must
be in the shaded area in Figure A of Exhibit 20.3. The shaded area is bounded below by the
horizontal axis, because the value of the option must be greater than $0, and it is bounded
above by the line that slopes upward at a 45-degree angle, because an option value greater than

20.2 Option Valuation 647

Figure A. Possible values with first two limitsValue of Call Option EXHIBIT 20.3
Possible Values of a Call
The first two limits tell usValue of Call Option Option Prior to Expiration
that the value of a call
Value of Call Option option prior to expiration The value of a call option:
must fall within this (1) must be greater than
shaded area. or equal to $0 (horizontal
axis) and (2) cannot be
$0 greater than the value of
Current Value of the underlying asset
Underlying Asset (45 degree line).

Figure B. Possible values with all four limits In addition to the two limits
illustrated in Figure A, the
The four limits tell us that value of a call option prior to
the value of a call option expiration: (3) will never be
prior to expiration will less than the value of the
actually fall within this option if it were exercised
shaded area. immediately where (4) the
value of the option is
$0 calculated using the present
Present Value of value of the exercise price,
Exercise Price discounted from the
Current Value of expiration date at the risk-free
Underlying Asset rate. These conditions are
both illustrated by the lower
Figure C. Typical payoff function for call option prior to expiration 45 degree line.

Value of call This figure shows the typical
option prior to relation between the value of
a call option prior to
expiration expiration and its value at
expiration. The value of the
$0 option prior to expiration is
Exercise farthest from the value of the
Price option at expiration when the
price of the underlying asset is
Current Value of near the exercise price.
Underlying Asset

There are two other limits on the value of a call option prior to expiration, and these limits
are somewhat more subtle. First, the value of a call option prior to the expiration date will
never be less than the value of that option if it had to be exercised immediately. This is true
because there is always a possibility that the value of the underlying asset will be greater than
it is today at some time before the option expires. Of course, it is possible that the value will be
lower, but since the value of the option cannot be less than $0 and there is no limit on how high
it can go, the expected effect of an increase in the value of the underlying asset on the value of
the option is greater than the expected effect of a decrease. The bottom line is that, prior to
expiration, the value of a call option will be greater than the value represented by the solid red
line in Figure A of Exhibit 20.1 (in the previous section of this chapter).1

The fourth and final limit arises because of the time value of money. When we consider
the value of a call option at some time prior to expiration, we must compare the current value

1Even if the value of the option ever fell below the line to the right of the exercise price in Figure A of Exhibit 20.1, it
would not stay there. This is because investors would be able to make an instant profit by buying the option, exercising
it to get the underlying asset, and then selling the underlying asset. Such trading by investors would drive the price of

648 CHAPTER 20 I Options and Corporate Finance

of the underlying asset with the present value of the exercise price, discounted at the risk-free
rate. We would be comparing apples and oranges if we did not do this. The present value of the
exercise price is the amount that an investor would have to invest in risk-free securities at any
point prior to the expiration date to ensure that he or she would have enough money to exer-
cise the option when it expired. Thus, when we compare the value of a call option prior to ex-
piration with the value at expiration, represented by the solid red line in Figure A of Exhibit
20.1, we must use the present value of the exercise price to draw the line. The shaded area in
Figure B of Exhibit 20.3 illustrates the possible values for a call option prior to expiration under
all four of the limits we have discussed.

In practice, we find that, prior to expiration, call options have a shape that is very similar
to the one illustrated by the dotted line in Figure C of Exhibit 20.3. Notice that this dotted line
approaches $0 as the value of the underlying asset gets very small relative to the exercise price.
This makes sense because, with a very low asset value, it becomes highly unlikely that the
owner of the option will ever choose to exercise it.

On the right side of the dotted line, you can see that the value of a call option prior to ex-
piration approaches the value of the call option at expiration. This is because when the current
value of the underlying asset is far to the right of the kink in the option’s payoff function, the
probability that this value will fall below the exercise price is very small. In other words, the
expected effect of an increase in the value of the underlying asset on the value of the option is
no longer much greater than the expected effect of a decrease.

Finally, notice that the dotted line is furthest above the value of the call option at expira-
tion when the price of the underlying asset is near the exercise price. At the exercise price, the
expected effect of an increase in the value of the underlying asset on the value of the option
exceeds the expected effect of a decrease by the greatest amount.

Variables That Affect Option Values

Five variables affect the value of a call option prior to expiration. Four of them are related to
the following questions:

1. How likely is it that the value of the underlying asset will be higher than the exercise price
the instant before the option expires?

2. How far above the exercise price might it be?

The first two variables are relatively easy to understand. They are the current value of the
underlying asset and the exercise price. The higher the current value of the underlying asset, the
more likely it is that the value of the asset will be above the exercise price when the call option
nears expiration. Furthermore, the higher the current value of the asset, the greater the likely
difference between the value of the asset and the exercise price. This means that, holding the
exercise price constant, investors will pay more for a call option if the underlying asset value is
higher, because the expected value of the option as it nears expiration is higher.2 For example,
suppose that you are considering purchasing a three-month American call option on a share
of IBM stock with an exercise price of $150. You should be willing to pay more for this option
if the current price of IBM stock is $155 than if it is $150.

The opposite relation applies to the exercise price. That is, the lower the exercise price, the
more likely that the value of the underlying asset will be higher than the exercise price when
the option nears expiration. In addition, the lower the exercise price, the greater the likely dif-
ference between these two amounts. Thus, the lower the exercise price, the more valuable the
option is likely to be at expiration. Of course, if the option is expected to be more valuable at
expiration, it will also be more valuable at any point prior to expiration. Returning to our IBM

2We are focusing in this discussion on what the value of the underlying asset is likely to be immediately before the
option expires because it does not generally make sense to exercise an option before then as long as there is a chance
that the value of the underlying asset could increase further. An exception is when the value of the underlying asset is
not expected to be higher as the expiration of the option nears because value is being distributed to the owners of the
underlying asset (for example, through dividend payments). In a situation like this, it can be appropriate to exercise
a call option immediately before such a payment. There are also situations where it is advantageous to exercise a put
option early. Such situations can arise if it is very likely that the option will be exercised at expiration. When this hap-
pens, the value received from exercising the option today can exceed the present value of the amount that is expected

example, we see that a call option with an exercise price of $145 is worth more than a call op- 20.2 Option Valuation 649
tion with an exercise price of $150.
You can read about what
We turn next to two variables that affect the value of call options in somewhat more subtle affects the values of
ways. These variables are the volatility of the value of the underlying asset and the time until the financial options and
expiration of the option. To understand how these factors affect the value of a call option, recall how they are traded at
from Figure C of Exhibit 20.3 that the payoff function for a call option prior to expiration is not the web sites for the
symmetric. If the value of the underlying asset is well above the exercise price, then the value Chicago Board Options
of the option varies in much the same way as the value of the underlying asset. However, if the Exchange (CBOE) at
value of the underlying asset is well below the exercise price, then the value of the option ap- http://www.cboe.com/
proaches $0 but changes at a much lower rate than the value of the underlying asset changes. It and the International
does not matter if the underlying asset value is just a little bit below the exercise price or is Securities Exchange
completely worthless—a call option cannot be worth less than $0. (ISE) at http://www
.iseoptions.com/.
To show how the volatility of the underlying asset value affects the value of an option, we
will consider a call option on an underlying asset that has a value exactly equal to the exercise
price of the option. The value of this option will increase more when the value of the underly-
ing asset goes up than it will decrease when the value of the underlying asset goes down. Let’s
suppose that the value of the underlying asset is equally likely to go up or down. In this case,
the further the value of the asset is likely to move (the greater its volatility), the higher the value
of a call option on this asset will be. In other words, the greater the volatility of the underlying
asset value, the higher the value of a call option on the asset prior to expiration.

In our IBM example, suppose the exercise price for a call option on IBM stock is $150, the
current price of the stock is $150, and the option expires in one year. Further suppose that
the standard deviation, ␴, of the return on the IBM stock is 30 percent per year. Recall from the
discussion in Chapter 7 that with a standard deviation of 30 percent, there is a 5 percent chance
that the IBM stock price will change by more than 58.8 percent (1.96 standard deviations ϫ 30
percent ϭ 58.8 percent) by the time the option expires. In other words, there is a 5 percent
chance that the IBM stock price will be less than $61.80 [$150 ϫ (1 Ϫ 0.588) ϭ $61.80] or
greater than $238.20 [$150 ϫ (1 ϩ 0.588) ϭ $238.20] in a year. If, instead of 30 percent, the
standard deviation of IBM stock were 40 percent per year, there would be a 5 percent chance
that the price would be below $32.40 or above $267.60. (You should check these numbers to
make sure you know how they are calculated.) As you can see, this higher standard deviation
means the stock price is more volatile. Investors will pay more for an option on a stock that has
a more volatile price, because the potential change in the price is greater.

The time until the expiration affects the value of a call option through its effect on the
volatility of the value of the underlying asset. The greater the time to maturity, the more
the value of the underlying asset is likely to change by the time the option expires. For
example, let’s return once again to the IBM example. Suppose that the option expires in two
years rather than in one year. People who study statistics have found that the standard devia-
tion of the return on an asset increases over time by the square root of n, where n is the number
of periods. Thus, if the standard deviation of the return on IBM stock is 30 percent per year, the
standard deviation over two years will be:

s2 years ϭ s ϫ 1n21/2 ϭ 30 ϫ 12 years21/2 ϭ 30 ϫ 1.414 ϭ 42.42%

Clearly, then, a two-year option will be worth more than a one-year option if all other charac-
teristics of the options are the same.

We’ve now discussed four of the five variables that affect the value of an option. The fifth
variable is the risk-free rate of interest. The value of a call option increases with the risk-free
rate. Exercising a call option involves paying cash in the future for the underlying asset. The
higher the interest rate, the lower the present value of the amount that the owner of a call op-
tion will have to pay to exercise it.

The Binomial Option Pricing Model

In this section, we use a simple model to show how we can calculate the value of a call option
at some point before the expiration date. This model assumes that the underlying asset will
have one of only two possible values when the option expires. The value of the underlying asset
will either increase to some value above the exercise price or decrease to some value below the

650 CHAPTER 20 I Options and Corporate Finance

arbitrage To solve for the value of the call option using this model, we must assume that investors
buying and selling assets in a have no arbitrage opportunities with regard to this option. Arbitrage is the act of buying
way that takes advantage of and selling assets in a way that yields a return above that suggested by the Security Market
price discrepancies and yields Line (SML), which we discussed in Chapter 7. In other words, the absence of arbitrage op-
a profit greater than that which portunities means that investors cannot earn a return that is greater than that justified by the
would be expected based systematic risk associated with an investment. As an example of an arbitrage opportunity,
olely on the risk of the suppose that the stock of a particular company is being sold for a lower price in one country
ndividual investments than in another country. An investor could simultaneously buy the stock in the country
where it is less expensive and sell it in the country where it is more expensive. Assuming that
the profit exceeds any transaction costs, the investor would earn an instantaneous risk-free
profit. Since it is instantaneous, this profit would, by definition, be above the SML because
the SML would predict that the expected return on a risk-free investment is zero if the hold-
ing period is zero.

To value the call option in our simple model, we will first create a portfolio that consists of
the asset underlying the call option and a risk-free loan. The relative investments in these two
assets will be selected so that the combination of the asset and the loan have the same cash
flows as the call option, regardless of whether the value of the underlying asset goes up or
down. This is called a replicating portfolio, since it replicates the cash flows of the option. The
replicating portfolio must have the same value as the option today, since it has the same cash
flows as the call option in all possible future outcomes. If the replicating portfolio did not have
the same value as the option, an investor could construct an arbitrage portfolio by buying the
cheaper of the two and selling the more expensive of the two. Such trading would eventually
drive the values of the option and the replicating portfolio together.

To see how a replicating portfolio is constructed, consider an example. Suppose that
the stock of ABC Corporation currently trades for $50 and that its price will be either $70
or $40 in one year. We want to determine the value of a call option to buy ABC stock for
$55 in one year. First, notice that the value of this option is $15 if the stock price goes up to
$70 ($70 Ϫ $55 ϭ $15) and that it is $0 if the stock price goes down to $40, since the option
will not be exercised. Suppose also that the risk-free rate is 5 percent.

We can construct a portfolio consisting of x shares of ABC Corporation stock and a
risk-free loan with a value of y dollars that produces a payoff of either $70 or $40. The risk-
free loan may involve either borrowing or lending, as you will see. For each risk-free dollar
we lend, we know that we will receive $1.05 regardless of what happens to the price of ABC
stock. In the same way, if we borrow $1, we will owe $1.05 at the end of the year. The value
of the stock, the risk-free loan, and the option today and at expiration can be illustrated
as follows.

Today Stock (x) Risk-Free Loan (y) Option
$50 $1 ?

Expiration $40 $70 $1.05 $1.05 $0 $15

The value of each asset when the stock price goes up to $70 is shown on the right arrow, and
the value when the stock goes down to $40 is shown on the left arrow. Notice that we do not
know the value of the option today—that is what we are trying to calculate.

We can write two equations that define the replicating portfolio that we want to construct:

$15 ϭ 1$70 ϫ x2 ϩ 11.05 ϫ y2
$0 ϭ 1$40 ϫ x2 ϩ 11.05 ϫ y2

The first equation represents the case in which the stock price increases to $70, and the second
equation represents the case in which the stock price goes down to $40. The first equation says
that we want the portfolio to be worth $15 when the stock price increases to $70 and that the
$15 value will consist of x shares of stock worth $70 and a risk-free loan with a face value of y

20.2 Option Valuation 651

if the stock price falls to $40, we want the portfolio to be worth $0. In this case, the portfolio
will consist of x shares of stock worth $40 and a risk-free loan with a face value of y and a value
in one year of $1.05 per dollar of face value.

Since we have two equations and there are two unknowns, x and y, we can solve for the
values of the unknowns. Recall from your algebra class that we can solve for x and y by first
writing one equation in terms of either x or y and then substituting the result into the second
equation. For example, the first equation can be written in terms of x as follows:

$15 Ϫ 11.05 ϫ y2
x ϭ $70
Now, substituting into the second equation gives us:

$15 Ϫ 11.05 ϫ y2
$0 ϭ a$40 ϫ $70 b ϩ 11.05 ϫ y2
We can now solve this equation for y as follows:

$0 ϭ a$40 ϫ $15 b Ϫ a$40 ϫ 1.05 ϫ y ϩ 11.05 ϫ y2
$70 $70 b

$0 ϭ $8.5714 Ϫ 10.6 ϫ y2 ϩ 11.05 ϫ y2

$0 ϭ $8.5714 ϩ 0.45y

0.45y ϭ Ϫ$8.5714

Therefore:

y ϭ Ϫ$8.5714 ϭ Ϫ$19.05
0.45

Finally, substituting this value back into the first equation gives us the value of x:

$15 Ϫ 11.05 ϫ Ϫ$19.052
x ϭ $70

x ϭ $15 ϩ $20.00
$70

x ϭ 0.5

This tells us that the replicating portfolio consists of one-half share of ABC Corporation
stock (x ϭ 0.50) and a $19.05 risk-free loan (y ϭ Ϫ19.05). The negative value for y tells us that
we would borrow, rather than lend, $19.05 at the risk-free rate. If we buy one-half share of
stock and borrow $19.05, then in one year our replicating portfolio will have exactly the same
payoff as the call option with an exercise price of $55.

If the value of the stock declined to $40, we would own one-half share of stock worth $20,
and we would owe $19.05 ϫ 1.05 ϭ $20 on the loan. Since the value of the stock would exactly
equal the amount owed on the loan, the portfolio would have a total value of exactly $0. In
contrast, if the value of the stock increased to $70, the one-half share of stock would be worth
$35. Since we would still owe only $20 in this case, the portfolio would have a total value of
$15. Since these payoffs are exactly the same as those for the option, this portfolio must have
the same value as the option.

At this point, we know what the replicating portfolio is, and we know that the replicat-
ing portfolio must have the same value as the call option. Now all we have to do to estimate
the value of the call option is to figure out what the value of the replicating portfolio is. To
do this, we simply determine how much of our own money we would actually have to invest
to construct the replicating portfolio. In our example, we could use the $19.05 loan to help
purchase the stock, so we would not have to come up with all the money for the stock on
our own. In fact, since a share of ABC Corporation stock is currently worth $50, one-half
share of this stock would cost only $25. Therefore, we would have to come up with only
$5.95 ($25.00 Ϫ $19.05 ϭ $5.95) over and above the amount received from the loan to buy
the stock. Since $5.95 is the amount of money that we would actually have to invest to ob-
tain the replicating portfolio, it is the value of this portfolio and therefore the value of the

652 CHAPTER 20 I Options and Corporate Finance

The equation for calculating the value of the replicating portfolio, and therefore the value
of the call option, can be expressed as follows:

Value of the call option today ϭ C ϭ 1$50 ϫ x2 ϩ 11 ϫ y2
ϭ 1$50 ϫ 0.52 ϩ 11 ϫ Ϫ$19.052
ϭ $5.95

Notice, too, that the exercise price, the current price of the underlying stock, the possible
future prices of the underlying stock, and the risk-free rate are all that entered into our calcula-
tions. We did not even mention the probabilities that the stock price would go up or down at
any point. That is because the volatility of the underlying stock value is accounted for by how
far apart the two possible future values are. Similarly, the time to expiration is not directly
considered. However, the time to expiration affects how high and how low the stock price can
be when the option expires.3

This model may seem surprisingly simple. However, that is largely because we chose to illus-
trate a simple example. The model can be extended in several ways. For example, we can incorpo-
rate possible prices for the underlying asset between now and the expiration date of the option. The
underlying asset price might take one of two values one month (or day or hour) from now, and then
for each of those values there might be two possible values in the following month (day or hour),
and so on. Solving a model such as this requires us to work backwards from the expiration date to
find the value of the option at each intermediate date and price until we finally arrive at the value of
the option today. Most modern option pricing models are extensions of this type of model.

LEARNING APPLICATION 20.1 Valuing a Call Option
BY
DOING PROBLEM: You are considering purchasing a call option on the stock of Grote Agricul-
tural Company. Grote stock currently trades for $35 per share, and you predict that its
price will be either $25 or $50 in one year. The call option would enable you to buy
a share of Grote stock in one year for $30. What is this option worth if the risk-free rate is
4 percent?

APPROACH: The value of the option can be determined by computing the cost of
constructing a portfolio that replicates the payoffs from that option.

SOLUTION: With an exercise price of $30, the option will be worth $20 if the stock
price rises to $50 ($50 Ϫ $30 ϭ $20) and will be worth $0 if the stock price declines to
$25. Therefore, the replicating portfolio for this option can be determined from the fol-
lowing two equations:

$20 ϭ 1$50 ϫ x2 ϩ 11.04 ϫ y2
$0 ϭ 1$25 ϫ x2 ϩ 11.04 ϫ y2

Solving for x and y, we find that x ϭ 0.80 and y ϭ Ϫ$19.23. Therefore, the replicating
portfolio consists of 0.8 share of Grote stock and a $19.23 loan. Since a 0.8 share would
cost $28 (0.8 ϫ $35 ϭ $28), and $19.23 of this amount would be covered by the loan, this
replicating portfolio would cost $8.77 ($28.00 Ϫ $19.23 ϭ $8.77) to construct. Therefore,
the call option is worth $8.77.

Put-Call Parity

put-call parity To this point, our discussion has focused on call options. As mentioned earlier, this is possible
he relation between the value because there is a simple relation that enables us to calculate the value of a put option once we
of a call option on an asset know the value of a call option with the same exercise price. This relation is called put-call
and the value of a put option parity. The formula for put-call parity is:
on the same asset that has the
ame exercise price P ϭ C ϩ XeϪrt Ϫ V (20.1)

3There are other ways to solve the binomial pricing problem than by actually finding an equivalent portfolio. While they

20.2 Option Valuation 653

where P is the value of the put option, C is the value of the call option, X is the exercise price,
r is the risk-free rate, t is the amount of time before the option expires, and V is the current
value of the underlying asset. The term eϪrt is the exponential function that you can calculate
using the “ex” key on your calculator; it is simply a discount factor that assumes continuous
compounding. It is important to make sure that the r and t are both stated in the same units of
time (for example, months or years).

To see how this formula works, let’s consider the option on the stock of ABC Corporation
that we just valued. We know that C ϭ $5.95, X ϭ $55, r ϭ 0.05, t ϭ 1, and V ϭ $50. Substitut-
ing these values into the put-call parity formula and solving for P, we get

P ϭ $5.95 ϩ $55eϪ10.052112 Ϫ $50
ϭ $5.95 ϩ $52.32 Ϫ $50
ϭ $8.27

Notice that the variables used in this calculation are the same variables that determine the
value of a call option. This means that the same factors that affect the value of a call option also
affect the value of a put option. Notice, too, that the value of the put option ($8.27) is greater
than the value of the call option ($5.95) in this example. This will not always be true. However,
it is true in our example because the current stock price of $50 is below the $55 exercise price.

Valuing a Put Option APPLICATION 20.2 LEARNING
BY
PROBLEM: In Learning by Doing Application 20.1, we found that a call option on a
share of Grote Agricultural Company stock is worth $8.77 when the stock price is $35, the DOING
exercise price is $30, the risk-free rate is 4 percent, and the time to maturity is 1 year.
What is the value of a put option on a share of this stock if the exercise price and all other
variables have the same values?

APPROACH: Use the put-call parity relation, Equation 20.1, to calculate the value of a
put option.

SOLUTION: The value of the put option is as follows:

P ϭ C ϩ XeϪrt Ϫ V
ϭ $8.77 ϩ $30eϪ 10.042112 Ϫ $35
ϭ $8.77 ϩ $28.82 Ϫ $35
ϭ $2.59

Note that the value of the put option is less than the value of the call option in this ex-
ample. This is because the current price of the stock is above the exercise price.

Valuing Options Associated with the Financial

Securities That Firms Issue

In the chapter preview we stated that financial options are often included in the financial
securities that firms issue and that they make the valuation of those securities more compli-
cated. A detailed discussion of the valuation of financial securities with options is beyond
the scope of this chapter. However, because such options are quite common, it is important
that you have some intuition concerning how they affect security values. The key principle
that we use in valuing securities with options is known as the principle of value additivity. It
states that if two independent assets are bundled together, the total value of both assets
equals the sum of their individual values. In other words, the value of a financial security
with an option equals the value of the same security without the option, plus the value of the
option. To illustrate this idea, let’s consider a few of the many options that are commonly
observed in financial securities.

Financial options are often added to the securities that firms issue because doing so is

654 CHAPTER 20 I Options and Corporate Finance

issue debt, they must be concerned about the amount of cash required to make interest and
principal payments. If these payments are too great, the company’s operations might not gen-
erate enough cash to both service the debt and fund the company’s growth. One way to reduce
the interest payments on debt is to make it convertible into common stock.

To see how this works, consider the convertible bonds that we described in Chapter 8.
Suppose that a 20-year vanilla bond issued by a particular company must have coupon pay-
ments of $80 per year, or 8 percent, in order to sell for its par value of $1,000. Further suppose
that management of that company must raise $50 million today and only expects to have
enough cash to pay interest of $3 million per year, or 6 percent, on the $50 million.

One way to reduce the amount of interest that the firm must pay on the bonds is to make
them convertible into the company’s stock. For example, if the company’s stock is currently
trading at $40 per share, the bond might be structured so that buyers have the option (right,
but not obligation) to convert each bond into 20 shares of stock. With this arrangement,
each bond includes a call (conversion) option with an exercise price of $50 per share
($1,000/20 shares ϭ $50 per share). The exercise price of the conversion option is above the
current stock price. However, since there is a chance that the stock price will go above $50
before the debt matures in 20 years, this call option has a value which can be calculated using
the binomial option pricing model.

When a conversion option is included with a bond, investors will be willing to accept a
lower interest rate. How much lower depends on the value of the option. If the company
wants to sell the convertible bonds at their par value of $1,000, the present value of the inter-
est and principal payments plus the value of the conversion option must equal $1,000. In the
example above, if the bonds are going to pay 6 percent, the conversion option must be worth
$197.30. This is because the valuation methods discussed in Chapter 8 tell us that a 20-year
bond paying a coupon of 6 percent is only worth $802.70 if the market requires a coupon
rate of 8 percent (you might check this number to confirm that you understand the bond
valuation concepts from Chapter 8). If a conversion option with an exercise price of $50 is
worth more or less than $197.30, then management will have to adjust the exercise price
upward or downward until the total value of the 6 percent bond plus the conversion option
equals $1,000.

Convertible preferred stock provides another common example of a financial security that
has an option associated with it. This type of preferred stock, which is typically sold to venture
capitalists, for example, is convertible into the common stock of the company at a prespecified
exercise price. Recall from Chapter 9 that regular preferred stock with no maturity can be val-
ued using the zero-growth dividend model, Equation 9.2:

P0 ϭ D
kps

For example, if the preferred stock pays an annual dividend, D, of $10 and the required rate of
return, kps, is 10 percent, then the value of the preferred stock is $100 ($10/0.10 ϭ $100). If this
preferred stock is made convertible into the company’s common stock, its value will be greater
than $100 by an amount that equals the value of the conversion option. The company will get
a higher price for convertible preferred stock because it is selling investors both regular pre-
ferred stock plus a conversion option.

Convertible bonds and preferred shares are not the only types of securities that firms
issue with options attached to them. Another common transaction where managers sell fi-
nancial securities with options is when they bundle options to purchase a company’s com-
mon stock with common shares that are being sold in an initial public offering (IPO). When
this happens, for each 100 shares that an investor purchases, he or she also receives options
(which are called warrants in these instances) to purchase additional shares, on or before a
specified future date, for a price that is higher than the IPO price. For example, if the shares
are expected to sell for $10 each in the IPO, the investor might have the option to purchase
a certain number of shares at any time in the next five years for $15 per share. Why would
the managers of a firm bundle options with stock in an IPO? One reason is to reduce the
number of common shares that must be sold at the IPO price in order to raise the amount of
money that the firm needs. As was the case with convertible bonds and preferred stock, since
the options have value, investors will pay a higher price for the package of stock plus options

20.3 Real Options 655

> BEFORE YOU GO ON

1 . What are the limits on the value of a call option prior to its expiration date?

2 . What variables affect the value of a call option?

3 . Why are the variables that affect the value of a put option the same as those
that affect the value of a call option?

20.3 REAL OPTIONS

Many investments in business involve real options—options on real assets. Unfortunately, as LEARNING OBJECTIVE 3
we mentioned earlier, NPV analysis does not adequately reflect the value of these options.
While it is not always possible to directly estimate the value of the real options associated with real option
a project, it is important to recognize that they exist when we perform a project analysis. If we An option for which the
do not even consider them, we are ignoring potentially important sources of value. In this sec- underlying asset is a real asset
tion, we provide an overview of the types of real options commonly associated with real invest-
ments. As you read this section you should note that the first three types of real options—options
to defer investments, make follow-on investments, and change operations—are call options
while the fourth type of real option—the option to abandon a project—is a put option.

Options to Defer Investment You can find a list of
Web sites with
In the chapter opener, we used three large casino/hotel/condo projects in Las Vegas to illus- information about real
trate some real options that are commonly available to business managers. These include the options at http://www
option to suspend or defer completing the investment. Real estate development projects can .real-options.com/
often be suspended if the expected cash flows decline or become less certain. The expected cash resources_links.htm.
flows from all three Las Vegas projects declined and became less certain when the economy
went into recession in December 2007. In response, the managers of Echelon Place decided
to exercise the option to defer completing construction while the managers of Fontainebleu
and The Cosmopolitan decided to press ahead. By suspending construction, the managers of
Echelon Place gave themselves the opportunity to assess the severity of the recession before
investing additional money. This is equivalent to waiting to see what happens to a stock’s price
before deciding whether to exercise a financial call option on it. In the end, the managers of
Echelon place decided to delay the project even more than initially anticipated. Their decision
almost certainly saved the owners of that project a lot of money. Of course, the owners of
Fontainebleu and The Cosmopolitan lost their investments when those projects filed for
bankruptcy.

The earlier an investment is deferred, the greater the potential benefit from exercising that
option. It is relatively rare for a real estate project to be suspended once construction has pro-
gressed as far as it had with the Las Vegas projects. While developers typically have the right to
do this, they tend not to even begin construction unless they are highly confident that the proj-
ect will be completed. Instead, developers often purchase deferral options that can be exercised
before construction begins. Specifically, they purchase options on properties that they might
want to develop in the near future. For example, a developer might pay a landowner $100,000
for a one-year option to purchase a property at a particular price. By accepting the payment, the
landowner agrees not to sell the property to anyone else for a year. Such an option provides
the developer with time to make a final decision regarding whether or not to actually purchase
the land and proceed with a project. While the underlying asset for a financial option might
be a share of stock, the underlying asset for the developer’s option is land. Since the developer
will still have to buy the land if he or she decides to proceed with the project, the cost of the
option reflects a cost of being able to collect more information before making a final decision.

Another common example of an option to defer investment is found in the oil industry.
Many oil companies own drilling rights on properties that are expected to contain oil deposits,
but that have not yet been developed. In these situations, the oil companies have the option to
wait and see what happens to oil prices before deciding whether to invest in developing the

656 CHAPTER 20 I Options and Corporate Finance

field is expected to produce, while the exercise price is the amount of money that the company
would have to spend to develop it (drill the well and build any necessary infrastructure). Just
as the value of a share of stock might go up or down, the value of the cash flows produced by
the oil field might increase or decrease with the price of oil.

The value of an option to defer investment is not reflected in an NPV analysis. Recall that
the NPV rule tells us to accept a project with a positive NPV and to reject one with a negative
NPV. NPV analysis does not allow for the possibility of deferring an investment decision (or
deferring completion of a project once it is underway). It assumes that we invest either now or
never. However, if we have the option of deferring an investment decision, it may make sense
to do so. After all, a project that has a negative NPV today might have a positive NPV at some
point in the future. The price of the product may increase, production costs may decline, or the
cost of capital may go down, making the project attractive. We need not assume that an invest-
ment that is unattractive today will never be attractive.

Real options are Options to Make Follow-On Investments
considered by NASA
when space systems and Another very important type of real option is an option to make follow-on investments. Some
other investments are projects open the door to future business opportunities that would not otherwise be avail-
evaluated. See the able. For example, until the late 1990s, Dell, Inc., focused on selling computers to businesses.
following page on the Although the company sold computers to individuals for home use, it did not focus on that
NASA Web site for market segment. In the late 1990s, Dell decided to target the home personal computer mar-
references to additional ket and introduced a low-price, bare-bones computer. At first glance, this did not look like a
readings in this area: very good move, because the low-end home computer business has small profit margins.
http://ceh.nasa.gov/ However, the move created options for a wide range of follow-on investments. By moving
webhelpfiles/Real_ into the home computer market, Dell established relationships with many individual con-
Option_Valuation.htm. sumers. These relationships, in turn, made it feasible for Dell to later move into new areas,
such as the sale of cameras, TVs, MP3 players, and other consumer electronics goods. In
other words, investing in the home computer business provided Dell with options to enter
other consumer product markets.

Another example of an option to make follow-on investments concerns an investment in a
new technology that can be extended to other products. For instance, in the early 1990s, Boeing
Company invested in a computer-aided aircraft design system as part of the development of its
Boeing 777 aircraft. This system allowed the company to complete much more of the design work
for a new aircraft on a computer before building a prototype, thereby lowering the cost of design-
ing and building a new aircraft. While the cost of the new system and the associated facilities—
over $1 billion—was relatively high compared with the cost of the 777 project, the investment
provided benefits that extended well beyond that project. For example, the technologies could be
used in the design of other new aircraft, both civilian and military. By reducing the cost of devel-
oping new aircraft, the design system had the potential to make projects economically attractive
that would not have been attractive otherwise.

Options to make follow-on investments are inherently difficult to value because, at the
time we are evaluating the original project, it may not be obvious what the follow-on projects
will be. Even if we know what the projects will be, we are unlikely to have enough information
to estimate what they are worth. Of course, this makes it impossible to directly estimate the
value of any option associated with them. Nevertheless, it is important for managers to con-
sider options to make follow-on investments when evaluating projects. Doing so is a central
part of the process of evaluating projects in the context of the overall strategy of the firm. Proj-
ects that lead to investment opportunities that are consistent with a company’s overall strategy
are more valuable than otherwise similar projects that do not.

Options to Change Operations

In addition to options to defer investment and options to make follow-on investments,
which are real options related to the investment decisions themselves, there are also real op-
tions that are related to the flexibility managers have once an investment decision has been
made. These options, which include the options to change operations and to abandon a proj-
ect, affect the NPV of a project and must be taken into account at the time the investment

20.3 Real Options 657

In an NPV analysis, we discount the expected cash flows from a project. We often consider
several alternative scenarios and use our estimates of the probabilities associated with those
scenarios to compute the expected cash flows. While this sort of analysis does consider alterna-
tive scenarios, it does not fully account for the fact that once a project has begun, the managers
at a company have options to change operations as business conditions change. This means that
there is value associated with being able to change operations that is not fully reflected in a
scenario analysis.

The changes that managers might make can involve something as simple as reducing out-
put if prices decline or increasing output if prices increase. Businesses do this all the time in
response to changing demand for their goods and services. At the extreme, managers might
temporarily suspend operations entirely if business conditions are weak. This is quite common
in the auto industry, where we often hear of plants being temporarily shut down during peri-
ods of slow auto sales. Other changes in operation can involve fundamentally altering the way
in which a product is produced, as when a new production technology becomes available,
making the old technology uncompetitive.

Having the flexibility to react to changing business conditions can be very valuable. Since we
do not know precisely how conditions are likely to change it can be difficult to estimate just how
valuable this flexibility is. Nevertheless, we can see that managers do recognize the importance of
flexibility by observing how they structure projects. For example, most modern office buildings
do not have permanent internal walls. Not having permanent walls provides flexibility in config-
uring the offices and work spaces in the building. If more people must be put into a building than
originally anticipated, the work spaces can be compressed to fit them. If the company finds that
it does not need all of the space, having a flexible interior makes it easier to change things so that
the excess space can be leased. Similarly, when a company plans to build a new manufacturing
facility, it often acquires more land that is immediately needed and designs the facility to accom-
modate additional production capacity if demand for its products is greater than expected.

Building flexibility into a project costs money, but this can be money well spent if things
change unexpectedly. The flexibility to expand, scale back, or temporarily shut down a project
or to change the methods or technology employed in a project are all options that managers
should consider when evaluating projects. Projects with more flexibility in these dimensions
are inherently more valuable.

Options to Abandon Projects

A project can also be terminated if things do not go as well as anticipated.4 In other words,
management often has an option to abandon a project. The ability to choose to terminate a
project is a bit like a put option. By shutting down the project, management is saving money
that would otherwise be lost if the project kept going. The amount saved represents the gain
from exercising this option.

As with flexibility, we can see that managers recognize the importance of having an option
to abandon a project by observing the way they design projects. Consider, for example, that
most industrial buildings are built like big boxes that can be easily reconfigured as manufactur-
ing spaces, warehouses, or even retail outlets, depending on which use is most valuable. Sup-
pose a company is building a facility to use as a warehouse. If the building is only able to ac-
commodate a warehouse, it might end up sitting empty for long periods of time—for example,
if the area has excess warehouse space at some point in the future. Designing the building so
that it can be reconfigured relatively inexpensively for some other use increases the likelihood
that the building will remain fully utilized in the future.

Concluding Comments on NPV Analysis

and Real Options

We have stated that NPV analysis does not account for real options very well. This is true
because the riskiness of a project that has real options associated with it varies with time, and
the appropriate discount rate varies with the risk. For example, in order to use NPV analysis

4An exception exists where a contractual agreement prevents the project from being terminated without payment of a

658 CHAPTER 20 I Options and Corporate Finance

to value an option to expand operations, we would not only have to estimate the expected
value of all the cash flows associated with the expansion but would also have to estimate the
probability that we would actually undertake the expansion under alternative future scenar-
ios and determine the appropriate rate(s) at which to discount the incremental cash flows
from the expansion back to the present. Furthermore, the discount rate for the original proj-
ect cash flows could change with the expansion.

In some cases, we can incorporate the value of a real option into an investment analysis by
valuing the option separately and then adding this value to the NPV estimate. When we do
this, we value the real option using valuation methods similar to those used to value financial
options, as illustrated in Section 20.2.

DECISION The Value of Real Options
MAKING
SITUATION: You work for a company that manufactures cardboard packaging for
EXAMPLE 20.2 consumer product companies under long-term contacts. For example, your company
manufactures the boxes for several popular cereal and aspirin products. You have just
won a large five-year contract to produce packaging materials for a company that sells
furniture on the Internet. Since this contract will require you to produce much larger
boxes than you currently can produce, you must purchase some new equipment. You
have narrowed your choices to two alternatives. The first is a capital-intensive process
that will cost more up front but will be less expensive to operate. This process requires
very specialized equipment that can be used only for the type of packaging that your
furniture client needs. The second alternative is a labor-intensive process that will
require a smaller up-front investment but will have higher unit costs. This process in-
volves equipment that can be used to produce a wide range of other packages. If the
expected life of both alternatives is 10 years and you estimate the NPV to be the same
for both, which should you choose?

DECISION: You should choose the labor-intensive alternative. Your contract is only for
five years, and there is a chance that it will not be renewed before the equipment’s useful
life is over. If the contract is not renewed, it will be easier to convert the labor-intensive
equipment to another use. In other words, the labor-intensive alternative gives you the
added value of having the option to abandon producing packaging for furniture.

> BEFORE YOU GO ON
1 . What is a real option?
2 . What are four different types of real options commonly found in business?
3 . Is it always possible to estimate the value of a real option? Why or why not?

20.4 AGENCY COSTS

LEARNING OBJECTIVE Agency conflicts arise between stockholders and lenders (creditors and bondholders) and
between stockholders and managers because the interests of stockholders, lenders, and man-
agers are not perfectly aligned. In fact, their interests can differ greatly. One reason is that the
claims that they have against the cash flows produced by the firm have payoff functions that
look like different types of options. We now discuss how these payoff functions lead to agency

20.4 Agency Costs 659

Agency Costs of Debt

In Chapter 16, we discussed agency costs that arise in a company that uses debt financing.
We noted that these costs occur because the incentives of people who lend to a company
differ from those of the stockholders. If you were to carefully reread those discussions
now, you might notice that the problems we discussed arise because the payoff functions
for stockholders and lenders differ like those for the different options we have been
discussing.

To understand why this is the case, consider a company that has a single loan outstanding.
This loan will mature next year, and all of the interest and principal will be due at that time.
Now, consider what happens when the debt matures. On the one hand, if the value of the com-
pany is less than the amount owed on the debt, the stockholders will simply default, and the
lenders will take control of the assets of the company. The stockholder claims will be worth $0
in this case. If, on the other hand, the value of the company is greater than the amount owed
on the loan, the stockholders will pay off the loan and retain control of the assets. In this case,
the stockholder claims will be worth the difference between the value of the firm and the
amount owed to the lenders.

In other words, the payoff function for the stockholders looks exactly like that for the
owner of a call option, where the exercise price is the amount owed on the loan and the under-
lying asset is the firm itself. If the value of the firm exceeds the exercise price, the stockholders
will choose to exercise their option; and if it does not exceed the exercise price, they will let
their option expire unexercised. Figure A of Exhibit 20.4 illustrates the payoff function for the
stockholders in this simple example.

The payoff function for the lenders in our example is illustrated in Figure B of Exhibit 20.4.
If the value of the firm is less than the amount owed, the lenders receive only the assets of the
firm; and if the value of the firm is greater than the amount owed, the lenders receive only the
amount owed. One way to think about the payoff function for the lenders is that when they
lend money to the firm, they are essentially selling a put option to the stockholders.5 This op-
tion gives the stockholders the right to “put” the assets to the lenders with an exercise price that
equals the amount they owe. When the value of the firm is less than the exercise price, the
stockholders will exercise their option by defaulting. Of course, the stockholders are able to
default and walk away only because our bankruptcy laws limit their liability to the amount that
they have invested in the company.

The Dividend Payout Problem

Knowing that debt and equity claims are like options in which the underlying asset is the firm,
we can use the intuition gained from the discussion of the determinants of option value to bet-
ter understand the agency costs of debt. The incentives that stockholders of a leveraged firm
have to pay themselves dividends arise because of their option to default. If a company faces
some realistic risk of going bankrupt, the stockholders might decide that they are better off
taking money out of the firm by paying themselves dividends. This situation can arise because
the stockholders know that the bankruptcy laws limit their possible losses. If the firm goes
bankrupt and the lenders end up receiving, for example, 50 percent rather than 80 percent of
what they are owed, it will make no difference to the stockholders, who will get nothing from
the liquidation of the company’s assets in either case.

The Asset Substitution Problem

In Chapter 16, we saw that when bankruptcy is possible, stockholders have an incentive to
invest in very risky projects, some of which might even have negative NPVs. Stockholders
have this incentive because they receive all of the benefits if things turn out well but do not
bear all of the costs if things turn out poorly. Since equity claims are like call options on the
assets of the firm, this asset substitution problem should not be surprising. We pointed out

5This payoff function is actually like that from the combination of selling a put option and buying a risk-free loan.
Lenders receive the face value of the loan from the risk-free bond, but they might have to pay some or all of that value
in losses on the put option. Since the risk-free loan payout is unaffected by changes in the value of the firm, it does not

660 CHAPTER 20 I Options and Corporate Finance

EXHIBIT 20.4 Figure A. Stockholder payoff function
Payoff Functions for
Stockholders and Lenders Value of Equity When the value of the firm is below the face
value of the debt, the stockholders default
The equity in a leveraged and the equity is worth $0.
corporation is like a call option
on the underlying assets of the When the value of the firm is above the
firm. The stockholders exercise face value of the debt, the stockholders
their option by paying off the repay the debt and the equity is worth the
debt if the firm is worth more difference between the firm value and the
than the face value of the debt face value of the debt.
when the debt matures. If the
value of the firm is lower than $0
the face value of the debt, the
stockholders can default (let Face Value Firm Value
their option expire) without
incurring losses beyond their of Loan
investment in the firm.
Figure B. Lender payoff function
The lenders’ payoff function is
like that for the seller of a put Value of Loan When the value of the firm is below the face value of the
option. They have effectively debt, the stockholders default and the lenders receive the
agreed to purchase the firm for value of the firm.
an amount that equals the face
value of the firm’s debt, at the When the value of the firm is above the
discretion of the stockholders. face value of the debt, the stockholders
repay the debt and the lenders receive
the face value of the debt.

$0

Face Value Firm Value

of Loan

earlier in this chapter that the more volatile the value of the underlying asset, the more valu-
able a call option on that asset will be. Stockholders of leveraged firms know this and there-
fore have an incentive to invest in risky projects that increase the overall volatility of the
value of their companies’ assets.

Lenders, in contrast, do not want the firm to invest in high-risk projects. As you can see
from their payoff function in Exhibit 20.4, the lenders bear costs as the value of the firm drops
below the amount they are owed but do not benefit at all as the value of the firm’s assets in-
creases above that amount. Lenders to companies that are worth more than they are owed can
only expect to lose when a project increases the overall riskiness of a company’s assets.

The Underinvestment Problem

Chapter 16 also explained that stockholders have incentives to turn down positive NPV proj-
ects when all of the benefits are likely to go to the lenders. You can see how this underinvest-
ment problem arises from the differences in the payoff functions in Exhibit 20.4. Suppose that
the company will owe $10 million when the loan matures, that the company is currently worth
$5 million, and that the loan matures next week. This company is financially distressed because
its assets are not even worth as much as its outstanding debt—so it is unlikely to have enough
money to finance new investments. Now suppose that management identifies a positive NPV
project that would require a $3 million investment and that has a positive NPV of $1 million
that will be realized before the debt payment must be made. Management would have a hard
time convincing the stockholders to invest an additional $3 million in the firm, because even
if the investment turns out to be worth $4 million, all of the money will go to the lenders. The
stockholders have a strong incentive to turn down this positive NPV project.

Agency Costs of Equity

Many of our discussions assume that managers act in the best interests of the stockholders.
Since managers are hired to manage the firm on behalf of the stockholders, this might appear

the stockholders’ best interest. This is because the payoff function for a manager can be quite 20.4 Agency Costs 661
different from that for stockholders. In fact, a manager’s payoff function can look a lot like a
lender’s payoff function. Face Value
of Loan
To see how this is possible, consider the connection between managers’ personal wealth
and the performance of the companies for which they work. The present value of a manager’s Firm Value
future earnings is a large part of his or her overall wealth. If a company gets into financial dif-
ficulty and a manager is viewed as responsible, that manager could lose his or her job and find
it difficult to obtain a similar job at another company. Of course, the most obvious way for a
company to get into financial difficulty is to default on its debt. So as long as a company is able
to avoid defaulting on its debt, a manager has a reasonable chance of retaining his or her job.
Once the firm defaults, the chances of job loss increase dramatically. In addition, researchers
have found that senior managers of financially distressed large public companies who lose
their jobs find it difficult to obtain similar jobs afterwards.6 We might also expect that the
worse the company’s financial distress, the worse the manager’s future employment prospects
and the lower the present value of the compensation that he or she can expect to receive in the
future. If this is so, when the value of a firm is less than the amount it owes, the payoff function
for a manager will look something like that for the lender in Figure B of Exhibit 20.4—it will
slope downward as the value of the firm decreases.

On the positive side, we would expect the present value of a manager’s future earnings to
increase with the value of the firm when this value is above the amount that the company owes
to its lenders. Managers will receive larger bonuses and larger pay raises, and any stock or op-
tions that they receive will be more valuable. However, these increases will not be nearly as
large as those for stockholders. The stockholders are not likely to give the managers a large
proportion of any increase in firm value. The net result is that the payoff function for managers
can look something like the one in Exhibit 20.5.

The fact that the payoff function for a manager resembles that for a lender means that
managers, like lenders, have incentives to invest in less risky assets and to distribute less value
through dividends and stock repurchases than the stockholders would like them to. These
tendencies are reinforced by the fact that managers are individuals who do not hold diversified
portfolios, since most of their wealth is tied to the performance of their firms. Managers tend
to make conservative investment, financing, and payout decisions because the personal cost to
them of failure can be very great.

Boards of directors understand how the incentives of managers differ from those of
stockholders. Consequently, boards put a great deal of effort into designing compensation
plans that make the payoff functions for managers look as much as possible like those of stock-
holders. Ultimately, this is a key to minimizing agency conflicts between stockholders and the
managers that represent them.

EXHIBIT 20.5 Value of Manager's
Representative Payoff Function for a Manager Future Compensation

The payoff function for a manager with a typical compensation arrangement is $0
more similar in shape to the payoff function for a lender than for a stockholder.
While a stockholder’s payoff function is flat to the left of the face value of the loan,
the value of the manager’s compensation is downward sloping, much like the
payoff for a lender. When the value of the firm is greater than the face value of the
loan, the value of the manager’s compensation does not increase as much as the
value of the firm’s shares (the line in the payoff function is not as steep). Because
managers’ payoff functions differ from those for stockholders, managers have
incentives to take actions that are not in the best interests of stockholders.

> BEFORE YOU GO ON
1 . What do the payoff functions for stockholders and lenders look like?
2 . What does the payoff function for a typical manager look like?

662 CHAPTER 20 I Options and Corporate Finance

20.5 OPTIONS AND RISK MANAGEMENT

LEARNING OBJECTIVE We have discussed options that are bundled with financial securities, how options found in
real investments can have value, and how the option-like payoff functions of stockholders,
lenders, and managers contribute to agency conflicts. Another place in which options are fre-
quently encountered in corporate finance is in the management of risk. Risk management in-
volves hedging, or reducing the financial risks faced by a firm. Options, along with other de-
rivative securities, such as forwards, futures, and swaps, are used to reduce risks associated
with commodity prices, interest rates, foreign exchange rates, and equity prices.

To see how risks can be managed using options, consider an oil company that is producing
and selling oil to refiners. Suppose that the price of West Texas Intermediate (WTI) crude oil
has recently risen above $90 per barrel and the company wants to make sure that, even if prices
drop below $85 per barrel, it will receive at least $85 per barrel for each barrel of WTI that it
sells during the next three months. If the company plans to sell 100,000 barrels of oil in the
next three months, the financial managers can hedge the price risk by purchasing put options
on 100,000 barrels of oil with an exercise price of $85 per barrel plus the cost of the options.
The maturity dates on the options must be selected to match the timing of the company’s oil
output over the next three months. In addition, the actual exercise prices on the options must
be slightly greater than $85 to account for the premiums that the company pays to purchase the
options. This will ensure that the company actually receives $85 per barrel after paying for the
options.

One interesting benefit of using options in this way is that they provide downside protec-
tion but do not limit the upside to the company if oil prices continue to increase. Put options
give the company the right to sell its oil for the exercise price if WTI prices fall, but because
there is no obligation to sell, the company can still benefit if oil prices increase. As discussed
earlier, this is just like buying insurance. In fact, many insurance contracts are really little more
than specialized put options.

In addition to using options and other derivative securities to manage commodity price
risks, as in the oil company example, companies can use these securities to manage risks
associated with changing interest rates. Large swings in interest rates can cause a great deal of
volatility in the net income of a highly financially leveraged company whose managers rely on
floating-rate debt. As interest rates go up and down, the company’s interest expense also goes
up and down. Furthermore, under certain circumstances, such volatility can actually increase
the company’s taxes. Needless to say, all of this can cause problems for managers with Wall
Street analysts.

Options can also be used to manage risks associated with foreign exchange rates. For ex-
ample, as we discuss in Chapter 21, the revenues that a U.S. company reports can be strongly
affected by changes in exchange rates if the company manufactures products in the United
States and has significant overseas sales. If the dollar strengthens against foreign currencies, for
example, the company will have to increase the overseas prices of its products in order to maintain
the same dollar prices per unit. This, in turn, can prompt consumers in overseas markets to pur-
chase fewer of the company’s products. By using options and other derivative securities to
protect against exchange rate movements, managers can limit declines in revenues that occur
because of such movements.

Finally, options can be used to manage risks associated with equity prices. This is espe-
cially important to companies that have traditional defined-benefit pension plans, which pro-
vide retirees with guaranteed retirement payments. Companies are required to put money
aside to cover the costs of these payments, and this money is generally invested in stocks.
When the stock market declines significantly, these companies must replace any lost value with
new contributions, which must come from earnings. As you might expect, companies are very
interested in managing the risk that they will have to make such contributions.

> BEFORE YOU GO ON
1 . What is hedging?
2 . What types of risks can options be used to manage?

Self-Study Problems 663

S um m a ry of Learning Objectives

1 Define a call option and a put option, and describe the to make a decision to accept or reject a project at a particular
payoff function for each of these options. point in time. It is not designed to incorporate potential value as-
sociated with deferring the investment decision. Incorporating
An option is the right, but not the obligation, to buy or sell an asset the value of the other real options into an NPV framework is
for a given price on or before a specific date. The price is called the technically possible but would be very difficult to do because the
exercise or strike price, and the date is called the exercise date or rate used to discount the cash flows would change over time with
expiration date of the option. The right to buy the asset is known as their riskiness. In addition, the information necessary to value
a call option. The payoff from a call option equals $0 if the value of real options using the NPV approach is not always available.
the underlying asset is less than or equal to the exercise price at ex-
piration. If the value of the underlying asset is greater than the exer- 4 Describe how the agency costs of debt and equity are
cise price at expiration, then the payoff from a call option is equal to related to options.
the value of the asset value minus the exercise price. The right to sell
the asset is called a put option. The payoff from a put option is $0 The chapter discusses two principal classes of agency conflicts.
if the value of the underlying asset is greater than or equal to the The first is between stockholders (owners) and lenders. When
exercise price at expiration. If the value of the underlying asset is less there is a risk of bankruptcy, stockholders may have incentives to
than the exercise price, then the payoff from a put option equals the increase the volatility of the firm’s assets, turn down positive
exercise price minus the value of the underlying asset. NPV projects, or pay out assets in the form of dividends. Stock-
holders have these incentives because their payoff functions look
2 List and describe the variables that affect the value of like those for the owner of a call option.
an option. Calculate the value of a call option and of a
put option. The other principal class of agency conflicts is between
managers and stockholders. Managers tend to prefer less risk
The value of an option is affected by five variables: the current than stockholders. They also prefer to distribute fewer assets in
price of the underlying asset, the exercise price of the option, the the form of dividends because their payoff functions are more
volatility of the value of the underlying asset, the time left until like those of lenders than those of stockholders. These prefer-
the expiration of the option, and the risk-free rate. ences are magnified by the fact that managers are risk-averse
individuals whose portfolios are not well diversified.
Section 20.2 describes how to calculate the values of call
and put options, both at expiration and at some point before 5 Explain how options can be used to manage a firm’s
the expiration date. exposure to risk.

3 Name some of the real options that occur in business A company can adjust its exposure to risks associated with com-
and explain why traditional NPV analysis does not ac- modity prices, interest rates, foreign exchange rates, and equity
curately incorporate their values. prices by buying or selling options. For example, a company that
is concerned about the prices it will receive for products that will
Real options that are associated with investments include options be delivered in the future can purchase put options to partially or
to defer the investments, make follow-on investments, change opera- totally eliminate that risk.
tions, and abandon projects. Traditional NPV analysis is designed

S um m a ry of Key Equations

Equation Description Formula
20.1 Put-call parity P ϭ C ϩ XeϪrt Ϫ V

Self-Study Problems

20.1 Of the two parties to an option contract, the buyer and the seller, who has a right and who has an
obligation?

20.2 The stock of Augusta Light and Power is currently selling at $12 per share. Over the next year
the company is undertaking a new electricity production project. If the project is successful, the
company’s stock is expected to rise to $24 per share. If the project fails, the stock is expected to fall
to $8 per share. The risk free rate is 6 percent. Calculate the value today of a one year call option
on one share of Augusta Light and Power with an exercise price of $20.

20.3 ADCAP International is a U.S.-based company which sells its products primarily in overseas mar-
kets. The company’s stock is currently trading at $50 per share. Depending on the outcome of U.S.
trade negotiations with the countries to which ADCAP exports its products, the company’s stock
price is expected to be either $65 or $30 in six months. The risk free rate is 8 percent per year. What

664 CHAPTER 20 I Options and Corporate Finance

20.4 Your company is considering opening a new factory in Europe to serve the growing demand for
your product there. What real options might you want to consider in your capital budgeting analy-
sis of the factory?

20.5 Your firm, which uses oil as an input to its production processes, hedges its exposure to changes
in the price of oil by buying call options on oil at today’s price. If the price of oil goes down by the
time the contract expires, what effect will that have on your company?

Solutions to Self-Study Problems

20.1 The buyer (owner) of the option has the right to exercise the option but is not required to do
so. The seller (or writer) of the option is obligated to take the other side of the transaction if the
option owner decides to exercise it.

20.2 First determine the payoffs for the stock, a risk free loan, and the call option under the two possible
outcomes. In one year, the stock price is expected to be either $8 or $24. The loan will be worth
$1.06 regardless of whether the project is successful. If the project fails, the stock price will be less
than the exercise price of the call option. The option will not be exercised, and will be worth $0. If
the project is successful, the stock price will be higher than the exercise price of the call option. The
option will be exercised and its value will be the difference between the stock price and the exercise
price, $4.

Stock (x) Risk-Free Loan (y) Call Option

Today $12 $1 ?

Expiration $8 $24 $1.06 $1.06 $0 $24 Ϫ $20 = $4

The stock and loan can be used to create a replicating portfolio which has the same payoff as the
call option:
1$8 ϫ x2 ϩ 11.06 ϫ y2 ϭ $0
1$24 ϫ x2 ϩ 11.06 ϫ y2 ϭ $4

Solving the two equations yields: x ϭ 0.25, y ϭ Ϫ1.887

The value of the call option is the same as the current value of this portfolio:
($12 ϫ 0.25) ϩ ($1 ϫ Ϫ1.887) ϭ $1.11

20.3 Here we solve directly for the value of the put option. First we determine the payoffs for the stock,
a risk free bond, and the put option under the two possible outcomes. To determine payoff of the
bond six months from now, now we must calculate the six-month risk free interest rate given the
one year risk free rate in the problem statement:
Six month risk free rate ϭ (1 ϩ 0.08)1/2 Ϫ 1 ϭ 1.039, or 3.9%

The payoffs are therefore:

Stock (x) Risk-Free Loan (y) Put Option

Today $50 $1 ?

Expiration $30 $65 $1.039 $1.039 $40 Ϫ $30 = $10 $0

Now we can use the stock and bond to create a replicating portfolio, which will give the same
payoff as the put option:
1$30 ϫ x2 ϩ 11.039 ϫ y2 ϭ $10
1$65 ϫ x2 ϩ 11.039 ϫ y2 ϭ $0

Solving the two equations we determine x ϭ Ϫ0.286, y ϭ 17.87

The value if the put option is the same as the current value of this portfolio:

Critical Thinking Questions 665

Alternatively, you could solve this problem by calculating the value of a call option with an exercise
price of $40 par share and then using the put-call parity relation. The value of the call option is
$15.09 and value of the associated put option calculated using the put-call parity relation is $3.52.
The difference ($3.58 vs. $3.52) is due to rounding and the compounding assumption for the dis-
count rate.

20.4 Several significant real options might be associated with the factory. First, by having a factory in
Europe, and the employees and management associated with it, your company might be better
positioned to introduce products to the European markets. In addition, you will have options to
change operations, to sell the factory, or to simply abandon the project.

20.5 The effect on your company of the decline in the price of oil will be to increase earnings. This
is because the oil is an input to your production process, and a drop in prices will reduce your
expenses. If the price of oil goes down, you would let the call option expire without exercising it.
Of course, the benefit your company receives from the drop in oil prices would be reduced by the
amount that you paid to purchase the option.

Critical Thinking Questions

20.1 Options can be combined to create more complicated payoff structures. Consider the combina-
tion of one put option and one call option with the same expiration date and the same strike
price. Draw the payoff diagram and describe what the purchaser of such a combination thinks
will happen before expiration.

20.2 A writer (seller) of a call option may or may not actually own the underlying asset. If he or she
owns the asset, and therefore will have the asset available to deliver should the option be exer-
cised, he or she is said to be writing a covered call. Otherwise, he or she is writing a naked call
and will have to buy the underlying asset on the open market should the option be exercised.
Draw the payoff diagram of a covered call (including the value of the owned underlying asset)
and compare it with the payoff of other options.

20.3 An American option will never be worth less than a European option. Evaluate this statement.

20.4 Explain why, in the binomial pricing theory, the probabilities of an upward move versus a down-
ward move are not important.

20.5 Like all other models, the binomial pricing model is a simplification of reality. In this model, how
do we represent high volatility or low volatility of the value of the underlying asset?

20.6 What kinds of real options should be considered in the following situations?
a. Wingnuts R Us is considering two sites for a new factory. One is just large enough for the
planned facility, while the other is three times larger.
b. Carousel Cruises is purchasing three new cruise ships to be built sequentially. The first
ship will commence construction today and will take one year to build. The second will
then be started. Carousel can cancel the order for a given cruise ship at any time before
construction begins.

20.7 Future Enterprises is considering building a factory that will include an option to expand opera-
tions in three years. If things go well, the anticipated expansion will have a value of $10 million
and will cost $2 million to undertake. Otherwise, the anticipated expansion will have a value of
only $1 million and will not take place. What information would we need in order to analyze this
capital budgeting problem using the traditional NPV approach that we would not need using
option valuation techniques?

20.8 Corporations frequently include employee stock options as a part of the compensation for
their managers and sometimes for all of their employees. These options allow the holder
to buy the stock of the company for a preset price like any other option, but they are usu-
ally very long lived, with maturities of 10 years. The goal of stock option plans is to align
the incentives of employees with those of stockholders. What are the implications of these
compensation plans for current stockholders?

20.9 You are a bond holder of ABC Corp. Using option pricing theory, explain what agency concerns
you would have if ABC were in danger of bankruptcy.

20.10 A bond covenant is a part of a bond contract that restricts the behavior of the firm, barring it
from taking certain actions. Using the terminology of options, explain why a bond contract
might include a covenant preventing the firm from making large dividend payments to its stock-
holders.

666 CHAPTER 20 I Options and Corporate Finance

Questions and Problems

BASIC > 20.1 Option characteristics: What is an option?

20.2 Option characteristics: Explain how the payoff functions differ for the owner (buyer) and the
seller of a call option. Of a put option.

20.3 Option payoffs: What is the payoff for a call option with a strike price of $50 if the stock price
at expiration is $40? What if the stock price is $65?

20.4 Option payoffs: What is the payoff for a put option with a strike price of $50 if the stock price
at expiration is $40? What if the stock price is $65?

20.5 Option valuation: What are the five variables that affect the value of an option, and how do
changes in each of these variables affect the value of a call option?

20.6 Option valuation: Assuming nothing else changes, what happens to the value of an option as
time passes and the expiration date gets closer?

20.7 Option valuation: What does the seller of a put option hope will happen?

20.8 Option valuation: What is the value of an option if the stock price is zero? What if the stock
price is extremely high (relative to the strike price)?

20.9 Option valuation: Like owners of stock, owners of options can lose no more than the amount
they invested. They are far more likely to lose that full amount, but they cannot lose more. Do
sellers of options have the same limitation on their losses?

20.10 Option valuation: What is the value at expiration of a call option with a strike price of $65 if
the stock price is $1? $50? $65? $100? $1,000?

20.11 Option valuation: Suppose you have an option to buy a share of ABC Corp. stock for $100.
The option expires tomorrow, and the current price of ABC Corp. is $95. How much is your
option worth?

20.12 Option valuation: You hold an American option to sell one share of Zyther Co. stock. The op-
tion expires tomorrow. The strike price of the option is $50, and the current stock price is $49.
What is the value of exercising the option today? If you wanted to sell the option instead, about
how much would you expect to receive?

20.13 Real options: What is the difference between a financial option and a real option?

20.14 Real options: List and describe four different types of real options that are associated with
investment projects.

20.15 Agency costs: How are options related to the agency costs of debt and equity?

I N T E R M E D I AT E > 20.16 Option valuation: Suppose that you own a call option and a put option on the same stock and

that these options have the same exercise price. Explain how the relative values of these two op-
tions will change as the stock price increases or decreases.

20.17 Other options: A callable bond is a bond that can be bought back by the bond issuer before
maturity for some pre-specified price (normally a small amount above face value) at the discre-
tion of the bond issuer. How would you go about finding the value of such a bond? Would the
bond be worth more or less than an equivalent noncallable bond?

20.18 Other options: A convertible bond is a bond that can be exchanged for stock at the discretion
of the bondholder. How would you go about finding the value of such a bond? Would the bond
be worth more or less than an equivalent nonconvertible bond?

20.19 Option valuation: The seller of an option can never make any money from a change in the value
of the underlying asset; he or she can only hope that the option will not be exercised and that and
he or she will not lose any money. Given that this is the case, why do people sell options?

20.20 Option valuation: The stock of Socrates Motors is currently trading for $40 and will either rise
to $50 or fall to $35 in one month. The risk-free rate for one month is 1.5 percent. What is the
value of a one-month call option with a strike price of $40?

20.21 Option valuation: Again assume that the price of Socrates Motors stock will either rise to $50
or fall to $35 in one month and that the risk-free rate for one month is 1.5 percent. How much is
an option with a strike price of $40 worth if the current stock price is $45 instead of $40?

20.22 Option valuation: Assume that the stock of Socrates Motors is currently trading for $40 and
will either rise to $50 or fall to $35 in one month. The risk-free rate for one month is 1.5 percent.