10A.2Spot Rates, Annuity Rates, and ParRates

This subsection discusses how to convert par rates to spot rates and annuity rates, and vice

versa. First, it provides a general rule for comparing the three types of term structures, which

are graphed in Exhibit 10A.2.

If the term structure of one of the three yield curves (spot, annuity, or par) is consistently

upward or downward sloping, the other two yields will slope in the same direction. The

spot yield curve will have the steepest slope, the par curve will have the second steepest

slope, and the annuity yield curve will have the gentlest slope. If the yield curve is at, all

three yield curves will be identical.²

To simplify the analysis of how to translate one type of yield into another, assume that the

yields are compounded annually and computed only at years 1, ..., T.Use interpolation to gen-

erate approximate yields for fractional years (for example, 1.5 years or 2.3 years).

Using Spot Yields to Compute Annuity Yields and ParYields.Computing annuity yields

from spot yields is relatively straightforward. The annuity yield for year tis found by assuming

a cash payment of a xed amount for years 1, 2, . . . , t—for example, $1 is paid at each of those

years. This is like a portfolio of tzero-coupon bonds, each with a $1 face value. Using spot yields,

compute the present value of each zero-coupon bond and sum up the tpresent values. If the pres-

ent value is P,nd the internal rate of return for the cash ows: P,1, 1, . . . , 1. This is the

annuity yield-to-maturity for date t. Repeat this process for t 1, t 2, t 3, . . . , and t T.

To compute par yields from spot yields, strip off $100, the principal, from the nal payment

of the par bond, assumed for purposes of illustration to have a face value of $100. This leaves

an annuity with a maturity identical to the par bond and a cash ow equal to the bond’s coupon.

For a $100 bond, this coupon is the coupon rate on a par bond in percentage terms. To identify

this coupon, note that the annuity’s present value equals $100, the present value of the par bond,

less the present value of $100 paid at maturity, discounted with the zero-coupon rate for that

²This relation between the slopes is due to the different timing and size of the cash ows (see

Chapter 23) of the three types of bonds.

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Chapter 10Investing in Risk-Free Projects	365
EXHIBIT10A.2Three Types of Yield Curves

	Spot (Zero Coupon) Curve
10%
Yield
7%

10	20 30 Maturity (years)

Par Curve

10%

Yield

7%

10	20 30 Maturity (years)

Annuity Curve

10%

Yield

7%

10	20 30 Maturity (years)

maturity. Thus, the bond’s coupon is the ratio of the present value of this annuity to the present

value of a $1 annuity of the same maturity. Since, for par bonds, the coupon rate and yield to

maturity are the same on coupon payment dates (see Chapter 2), this ratio is the par yield to

maturity, in percentage terms.

Example 10A.1 demonstrates how to compute annuity yields and par yields from spot yields.

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366Part IIIValuing Real Assets

Example 10A.1:Computing Annuity Yields and ParYields from Spot Yields

Assume that spot yields, compounded annually, are 7 percent, 8 percent, and 9 percent for

years 1, 2, and 3, respectively.Compute (1) the annuity yields and (2) par yields for years

1, 2, and 3, assuming annual cash ows for these bonds.

Answer:(1) The annuity yield for year 1 is 7 percent because a one-year annuity with

annual payments has only one payment and thus behaves like a zero-coupon bond.The

annuity yield for year 2 is found by rst computing the present value of a two-year annuity.

With $1 payments, such a bond has a value of

$1$1

$1.7919

1.08²

1.07

The yield to maturity of a bond with a cost of $1.7919 and payments of $1 at year 1 and 2

is 7.65 percent.This is the point on the annuity yield curve for year 2.The annuity yield for

year 3 is the yield to maturity of a bond with payments of $1 in years 1, 2, and 3, and a

cost of

$1$1$1

$2.5641

1.08²³

1.071.09

This is 8.28 percent.

(2) The one-year par yield is the same as the zero-coupon yield because its cash ows

are the same as that of a zero-coupon bond.To compute the two-year par yield, assume a

face value of $100.Stripping the principal off the end of the bond leaves an annuity with a

present value of

$100

$100 $14.266118

1.08²

Since a $1 annuity of two years’maturity has a present value of $1.7919, both the coupon

rate and the yield to maturity (in percent) of the two-year par bond must be

$14.266118

7.96%

$1.7919

Repeating the process for stripping $100 off the last payment of the three-year par bond

leaves a present value of

$100

$100 $22.781652

1.09³

Dividing this by the present value of a $1 annuity with a maturity of three years gives the

coupon and yield to maturity of the three-year par bond in percentage terms

$22.781652

8.88%

$2.5641

Computing Spot Yields from ParYields and Annuity Yields.There are many ways to com-

pute the spot yield curve from the par yield curve. Perhaps the easiest way to proceed is to move

sequentially from the shortest maturity to the longest. With annual compounding and annual

coupons, the par bond with the earliest maturity is a zero-coupon bond. The spot yield is iden-

tical to the par yield in this case. The two-year spot yield is computed from a long position in

the two-year par bond and a short position in the one-year par bond. Because the one-year par

bond is a zero-coupon bond, it is possible to form a portfolio that looks like a zero-coupon bond

with two years’maturity by weighting the two par bonds properly. The yield of this properly

weighted portfolio is the spot yield for two years’maturity. Given the one- and two-year spot

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Chapter 10

Investing in Risk-Free Projects

367

rates, it is possible to form a portfolio consisting of a long position in a three-year par bond and

short positions in one- and two-year zero-coupon bonds that look like a three-year zero coupon

bond. The yield of this bond is the three-year spot rate. Proceed sequentially in this manner as

far out into time as necessary.

Example 10A.2 illustrates the procedure described above.

Example 10A.2:Computing Spot Yields from ParYields

Compute the spot yield curve, assuming annually compounded yields, for years 1, 2, and 3,

given par curve yields as follows:

	Maturity in Years
	123
Par yield	9%8%7%

All par yields assume coupons paid annually and all yields are compounded annually.

Answer:The one-year spot yield is 9 percent because the one-year par bond is the same

as a zero-coupon bond.

The two-year par bond has an 8 percent coupon paid at year 1.As a par bond, its face

value is the same as its market value.Assume a face value of $100, although any value will

do.The cash ows of this bond are $8 at year 1 and $108 at year 2.The one-year par bond

has a cash ow of $109 at year 1.A long position in the two-year $100 par bond and a

short position in 8/109 of the one-year $100 par bond has a future cash ow at year 2 in

the amount of $108 and no future cash ows at other dates.Thus, the position is equiva-

lent to a two-year zero-coupon bond and, since it has a value of $92.66055 ( $100

(8/109)$100), the yield of this bond, r,satises the equation

$108

$92.66055

r)²

(1

implying r 7.96 percent.Alternatively, knowing that the appropriate discount rate for the

$8 coupon of the par bond at year 1 is 9 percent, stripping that coupon leaves a bond with

a value of $100 $8/1.09 $92.66055, which as seen above implies a 7.96 percent yield.

To nd the year 3 spot rate, strip the two $7 coupons at years 1 and 2 from the par bond.

This leaves a bond with a value of

$7$7

$100 $87.5722

1.09(1.0796)²

With a future payoff at year 3 in the amount of $107, and no future cash ows at other dates,

the stripped bond now has the payoff of a zero-coupon bond with a 3-year spot yield rthat

solves

$107

$87.5722

(1r)³

implying r 6.91 percent.

To compute zero-coupon yields from annuity yields, strip all payments from the annuity

except the last one. If the annuity yields are known, the present value of the stripped cash ows

and the original annuity is known. This leaves a single future cash ow. The yield to maturity

of this cash ow is the spot yield for that maturity.

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368Part IIIValuing Real Assets

Example 10A.3 illustrates how to compute spot yields from annuity yields.

Example 10A.3:Computing Spot Yields from Annuity Yields

If the annuity yields are 7 percent, 7 percent, and 8 percent for years 1, 2, and 3, respec-

tively, compute the spot yields.Assume annual cash ows and annual compounding.

Answer:The rst two spot yields are both 7 percent.The rst is 7 percent because the

annuity is a zero-coupon bond.The second is 7 percent because the annuity yield would be

higher (lower) than 7 percent if the spot yield was higher (lower) than 7 percent.(You can

prove this with the method we now use.)

The present value of the two-year annuity at 7 percent, using $1 cash ows, is about

$1.81.The present value of the three-year annuity at 8 percent is about $2.58.The difference

between the values of the two- and three-year annuities is $0.77.For the present value of

$1 paid in three years to be $0.77, the 3-year spot yield has to be about 9.15 percent.

Why Spot Yields Should Be Computed from ParYields.The term structure of risk-free

interest rates given by the Treasury yield curve is generally computed from on-the-run Treasury

bonds. You can think of on-the-run bonds as newborns. They become slightly less interesting to

many investors as they become toddlers, and traders begin to lose interest in them as they become

children and young adults. As these bonds age, many of them get tucked away into pension

accounts and insurance funds to pay off future liabilities. As the actively traded supply of these

bonds begins to diminish, it becomes more difcult to use the bonds for short sales. Moreover,

the lack of active trading means that investors are never sure whether the price they pay is fair,

because the last trading price they observe is often stale.

For this reason, it is useful to focus on the on-the-run bonds to compute a Treasury yield.

The prices are fresh, the trading is active, and the opinions of many bond market participants

have gone into determining their yields. Even though many zero-coupon Treasury bonds—they

are known as Treasury strips—are available, it is still better to infer spot yields from on-the-run

par bonds than to compute them directly from the Treasury strips.

While we generally agree with this approach, there is one major caveat. On-the-run Treasury

securities do not exist at every maturity. Currently, on-the-run notes and bonds exist for matu-

rities of years 1, 2, 3, 4, 5, 7, 10 and 30 (with the 30-year bond likely to be dropped in 2001).

What about years 6, 8, 9, and 11–29? Asimilar problem arises if one needs a zero-coupon rate

for a fractional horizon such as 3.5 years.

The general procedure to use when maturities for on-the-run securities are missing is to inter-

polate par yields for the missing maturities and to compute spot rates from the interpolated par

yields. While traditional practice typically uses linear interpolation to ll in the missing date,

clearly superior nonlinear interpolation procedures also exist which make the yield curve appear

to be smooth. These techniques are beyond the scope of this text.

Key Terms forAppendix 10A

annuity term structure	363	par yield curve363
LIBOR term structure	363	spot term structure363

Exercises forAppendix 10A

10A.1.	Azero-coupon bond maturing two years from	10 percent (annual compounding). Both bonds
	now has a yield to maturity of 8 percent (annual	have a face value of $100.
	compounding). Another zero-coupon bond with	a.What are the prices of the zero-coupon bonds?
	the same maturity date has a yield to maturity of

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Chapter 10

Investing in Risk-Free Projects

369

	b.Describe the cash ows to a long position in	respectively 4.5 percent, 5 percent, and 5.25
	the 10 percent zero-coupon bond and a short	percent. Assume annual compounding for all rates
	position in the 8 percent zero-coupon bond.	and annual payments for all bonds.
	c.Are the cash ows in part bindicative of	10A.3.Compute spot yields and annuity yields for years
	arbitrage?	1, 2, 3, and 4 if par yields for years 1, 2, 3, and 4
	d.Suppose the 10 percent bond matured in 3 years	are, respectively, 4.5 percent, 5 percent, 5.25
	rather than 2 years. Is there arbitrage now?	percent, and 5.25 percent. Assume annual
10A.2.	Compute annuity yields and par yields for years	compounding for all rates and annual payments
	1, 2, and 3 if spot yields for years 1, 2, and 3 are	for all bonds.

References and Additional Readings forAppendix 10A

Dufe, Darrell. “Special Repo Rates.” Journal of FinanceGrinblatt, Mark, with Bing Han. “An Analytic Solution

51 (1996), pp. 493–526.for Interest Rate Swap Spreads.” Review of

International Finance(2001), forthcoming.

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CHAPTER
11
	Investing	in	Risky	Projects

Learning Objectives

After reading this chapter you should be able to:

1.Estimate the cost of capital with the CAPM, APT, and dividend discount models.

2.Understand how to implement the comparison approach with the risk-adjusted

discount rate method and know its shortcomings.

3.Understand when to use comparison firms and when to use scenarios to obtain

present values.

4.Discount expected future cash flows at a risk-adjusted rate and know when to

discount the certainty equivalent at a risk-free rate.

5.Identify the certainty equivalent of a risky cash flow, using equilibrium models,

risk-free scenarios, and forward prices.

Thermo Electron (http://www.thermo.com/index.html), a Boston-area company listed

on the New York Stock Exchange, has followed the general strategy of starting new

lines of business and turning them into independent public companies with their own

publicly traded shares. Thermo Electron benefits from this strategy in two ways.

First, by taking its projects public, Thermo Electron learns how financial market

participants value its investment projects, which helps the company’s managers make

better capital allocation decisions. Second, since the compensation of the managers

who run the various divisions is based on the stock price of the division, the

managers are highly motivated to make decisions that maximize share price.

In the last two chapters, we have emphasized that a manager should adopt a project

if the cost of investing in the project is less than the cost of a portfolio of financial

assets that produces approximately the same future cash flow stream. With any pur-

chase a firm makes, value is enhanced by buying from the cheapest sources. Capital

investments are no different. Managers should buy their future cash flow streams from

the cheapest source—whether it is the real asset or the financial markets.

This approach to project evaluation is also equivalent to asking how the cash flows

of a corporate project would be valued by a competitive capital market if the project

could be spun off and sold as a separate entity. If the separate entity has its own traded

370

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Chapter 11

Investing in Risky Projects

371

stock, it is easy to understand how to choose value-creating real investments: The firm

creates value by taking all real investment projects that cost less than what the project

can generate from the sale of stock in the project. In this simple world where all invest-

ment projects are sold to the capital markets, the criterion for accepting an investment

project is the same as that used with any other product a firm might sell. IBM would

not sell a computer if the costs were higher than the selling price. Likewise, the com-

pany would not create a new division if the costs of doing so were greater than the

expected proceeds from taking the division public with a new issue.

Thermo Electron, described in the opening vignette, obviously views the benefits

of its approach to real asset investment as exceeding the costs of taking each of its

divisions public, which can be substantial (see Chapter 3). However, Thermo Electron

is unusual in this regard. We are unaware of any other firm that follows this strategy.¹

Other firms attempting to address the question. “How much would the financial mar-

kets pay for the future cash flows of this investment project?” must recognize that since

the sale of stock in the project is only hypothetical, answering this question is much

more difficult than it is for Thermo Electron. How to answer this question, in a prac-

tical way, without selling your project or division, is the subject of this chapter.

In most U.S. firms, managers value risky investments in much the same way that

they value riskless investments. First, the manager forecaststhe future cash flows of

the investment and then discounts them at some interest rate. This task, however, is

much more difficult for risky investments because (1) the manager needs to know what

it means to forecast cash flows when they are risky, and (2) identifying the appropri-

ate discount rate is more complex when cash flows are risky.

One of the lessons of this chapter is that the way one forecasts cash flows deter-

mines the discount rate used to obtain present values. This chapter considers two ways

to “forecast” a cash flow:

1.Forecast the expected cash flow.

2.Forecast the cash flow adjusted for risk (defined shortly).

First, consider the case (see point 1) where the manager forecasts a stream of

expected cash flows for the proposed project. Each expected cash flow, generated as

the probability-weighted sum of the cash flow in each of a variety of scenarios, could

then be discounted with a risk-adjusted discount rate determined by a risk-return model

like the CAPM or the APT. Obtaining PV’s in this fashion is known as the risk-

adjusted discount rate method.

An outline of a traditional valuation with this approach is to

•	Estimate the expected future cash flows of the asset
•	Obtain a risk-adjusted discount rate and discount just as we did in the previouschapter to find the present value.

Since the expected cash flows are generated by assets that we are trying to value,

it is important to find their present values by discounting them at the expected returns

of assetsthat are comparable to the assets being valued. These typically are the assets

of publicly traded firms in the same line of business as the asset being valued. The

problem is that even the assets of publicly traded firms—for example, their factories

and machines—are rarely traded. Instead, financial markets trade claims on the cash

¹Other firms undertake the same type of equity carve-outs described for Thermo Electron, but none

do it with the same frequency or the same intent.

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372Part IIIValuing Real Assets

flows of these assets. Examples of these claims are the debt and equity of publicly

traded firms. Hence, one has to start with the debt and equity claims on assets and

somehow convert them to an asset expected return to obtain the appropriate risk-

adjusted discount rate. The first step is to identify the betas of these claims. Then one

typically

a)Uses a formula to convert the equity and debt betas of comparable publicly

traded firms to asset betas to undo the confounding effect of leverage on equity

betas. (This process is known as unlevering the beta.)

b)Applies an asset pricing model to convert the asset beta to an asset expected

return which is then used as the risk-adjusted discount rate.

Alternatively, one can

a)Use an asset pricing model to convert the equity beta to an equity expected

return.

b)Use a formula to convert the equity expected return (and sometimes the debt

return) of comparable publicly traded firms to asset expected returns to undo the

confounding effect of leverage on equity betas. (This process is typically known

as unlevering the equity return.) The asset expected return is then used as the

risk-adjusted discount rate.

(These steps become even more complex in the presence of corporate taxes, a topic we

defer to Chapter 13.)

As noted above (see point 2), there is another case where managerial forecasts of

cash flows are adjusted for risk. As we will see, this does not necessarily present prob-

lems when managers are fully aware of the way they are forecasting and understand

its implications for the discount rate. However, managers are often unaware of what

they are forecasting or how they are adjusting for risk. For example, managers often

think they are forecasting expected cash flows when, in reality, their forecasts are not

true expectations because they ignore possible events that potentially can make the cash

flows zero.

As an illustration of this, consider an energy company, deciding whether to invest

hundreds of millions of dollars on a pipeline running through a developing country,

knowing that it has a 5 percent chance of losing the entire investment in the event of

a change in government. The managers of some energy companies will ignore this

threat when they calculate the expected cash flows, but they will increase the dis-

count rate to reflect the higher risk. Instead of employing this ad hoc adjustment to

the discount rate, one should adjust the expected cash flows to reflect the possibil-

ity of the unfortunate outcome and discount expected cash flows at the appropriate

risk-adjusted rate.

In other situations, managers account for the riskiness of a cash flow by reporting

a conservative “expected cash flow,” that is, they place too much weight on the bad

outcomes and too little on the good ones. This procedure, while generally misapplied,

has some merit. In some instances, the conservative forecasted cash flow can be viewed

as a certainty equivalent, or hypothetical riskless cash flow that occurs at the same

time and which has the same present value as the risky cash flow being analyzed.²

Valuing a stream of certainty equivalents with the certainty equivalent method

involves discounting at the risk-free rate. Thus, for valuation purposes, certainty equiv-

alent cash flows can be treated as if they are certain.

²Chapter 7 introduced forward prices for zero-cost financial contracts. The certainty equivalent can be

thought of as the preferred terminology for the forward prices of real asset cash flows.

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373

The choice of whether to obtain a present value by discounting an expected cash

flow at a risk-adjusted discount rate or by discounting a certainty equivalent cash flow

at a risk-free rate depends largely on the information available to value the project. If

traded securities exist for companies that have the same line of business as the project,

then the beta risk of these comparison securities can be used to identify the appropri-

ate discount rate for the project’s expected cash flows. In this case, if the expected cash

flows are easier to identify than the certainty equivalents of the cash flows, discount-

ing expected cash flows will be the preferred method.

If it is not possible to identify the beta risk of comparison securities (for example,

because securities in the same line of business are nonexistent), then beta risk needs to

be identified from scenarios that allow a statistician to estimate it. In the latter setting,

the certainty equivalent method is generally more convenient. In addition, it is some-

times easier to identify certainty equivalent cash flows than expected cash flows, par-

ticularly when forward or futures prices exist for commodities that are fundamental to

the profitability of the project: In this case, discounting certainty equivalents at a risk-

less rate of interest would be the preferred method of identifying the present values of

real assets.

When applied properly, discounting expected cash flows at a risk-adjusted discount

rate and certainty equivalent cash flows at a risk-free discount rate provide the same

value for the project’s future cash flows. They are equivalent because both methods are

applications of the general valuation principle used throughout the text: Identify a track-

ing portfolio and use its value as an estimate of the value of the asset’s future cash

flows. Real asset investment should be undertaken only in situations where the finan-

cial investments that track the future cash flows of the real asset have a value that

exceeds the project’s cost.

The approach to valuing risky real assets presented in this chapter is eminently

practical. For example, numerous corporations and banks use the two valuation method-

ologies. Moreover, the chapter analyzes in depth a great number of implementation

issues and obstacles.³