Implementing the Risk-Free Scenario Method in a Multiperiod Setting

The risk-free scenario method avoids many of the problems faced by more traditional

methods in a multiperiod setting (see Section 11.5). To illustrate the multiperiod use of

the risk-free scenario method, assume that yields on one-year, five-year, and 10-year

²⁵The

major drawback to the risk-free scenario method is that it will only provide the true present

value if the distribution of the return of the tangency portfolio and the cash flow belong to certain

families of distributions, including the bivariate normal distribution. If the distribution of the cash flow

and the return of the tangency portfolio is one in which the conditional expectation is nonlinear, the

forecast under this scenario is not the same as the certainty equivalent.

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Chapter 11

Investing in Risky Projects

411

risk-free zero-coupon bonds are respectively 5, 6, and 7 percent. Consider a computer

operating system designed by Microsoft; it has a life of 10 years and will have three

major versions: version 8.0 (sold at the end of year 1), 9.0 (sold at the end of year 5),

and version 10.0 (sold at the end of year 10). For simplicity, assume that so much soft-

ware pirating is going on between these major revisions of the software product that

the cash flows between revisions are essentially zero.

To obtain the present value of the three future cash flows, it is necessary to obtain

estimates of the year 1, year 5, and year 10 cash flows under their respective risk-free

scenarios. These cash flow estimates are not easily obtained but, as we show below,

are probably no more difficult to obtain than estimates of the year 1, year 5, and year

10 expectedcash flows.

To obtain the risk-free scenario estimate for year 1, envision an estimate of the

cash flow under a scenario in which all assets, with dividends reinvested, are expected

to appreciate by 5 percent, the one-year risk-free rate. The present value of the year

1 cash flow is that estimate discounted back one year at a rate of 5 percent. To obtain

the risk-free scenario estimate for year 5, envision what the year 5 cash flow would

be if all assets, with dividends reinvested, are expected to appreciate at a rate of 6

percent per year for these 5 years, the yield on a five-year risk-free bond. The pres-

ent value of the year 5 cash flow is that estimate discounted back five years at a rate

of 6 percent. To obtain the risk-free scenario estimate for year 10, envision what the

year 10 cash flow would be if all assets, with dividends reinvested, are expected to

appreciate at a rate of 7 percent per year for these 10 years, the yield on a 10-year

risk-free bond. The present value is this estimate, discounted back 10 years at a rate

of 7 percent per year.

When the mean-variance efficient (that is, tangency) portfolio appreciates at the

risk-free rate, all securities are expected to appreciate at the risk-free rate, including the

stock of Microsoft. Thus, a reasonable procedure for estimating the cash flows for the

risk-free scenarios at the three horizons is to forecast the cash flow as a multiple of

Microsoft’s future stock price and compute what the price of Microsoft’s stock and the

project cash flow would be when Microsoft’s stock appreciates at a risk-free rate.²⁶

This forecast will be the true certainty equivalent if the error in the cash flow fore-

cast is distributed independently of the return of the mean-variance efficient portfolio

for that horizon. For simplicity, assume that Microsoft will not pay any dividends over

the next 10 years. For companies that pay dividends, forecast the stock value with all

dividends reinvested.

If Microsoft is currently trading at $100 a share, it will trade at $105 a share one

year from now in a risk-free scenario, given a risk-free rate of 5 percent per year. Over

a five-year period, Microsoft will trade at $133.82 if it appreciates at the five-year risk-

free rate, 6 percent per year. In 10 years, it will trade at $196.72 if it appreciates at the

10-year risk-free rate, 7 percent per year.

Assume that Microsoft’s managers believe the operating system is expected to

generate cash equal to 10 million times Microsoft’s stock price per share. Hence,

the new operating system is expected to generate $1.05 billion at the end of year 1

if Microsoft stock with dividends reinvested is then selling at $105 a share, $1.3382

billion at the end of year 5 if Microsoft stock sells for $133.28 a share at that point,

and $1.9672 billion at the end of year 10 if Microsoft stock sells for $196.72 at that

point.

²⁶This

multiple could differ for different horizons, but in the example we will assume it does not

change with the horizon.

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412Part IIIValuing Real Assets

The present value of the operating system is then

$1.05 billion$1.3382 billion$1.9672 billion

PV

1.05(1.06)⁵¹⁰

(1.07)

$1.05 billion$1.3382 billion$1.9672 billion

1.051.33821.9672

$3 billion

Example 11.10 presents another illustration of the risk-free scenario method.

Example 11.10:Multiperiod Valuation with the Risk-Free Scenario Method

Omegatron, a game software company, wants to value the cash flows of its new Doombo

game at years 5 and 10.Assume the following:

•	Cash flow forecast errors are noise;that is, they are distributed independently of everything.

•	The year 5 cash flow of Doombo has an expected value of $39 million if its five-yearstock return, with dividends reinvested in Doombo stock, is 30 percent over the fiveyears.

•	The year 10 cash flow of Doombo has an expected value of $80 million if aninvestment in its stock (with all dividends reinvested) doubles over 10 years.

•	At date 0, $1.00 buys $1.30 in face value of a risk-free, five-year zero-coupon bond.
•	At date 0, $1.00 buys $2.00 in face value of a risk-free, ten-year zero-coupon bond.

What are the present values of the two cash flows?

Answer:Applying the certainty equivalent formula, the present value of the year 5 cash

flow is

$39 million

$30 million

1.30

and the year 10 cash flow’s present value is

$80 million

$40 million

2.00

The cash flow estimate in Example 11.10 is trickier than it may seem at first. The

long-term appreciation in the stock is assumed to equal the appreciation of a risk-free

security. This does not mean that the stock has to appreciate year by year at the same

rate as the risk-free security. Like the tortoise and the hare, the stock can start off faster

than the risk-free security, then slow down, or vice versa, just as long as they end up

in the same place at the same time. In a risk-free scenario, the manager knows that the

geometric mean of the stock return is the risk-free rate, but he does not know the pat-

tern of short-horizon returns by which that geometric mean is achieved. Each pattern

could generate a different project cash flow. In this case, it is important to analyze and

weigh the likelihood of paths in order to arrive at the expected cash flow under the

scenarios in which the stock’s geometric mean return is the risk-free rate.

The firm’s own stock price is not the only candidate to use for a risk-free scenario;

other traded securities or portfolios of securities are perfectly adequate substitutes.

Generally, using more securities and portfolios makes it more likely that the cash flow

forecast error will be distributed independently of the return of the mean-variance effi-

cient portfolio.

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Chapter 11

Investing in Risky Projects

413

The ease with which the risk-free scenario method is applied in a multiperiod set-

ting gives it a major advantage over the risk-adjusted discount rate method or the tra-

ditional certainty equivalent method. When it can be applied, we believe that the risk-

free scenario method generates better approximations to the true present values than

those estimated with more traditional methods.

Providing Certainty Equivalents without Knowing It

In many instances, the cash flow for the risk-free scenario is provided unwittingly by ana-

lysts or managers. This situation usually arises when the manager wants to be conserva-

tive in his forecast, knowing that the cash flow is risky and the forecast is imprecise.

For example, consider a financial analyst working at a hypothetical company that

we will call Elliot Hand Tools. The engineers have designed a new hand drill and have

calculated the costs of setting up a plant to manufacture this product. After the engi-

neers calculate the manufacturing costs per unit, the market research department esti-

mates a projected selling price and the number of units that they think Elliot Hand

Tools can sell. Based on all of this information, the financial analyst forecasts a stream

of future cash flows and then evaluates whether the company should go through with

the project.

To discount the cash flow stream, the financial analyst has to know a bit about how

the cash flows were estimated. If the engineers and marketing researchers decide to

give conservative estimates because the cash flows are risky, then the cash flow stream

may be better thought of as a certainty equivalent that should be discounted at the risk-

free rate. The analyst would not want to discount such risk-adjusted cash flows at a

risk-adjusted discount rate. However, the analyst must also be aware that the conser-

vative estimates of the engineers and marketing researchers are unlikely to be the pre-

cise certainty equivalent.²⁷