11.7Obtaining Certainty Equivalents with Risk-Free Scenarios

The CAPM and APTimplementations of the certainty equivalent method require

knowledge of the composition of a portfolio that tracks the evaluated investment’s cash

flow. Moreover, to obtain the certainty equivalent of the cash flow, the manager must:

1.Compute two items—the expected cash flow and its adjustment for risk based

on the cash flow’s covariance with the return of the tangency portfolio

multiplied by the risk premium of the tangency portfolio.

2.Obtain the difference between these two items.

In this section, we present an alternative approach for computing certainty equivalents.

ADescription of the Risk-Free Scenario Method

An alternative computational approach to the certainty equivalent, which we call the

risk-free scenario method, provides the manager with a simple way to estimate the cer-

tainty equivalent cash flow. The risk-free scenario methodgenerates the certainty

equivalent with a typically conservative cash flow forecast under a scenario where all

assets are expected to appreciate at the risk-free rate. In other words, the certainty equiv-

alent cash flow is assumed to be the expected cash flow in situations where the tan-

gency portfolio return equals the risk-free rate.

Distributions forWhich the Risk-Free Scenario Method Works.This method

works when the returns of the tangency portfolio and the future cash flows of the proj-

ect have specific distributions. Specifically, it must be a distribution where the expec-

tation of the future cash flow, given the return of a mean-variance efficient portfolio,

is a linear function of the return of the tangency portfolio.

Algebraically, this can be expressed as follows

E˜ given the return R) abR

(C TT

The values of aand b, the intercept and cash flow beta, respectively, do not change

for different outcomes of the return of the tangency portfolio. This is basically the

assumption of linear regression and it is satisfied by, among other distributions, the

normal distribution. The key feature of this distributional assumption is that the error

in the cash flow forecast is distributed independently of the tangency portfolio’s return.

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Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

Chapter 11

Investing in Risky Projects

409

Inputs forthe Risk-Free Scenario Method.The risk-free scenario method uses as

its input an estimate of the project’s cash flows, assuming that the return of the tan-

gency portfolio, and thus the tracking portfolio, equals the risk-free return. In other

words, instead of asking the engineers and marketing research managers to estimate

the expected cash flows of a project, the analyst asks them to come up with what they

think the cash flows would be in a scenario where the tracking portfolio (a combina-

tion of the market portfolio and a risk-free asset if the CAPM holds) has a return that

equals the risk-free rate. As shown below, eliciting this kind of information is useful

because, under the conditions noted above, these conditional expected cash flows—

that is, expected cash flows conditional on the tracking portfolio return equaling the

risk-free return—can be viewed as the cash flow’s certainty equivalent. To see this,

regress the actualexcess returns of any zero-NPVinvestment (return ˜less the risk-

r

free rate) on the actual excess return of the tangency portfolio. The resulting equa-

tion is

˜ r (R˜˜	(11.7)
r r)
fTf

where (for each particular outcome of the return of the tangency portfolio)

E(˜)0

Equation (11.7) indicates that high beta investments are expected to outperform low

beta investments in scenarios where the tangency portfolio return exceeds the risk-free

return. The opposite is true in scenarios where the risk-free return exceeds the tangency

portfolio’s return. However, when the tangency portfolio return equals the risk-free

return, the expected returns of all zero-NPVinvestments are equal to (alpha), irre-

spective of their betas. Moreover, it is possible to show that the intercept () is 0 in

equation (11.7) by first calculating the expected values of both sides of the equation

and noting, from the familiar risk-expected return equation first developed in Chapter

5, that

rr (R r)	(11.8)
fTf

Hence, letting 0, as implied by equation (11.8), the expectation of the left-hand

side of equation (11.7) in the risk-free scenario is 0, and thus

E(˜ given ˜r)r

rR

Tff

Obtaining PVs with the Risk-Free Scenario Method.This analysis demonstrates

that when the return of the tangency portfolio equals the risk-free return, all zero-NPV

investments are expected to appreciate at the risk-free rate. In this risk-free scenario, a

project’s expected future value is

E(˜ given risk-free scenario) (1r)

PV

Cf

Thus, once the product (1 r)

PVis estimated, the analyst can determine the PV

f

by discounting the expected cash flow for the risk-free scenario,

	E(˜ given risk-free scenario),
	C
at the risk-free rate.

Result 11.8

(Estimating the certainty equivalent with a risk-free scenario.)If it is possible to estimatethe expected future cash flow of an investment or project under a scenario where all secu-rities are expected to appreciate at the risk-free return, then the present value of the cashflow is computed by discounting the expected cash flow for the risk-free scenario at therisk-free rate.

Grinblatt834Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw834Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

410Part IIIValuing Real Assets

There is no need to estimate betas or to identify the tangency portfolio used in the

tracking portfolio if it is possible to forecast the future cash flow under a scenario where

all securities are expected to appreciate at the risk-free rate. Moreover, the task of esti-

mating the cash flows for all possible scenarios and weighting by the probabilities of

the scenarios is eliminated with the risk-free scenario method. The only scenario where

cash flow forecasts are needed is one in which all securities are expected to earn the

risk-free return. Because the tangency portfolio return equals its expected return in the

average scenario, and the tangency portfolio’s expected return is larger than the risk-

free return, the risk-free scenario is more pessimistic than the average scenario.

An Illustration of How to Implement the Risk-Free Scenario Method.Example

11.9 illustrates how to implement the risk-free scenario method.

Example 11.9:Valuation with the Risk-Free Scenario Method

The McGirwin Company is evaluating a project with a one-year life that has an uncertain

cash flow at the end of the first year.Its managers estimate that the project will generate a

cash flow of $100,000 at the end of year 1 under a scenario where all securities are expected

to earn the risk-free return of 5 percent per year.What is the present value of this risky proj-

ect? For what costs should the project be accepted or rejected?

Answer:$100,000 is the certainty equivalent of the future cash flow.Discounting this at

a rate of 5 percent yields $100,000/1.05 or $95,238.Therefore, if the project costs less than

$95,328, McGirwin managers should accept it.If it costs more than $95,328, they should

reject it.

Advantages of the Risk-Free Scenario Method.As a practical matter, the advan-

tage of employing the risk-free scenario method is obvious. In the risk-free scenario,

investors expect the stock held by shareholders in the manager’s own firm and the stock

in all other firms in the industry to appreciate at the risk free rate (with dividends rein-

vested). For this moderately pessimistic scenario, the manager may find it easier to esti-

mate the future cash flow of the project than to estimate both its expected value over

all scenarios and its covariance with the tangency portfolio, assuming that it is possi-

ble to even identify the tangency portfolio.

In theory, the present value obtained with the risk-free scenario method should be

the same as that obtained with the traditional certainty equivalent method. In practice,

however, there is no reason for these methods to generate either identical certainty

equivalents or identical present values because the estimates of cash flows for risk-free

scenarios and estimates of cash flow betas for traditional certainty equivalent

approaches are imperfect.²⁵