Identifying the Certainty Equivalent from Models of Risk and Return

˜C˜C˜

Denote CE(C)as the certainty equivalent of uncertain future cash flow and E()as

the cash flow mean, Result 11.6 describes how to compute certainty equivalents from

a risk-expected return model.

Result 11.6

To obtain a certainty equivalent, subtract the product of the cash flow beta and the tangencyportfolio risk premium from the expected cash flow; that is

CE(˜)E(˜) b(R r)

CC

Tf

where

cov(˜R˜)

C,

b^T

2

T

The Cash Flow Beta and Its Interpretation.Result 11.6 adjusts for risk with the

cash flow beta, denoted as b.The cash flow betais the covariance of the future cash

flow(not the return on the cash flow) with the return of the tangency portfolio, divided

by the variance of the return of the tangency portfolio; that is

cov(˜ ˜

C,R)

b^T

2

T

This risk measure is the amount of the tangency portfolio that must be held to track,

as best as possible, the future cash flow. In contrast to the return beta ( ), which is

used with the risk-adjusted discount rate method, the cash flow beta (b) can be com-

puted directly after forming scenarios, as we will illustrate shortly. Since obtaining this

cash flow beta does not require prior knowledge of the present value, the certainty

equivalent is a superior vehicle for identifying present values when return and cash

flow estimation in scenarios is the only method available for generating risk measures.

The Certainty Equivalent Present Value Formula and Its Interpretation.To

obtain the present value, discount the certainty equivalent at the risk-free rate. Com-

bining this finding with Result 11.6 generates the following result:

Result	11.7	(The certainty equivalent present value formula.) PV,the present value of next period’s cash
		flow, can be found by (1) computing E(˜)the expected future cash flow and the beta of
		C
		the future cash flow, (2) subtracting the product of this beta and the risk premium of the
		tangency portfolio from the expected future cash flow, and (3) dividing by (1the risk-
		free return); that is

E(˜) b(R r)

C

PV^Tf

1r

f

Grinblatt824Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw824Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

Chapter 11

Investing in Risky Projects

405

Thus, the certainty equivalent present value formula first adjusts for the risk-premium

component and then for the time value of money. To compute the net present value,

subtract the initial cost of the project, C, from this present value.

0

One interpretation of the certainty equivalent formula in Result 11.6 comes from rec-

ognizing that b,the cash flow beta, is the tracking portfolio’s dollar investment in the

tangency portfolio. The tangency portfolio earns an extra expected return (that is, a risk

premium) because of risk. Specifically, R ris the future additional amount earned

Tf

per dollar invested in the tangency portfolio because of the tangency portfolio’s system-

atic (or factor) risk. For an investment of bdollars in the tangency portfolio, the addi-

tional expected cash flow (in dollars) from the project’s systematic (or factor) risk is thus

b(R r)

Tf

˜

Hence, subtracting b(R r)from the expected cash flow E(C)yields

Tf

˜

E(C) b(R r)

Tf

This represents the cash flow that would be generated if the project had a cash flow

beta of zero or, alternatively, if the future cash flow were risk free.

An Illustration of a Present Value Computation When the Cash Flow Beta Is

Given.Example 11.7 illustrates how to compute present values given cash flow betas.

Example 11.7:Computing the Cost of Capital

Each share of Hot Shot Computer Corp (HSCC), a wholly owned subsidiary of Novel, Inc., first

seen in Example 11.1, has a cash flow beta (b) of $10.91 when computed against the tan-

gency portfolio.One year from now, this subsidiary has a .9 probability of being worth $10 per

share and a .1 probability of being worth $20 per share.The risk-free rate is 9 percent per

year.The tangency portfolio has an expected return of 19 percent per year.What is the pres-

ent value of HSCC, assuming no dividend payments to the parent firm in the coming year?

Answer:The expected value of HSCC one year from now is

$11 per share.9($10).1($20)

The numerator in the certainty equivalent formula, the certainty equivalent, is thus

$9.91$11 $10.91(.19 .09)

The subsidiary’s present value is its certainty equivalent divided by 1 plus the risk-free rate

or approximately

$9.91 per share

$9.09 per share

1.09

Cash Flow Betas and Return Betas.The answer in Example 11.7 is identical to the

answer given in Example 11.1 because Example 11.7 uses a cash flow beta consistent

with the return beta from Example 11.1. Note that in Example 11.1, is the beta of a

comparison financial security, which is a zero NPVinvestment. The cash flow beta

b$9.09 ; that is, the ratio of the cash flow beta to the return beta (as computed for

a cost of $9.09) equals the project’s present value

b

PV

This equation is not valid if PV 0 (and expected cash flow is non-negative).

Grinblatt826Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw826Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

406Part IIIValuing Real Assets

The CAPM, Scenarios, and the Certainty Equivalent Method

The last subsection suggested that scenarios provide one way to identify cash flow betas

and present values with the certainty equivalent method. Example 11.8 illustrates how

to implement this scenario method, assuming that the market portfolio is the tangency

portfolio.

Example 11.8:Present Values with the Certainty Equivalent Method

The Adonis Travel Agency, examined in Examples 11.5 and 11.6, wishes to estimate the

present value of the cash flow from purchasing 10 new airline reservation computers.The

new computers, which are faster than the current ones in place at the agency, are expected

to increase the number of reservations each agent can handle.For simplicity, assume that

all the additional cash flows associated with the increase in booking capacity are received

one year from now.The size of the increase is tied to the state of the economy.Over the

next year, three possible economic scenarios are considered, which are described in the fol-

lowing table, taken from Example 11.5:

	Market	Incremental Cash
Outcome	ProbabilityReturn (%)	in One Year

	3
Recovery		25%	$150,000
	4

	3
Recession		1	35,000
	16

	1
Depression		15	5,000
	16

What is the present value of the additional cash flow one year from now if the risk-free return

over the next year is 8.625 percent and the CAPM determines the expected returns of traded

securities?

Answer:The risk premium of the market portfolio is

331

R r(.25)( .01)( .15) .08625.09

M^f41616

while the variance of the market return (computed in Example 11.5) is .017236.

The expected incremental cash flow is

331

$119,375($150,000)($35,000)($5,000)

41616

The covariance of the cash flow with the return of the market portfolio

331

cov (˜,˜)$1,000(150)(.25)(35)( .01)(5)( .15) ($119,375)(.17625)

CR

^m41616

$28,012.5 $21,039.84375$6,972.65625

$6,972.65625

which generates a cash flow beta of .Substituting these values into the cer-

.017236

tainty equivalent formula leaves a present value of approximately

$119,375 $6,972.65625(.09)

.017236

PV$76,379

1.08625

Grinblatt828Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw828Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

Chapter 11

Investing in Risky Projects

407

Example 11.8 is based on the same numbers as Example 11.5, where, using the

risk-adjusted discount rate method, we computed an erroneous present value for the

Adonis Travel computer cash flow of $82,311. The latter number was too high because

negative NPVprojects have underestimated return betas.

Example 11.8 demonstrates that the certainty equivalent method gives the true pres-

ent value of $76,379. The last section and Example 11.6, using the same travel agency,

emphasized the importance of knowing this true present value for mutually exclusive

projects.

The APT and the Certainty Equivalent Method

To obtain the certainty equivalent in the one-factor APT, subtract from the expected

future cash flow the product of

1.the factor loading of the future cash flow and

2.the risk premium of the factor.

If there is more than one factor, sum these products over all factors and then subtract.²⁴

Then discount this certainty equivalent at the risk-free rate to obtain the present value;

that is

˜bb. . .b)

E(C) (

PV¹122KK

1r

f

where b(j1, . . . , K) is the factor loading of the future cash flow(not the cash

j

flow return) on the jth factor. The symbol brepresents the number of dollars invested

j

in the jth factor portfolio that best tracks the cash flow of the project. The amount sub-

tracted from E(C˜in the numerator of this ratio is thus the additional expected cash

)

flow arising from the factor risk of the project.

The Relation between the Certainty Equivalent Formula and the Tracking Portfolio Approach

Recall the Hilton casino illustration from Sections 11.1 and 11.2, which ascribed a pres-

ent value of $10.0 million to an expected cash flow of $11.3 million from Louisiana

gambling. Hilton’s tracking portfolio for the casino consisted of $5 million invested in

the market portfolio, which has a risk premium of 14 percent (20% 6%), and

$5million invested in a risk-free asset, which has a return of 6 percent. As suggested

in the previous subsection, the amount invested in the tangency portfolio (in this case,

the market portfolio) is the cash flow beta (b). Hence, the Hilton casino cash flow beta

is $5 million. Using this cash flow beta in the certainty equivalent formula (see Result

11.6) yields a certainty equivalent (the numerator) of

$11.3 million $5.0 million

.14 $10.6 million

and thus a present value (see Result 10.7) of

$10 million $10.6 million/1.06.

In practice, one first obtains the cash flow beta, $5 million, from scenarios, and

only then is it possible to recognize this as the amount of the tracking portfolio invested

²⁴In

the multifactor case, the cash flow factor loading will be its multiple regression coefficient

against the factor and it will be its covariance with the factor divided by the factor variance only if the

factors are uncorrelated.

Grinblatt830Titman: Financial	III. Valuing Real Assets	11. Investing in Risky	© The McGraw830Hill
Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

408Part IIIValuing Real Assets

in the market portfolio. To keep the tracking as close as possible, the expected cash

flow from the tracking portfolio of financial securities must be the same as the expected

cash flow from the casino. (Can you explain why?) Hence, in deriving this tracking

portfolio, it is important to know that the expected cash flow from the casino was $11.3

million. Then, solving for the risk-free investment, x,in combination with a $5 million

investment in the market portfolio, yields an expected future cash flow of $11.3 mil-

lion, which pins down the risk-free investment. Algebraically, xsolves

$11,300,000 x(1.06) $5,000,000 (1.2), or

x$5,000,000

The certainty equivalent method gives the same present values as the tracking port-

folio approach used earlier. Indeed, the certainty equivalent is derived from the track-

ing portfolio approach as evidenced by the fact that cash flow beta bis the tracking

portfolio’s expenditure on the market (or tangency) portfolio.