## B An Application of the Certainty Equivalence Method The CAPM Cost of Capital For a Non Traded Good

The opportunity cost You are asked to advise a firm on its appropriate cost of capital. The owners of this firm are ^^ddlrOS very wealthy and widely diversified, so that their remaining portfolio is similar to the market e corporation. portfolio. (Otherwise, our investor's opportunity cost of capital may not be well represented by the CAPM—and therefore, the calculations here are not relevant for the typical cash-strapped entrepreneur.) To make this a more realistic and difficult task, this firm is either privately held or only a division, so you cannot find historical public market values, and there are no obvious publicly traded comparable firms. Instead, the firm hands you its historical annual cash flows:

Year 1999 2000 2001 2002 2003 2004 Average

S&P500 +21.4% -5.7% -12.8% -21.9% +26,4% +9.0% +2.7%

Cash Flows $8,794 $5,373 $8,397 $6,314 $9,430 $9,838 $8,024

In an ideal world, this is an easy problem: you could compute the value of this firm every year, then compute the beta of the firm's rate of return with respect to the market rate of return, and plug this into the CAPM formula. Alas, assessing annual firm value changes from annual cash flows is beyond my capability. You can also not presume that percent changes in the firm's cash flows are percent changes in the firm's value—just consider what would happen to your estimates if the firm had earned zero in one year. All this does not let you off the hook: what cost of capital are you recommending? Having only a time series of historical cash flows (and no rates of return) is a very applied and not simply an obscure theoretical problem, and you might first want to reflect on how difficult it is to solve this problem without the certainty equivalence formula.

First, we have to make our usual assumption that our historical cash flows and market rates of Let us attempt to value returns are representative of the future. To solve our problem, we begin by computing the beta this. of the firm's cash flows with respect to the S&P500. This is easier if we work with differences from the mean,

Year 1999 2000 2001 2002 2003 2004 Average

S&P500 +0.187 -0.084 -0.155 -0.246 +0.237 +0.063 0

Cash Flows +$770 -$2,651 +$373 -$1,710 +$1,406 +$1,814 $0

To compute the covariance of the S&P500 returns with our cash flows, we multiply these and take the average (well, we divide by N - 1, because this is a sample, not the population, but it won't matter in the end),

CoVcF,rM =

(+0.187) ■ ( + $770) + (-0.084) ■ (-$2,651) + ■ ■ ■ + (+0.063) ■ ( + $1,814)

and compute the variance

0.0373

The cash flow beta is the ratio of these, CovcF,rM $235.4

Var(rM) 0.03734

It is easiest now to proceed by considering the historical mean cash flow of $8,024. We need an assumption of a suitable equity premium and a suitable risk-free rate. Let us adopt 3% and 4%, respectively. In this case, the value of our firm would be

$245

CF,rM

The certainty equivalence formula tells us that because our firm's cash flows are correlated with the market, we shall impute a risk discount of $245. We can translate this into a cost of capital estimate—at what discount rate would we arrive at a value of $7,546?

file=invest-capm-c.tex

We now have an estimate of the cost of capital for our cash flow for next year. We can also translate this into an equivalent returns-based market-beta, which is

Are we close? Now I can reveal who the firm in this example really was—it was IBM. Because it is publicly traded, we can see how our own estimate of IBM's cost of capital and market beta would have come out if we had computed it from IBM's annual market values. Its rates of return were

Year 1999 2000 2001 2002 2003 2004 Average

IBM's Rate of Return +17.5% -20.8% +43.0% -35.5% +20.5% +7.2% +5.3%

If you compute the market-beta of these annual returns, you will find an estimate of 0.7—very close to the estimate we obtained from our cash flow series. (por IBM, this is a very low market-beta estimate. If we used monthly cash flows or monthly stock returns, we would obtain a considerably higher market-beta estimate.)

Solve Now!

Q 13.42 A firm reported the following cash flows:

Year 1999 2000 2001 2002 2003 2004 Average

S&P500 +21.4% -5.7% -12.8% -21.9% +26,4% +9.0% +2.7%

Cash Plows +$2,864 +$1,666 -$1,040 +$52 +$1,478 -$962 +$997

(Note that the cash flows are close to nothing in 2002 and even negative in 2004, the latter preventing you from computing percent changes in cash flows.) What cost of capital would you recommend for this firm?

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