## Capm As A Pricing Formula

The CAPM is a pricing model. However, the standard CAPM formula does not contain prices explicitly—only expected rates of return To see why the CAPM is called a pricing model we must go back to the definition of return

Suppose that an asset is purchased at price P and later sold at price Q The rate of return is then r = (Q ~ P)/P. Flere P is known and Q is random. Putting this in the CAPM formula, we have

Solving for P we obtain

This gives the price of the asset according to the CAPM We highlight this important result:

Pricing form of the CAPM The price P of cm asset with payoff Q is

This pricing formula has a form that very nicely generalizes the familiar discounting formula for deterministic situations. In the deterministic case, it is appropriate to discount the future payment at the interest rate /y, using a factor of 1/(1 + /y) In the random case the appropriate interest rate is '/-+(0/— ry), which can be regarded as a risk-adjusted interest rate

Example 7.5 (The price is right) Gavin Jones is good at math, but his friends tell him that he doesn't always see the big picture. Right now, Gavin is thinking about investing in a mutual fund. This fund invests 10% of its funds at the risk-free rate of 7% and the remaining 90% in a widely diversified portfolio that closely approximates the market portfolio, which has an expected rate of return equal to 15% One share of the mutual fund represents $100 of assets in the fund Having just studied the CAPM, Gavin wants to know how much such a share should cost.

Gavin figures out diat the beta of the fund must be 90 The value of a share after 1 year is expected to be 10 x 1.07 + 90 x 1.15 — 114 20. Hence, according to (7.6),

114 20

Yes, the price of a share will be equal to the value of the funds it represents. Gavin is reassured (but suspects he could have figured that out more simply)

Example 7.6 (The oil venture) Consider again, as in Example 7 2, the possibility of investing in a share of a certain oil well that will produce a payoff that is random because of the uncertainty associated with whether ot not there is oil at that site and because of the uncertainty in future oil prices The expected payoff is $1,000 and the standard deviation of return is a relatively high 40%. The beta of the asset is (5 — .6, which is relatively low because, although the uncertainty in return due to oil prices is correlated with the market portfolio, the uncertainty associated with exploration is not The risk-free rate is >y — 10%, and the expected return on the market portfolio is ,17. What is the value of this share of the oil venture, based on CAPM? (Recall that earlier it was stated that the offered price was $875 ) We have immediately

and a does not enter the calculation .

The venture may be quite risky in the traditional sense of having a high standard deviation associated with its return But, nevertheless, it is fairly priced because of the relatively low beta

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