## The Certainty Equivalent Present Value Formula and Its Interpretation To

obtain the present value, discount the certainty equivalent at the risk-free rate. Combining 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(C) the expected future cash flow and the beta of 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 (1 + the risk-free return); that is

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, — Co, from this present value.

One interpretation of the certainty equivalent formula in Result 11.6 comes from recognizing 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, RT - rf is the future additional amount earned per dollar invested in the tangency portfolio because of the tangency portfolio's systematic (or factor) risk. For an investment of b dollars in the tangency portfolio, the additional expected cash flow (in dollars) from the project's systematic (or factor) risk is thus b(RT - rf)

Hence, subtracting b(RT - rf) from the expected cash flow E(C) yields

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.