Question:

Everything here is simple one-period. Assume a firm generates $100 million in cash flow with volatility of $20 million. The riskless rate is 4%; the equity premium is also 4%. The firm is evaluating a LARGE PROJECT: investment of $50 million with end-of-single-period payoff = $70 million. The project will have cash flow volatility of $10 million. The project has a beta (with respect to market) of 1.6 and a correlation with the firm’s cash flow of 0.5.

(i) What is the project’s cost of capital (COC) using CAPM?
(ii) Based on the project’s COC, what is the net present value (NPV) of the project?
(iii) The NPV is positive. Therefore, shouldn’t the firm necessarily invest in a positive NPV project?
(iv) After investing in the project (i.e., the pre-project firm becomes firm plus project), what is the volatility of the new firm’s cash flow?
(v) What is the new firm’s cash flow at risk (CFaR) with 99% confidence?
(vi) If the firm’s cost of CFaR is $0.80 per dollar of CFaR, what is the project’s NPV adjusted for CFaR?

Answer:

Here is an EditGrid worksheet that performs the calculations.
(i)
If we employ the CAPM, the project’s cost of capital is given by = riskless rate + (beta)(equity premium). In this case, COC = 10.4% = 4% + (4%)(1.6).

(ii)
The project NPV is given by (payoff)/(1+COC) - (investment). In this case, $70 MM/(1+10.4%) - $50 MM = $13.41. Note this follows Stulz. Alternatively, you could assume a one-year period and discount continuously: $70*EXP(-10.4%)-$50 = $13.09.

(iii)
No, because the analysis so far does not consider risk (or assumes risk is unchanged). If the project is sufficiently risky, it will not be worthwhile.

(iv)
Variance of the firm’s cash flow including the project is given by application of the mean-variance portfolio formula:
Variance [cash flows] = [Variance of firm cash flows] + [Variance of project cash flows] + (2)(firm cash flow volatility)(project cash flow volatility)(Correlation). In this case,

Variance [firm plus project CF] = ($20)^2 + ($10)^2 + (2)(20)(10)(0.5) = $700, so
Volatility [firm plus project CF] = SQRT(700) = $26.5 MM

(v)
CFaR @ 99% = $26.5 MM CF volatility * NORMSINV(99%) = 61.5 MM

(vi)
Project NPV, adjusted for CFaR = $13.41 - ($0.80 cost of CFaR/dollar)($61.5 - $46.5 MM) = +$1.4 million. Therefore, the project is worth undertaking.

Note this last step: the project increased firm CFaR from $46.5 to $61.5 MM. Given $0.80 marginal cost of CFaR, the dollar cost of the increase in CFaR = ($0.80 cost of CFaR/dollar)($61.5 - $46.5 MM). That gets subtracted from the project NPV.

Hi Samuel,

I add the XLS above and here.

Re (iv), please note the 2nd tab, I used the weights to show the answer is the same. The normal formula (with weights) is derived from a simpler property of variance: variance (a + b) = variance (a) + variance (b) + 2*Covariance(A,B). So, it is just shorter to use that, as above; however, as you can see from the 2nd tab in the worksheet above, we can “take the long way around” to the same result.

Re (v), $46.5 million is the firm’s before-new-project CFaR = $20 MM volatility * 2.33 normal deviate @ 99% confidence

David