Stulz chapter -2, Exercises 6,7 and 8 (all three related)

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David, I'm having trouble with the following problems from Stulz.

Q6. A firm has an expected cash flow of $500 million in one year. Beta is 0.8.Rf=5% and Risk premium on thew market if 6%. What is the PV of the cash flow? If beta doubles what happens to the cash floow.

Q7. Using the data above, consider the impact of hedging the cash flow against systemic risk. If management wants to reduce the systematic risk to zero, how could it do so? How much would the firm have to pay investors to bear the systematic risk of the cash flow.

Q8. Consider the situation in Q6. To hedge the firm's systematic risk, management has to pay investors to bear this risk. Why is it that the value of the firm for shareholders does not fall when the firm pays other investors to bear the cash flow's systematic risk?

I need your help with the answers:
Q6 is straight forward. 500/[1+5%+0.8*6%]=455.4 million

However, I'm not very sure about my approach to Q7. Need your inputs.
If beta=0, firm value is 476.2 million. Thus the cost of hedging should be 476.2-455.4=20.8 million (is it implied from the hedging irrelevance)

And in Q8, I'm lost.

Alan

 

[David Harper CFA FRM]

Hi Alan,

I agree re Q6. His Chapter 2 theory is that risk management cannot add value (no free lunch); on the systematic risk, the cost to hedge out beta (in theory) equals the benefit. So i think he's looking for (I'm not a fan of Stulz Ch 2 and 3, i think they are terribly written and I'm not alone in suggesting some of his arguments are circular, fwiw).

Q7. Firm eliminates systematic risk by taking a short position in a market index futures contract (see p 41. "shorting the market"). Under the theory here (normal backwardation. Same as Hull), the short expects to lose on the futures contract, just as the long must expect a gain in order to assume the systematic risk. I'm bootstrapping Stulz, but this won't change firm value of $455.4, so forward price, F(0), must be 455*(1+5%) = $478.1. So firm "pays" investors E[S(t)] - F(0) = 500 - 478.1 = $21.86 (in FV terms) by way of expected loss on the futures contract shorted in order order to bring the firm to zero beta.
again, E(future firm value) = E(St) = $500; i.e., my read of Stulz is that this firm's beta doesn't really change, rather, he brings the net beta to zero with the ADDITION of the futures contract
So we have E(St) = 500 and F(0) = $455.4

Under theory of backwardation: the LONG future (i.e., the "investor" who will buy this systematic risk) expects to profit by E(St) - F = $21.7
The SHORT futures (the firm) expects to lose by F-E(St): the cost to hedge by transferring positive systemic risk to investors

Q8. Tautological (IMO) but Stulz is saying that Firm value = Future cash flow / discount rate, and by paying investors we just lowered the future cash flow by $21.7. Why doesn't that lower firm value? because our (effective = firm + future contract) beta = 0, so we reduce the denominator also. No free lunch.

Hope that helps, David

 

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Thanks David for the explanation.

I'm also not comortable with these two chapters

Just one thing. The cost of transfering the risk is E(St)-F0 = 500-478.1 and not 478.1-455.4 (I think that's what you meant in your post above)

Thanks a lot.

Alan

 

[David Harper CFA FRM]

Hi Alan,

Yes, thank you for the observant correction, I should have said cost in future value terms = E(St) - F0, just as you have. Fixed in my original. Thanks!

David