Friday, April 22, 2011

Corporate finance brainteaser

Warning: Original content by blog author ahead

It's standard corporate finance practice to calculate the value of future cashflows as the expected cashflow divided by a risk-adjusted discount rate.

Suppose you have a project that delivers one year from now a $1 billion gain with 50% probability and a $1 billion loss with 50% probability, so that its expected cashflow is 0. This is an extremely unattractive proposition given its high risk and zero expected reward, but following the usual practice tells us that the value of this cash flow package is 0, since 0/(1+r) = 0 for any r that is not -1. So the discounted cash flow method tells us that we are indifferent between having this gamble and not having this gamble!

How do we reconcile this result with the fact that we would actually be willing to pay a large sum to avoid taking on this gamble?

Blog comments are temporarily open, or you can email me (my address is on my personal homepage). Prize for a correct answer is immortal glory via mention on this blog.

UPDATE: You can find a hint here.


Basil Seal said...

the risk adjusted discount rate rates wins and loses equally, provided they fall at the same time, so it must be something else.

perhaps it's because the benefits and costs are measured linearly and hence with risk neutrality, rather than risk aversion?

Josh said...

You're "investing" (putting at risk) 1 billion and getting back 1 in expectation. If you are averse, Npv=-1+1/(1+r)<0

In general, you should always account for capital that is at risk (and when it is returned and no longer at risk) in cash flow forecasts.

Jess Austin said...

I would start by breaking this into two potential investments: a .5 chance at $1B and a .5 chance at $-1B. If you're going to adjust the expectation for risk in the way you propose, why wouldn't the discount go the other direction for a loss? So the risk-adjusted expectation might be:

$0.5B/(1+r) - $0.5B/(1-r)

which unless I'm braindead simplifies to:


It would take an extremely well-capitalized investor for the r to remain constant over a $2B range! You've set p=0.5 in this question. The generalization would be, for a given r, what does p have to be to make the investor at least indifferent to the investment? I'm thinking:

p >= (r+1)/2

Jane said...

Discounting by the risk-adjusted rate is appropriate for companies that only worry about risk correlated with the market. If they and their investors face perfect capital markets and no bankruptcy concerns, then only market risks matters. Firms and executives should be fine with or without this project, as they diversify away such idiosyncratic risk. Strong assumptions, but that's mainstream finance theory.

Daniel said...

Two points:
1. In the textbook world that Jane mentions above, perfect capital markets mean that borrowing and lending is frictionless. You can borrow any amount (even $1 billion) provided you pay the discount rate r.

2. Invest in projects in descending order of NPV. This project would only happen after all other positive NPV projects were undertaken.

BC said...

Hi, two guesses:

1) Since you mentioned willing to pay a large sum, any insurance premium that is priced better than the stated probability (<50%) would give me a positive expected value.

2) Didn't include T0 investment. The cost of carry (discount rate) will give you a negative return, hence this project is rejected.

rahrah said...

In order to answer this question you have to be a believer in interdisciplinary research, like in the University of California system. You have to draw upon principals of behavioral finance. All that aside, basically its the principal of loss aversion. people will prefer 5 dollars at 100% probability rather than 50% of $20 gain and 50% chance of 9 dollar loss even though the latter has a higher expected value. loss aversion is also seen in premiums paid for put options during volatility and negative real rates on treasurines during the financial crisis.