Math Help - Expected Value and Utility - Simple yet very confusing

Hey there,
I've been working on the following problem for a while now and I have done as much as I could, but I need your help to go any further.

I used to love statistics and never thought I'd be asking for help, but rather figure out on my own but it has been 3 years since I've taken any math course/calculus.

Suppose that you are risk averse, and that your utility of money is given by the log utility function:
u(w) = log w , where w is money (or wealth). Remember, log means the natural logarithm.

Consider a simple decision problem where you, as an investor, must decide how much of your wealth to allocate between a risky asset and a riskless asset. The riskless asset carries no risk, but no return—for example, stuffing cash under your mattress. The risky asset will return at rate rg with probability p (a good outcome), and rate rb with probability 1-p (a bad outcome). Therefore if your beginning wealth is w and if you invest x in the risky asset, where x could be anything between 0 and w, your two possible outcomes are to end up with either w + rgx or w + rbx.

Part A:Suppose that p = 0.5, rg = 0.1, and rb = - 0.1. Regardless of your current wealth w, how much money will you allocate to the risky asset?


My answer: .5(.1) + .5(-.1) = 0 = expected return of risky asset. The return on riskless asset is also 0, so a risk averse investor will allocate nothing to the risky asset.


Part B: Suppose that your current wealth is w = 100; that rg = 0.1 and rb = - 0.1; and that p = 0.52 (so that the risky investment carries a positive expected return, but just barely). How much money x will you allocate to the risky asset if you wish to maximize expected utility?


My answer: .52(100+.1x) + .48(100-.1x) = 100 + .004x = return. So in this case the return on risky asset is .004 and the return on riskless is still 0, as stated in problem. I said to maximize utility, you would allocate all the 100 to the risky asset. But I have a hunch its wrong b/c I need to use the log(w) function (utility of money) somehow, that's stated in the problem. Maybe take log of (100+.004x)..i am thrown off.

Before I fully understand how that utility function plays a role in Part B, there is no way I can start Part C.

Part C: Imagine that the government levies a tax t on returns from all risky assets. Your after-tax wealth will therefore either w +(1 - t ) *rg x or w +(1 -t ) *rb x, with probabilities p and 1 -p, respectively. (A tax on a negative return is like a subsidy—just like, for example, the subsidy the IRS gives by allowing people to deduct investment losses from their taxable income.)

1. Suppose that the tax rate is t = 0.1, and that the numbers are otherwise the same from Part B. If you
wish to maximize expected utility, how much of your money x will you allocate to the risky asset now?
What is the expected utility of this optimal allocation? Is it better or worse than the expected utility of
the optimal allocation in the “no-tax” regime?

2. Now suppose that the tax rate is even higher: t = 0.25, and that the numbers are otherwise the same
from Part B. How much of your money x will you allocate to the risky asset now? What is the expected
utility of this optimal allocation? Is it better or worse than the expected utility of the optimal allocation
from Parts B and C1?

3. As a general rule, if investors are risk averse, how does the optimal level of investment in a risky asset
change when you tax its return? How does the tax rate affect the expected utility that investors will
enjoy, assuming that they optimize their expected utility for whatever tax rate the government sets? You
can provide “qualitative” answers and not necessarily quantitative ones, but make sure to explain your
reasoning.

Derive an equation that characterizes your optimal x (allocation to the risky asset) as
a function of p, rg , rb , w, and the tax rate t .This requires some basic calculus. If you can’t solve for x, write down a correct equation that x must satisfy in order to be optimal, and simplifying
it as much as you can.

Thanks for all the help & I'll continue to report my progress on this problem.

I may look at this more later, but in part (a) you should be maximizing the expected value of log(w). The underlying idea behind this stuff is that the goal of the investor is to maximize their expected utility.

I am not sure if I should be using the log(w) function in Part A at all since it mentions nothing about utility. But for Part B, I do know that you need to maximize expected utility but i just don't know how to do that.

Originally Posted by recursive83

I am not sure if I should be using the log(w) function in Part A at all since it mentions nothing about utility. But for Part B, I do know that you need to maximize expected utility but i just don't know how to do that.

If you assume that expected utility is what matters, then literally every question is going to be tied to the utility function in one way or the other - for example, risk aversion can be derived from the utility function (i.e. assuming the investor is risk averse given the log utility function is redundant - if I recall correctly, risk aversion is immediate from the utility function being convex). Part (a) is fine since just averaging the numbers alone tells you that investing is bad, but you might want to try to derive the same result by appealing the utility function, as practice. You would want to maximize

in X. This may give you some ideas on how to approach the other problems. I may look at the other problems a little later.

EDIT: This should help you with (b). In general, to choose X, you want to maximize

in X.

Part (c) just puts a twist on this, but they give you the expression for your post-investment wealth.