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I came across this problem today:

Calculate the log-utility optimal fraction of your capital to bet on a fair coin flip where you win $x$ on heads and lose $y$ on tails.

What is the meaning of log-utility in log-utility optimal fraction? Is this an portfolio management term or a statistics term? Apologies if this is in the wrong SE, please close this if it’s irrelevant!

user107224
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    Log utility on value $c$ is a function $u(c) = \log c$. (It doesn't matter what base it is, since these are all equivalent up to a constant factor.) It represents constant relative risk aversion. It's a term from economics. – Brian Tung Nov 26 '18 at 01:56
  • @BrianTung How are they equivalent? For example, when throwing a coin, head gives $2 and tail gives $1. Then $\frac{\log2+\log1}{2}$ is obviously different from $\frac{\ln2+\ln1}{2}$ – Interstigation May 06 '21 at 21:17
  • @CELLSecret: The log-utility is different, but all the utilities change by the same constant factor ($\ln 10$, in this case), and the optimal fraction is unchanged by this scaling factor. That's what I mean. Sorry that was unclear. – Brian Tung May 06 '21 at 22:31
  • Oh I see. How do we determine if we would throw the coin again? If I have a question that states this: You have 3 chances of throwing a dice. At the end of each throw, you can choose to keep the money or keep throwing. The money you are given is determined by the last throw's point value, that is, 1 for $1 and 2 for $2 etc. Assume you are risk averse and have log-utility, what is the expected amount of money? Which base do I use here? (also I'm glad that you are still using StackExchange frequently, wasn't really expecting you to answer that quick lol) – Interstigation May 06 '21 at 23:16

1 Answers1

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If the initial capital is $1$ unit of wealth and the fraction bet is $f$, then the capital after betting on the coin toss is

$$W_1 = 1 + f\xi$$

where $\xi$ is a binary random variable. Assuming a fair coin and WLOG $x,y > 0$ enumerated in wealth units , we have $$P(\xi = x) = P(\xi = -y) = 1/2$$

The expected capital is

$$E(W_1) = E(1 + f\xi) = 1+ f(x-y)/2$$

If $x >y$, you have an edge and the optimal fraction is $f^* = 1$ to maximize expected capital. However, if you are risk-averse you might not play to avoid a $50 \%$ chance of losing $y$.

A utility function is a construct that assigns preferences to random outcomes (gains and losses). Specifically log utility was introduced by Bernoulli to resolve the St. Petersburg paradox

The optimal fraction with log utility is obtained by maximizing

$$E( \log W_1) = \frac{1}{2} \log(1 + fx) + \frac{1}{2} \log (1 - fy)$$

subject to the constraint $0 \leqslant f \leqslant 1$.

RRL
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