Kelly Betting: Why Maximize Expected Log Wealth

# Why maximize expected growth rate?


# Why maximize expected growth rate?












It seems to me that the optimality of the Kelly Criterion relies on the assumption that it is in an investor's best interest to maximize his portfolio's expected growth rate. Why would he care what the expected growth rate is? Shouldn't his objective be to maximize his expected terminal wealth, or rather the utility of his terminal wealth?

## Answer by RRL (score 8)

https://quant.stackexchange.com/a/14377

The Kelly criterion is just one approach to portfolio construction (or bet sizing) that considers the risk-return tradeoff. There are many possible strategies (static or dynamic) that incorporate other criteria such as the maximum drawdown, probability of ruin, etc.

As pointed out by @John, Kelly is maximizing the log of wealth, which is equivalent to saying utility is the log of wealth.

Consider the coin tossing analogy where you bet on a sequence of biased coin tosses.

On the $k$th toss you bet $B_k$ dollars and win $B_k$ with probability $p > 1/2$ or lose $B_k$ with probability $1-p < 1/2$. After $n$ trials your expected wealth is

$$E(W_n) = W_0 + \sum_{k=1}^{n} (2p-1)B_k$$.

Since the game has a positive expectation you would maximize the expected value by betting all your available wealth on each trial. However, the bold strategy has probability of ruin $1-p^n$ and $\lim_{n \rightarrow \infty}(1-p^n)=1$. So maximizing expected terminal wealth has for most people an unacceptable downside risk.

The standard Kelly approach is to bet a fixed fraction $f$ of wealth at each trial.

After each trial, the total wealth assuming even odds payoffs is

$$W_n= W_0\prod_{k=1}^n(1+fX_k),$$

where $X_1,X_2,\ldots$ are i.i.d. binary random variables with $P(X_k=1)=p$ and $P(X_k=-1)=1-p$. Then

$$\log W_n= \log W_0+\sum_{k=1}^n\log (1+fX_k),\\\ E\left[\log \left(\frac{W_n}{W_0}\right)^{1/n}\right]= p\log (1+f)+(1-p)\log(1-f),$$

and the expected value is maximized in favorable games $(p > 1/2)$ when

$$f =f^*= 2p-1.$$

Hence, the Kelly criterion maximizes both the expected logarithm of terminal wealth and the expected "growth-rate".

In the original paper, Kelly addressed a problem in the transmission of information and used the gambling analogy without reference to portfolio construction or utility. I would surmise that the strategy was proposed because of a number of desirable asymptotic properties.

The foundation of the strategy is the notion of fixed-fraction or proportional betting. If there is no positive lower bound on the bet size (i.e., infinitely divisible capital) then if $p > 1/2$ and $0 < f < 1$ there is zero probability that $W_n = 0$. Defining the growth rate as

$$G_f := E\left[\log \left(\frac{W_n}{W_0}\right)^{1/n}\right]= p\log (1+f)+(1-p)\log(1-f),$$ then if $G_f >0$ the Strong Law of Large Numbers implies that, almost surely,

$$\lim_{n \rightarrow \infty} \log \left(\frac{W_n}{W_0}\right)^{1/n}= G_f > 0,\\ \lim_{n \rightarrow \infty}W_n = \infty.$$

Finally, the optimal-$f$ strategy which maximizes $G_f$ and $E[\log W_n]$ has an expected time to reach a specified goal that is asymptotically less than any other strategy (including non-proportional strategies).