Why does Kelly maximise $E[\log\space G]$ rather than simply $E[G]$?

Given $$G=\frac {C_n} {C_0}$$, with $$C_0$$ the initial capital and $$C_n$$ the final capital after $$n$$ trades,

the Kelly criterion derives the optimal fraction of capital to invest in each trade, by maximising the expectation $$\mathbb{E}[\log G]$$.

Why did Kelly choose to maximise $$\mathbb{E}[\log G]$$ rather than simply maximising $$\mathbb{E}[G]$$? What is the rationale behind this choice?

• G is a product of terms, so it is convenient to take the log... makes it into a sum. Jan 7 '19 at 21:55
• @noob2 it's more subtle than that. In my simulations maximising $E[log G]$ leads to $f^*=0.1$ while maximising $E[G]$ leads to $f^*=1$. If your G values are 1,10,100 then $E[log G]=2$ while $E[ G]=37$. It's a whole different function, not just simply more practical. Jan 7 '19 at 22:05
• In fact, the paper by Thorp I had linked in a comment on your other recent question regarding the Kelly criterion, quant.stackexchange.com/q/43337/34134 , provides one explanation: maximizing wealth proper leads to bold plays which leads to, almost surely, financial ruin even when the game is fair, iirc but do actually peruse this time. Further, he quotes a result (and reference) that maximizing log wealth asymptotically dominates any other strategy in a certain sense. There are many other useful facts—like the one mentioned by noob2, etc. Jan 8 '19 at 0:18
• @LoveTooNapp29 thanks! yes this time I will read the whole paper :) I agree that maximising $E[Log G]$ leads to a more conservative behaviour than maximising $E[G]$ - the other advantages are practical, but secondary. Jan 8 '19 at 7:21

5 Answers

Maximizing $$E[\log(G)]$$ which corresponds to a concave utility function is a subtle way of incorporating risk aversion in the utility.

Maximizing $$E[G]$$ is basically saying that you have linear utility which corresponds to infinite risk appetite because as soon as you have positive expectation you are willing to bet as much capital as possible no matter the variance of the resulting portfolio. In practice it can easily lead to ruin almost surely if the game is played long enough although the linear utility function is blind to that.

On the other hand, since $$\log$$ is concave, the second order term ensures that there will be a balance found between expectation and variance.

• Yes, risk aversion in the context of expected utility theory can be used, but this answer doesn't get to the particular properties of the expected $\log$ objective in contrast to the expectation of other concave functions. Jan 9 '19 at 3:10
• @MatthewGunn from a trading perspective i see $\log$ just as some particular concave utility functional that happens to have a nice interpretation. All choices of concave utility will lead to avoiding being ruined almost surely. Therefore avoiding ruin is not by itself a justification of choosing the particular functional $\log$ vs another concave utility. Even if Kelly originally chose $\log$ to illustrate his idea i believe the essence of the idea appears when you interpret it in a quadratic utility context.
– Ezy
Jan 9 '19 at 12:04
• All choices of concave utility do not avoid ruin. Consider $u(x) = x^{0.9}$. It is concave: $u''(x) = -.09x^{-1.1} < 0$. You will find it leads to similar results to the risk neutral case of $u(x) = x$. Consider a simple case where you win $2$ times your bet with probability $\frac{1}{2}$. Utility $u(x) = x^.9$ leads you to bet $\approx 99.7\%$ of your wealth each time and converge on 0 wealth almost surely in the long run. May 9 '19 at 14:52

The short answer is that:

• Maximizing the expected logarithm leads to more wealth almost surely in the long run.
• In contrast, maximizing expected return can easily lead to going broke almost surely in the long run!

Maximizing expected return results in betting everything on your highest expected return investment. Repeatedly doing that over time typically leads to polarized outcomes: you become fantastically wealthy or (more likely) destitute.

Discussion of growth optimal portfolios

Kelly's original problem was related to repeated betting on binary outcomes, but the expected log wealth objective of his optimization problem connects more generally to what's called a growth optimal portfolio. Let $$\mathcal{S}$$ denote the feasible set of portfolio returns (where each return $$R \in \mathcal{S}$$ is a random variable denoting a gross return like $$\frac{X_{t+1}}{X_t}$$). The idea is that by choosing different investments, we can obtain any return $$R \in \mathcal{S}$$ for our overall portfolio. A return $$R^* \in \mathcal{S}$$ is called a growth optimal portfolio return if it solves:

$$\max_{R \in \mathcal{S}} \operatorname{E}[\log R]$$

Like many good ideas, the objective $$\operatorname{E}[\log R]$$ can be justified philosophically in multiple ways:

As @Dave Harris in his answer points out, it is the intuition of the latter, the attractive properties of maximizing the geometric mean, that initially motivated the expected log objective.

Note also the geometric mean return is the exponential of the (arithmetic) mean logarithmic return: $$\left(\prod_{i=1}^t R_i \right)^\frac{1}{t} = \exp\left( \frac{1}{t} \sum_{i=1}^t \log R_i \right)$$ When you maximize the log return or log wealth, you're maximizing the expected geometric growth rate.

In the long run, a growth optimal portfolio almost surely has higher wealth

Assume an IID setting where we repeatedly choose a stochastic return $$R$$ (from a set $$\mathcal{S}$$). Log wealth at time $$t$$ is given by the $$\log W_t = \sum_{i=1}^t \log R_i$$. Log wealth for the growth optimal portfolio is $$\log W^*_t = \sum_{i=1}^t \log R^*_i$$. We have $$\operatorname{E}[\log R^*] \geq \operatorname{E}[\log R]$$ by the optimality of $$R^*$$. Hence by the strong law of large numbers:

$$\lim_{t \rightarrow \infty} \frac{1}{t} \left( \log W^*_t - \log W_t\right) \geq 0 \text{ almost surely}$$

Log is a monotonic transformation so if log wealth is almost surely higher, wealth is almost surely higher. Something to realize too is that if $$\operatorname{E}[\log R] < 0$$, you will go broke almost surely over the long run.

Maximizing expected return has deeply problematic consequences

Perhaps unintuitively, maximizing expected return can easily lead to wealth heading towards $$0$$ in the repeated context instead of growing towards $$\infty$$.

For example, in the classic Kelly problem the problem is to choose what fraction $$f \in [0, 1]$$ of wealth to allocate to a risky, binary bet. Imagine you gain $$bf$$ with probability $$p$$ and lose $$f$$ with probability $$1-p$$. Maximizing expected return gives the problem:

$$\max_{f \in [0, 1]} p(1 + b f) + (1-p)(1-f)$$

This is equivalent to:

$$\max_{f \in [0, 1]} (p(b+1) - 1)f$$

Thus $$f^* = 1$$, you bet all your wealth, if the wager has positive expectation (i.e. $$p(b+1) > 1$$). In the limit as $$t \rightarrow \infty$$, this leads you to go broke almost surely! After $$t$$ bets, you have $$1 - p^t$$ probability of having 0 wealth. The expected log return is $$- \infty$$.

Kelly criterion:

Maximizing the expectation of the logarithm gives the problem: $$\max_{f \in [0, 1]} p \log (1 + b f) + (1-p) \log (1-f)$$

Assuming the bet has positive expectation, the constraints $$f \in [0, 1]$$ don't bind and the solution is given by the first order condition:

$$\frac{pb}{1 + bf} = \frac{1-p}{1 - f}$$

which simplifies to

$$f^* = \frac{p(b+1) - 1}{b}$$

A note of caution on growth optimal portfolios

While the growth optimal portfolio has many attractive properties, at least two general points of caution are in order:

• Log utility isn't magic. Other specifications, possibly with higher risk aversion, can be justified.
• Estimation errors can lead the estimated growth optimal portfolio to be substantially more risky than a naive analysis might suggest.

Models over the joint distribution of security returns are highly imprecise. You can get into extremely dangerous garbage in, garbage out problems when computing some notion of an optimal portfolio based upon highly uncertain inputs.

The original paper was concerned with optimizing the long run geometric return. In fact, he does not explicitly optimize either $$\mathbb{E}(G)$$ or $$\mathbb{E}(\log(G))$$. He also assumes the probabilities are known. The expectation is implicit in his assumption that $$G=\lim_{N\to\infty}\frac{1}{N}\log\left(\frac{V_N}{V_0}\right).$$

He notes that $$V_N=(1+\mathcal{l})^W(1-\mathcal{l})^LV_0.$$ The use of logs is from the obvious multiplicative relationship. Since $$V_N$$ is the only controllable variable and its structure is multiplicative, taking the logs simplifies the calculus with no loss of generality. This obviously reduces to $$G=\frac{W}{N}\log(1+\mathcal{l})+\frac{L}{N}\log(1-\mathcal{l})$$

The introduction of logarithms has only to do with the fact he is optimizing the long run geometric growth rate.

A quote from the original paper may give you an answer to your question as it has nothing to do with risk aversion, at least in Kelly's mind.

The gambler introduced here follows an essentially different criterion from the classical gambler. At every bet he maximizes the expected value of the logarithm of his capital. The reason has nothing to do with the value function which he attached to his money, but merely with the fact that it is the logarithm which is additive in repeated bets and to which the law of large number applies. Suppose the situation were different; for example, if the gambler's wife allowed him to bet one dollar each week but not to reinvest his earnings. He should then maximize his expectation (expected value of capital) on each bet. He would bet all his available capital (one dollar) on the event yielding the highest expectation.

This is indeed a very good question!

There were (and still are) very hefty debates, where even academic champions like Paul Samuelson were involved!

A very good starting point to get some main arguments is the following chapter 4 from the book "Fortunes Formula" by William Poundstone:

https://books.google.de/books?id=xz4y3u-qM04C&lpg=PA179&dq=fortunes%20formula%20part%20four&pg=PA179#v=onepage&q&f=false

The gist is that "in the long run" using the Kelly criterion would ensure an optimal outcome growth-wise "on average". But as we all know "in the long run we are all dead" and "on average" doesn't mean that it is the best outcome for You. In general the Kelly criterion is criticized for taking too much risk (i.e. generating too much volatility) which is why methods like e.g. "Half Kelly" are being used.

In a way it is not only a mathematical debate but touches also on more philosophical and psychological points. As I said, it is a very interesting question which can lead you down all kinds of different paths.

For me it was one of the starting points of QuantFinance and I am still fascinated by it.

• Thanks @vonjd my question arised while exactly reading that book 😋 indeed maximizing E[G] leads to a even more aggressive behavior than Kelly and Kelly itself seems too aggressive. So yes there sure is a psychological perspective: what profit distribution makes you sleep better at night. Jan 9 '19 at 13:12
• (+1) Part of argument too for "Half Kelly" etc... is how model or estimation error interacts with Kelly bet sizing. In practice, your estimated Kelly bet sizing (or estimated growth optimal portfolio) may differ significantly from the true, optimal fractions because of severe difficulties in estimation. There's also asymmetric consequences for betting above or below the Kelly fraction: betting above the Kelly fraction leads to lower expected log growth and massively more volatility. If you can't exactly see the cliff edge, taking a few steps back with "Half Kelly" may make a lot of sense. Jan 9 '19 at 16:13

Kelly maximises the geometric return in each period. I believe this is equivalent to maximising expectation of log wealth in the next period.

1. If you maximise the arithmetic return, your expected wealth will be higher. However the actual return you will experience is -100% i.e. you will go bust with probability 1.

2. If you maximise the geometric return, then with probability 1 you will achieve a higher wealth than anyone following any other strategy.

These statements may seem contradictory, but they are not, for more debate see the St Petersburg Paradox.

Both statements are only true in the limit of an infinite series of bets. I have seen the credit for point 2 given to Leo Breimann's 1960 paper Investment policies for expanding businesses optimal in a long‐run sense, but as Matthew Gunn notes above, the proof is a trivial consequence of the law of large numbers.

In a nutshell, the final wealth is a monotonically increasing function $$f^+$$ of the average log return $$W = W_0 (1+r_1)(1+r_2) ... (1+r_n) = W_0e^{\sum_n{ln(1 + r_i)}} = f^+\left(\sum\limits_n{ln(1 + r_i)}\right)$$

Since this is an arithmetic sum, the law of large numbers applies to it (therefore maximising $$\mathbb{E}(ln(1+r_i))$$ maximises the sum), and since the function is monotonically increasing, maximizing the sum maximises final wealth $$W$$. Note also that $$r_i$$ here are random variables.