# What is the optimal strategy when there is an equal chance for gain or loss but the size of the potential gain is larger?

I'm investigating a situation where the chance for gain or loss is the same, but the amount gained is greater than the amount that is lost. For example, the gain would be about 30% of the trade amount, and the loss would be 23% of the trade amount. While there is slightly more to it than that, that is the core of it -- random/even chance for hitting the gain or the loss the way the trade is structured, and approximately the percentages indicated. Please note either the gain or loss will be reached.

If one has amount A to invest, what considerations need to be taken into account to make a situation like this profitable, or is it not possible for it to be profitable (e.g. due to many successive losses)?

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## 1 Answer

This is practically a textbook case begging for the Kelly criterion.

In your specific example, the optimal trade size is $f^*A$, where $f^*$ maximizes the average rate of return $$\mathbb{E}[\log (X)]=0.5\log(1+0.3f)+0.5\log(1-0.23f).$$ Here $f$ is the fraction of the current capital to trade. A straightforward calculation yields that $$f^*=\frac{0.3-0.23}{2\times 0.3\times 0.23}\approx 0.5072$$

In general, if you expect to gain $gX$ with probability $p$ or lose $lX$ with probability $q$ on a trade of the size $X$, then the optimal (Kelly) bet is $$f^*=\frac{pg-ql}{gl}.$$

Some caveats might be worth noting.

• The Kelly framework assumes that sequential trades are (sufficiently) independent.
• Since the exact payoffs $g$, $l$ and probabilities $p$, $q$ are typically not known, it is safer to bet less than Kelly. Betting double the optimal Kelly bet reduces the growth rate of capital to zero (see e.g. "Good and Bad Properties of the Kelly Criterion" by Bill Ziemba).
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Thanks much for your detailed answer.  I am surprised at how high the ratio (0.5072) came out with these numbers. – Ray Jan 10 '12 at 1:12
Related, do you by chance know of any formulae that describe  "dollar-cost-averaging" for short term applications.  E.g. if an instrument varies approximately 20% a day in price, are there formulae that describe the effect of purchasing this instrument perhaps 10-12 times over 3 days at regular intervals, always spending the same amount, to average one's price down? Of course not every trade will succeed, as the item purchased may not rise back up, but is there formula that can help understand this better?  Thanks much. – Ray Jan 10 '12 at 1:27
@Ray: Thanks for your comments. I cannot give you a precise reference at the moment although this should be rather standard stuff in the theory of portfolio management. You might want to post this as a separate question. – olaker♦ Jan 10 '12 at 20:27
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