General form for the return R:

$$R = \Pi_i (1+xa_i)^{p_i}$$

Optimal conditions:

$$\partial_x R = \sum_i \frac{p_i a_i}{1+xa_i}\Pi_j ((1+xa_j)^{p_j} = R \sum_i \frac{p_i a_i}{1+xa_i} = 0$$

Specializing to previous case, where i runs from 1 to 2:

$$\frac{p_1 a_1}{1+xa_1} + \frac{p_2 a_2}{1+xa_2} = 0$$

$$(p_1 a_1)(1+xa_2) + (p_2 a_2)(1+xa_1) = 0$$

$$p_1 a_1+ x(p_1+p_2)a_1a_2 + p_2 a_2 = 0$$

$$x = -\frac{p_1 a_1+p_2 a_2}{a_1a_2} = -\frac{\left<a\right>}{a_1a_2}$$

Notice that when $p_1\rightarrow 1$, $x \rightarrow -\frac{1}{a_2}$, so as disaster becomes ever more unlikely, you would bet a proportion of your wealth up to the loss ratio. This is a reflection of the Kelly criterion's tendency to never allow anything to go to zero, under any circumstance.

We'll consider why this is undesirable in some future post. For now, Let's make most use of the formulation, and try to find good ways of summarizing win/loss ratios and frequencies to fit into the form above.

The terms for each outcome ($\frac{p_i a_i}{1+xa_i} =\frac{p_i}{a_i^{-1}+x}$)  sum to zero for the optimal $x$. Since $x > 0$, These terms are either monotonically increasing or decreasing with $x$ depending on the sign of $a_i$.

The question then is how one would represent the two groups of monotonically increasing and decreasing terms so has to figure out which $x$ they net out at. This will be covered in the next post.

Suppose you have a bet which loses money most of the time, but wins a massive amount now and then, how much money should you put on it?

Let's say the 1 time you win, you win $a for each dollar you bet, and the N times you lose, you lose $b for each dollar you bet. By the Kelly criterion, the geometric average rate of gain if you bet $x$ of your wealth would be

$$R = (1+xa)(1-xb)^n$$

Setting $\partial_x R = 0$, you get

$$x = \frac{a - nb}{(n+1)ab} = \frac{\left<\mathrm{arithmetic\ gain}\right>}{ab}$$

Suppose you are asked to flip a coin, and heads you win $3, and tails you lose $1---then $n=1, a=2,b=1$, and therefore $x = \frac{1}{2 \cdot 2} = \frac{1}{4}$, i.e., you should bet 25% of your wealth.

If you have a lottery ticket that has a 1 out of 5,000 chance of winning $10,000 that costs $1, and you are only allowed to buy one number, then $n=4999,a=9999, b=1$, and
$x = \frac{5000}{49999 \cdot 9999} \approx 10000^{-1}$ and you should only bet 0.01% of your wealth at a time.

Conversely, if you were selling a lottery ticket that had 1 out of 10,000 chance of winning $5,000 that cost $1, $n = 9999, a = -4999, b = -1$, and $x = \frac{5000}{9999\cdot4999} \approx 10000^{-1}$ and you should be trying to have about 0.01% of your wealth at stake.

Related: Do not play the lottery unless you are a millionaire

When I first found out about Case-Shiller futures, I was pretty excited at the possibility of buying a house, enjoying the cheap loan and tax benefits, and hedging out most of the risk.

Alas, the futures have no volume. As of today the open interest on the Feb 2010 New York Case-Shiller futures (NYMG10) is 2 (as in the first integer greater than 1).

It's difficult to make markets when the underlying and the instrument differ so much in liquidity? Granted, SP500 futures are more liquid than the basket of stocks too, but that gap is bridgeable. What differentiates a bridgeable from an unbridgeable gap? What implications does this have for the existence of noise traders?