The Kelly criterion is a technique for maximizing expected growth of assets by optimally investing a fixed fraction of your wealth in a series of investments. The idea has long been used in the world of gambling.

Example

You own a biased coin that will land heads up with probability p > . You find someone willing to bet any amount against you at evens. They are willing to bet any number of times. Clearly you can make a lot of money with this special coin. You start with $1,000. How much of this should you bet?

Long answer

Let's work with the above example. The first observation is that you should bet an amount proportional to how much you have. As you win and your wealth grows you will bet a larger amount. But you shouldn't bet too much. If you bet all $1,000 you will eventually toss a tail and lose everything and will be unable to continue. If you bet too little then it will take a long time for you to make a decent amount.

The Kelly criterion is to bet a certain fraction of your wealth so as to maximize your expected growth of wealth.

We use 0 to denote the random variable taking value 1 with probability p and —1 with probability 1 — p and f to denote the fraction of our wealth that we bet. The growth of wealth after each toss of the coin is then the random amount

The expected growth rate is

This function is plotted in Figure 2.5 for p = 0.55.

Figure 2.5: Expected return versus betting fraction.

This expected growth rate is maximized by the choice

This is the Kelly fraction.

A betting fraction of less than this would be a conservative strategy. Anything to the right will add volatility to returns, and decrease the expected returns. Too far to the right and the expected return becomes negative.

This money management principle can be applied to any bet or investment, not just the coin toss. More generally, if the investment has an expected return of x and a standard deviation a " x then the expected growth for an investment fraction of f is

which can be approximated by Taylor series

The Kelly fraction, which comes from maximizing this expression, is therefore

In practice, because the mean and standard deviation are rarely known accurately, one would err on the side of caution and bet a smaller fraction. A common choice is half Kelly.

Other money management strategies are, of course, possible, involving target wealth, probability of ruin, etc.

References and Further Reading

Kelly, JL 1956 A new interpretation of information rate. Bell Systems Technical Journal 35 917-926

Poundstone, W 2005 Fortune's Formula. Hill & Wang

Why Hedge?

Short answer

'Hedging' in its broadest sense means the reduction of risk by exploiting relationships or correlation (or lack of correlation) between various risky investments. The purpose behind hedging is that it can lead to an improved risk/return. In the classical Modern Portfolio Theory framework, for example, it is usually possible to construct many portfolios having the same expected return but with different variance of returns ('risk'). Clearly, if you have two portfolios with the same expected return the one with the lower risk is the better investment.

Example

You buy a call option, it could go up or down in value depending on whether the underlying go up or down. So now sell some stock short. If you sell the right amount short then any rises or falls in the stock position will balance the falls or rises in the option, reducing risk.

Long answer

To help to understand why one might hedge it is useful to look at the different types of hedging.

The two main classifications Probably the most important distinction between types of hedging is between model-independent and model-dependent hedging strategies.

• Model-independent hedging: An example of such hedging is put-call parity. There is a simple relationship between calls and puts on an asset (when they are both European and with the same strikes and expiries), the underlying stock and a zero-coupon bond with the same maturity. This relationship is completely independent of how the underlying asset changes in value. Another example is spot-forward parity. In neither case do we have to specify the dynamics of the asset, not even its volatility, to find a possible hedge. Such model-independent hedges are few and far between.

• Model-dependent hedging: Most sophisticated finance hedging strategies depend on a model for the underlying asset. The obvious example is the hedging used in the Black-Scholes analysis that leads to a whole theory for the value of derivatives. In pricing derivatives we typically need to at least know the volatility of the underlying asset. If the model is wrong then the option value and any hedging strategy could also be wrong.

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