18 Kelly Betting
In this lecture, we will describe Kelly Betting, which is a reasonable heuristic to decide how much of your money to invest in an asset that is profitable in expectation, but risky.
As a first example, consider the following situation:
This is a good bet in expectation: your expected wealth is
However, this is very risky: if you bet all of your money, you will lose everything with probability
To introduce Kelly betting, we will first introduce a sequential version of the Coin Flip situation above, and show how maximizing the most likely or median value of your wealth can lead to more reasonable choices decisions than maximizing your expected wealth. Then, we will introduce the Kelly Criterion in a more general setting. Finally, we will explain how to generalize Kelly betting to situations where you have to choose between multiple bets.
18.1 Maximizing the typical value of your wealth
Consider the following iterated version of the Coin Flip example above:
Let
if the coin comes up Headsx−αx+3αx=(1+2α)x if the coin comes up Tailsx−αx=(1−α)x
After 100 days, your wealth is
As in the previous example, the expected value of this quantity is greatest when
Instead of looking at the expectation of your wealth after 100 days, you could focus on its value in the typical case, by considering the modal or median value of your wealth. This corresponds to taking
You can then maximize this modal value over
Let us now consider a different bet:
The same reasoning as before shows that your wealth after 100 days is
The median and most likely value of this random variable is
In both these examples, we saw how maximizing the modal or median value of your wealth leads to more intuitive levels of risk-taking than maximizing the expected value. In the next section, we will introduce the Kelly Criterion in a more general setting.
TODO: include some simulations
18.2 Kelly Betting: the general case
18.2.1 Maximizing the Expected Logarithm
Suppose that:
- Your initial wealth is
dollarsS - Every day
:t - You invest some fraction
of your current wealthα - You receive
times the amount you paidXt
- You invest some fraction
Moreover, we make the following assumptions:
- The
are sampled independently from the same distribution:Xt Xtiid∼p almost surely: you cannot lose more than what you investedαXt>−1
Then, after
In general, we understand the limiting behavior of sums of random variables better than the limiting behavior of products. This motivates taking the logarithm, and forming the average:
As
We can then pick
18.2.2 The Kelly Criterion for binary bets
The following bet is a common use case of the Kelly Criterion:
- With probability
, you win and are returnedp times your investment, so you netb+1 times your investment.b - With probability
, you lose and receive nothing.q=1−p
In this case, the expected logarithm is
The derivative vanishes when:
which is equivalent to:
This fraction increases when when your probability of winning
As an example, consider the Iterated Biased Coin Flip example from the previous section. In that case,
18.3 Choosing between multiple bets
So far, we have considered cases where there is only one possible bet: you can either invest your money in this bet, or keep it. However, a more realistic situation is that at any point in time you have to choose between multiple bets. In these cases, how should you split your money between these bets?
For example, suppose you could invest a fraction
However, this optimization problem often becomes intractable as the number of possible bets becomes large. For example, think about the enormous number of possible investments a trader can make at any point in time.
One idea could be to use the Taylor approximation
which is maximized by
When choosing between
The formula becomes more complicated when the
which is maximized by