This is the second post in a four-part series exploring the Kelly criterion:
- The Kelly Criterion: Introduction
- The Kelly Criterion: Multiple Investment Opportunities (this post)
- The Kelly Criterion: Where does the logarithm come from?
- The Kelly Criterion: Comparison with Expected Values
After the last post introduced the Kelly criterion and its application in deciding how much to invest in a single gamble, we'll investigate whether Kelly can help us choosing between multiple investment opportunities.
We'll start with a mathematical model:
Simple, right? :-)
- are the available money before and after the first round, as before.
- is the risk-free return rate. This allows us to model opportunity costs (could put the money into a bank account instead of investing, ), or inflation (non-invested money loses value over time ).
- is the fraction of the available money invested in stock .
- is a random variable describing returns for stock .
The first term represents all the money not invested in any stock, being invested at the risk-free return rate . The second term represents the outcomes of the investments in individual stocks .
We will re-formulate a bit to simplify the expression:
We can further simplify by merging the and into the vectors and , with being a vector where all elements are :
Let's move from a single game round to rounds:
Now we are again in a position where we can follow Kelly's prescription: invoking the law of large numbers and finding the which maximizes the gain .
For large , we can approximate:
In this step we switched from the random variable to its outcomes . Each possible outcome occurs with probability . We justify this switch with the law of large numbers: the outcome will be observed proportionally to its probability, times (for large ). Consequently, we will have factors involving in the overall product, with iterating over all potential outcomes of .
Note that since is vector-valued, is a joint probability distribution.
At this point we are nearly finished. We are looking for the vector that maximizes . In analogy to last post, we arrive at:
This equation is not analytically solvable, but may be approximated as a quadratic programming problem as described in a paper by Vasily Nekrasov.
It should be obvious that the Kelly criterion is applicable in a wide range of scenarios, from gambling over investment decisions to whether to buy insurance. If you're interested in interactive plots that really helped me understand this material, you can find them in this Jupyter Notebook.
In the next post we'll discuss the origins of the logarithm in the Kelly formula.