This is the second post in a four-part series exploring the Kelly criterion:

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:

$$ V_1 = V_0 \left( (1+r_0)(1 - \sum_i l_i) + \sum_i l_i(1 + r_i) \right) $$

Simple, right? :-)

  • \(V_0, V_1\) are the available money before and after the first round, as before.
  • \(r_0\) is the risk-free return rate. This allows us to model opportunity costs (could put the money into a bank account instead of investing, \(r_0 = 0.004\)), or inflation (non-invested money loses value over time \(r_0 = -0.02\)).
  • \(l_i\) is the fraction of the available money invested in stock \(i\).
  • \(r_i\) is a random variable describing returns for stock \(i\).

The first term \((1+r_0) (1 - \sum_i l_i)\) represents all the money not invested in any stock, being invested at the risk-free return rate \(r_0\). The second term \(\sum_i l_i(1 + r_i)\) represents the outcomes of the investments in individual stocks \(i\).

We will re-formulate a bit to simplify the expression:

$$ \frac{V_1}{V_0} = (1+r_0) - \sum_i l_i (1+r_0) + \sum_i l_i(1 + r_i) = (1+r_0) + \sum_i l_i(r_i - r_0) $$

We can further simplify by merging the \(l_i\) and \(r_i\) into the vectors \(\vec l\) and \(\vec r\), with \(\vec 1\) being a vector where all elements are \(1\):

$$ \frac{V_1}{V_0} = (1+r_0) + \vec l \cdot (\vec r - r_0 \vec 1) $$

Let's move from a single game round to \(n\) rounds:

$$ \frac{V_n}{V_0} = \left( (1+r_0) + \vec l \cdot (\vec r - r_0 \vec 1) \right)^n $$

Now we are again in a position where we can follow Kelly's prescription: invoking the law of large numbers and finding the \(l\) which maximizes the gain \(V_n/V_0\).

For large \(n\), we can approximate:

$$ \frac{V_n}{V_0} \approx \prod_j \left( (1+r_0) + \vec l \cdot (\vec r_j - r_0 \vec 1) \right)^{p_j n} $$

In this step we switched from the random variable \(\vec r\) to its outcomes \(\vec r_j\). Each possible outcome \(\vec r_j\) occurs with probability \(p_j\). We justify this switch with the law of large numbers: the outcome \(\vec r_j\) will be observed proportionally to its probability, \(p_j n\) times (for large \(n\)). Consequently, we will have \(p_j n\) factors involving \(\vec r_j\) in the overall product, with \(j\) iterating over all potential outcomes of \(\vec r\).

Note that since \(\vec r_j\) is vector-valued, \(p_j\) is a joint probability distribution.

At this point we are nearly finished. We are looking for the vector \(\vec l_{opt}\) that maximizes \({V_n}/{V_0}\). In analogy to last post, we arrive at:

$$ \vec l_{opt} = \operatorname*{argmax}_{\vec l} \frac{V_n}{V_0} = \operatorname*{argmax}_{\vec l} \log \frac{V_n}{V_0} = \operatorname*{argmax}_{\vec l} \sum_j p_j \log \left( (1+r_0) + \vec l \cdot (\vec r_j - r_0 \vec 1) \right) $$

This equation is not analytically solvable, but may be approximated as a quadratic programming problem as described in a paper by Vasily Nekrasov.

Conclusion

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 a Jupyter Notebook containing several interactive plots that really helped me understand this material, you can find it here.

In the next post we'll discuss the origins of the logarithm in the Kelly formula.


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