When investing, we spend plenty of time thinking about which securities should we buy but we rarely wonder how much money should we allocate in each asset.

Although it does not seem like an important aspect, it is crucial when defining a strategy, up to the point that it can determine the whole performance of your portfolio.

In this sense, the **Kelly criterion** helps us selecting the optimal proportion such that we can maximize our expected returns. It is not surprise that successful investors such as Edward Thorp, Bill Gross, and Jim Simons among others have used this criterion in their investments.

## Introduction

In the previous post, we talked about the history of the Kelly criterion and some of its main characteristics when betting in coin tosses or investing in the stock market. In this post, we are going **to delve a little more** into this last one, the stock market.

## The formula

We will give some examples using the Kelly criterion for a one asset investment and a multiple asset portfolio.

So far I have read two different ways to obtain the Kelly fraction \(f^*\). In this section, we will see some of the main steps to obtain the formula. Refer to [1] and wikipedia for a better understanding.

### Through geometric Brownian motion

If we assume that the asset \(S\) follows a lognormal distribution, at time \(t\), \(S_t\) can be defined as:

$$

S_t = S_0 \exp((\mu – \frac{\sigma^2}{2})t + \sigma W_t)

$$

If we decide to invest a fraction \(f\) of our money in that asset, and invest the rest \(1 – f\) in a risk-free rate, such as Treasury Bills, then we would have an expected return of:

$$

E[f(\frac{S_1}{S_0} – 1) + (1 – f)r]

$$

Now, if we take logarithms of our expected return, in order to simplify the maximization problem, we will have \(G(f)\) which is the **expected logarithm returns**:

$$

G(f) = f \mu – \frac{(f\sigma)^2}{2} + ((1- f)r)

$$

Now if we maximize the logarithm of our expected returns \(G(f)\) we obtain the **Kelly fraction**.

### Thorp’s method

I won’t write the whole demonstration but, as shown by Throp in [1], we can start from a random variable \(X\) representing the return of an asset with \(E[X]=m\) and \(Var[X]=s^2\) following a Bernoulli with probability:

$$

P(X = m + s) = P(X = m – s) = 0.5

$$

With an expected return (taking into consideration an initial capital of \(V_0\), a given allocation \(f\) and a risk-free rate \(r\)) of:

$$

V(f) = V_0(1 + (1 – f)r + fX)

$$

We can demonstrate by dividing the time interval into n equal independent steps (to reach a continuous-time when \(n \to \infty\)) that the optimal allocation also knows as **Kelly fraction** is the same as the one obtained before:

$$

f^* = \frac{m – r}{s^2}

$$

This can seem familiar to you since is similar to the well-known Sharpe ratio, but instead of dividing by the standard deviation, we divide it by the variance.

So, to resume, the value \(f^*\) is **the optimal fraction**, of our total capital, that should be invested in specific security in order to maximize our expected returns.

Note that, if \(m > r + s^2 \) (this is \(f^* > 1\) ) then you will have to leverage. When \(m < r + s^2\) we will invest \(1 – f^*\) in Treasury Bills. For our study we will assume that if \(f^*<0\) we will invest all in Treasury Bills.

Before we start and for simplicity, I have made some assumptions.

- First, we live in a perfect world without commissions, transaction costs, and taxes.
- Second, we cannot short sell.
- Third, our trades don’t alter the market.

Having seen this introduction, we can start with the cool part!

## One asset

We have selected the S&P500 for the first example (we could use an ETF to replicate the S&P500).

In the following image, the evolution of the optimal \(f\) for our time interval is represented. For risky lovers, as we can see, the optimal \(f\) it is always higher than 1, meaning that you should invest more than your actual initial capital.

On the other hand, I was interested in seeing a graph with the predictive power of the optimal \(f\) with the next day returns. None, apparently.

I know that we do not all have the same level of risk; some will prefer to leverage while others don’t. So I have decided to set different maximum levels of leverage, from x1 to x5.

In the following graph, we see the evolution of the accumulated returns according to the different leverage strategies vs the benchmark.

As we can see, when we set the maximum leverage to 1, the Kelly strategy returns are equal to the benchmark returns. On the other hand, with greater leverages, you obtain greater and more volatile results up to the point where you reach maximum leverage of \(x5\), where, interestingly, you end up losing money!

## Multiple assets

Nevertheless, most of the time we are interested, in a portfolio of assets rather than just one, so in this example, we will use the Kelly criterion for multiples assets. For this example, I have randomly selected 15 different assets from the NASDAQ index.

To calculate the optimal Kelly criteria for each asset, it can be demonstrated that:

$$

F^* = C^{-1}(M – R)

$$

Where \(C\) is the covariance matrix and \(M – R\) the excess returns. Notice that, when the assets are not correlated, the optimal result is the same as for a single asset.

In the following graph, we can take a look at the correlation matrix. As we can see, none of the correlations is greater than 0.5 (which is already high enough). Nevertheless, we are going to calculate the optimal \(f\) assuming both, that they are correlated and uncorrelated.

The following graph shows the evolution of the optimal Kelly fraction for each asset (on the left we are assuming assets are correlated and on the right uncorrelated). As we can see, there are certain differences. Nevertheless, in both cases, there are assets whose optimal \(f\) is greater than two.

In order to limit the maximum leverage \(\alpha\), so that

\(f_1^* + … + f_{15}^* > \alpha\) we will do the following:

$$

f_i^{**} = \alpha \frac{f_i^*}{\sum_{i=1}^{15}f_i^*}

$$

Furthermore, we will need to compare our results with a benchmark, which will be calculated in two different ways:

- First we assume that our money is divided and invested in equally weighted parts and don’t touch them again. For example, if we have 1500€ we will invest 100€ in each security the first day, and leave it there.
- Second, we assume that our money is reinvested daily in equally weighted parts for all the securities. In this case, you would update the weights daily, so that all the securities have always the same amount of money.

As we can see, the reinvestment benchmark seems to perform better in terms of accumulated equity. However, I personally prefer the x1 leverage strategy, with lower volatility and constant growth.

## Conclusions

We have used the Kelly criterion principals to determine the amount of money invested in a determined security. Some of the main conclusions are.

- First of all, it is really interesting to see the effect of reducing or augmenting the amount of leverage. In most of the cases leverage equal to 5 has ended up in losses.
- On the other hand, a maximum leverage of 1, even though it doesn’t have the best performance in terms of cumulative returns, has the lowest volatility and constant returns.

Thanks for reading and see you soon in following posts!

## References

[1] Thorp, E. O. (2011). The Kelly criterion in blackjack sports betting, and the stock market. In The Kelly Capital Growth Investment Criterion: Theory and Practice (pp. 789-832).

[2] MacLean, L. C., Thorp, E. O., & Ziemba, W. T. (2010). Good and bad properties of the Kelly criterion. Risk, 20(2), 1

## related posts

Nice post. Almost as brilliant as the first part! 😉