1) Suppose we play a game with a independent identically distributed random sequence of outcomes (such as betting on a sequence of biased coin flips)

2) We have initial capital X_0. Our capital after the nth outcome is X_n.

3) Our betting strategy is to bet a fixed fraction of our capital on each outcome i.e the bet on outcome i, denoted as B_i = f X_{i-1} where f

can vary between zero and any positive number. If f>1, then we are leveraged and have to borrow capital in order to make the bet.

4) If the sequence of outcomes is a favorable game (i.e. E[X_n] > E{X_{n-1}]), then it is true that our capital increases exponentially with time. The rate after n outcomes is given by:

G_n(f) = (1/n)log(X_n/X_0) with expectation g_n(f) = E[G_n(f)] and limiting value g(f) = lim_{n -> infinity} g_n(f)

(I'm not going to discuss when/whether any expectations or limits exist ... lot of references on this if you're interested)

5) In some sense g(f) is average rate that our capital is expected to grow as we play the game. You can think of g(f) = log(1+r) where r is the yearly rate. The Kelly criterion proposes to choose f so that g(f) is maximized. I.e. we choose our bet size so that our expected rate of capital growth is maximized.

E.0 Thorp and others have tried to apply the Kelly criterion to various applications. Thorp famously applied it to bet sizing in blackjack when the player knows that he/she is getting a favorable bet (usually this knowledge comes about through card-counting). Thorp also applied this to the stock market ... In an example calculation, he approximated stock returns as truncated (2 sigma) Gaussian ... adjusted the mean and variance to coincide with empirical stock data, and applied the Kelly criterion. He found the optimal f to be 1.17, i.e. 17% leveraged stock position was optimal under his assumed truncated normal assumption.

If the assumed distributions are "nice", the usual formula for g(f) is

g(f) = E[log(1+fS)]

Where S is the return of the market over the riskless rate. One should note that the reason he used a truncated model, is that this formula doesn't exist for probability distributions with unbounded negative values .... such as the normal model.

I decided to do a similar study as the example given in Thorpe's paper except, I thought I would do the following:

a) I used 50 years of empirical data from the DFA matrix book from 1967 --- 2016 or REAL yearly returns for S&P500, Long Term Government Bonds (LTGB), and Short Term (One month) T-Bill (STTB) rates. I took these points to be my empirical distribution that I use to calculate g(f).

b) For the data presented, I assumed that our bet on each year is to have a portfolio of proportion f stock and (1-f) proportion LTGB. If f > 1, we borrow money at the STTB rate + 1 percent.

With these assumptions, we can compute g(f) based upon this last 50 years of data:

So, we see that the optimal f is something like 1.52. So based on the last 50 years of market data, the best strategy for growing your porfolio under the Kelly criterion is to be leveraged out around 50%.

I then did a Monte Carlo simulation for results to be expected over a 30 year investment horizon and computed the distribution of realized rates for values of f = .6, 1, 2. The spread (variance) of the results increase markedly with increasing f. For shorter investment periods the spread is of course going to be even greater.

I'm sorry for the poor quality, but if you look closely you can see the three distributions overlaid on one another. The more spread out the distributions also have slightly higher means.

I tried a variety of variants on the calculations .... things move around a little but not substantially

aa) I tried the cost of borrowing to be STTB + i, where i=0,1,1.5, 2, 3, 4 ... You have to get up to around 4 before you can get the optimal f = .8

bb) I tried assuming that LTGB and S&P500 returns were independent ... that made almost no change in the graphs

cc) I tried using just 25 years of past data and just 10 years of past data .... things move around a little but not qualitatively.

**TLDR VERSION: Using empirical data from the markets for the last 50 years, the Kelly criterion of maximizing the growth rate of your portfolio says that you should be leveraged about 50% to stocks (i.e. 150% of your portfolio should be stocks).**