# Calculating M in Kelly portfolio optimization

My Question

In: $$F^* = C^{−1}[M−R]$$ where $$M$$ is a vector of $$n$$ securities returns, is the log return, or arithmetic return, intended to be used for computing the drift rate $$M$$?

Background

Thorp writes (8.4) (see Page 34 row 18):

Consider first the unconstrained case with a riskless security (T-bills) with portfolio fraction $$f_0$$ and $$n$$ securities with portfolio fractions $$f_1,\cdots,f_n$$. Suppose the rate of return on the riskless security is $$r$$ and, to simplify the discussion, that this is also the rate for borrowing, lending, and the rate paid on short sale proceeds. Let $$C=[s_{ij}]$$ be the matrix such that $$s_{ij},i,j=1,\cdots,n$$, is the covariance of the $$i$$th and $$j$$th securities and $$M=(m_1,m_2,\cdots,m_n)^T$$ be the row vector such that $$m_i,i=1,\cdots,n$$, is the drift rate of the $$i$$th security.

continuing (Page 34 row 38)...

Then our previous formulas and results for one security plus a riskless security apply to $$g_\infty(f_1,...,f_n)=m−s^2/2$$. This is a standard quadratic maximization problem. Using(8.1)and solving the simultaneous equations $$∂g_\infty/∂f_i=0,i=1,...,n$$, we get $$F^∗=C−1[M−R]$$,

In section 8.2 of Thorps THE KELLY CRITERION IN BLACKJACK SPORTS BETTING,AND THE STOCK MARKET) table 7 (pg 31 row 27) shows mean log returns. Further down Thorp notes:

As a sensitivity test, Quaife used conservative (mean, std. dev.) values for the price relatives (not their logs) for BRK of (1.15, .20), BTIM of (1.15, 1.0) and the S&P 500 from 1926–1995 from Ibbotson (1998) of (1.125, .204) and the correlations from Table 7. The result was fractions of 1.65, 0.17, 0.18 and−1.00 respectively for BRK,BTIM, S&P 500 and T-bills. The mean growth rate was .19 and its standard deviation was 0.30

When switching between log normal returns vs arithmetic returns I find that $$F^*$$ leverages are higher when using arithmetic means compared to log normal mean returns for $$M$$ which seems counter intuitive to that being described as a more conservative estimation.

• He says its drift rate. He seems to mean the instantaneous return, like $\mu$ in GBM. For GBM the mean log-return is $\mu-0.5\sigma^2$. – fesman Jul 25 '20 at 7:01
• I think you may be right. If I recall the CAGR can be extrapolated by $r + F^T * C * F/2$ where $r$ is the risk free rate. I consistently get a ~6.3% difference in mean GBM future value (10k simulations) versus FV based on CAGR with GBM being higher after the same number of periods for both. I would have thought these would converge more closely but the difference I presume is attributed to GBM's diffusion. To note, this difference is the smallest when using drift for M based on log returns, like you suggested, and highest when using arithmetic returns and means for M. – oDUfrKeqea Jul 26 '20 at 16:59
• Thanks, I now wrote an answer. What do you think? – fesman Jul 26 '20 at 18:19
• Just to follow up, this does converge when sigma is set to zero in GBM. I missed a $*$ when calculating CAGR after de-levering the portfolio and multiplied by the time period instead of raising to the power of it in python. So I have confirmed that your answer is exactly right. Thank you for your help! – oDUfrKeqea Jul 26 '20 at 21:21

Thorp defines $$g_{\infty}$$ as the mean long run logarithmic portfolio return. He argues that this is maximized when the portfolio is set
$$F^{*}=C^{-1}(M-R)$$
Here is $$M$$ a vector of drift rates $$m_i$$ (these are usually denoted by $$\mu$$ in a Geometric Brownian motion). This is nowadays a fairly standard formula, see e.g. here https://faculty.chicagobooth.edu/john.cochrane/research/papers/portfolio_text.pdf (growth optimal portfolio is a special case of the CRRA model with $$\gamma=1$$).
The drift rate of security $$i$$ can be estimated e.g. as $$m_i=y_i+s_i^2/2$$, where $$y_i$$ is mean log-return and $$s_i^2$$ is variance of log-returns. The mean log-return is below the drift due to a "variance penalty".