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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.

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  • $\begingroup$ He says its drift rate. He seems to mean the instantaneous return, like $\mu$ in GBM. For GBM the mean log-return is $\mu-0.5\sigma^2$. $\endgroup$ – fesman Jul 25 '20 at 7:01
  • $\begingroup$ 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. $\endgroup$ – oDUfrKeqea Jul 26 '20 at 16:59
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    $\begingroup$ Thanks, I now wrote an answer. What do you think? $\endgroup$ – fesman Jul 26 '20 at 18:19
  • $\begingroup$ 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! $\endgroup$ – oDUfrKeqea Jul 26 '20 at 21:21
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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".

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