Mean-Variance Utility Function
Now that I am familiar with and enjoying calculations related to the Kelly criterion, let us consider another variant in which we define a utility function in terms of the mean and variance of the wealth and then consider the problem of maximizing it. An easy-to-understand example is the method of defining the expected utility function as a quadratic function:
\[ \small E[u(W)]= \mu_W-\frac{\gamma}{2}\sigma_W^2. \]
Here, \(\small \mu_W\) represents the expected return and \(\small \sigma_W^2\) represents the variance of the return. This has properties similar to the CRRA utility function (see the article below for more information on the CRRA utility function).
Let the utility function be:
\[ \small u(W) = \left\{ \begin{array}{ll}\frac{W^{1-\gamma}-1}{1-\gamma},& \quad \gamma \neq 1 \\ \log(W), & \quad \gamma=1 \end{array}\right. \]
and it can be approximated as:
\[ \begin{align*} \small u(W) & \small \approx u'(W)|_{W=1}(W-1) + \frac{1}{2}u^”(W)|_{W=1}(W-1)^2 \\ & \small = (W-1)-\frac{\gamma}{2}(W-1)^2 \\ & \small \approx \mu_W -\frac{\gamma}{2}\sigma_W^2 \end{align*} \]
by expanding it in a Taylor series around \(\small W=1\). The method of defining a utility function in terms of the expected value and variance of a wealth is a method commonly used in financial economics. In fact, when applying the Kelly criterion to stock investments, it is sometimes calculated by defining it as:
\[ \small u(W)= \mu_W-\frac{1}{2}\sigma_W^2 \]
(This method does have some troubles though. Readers who would like to learn more are referred to my paper, “On the Kelly Criterion in Stock Investment.”). Let us derive the optimal betting strategy in this case.
Optimal Betting Strategies for Mean-Variance Utility Functions
As with the general Kelly formula, let us consider the case where there is a house edge. In other words, we assume that one of the \(\small n\) choices will win, and the amount bet on each choice will be denoted as \(\small A_1,\cdots,A_n\). In this case, the odds of each option are given by
\[ \small \alpha_i = \frac{(1-\theta)\sum_{i=1}^n A_i}{A_i}. \]
Here, \(\small \theta\) is the house’s edge ratio. Note that in this case, the sum of the winning probabilities calculated backwards from the odds will not be 1, but will be:
\[ \small \sum_{i=1}^n q_i = \sum_{i=1}^n \frac{1}{\alpha_i} = \frac{1}{1-\theta} > 1. \]
Let the gambler’s estimated probability of winning be denoted as \(\small p_1,\cdots,p_n\) and the proportion of the funds he or she will keep be denoted as \(\small b\). In this case, the gambler’s optimization problem to maximize the mean-variance utility function can be expressed as:
\[ \begin{align*} \small \max_{b, f_1, \cdots, f_n} \; & \small E[u(W)] \\ \small \text{s.t.} \; & \small b + \sum_{j=1}^n f_j = 1, \; b \geq 0, \; f_j \geq 0 \; \forall j. \end{align*} \]
Here,
\[ \begin{align*} & \small E[u(W)]= \mu_W-\frac{\gamma}{2}\sigma_W^2 \\ & \small \mu_W = E[W] = \sum_{j=1}^n p_j (b+f_j\alpha_j) \\ & \small \sigma_W^2 = E[W^2]-E[W]^2 \\ & \small \quad \;\;\: =\sum_{j=1}^n p_j (b+f_j\alpha_j)^2 -\left(\sum_{j=1}^n p_j (b+f_j\alpha_j)\right)^2. \end{align*} \]
To simplify the calculations, we have defined it as a quadratic function of wealth rather than return, but it is clear that the results will be the same.
Let us solve the above optimization problem in the same way as the Kelly criterion. Introducing the slack variable \(\small k,\lambda_1,\cdots,\lambda_n \), defining the Lagrangian as:
\[ \small L(b, f_1,\cdots, f_n, k, \lambda_1,\cdots,\lambda_n) = E[u(W)] + k \left( 1-b- \sum_{j=1}^n f_j \right) + \sum_{i=1}^n \lambda_i f_i \]
and finding the Karush-Kuhn-Tucker (KKT) condition, we get
\[ \begin{align*} & \small \frac{\partial L}{\partial f_i} = p_i\alpha_i\left(1+\gamma \xi -\gamma(b+f_i\alpha_i)\right)-k + \lambda_i = 0, \quad \forall \; i \\ & \small \frac{\partial L}{\partial b} = \sum_{j=1}^n p_j(1+\gamma\xi-\gamma(b+f_j\alpha_j)) – k = 0 \\ & \small \frac{\partial L}{\partial k} = 1 -b -\sum_{j=1}^n f_j = 0 \\ & \small \lambda_if_i = 0, \quad \forall \; i. \end{align*} \]
Here,
\[ \small \xi = E[W] = \sum_{j=1}^n p_j (b+f_j\alpha_j) = b+\sum_{j=1}^n p_jf_j\alpha_j = b+\sum_{f_j>0}p_jf_j\alpha_j. \]
From the second equation of the KKT condition,
\[ \small k= \sum_{i=1}^np_i=1 \]
can be obtained (In fact, this formula is false. If the utility function is quadratic, it is possible that \(\small b<0\), so this must be taken into account as a constraint. For simplicity, we will assume that this formula is correct under the assumption that \(\small b \geq 0\) holds.). If we rearrange the first equation of the KKT condition for \(\small f_i>0\) with respect to \(\small f_i\), then
\[ \small f_i=\max\left\{- \frac{q_i^2}{\gamma p_i}-\left(b-\xi-\frac{1}{\gamma} \right)q_i, 0 \right\} \]
holds. Regarding the second equation of the KKT condition, if we aggregate separately the terms where \(\small f_i>0\) and the terms where \(\small f_i=0\), we get
\[ \small \sum_{f_i>0} \frac{1}{\alpha_i}+ (1+\gamma\xi-\gamma b)\sum_{f_i=0} p_j = 1. \]
Since
\[ \small \sum_{f_i=0} p_j = 1- \sum_{f_i>0} p_j, \]
we can obtain
\[ \small b+ \frac{1}{\gamma}\left(\frac{1-\sum_{f_i>0}\frac{1}{\alpha_i}}{1-\sum_{f_i>0}p_i}-1\right) = \xi. \]
By substituting into the equation for \(\small f_i\),
\[ \small f_i = \frac{1}{\gamma}\max\left\{- \frac{q_i^2}{p_i}+\left(\frac{1-\sum_{f_i>0}q_i}{1-\sum_{f_i>0}p_i}\right) q_i , 0 \right\} \]
can be obtained. Since
\[ \small b = 1- \sum_{f_i>0}f_i, \]
The amount of cash left on hand \(\small b\) can be calculated as:
\[ \small b= 1-\frac{1}{\gamma}\left[- \sum_{f_i>0}\frac{q_i^2}{p_i}+\left(\frac{1-\sum_{f_i>0}q_i}{1-\sum_{f_i>0}p_i}\right) \sum_{f_i>0}q_i \right]. \]
All that remains is to determine the range in which to add up \(\small p_i,q_i\) so that the cash on hands \(\small b\) is minimized. Without loss of generality, we reorder the subscripts to favor the gambler. That is, let assume \( \small p_1 /q_1 > p_2 /q_2 > \cdots > p_n / q_n \). In this case, by calculating \( \small t^{\ast} \) that satisfies
\[ \small t^{\ast} = \arg \min_t \left(1-\frac{1}{\gamma}\left[- \sum_{i=1}^t\frac{q_i^2}{p_i}+\left(\frac{1-\sum_{i=1}^tq_i}{1-\sum_{i=1}^tp_i}\right) \sum_{i=1}^tq_i \right]\right)\quad s.t. \;\; \sum_{i=1}^t q_i < 1 \]
(Here, \( \small \arg \min_t f(t) \) denotes the value of \( \small t \) that minimizes \( \small f(t) \).), the optimal betting ratio for the gambler can be calculated as:
\[ \begin{align*} & \small f_i = \frac{1}{\gamma}\max\left\{- \frac{q_i^2}{p_i}+\left(\frac{1-\sum_{i=1}^{t^\ast}q_i}{1-\sum_{i=1}^{t^\ast}p_i}\right) q_i , 0 \right\} \\ & \small b= 1-\frac{1}{\gamma}\left[- \sum_{i=1}^{t^\ast}\frac{q_i^2}{p_i}+\left(\frac{1-\sum_{i=1}^{t^\ast}q_i}{1-\sum_{i=1}^{t^\ast}p_i}\right) \sum_{i=1}^{t^\ast}q_i \right]. \end{align*} \]
The above is the optimal betting strategy for a gambler with a mean-variance utility function. When actual calculations are performed, it can be confirmed that for realistic \(\small \gamma\), a solution can be derived that approximately matches the result calculated using the CRRA-type utility function.
Fractional Kelly Strategy
An interesting property of the solution to this problem is that the stake ratio \(\small f_i\) is inversely proportional to the relative risk aversion \(\small \gamma\). For example, if the solution to \(\small \gamma=1\) is \(\small f^{\ast}_i\), then the solution \(\small f_i^{\gamma} \) for any \(\small \gamma\) can be calculated as:
\[ \small f_i^{\gamma} = \frac{1}{\gamma}f_i^{\ast}. \]
Since the solution for \(\small \gamma=1\) is approximately the same as the solution for the Kelly criterion, the strategy would be to bet an amount equal to the Kelly criterion betting ratio divided by a constant value. A strategy that multiplies such a Kelly criterion portfolio by a haircut is called the Fractional Kelly Strategy. Specifically, if \(\small \gamma =2\), it is called half Kelly, and if \(\small \gamma = 3\), it is called \(\small 1/3\) Kelly and so on. Since the mean-variance utility function was an approximation of the CRRA utility function, it can be seen that these strategies are in fact approximately identical to strategies that maximize the CRRA utility function, whose relative risk aversion is \(\small \gamma\). It is unclear whether this was derived purely from experience or whether it was theoretically conceived from the beginning with a CRRA-type utility function in mind.
Reference
[1] Ethier, Stewart N. The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer-Verlag, 2010.
[2] Hirano, Kaname, On the Kelly Criterion in Stock Investment, SSRN Paper, 2023.
[3] Huang, Chi-Fu and Robert H. Litzenberger, Foundations for Financial Economics, North Holland, 1988.
[4] Kelly, John L. A New Interpretation of Information Rate. Bell System Technical Journal, 1956.
[5] Thorp, Edward O., The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market. S.A. Zenios and W.T. Ziemba eds., Handbook of Asset Liability Management, Vol.1. Theory and Methodology, pages 385–428, 2006.
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