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I have recently watched two videos about Kelly Criterion for portfolio optimization, however one seem not to deduce it correctly (as people commented) and the another just don't show any deductions at all. Right after, I found this article.

As we can see, the unconstrained problem is proposed in equation 10 and its solutions lays right bellow, equation 12. But I'm interested in solving the constrained problem, equation 13. Thus, I tried this:

  1. First, we set the only constrain: $\vec{W} \vec{I}$, such that $\vec{W}$ is the vector of weights and the last satisfies $\vec{I} = \begin{bmatrix} 1 & 1 & \cdots & 1 \end{bmatrix}^{T}$;
  2. There's no short sellings here, thus $0 \leq w_i \leq 1$;
  3. We set the lagrangian $\mathcal{L}$ as follows:

\begin{equation}\tag{1} \mathcal{L}(\vec{W}, \lambda_1) = r + \vec{W}^{T} (\mathbb{E}[\vec{R}-r \vec{I}]) - \vec{W}^{T} \Sigma \vec{W} + \lambda_1(1-\vec{W}^{T} \vec{I}) \end{equation}

where $r$ is the risk-free asset return, $\vec{R}$ is the random vector of retuns of the risky assets, $\Sigma$ is the covariance matrix.

  1. Now, as usual, we maximize by calculating the gradient of $\mathcal{L}$ and setting it equal to zero. Thus, we arrive at

\begin{equation}\tag{2} \begin{bmatrix} 2 \Sigma & \vec{I} \\ \vec{I}^{T} & 0 \end{bmatrix} \begin{bmatrix} \vec{W} \\ \lambda_1 \end{bmatrix} = \begin{bmatrix} \mathbb{E}[\vec{R}]- r \vec{I} \\ 1 \end{bmatrix} \end{equation}

  1. Then we can solve it using QR factoratization and backwards substitution:

\begin{equation}\tag{3} QR \vec{x} = \vec{b} \\ R \vec{x} = Q^{T} \vec{b} \end{equation}

So, some things are still unclear to me:

  1. There's a mismatch between the expected value of the returns of the portfolio ($R_p$). In the first video, we can see (min 3) that, equivalently,

\begin{equation}\tag{4} \textrm{Kelly Criterion: } \mathbb{E}[R_p] = \mathbb{E}[\log(1+\vec{W} \vec{R})] \end{equation}

However, in the article, they do all calculations and deductions using $\mathbb{E}[R_p] = \mathbb{E}[\vec{W} \vec{R}] $, as Markowitz model uses. Any thoughts on this? Also, in the article, the authors said the model only works for normally distributed prices.

My bet is we use $\mathbb{E}[R_p] = \mathbb{E}[\vec{W} \vec{R}] $ for normally distributed prices and $\mathbb{E}[R_p] = \mathbb{E}[\log(1+\vec{W} \vec{R})]$ for lognormal distributed prices, since $\ln(\textrm{LogNormal}) = \mathcal{N}$.

  1. I have seen that there is a parabola associated with Kelly Criterion. Once solved (3), how do we get that nice parabola that everybody plots? I can't see how, since we get multiple weights. Honestly, for me, it seems it's going to look like the Markowitz model a lot more.

  2. In Markowitz model, we can parametrize the vector $\vec{b}$ in terms of the expected values that we want, namely $\mu_0$, to get the efficient frontier. Is there anything like that in Kelly's Criterion for portfolio optimization? If we parametrize $\vec{b}$ in (3), we get a curve of possible portfolio weights. So what is the difference between Kelly and Markowitz? Only how the expectation is defined?

I tried to find more information in the internet, but I had a really hard time, few sources explaining the math behind the model that could clarify such questions. Also, any book recommendations are welcome!

MAIN ONE: Is this deduction correct? Anything to be corrected or added would be appreciated!!

Thanks

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  • $\begingroup$ Re Question 3. Kelly assumes a specific utility function (log utility) so it gives a unique solution. Markowitz produces a frontier from which different users can select their preferred point based on personal risk/return tradeoff i.e. based on their individual utility function. (BTW Kelly solutions tend to be rather aggressive in risk and return). $\endgroup$
    – nbbo2
    Oct 23, 2021 at 13:01

2 Answers 2

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As correctly mentioned in the comments, Kelly tends to be more aggressive than MVA. Its main weakness is a focus on a single bet/security and ignoring serial correlation of returns, something which makes large drawdowns possible. An exposition of various features of both mean-variance and Kelly, including effects of trading costs and constraints, is given in my recent book.

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  • $\begingroup$ Thanks for your answer and recommendation! Sorry, but in the article the authors say it is possible to use the Kelly Criterion to portfolios. Are we talking about the same Kelly thing? $\endgroup$
    – Mr. N
    Oct 28, 2021 at 0:13
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In equation 1 the third item should be divided by 2. As you mentioned the Kelly maximizes log return. So that 2 in equation(2) should be 1. I think no parabola is possible. Just like a tangent portfolio there is just one Kelly portfolio which is on the efficient frontier of Markorwitz.

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