Choosing an Allocation to Maximize Median Wealth

Summary

The document poses an allocation problem for a risky investment with two possible returns: a gain or a loss, each occurring with a specified probability. It asks what portfolio fraction maximizes median wealth, using example gain and loss magnitudes, and considers both a zero interest rate and a positive interest rate. The investor is unsure whether the Kelly fraction, which is commonly framed around long-run growth, answers a median-wealth question.

The text contains no proposed solution, derivation, or evidence. It therefore serves as a problem statement rather than an explanation of an optimal strategy. A solution would need to define the investment horizon and how outcomes compound, clarify whether the loss input is expressed as a return or terminal-value multiplier, and account for the safe rate when comparing wealth paths. The document itself does not resolve how the median objective differs from maximizing expected wealth or logarithmic growth.

Key ideas

• The question asks for the portfolio share that maximizes median wealth under a two-outcome investment.
• It distinguishes the median objective from the Kelly criterion’s usual long-run growth framing.
• It considers how a positive interest rate may affect the allocation choice.
• The document supplies no solution, so horizon, compounding, and return definitions remain unspecified.

Tags

# Investment strategy to maxmize median wealth


# Investment strategy to maxmize median wealth












I came across this during an interview prep: Investment in an asset returns $x$ with a probability of $p$ and $-y$ with a probability $q=1-p$. What fraction of my portfolio should be allocated to this investment to maximize median wealth? Let's take $x = 0.4$ and $y = -0.3$. Also, consider two cases a) interest rate = $0$ and b) interest rate = $0.05$.

My initial approach is to use Kelly formula: $f=\frac{p.b-q}{b}$ but not sure what to use for $b=1.4/0.7=2$? Also, I am not sure how to approach median vs. mean? Lastly, I do not know how to account for interest rate. Would appreciate your thoughts. Thanks.

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