Deriving the Log-Optimal Allocation in the Samuelson Model
Summary
The document addresses how to identify the portfolio fraction that maximizes expected log wealth in a continuous-time Samuelson model with a zero safe rate. A risky allocation generates drift proportional to its expected return and volatility proportional to its exposure. Applying Itô’s formula to log wealth gives an expected growth rate equal to the allocation’s return contribution minus a quadratic volatility penalty.
Completing the square shows that this objective is maximized at the expected return divided by variance. The answer also clarifies an ambiguity in the question: maximizing expected log wealth directly is not the same statement as maximizing expected log wealth relative to a reference strategy, although the relative formulation has the same optimizer when the reference term is fixed. The derivation assumes constant allocation, return and volatility parameters over the horizon, and the stated diffusion model; it does not discuss constraints, time-varying opportunities, or estimation error.
Key ideas
- Itô’s formula turns the risky wealth process into a log-growth rate with a quadratic penalty for volatility.
- Expected log wealth is a concave quadratic function of the allocation under constant parameters.
- Completing the square identifies the maximizing allocation as expected return divided by variance.
- A fixed reference strategy changes the objective by a constant and does not change its optimizer.
- The result relies on the model’s constant-parameter assumptions and does not include allocation constraints.
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Full text
# optimal strategy problem (using Jensen's inequality)
# optimal strategy problem (using Jensen's inequality)
I have a strategy in Samuelson model with zero safe rate defined as $$Z_t^{\Pi}=\frac{X_t^{\Pi}}{X_t^{\rho}} \quad \quad (1)$$ where $$\frac{dX_t^{\Pi}}{X_t^{\Pi}} = \mu \pi dt + \sigma \pi \ dW_t \quad \quad (2)$$ $$ \frac{dX_t^{\rho}}{X_t^{\rho}} = \mu \rho dt + \sigma \rho \ dW_t \quad \quad (3)$$
what gives the following dynamic $$\frac{dZ_t^{\Pi}}{Z_t^{\Pi}} = (\mu -\sigma^2 \rho )(\pi - \rho) dt + \sigma (\pi - \rho)\ dW_t \quad \quad (4)$$
To prove that $\rho=\frac{\mu}{\sigma^2} $ is the optimal strategy for the $$\max_{\Pi} E [ logX_T^{\Pi}] \quad \quad (5)$$ Using (1), logarithmic property, Jensen's inequality and supermartingale property I can derive the below inequality $$ E \big{[} log(X_t^{\Pi}) \big{]} \leq E \big{[} log(X_t^{\rho}) \big{]} \quad \quad (6)$$
The question I have is how the inequality (5) implies that the optimal strategy is $\rho=\frac{\mu}{\sigma^2} $?
## Answer by Gordon (score 1, accepted)
https://quant.stackexchange.com/a/26049
I assume that the problem is $$\max_{\pi} E\left(\ln Z_T^{\Pi} \right).$$ Note that $\ln Z_t^{\Pi} = \ln X_t^{\Pi} -\ln X_t^{\rho}$. Moreover, \begin{align*} d\ln Z_t^{\Pi} &= d\ln X_t^{\Pi} -d\ln X_t^{\rho}\\ &=\Big[\big(\mu \pi - \frac{1}{2}\sigma^2 \pi^2\big) - \big(\mu \rho- \frac{1}{2}\sigma^2 \rho^2\big) \Big]dt + \sigma(\pi-\rho)dW_t. \end{align*} Then \begin{align*} E\left(\ln Z_T^{\Pi} \right) = \Big[\big(\mu \pi - \frac{1}{2}\sigma^2 \pi^2\big) - \big(\mu \rho- \frac{1}{2}\sigma^2 \rho^2\big) \Big]T. \end{align*} Consequently, \begin{align*} \max_{\pi}E\left(\ln Z_T^{\Pi} \right) &= \max_{\pi}\Big[\big(\mu \pi - \frac{1}{2}\sigma^2 \pi^2\big) - \big(\mu \rho- \frac{1}{2}\sigma^2 \rho^2\big) \Big]T\\ &=\max_{\pi}\Big[-\frac{1}{2}\sigma^2\big( \pi - \frac{\mu}{\sigma^2} \big)^2 + \frac{1}{2}\mu^2- \big(\mu \rho- \frac{1}{2}\sigma^2 \rho^2\big) \Big]T, \end{align*} which is a maximization problem for a quadratic function of $\pi$. It is then clear that the maximum is achieved at $\pi = \frac{\mu}{\sigma^2}$.
> EDIT
Consider the problem $\max_{\pi}E\left(\ln X_T^{\Pi} \right)$. Note that \begin{align*} d\ln X_t^{\Pi} = \big(\mu \pi - \frac{1}{2}\sigma^2 \pi^2\big)dt + \sigma \pi dW_t. \end{align*} Then, \begin{align*} \max_{\pi}E\left(\ln X_T^{\Pi} \right) &= \max_{\pi}\big(\mu \pi - \frac{1}{2}\sigma^2 \pi^2\big)T\\ &=\max_{\pi}\Big[-\frac{1}{2}\sigma^2\big( \pi - \frac{\mu}{\sigma^2} \big)^2 + \frac{1}{2}\mu^2 \Big]T, \end{align*} which is again a maximization problem for a quadratic function of $\pi$, and the maximum is achieved at $\pi = \frac{\mu}{\sigma^2}$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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