Why Single-Period Sharpe Ratio Does Not Determine Position Size

Summary

The question asks how to choose a position as a function of a return predictor to maximize expected Sharpe ratio. Returns are modeled as a predictor-linked component plus independent noise, and the question extends the objective across multiple trading periods. The response focuses on the single-period formulation and shows that its Sharpe ratio does not identify an optimal position function.

Conditional on a predictor value, multiplying the return by a position scales both expected profit and return volatility by the position’s magnitude. The position therefore cancels from their ratio, leaving a Sharpe expression determined by the predictor and noise scale. This conclusion is limited to the stated ratio and model: it does not solve the revised multi-period expected-Sharpe problem or account for trading costs, leverage constraints, or other portfolio objectives.

Key ideas

• In the single-period model, position size scales expected profit and volatility proportionally.
• That proportional scaling makes the conditional Sharpe ratio independent of the position function.
• The result does not determine an optimal multi-period position under the revised objective.
• Costs, constraints, and alternative objectives could change the position-sizing problem.

Tags

# Closed-form solution to optimal single assset position sizing with predicted returns


# Closed-form solution to optimal single assset position sizing with predicted returns












Say that I observe a predictor $w_t \sim N(0,\sigma_1)$ for the returns in a single asset over the next time interval:

$$ r_t = \alpha w_{t-1} + z_t $$ where $z_t \sim N(0,\sigma_2)$ is unobserved and independent of $w_{t-1}$.

I want to find the function $f$ that maps $w_t$ to a position in the asset which maximizes the expected Sharpe Ratio of the next $n$ trading returns: $$\max_f \mathrm{E}\left[\frac{\sqrt{n}\sum_{i=1}^n p_i}{n \sqrt{\sum_{i=1}^n \left(p_i - \overline{p_i}\right)^2}} \right]$$ where the profit at time step $t$ is $$p_t = r_t f(w_{t-1}) $$

Is there an easy way using something like stochastic control to find the optimal $f$ without any constraints on its structure?

EDIT: As pointed out by Chris, I had oversimplified the problem by only looking at a single-step. I've modified the description to make it n-step.

## Answer by Chris Taylor (score 2)

https://quant.stackexchange.com/a/25352

As you have defined the Sharpe ratio, it is independent of your position. You have $$ \mathrm{E}[rf(z_1)] = \alpha z_1 f(z_1) $$ and $$ \mathrm{Var}[rf(z_1)] = \sigma_2^2f(z_1)^2 $$ and hence $$ \frac{\mathrm{E}[rf(z_1)]}{\sqrt{\mathrm{Var}[rf(z_1)]}} = \frac{\alpha z_1}{\sigma_2} $$ independent of the function $f(z_1)$.

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