Constructing a Sharpe-style score from profit and loss variability

A return-to-risk score

The Sharpe construction is presented as a classic return-to-risk score that pairs average return with the variability of that return. Realized returns are compared with the variability of those returns.

That construction is distinguished from a measure of the chance of losing the original investment. Risk, in this score, is the variability of the profit-and-loss series, not the chance that the original stake is lost.

How the pieces are assembled

One stated arithmetic form subtracts a risk-free rate from average profit and loss, then divides by the standard deviation of profit and loss.

The risk-free rate is a baseline subtracted from average profit and loss before the result is scaled by dispersion. Average profit and loss uses mathematical expectation as the expected-value ingredient. The profit-and-loss standard deviation is the variability term in the denominator of the Sharpe construction.

Reading a sampled series

Estimator bias is the gap between an estimator's expected value and the quantity it is meant to recover. That gap matters when average profit and loss is treated as a mathematical expectation.

Noise is defined as price and volume fluctuations that can obscure a reading of market direction. A noisy signal is a series in which random influences cannot be set aside.

Shannon entropy is the expected information, or uncertainty, carried by each observation from a distribution. That expected information uses the same idea of mathematical expectation that appears when averaging profit and loss. A probability density function is a graph of the chance that a particular price datapoint occurs.

Editorial interpretation: TradersWeek treats those definitions as context for reading a sampled profit-and-loss series, not as extra arithmetic steps in the Sharpe construction.