Return risk

Standard deviation is how far returns sat from their average. A monthly series of 2%, 3%, 1%, 2%, 2% is tight. A series of −8%, 12%, −4%, 15%, −10% wanders. The second series has the higher standard deviation.

Two investments can share a 10% average return. One has a 5% standard deviation. The other has 25%. The averages match. The path of returns does not. A large gain adds to the standard deviation the same way a large loss does. The statistic measures variability.

Annualized, that width is realized volatility: how much the underlying already moved. Implied volatility is solved from an option price. The volatility surface lesson keeps those two questions apart.

Beta is how sensitive the investment’s returns were to a chosen benchmark, often the S&P 500. A beta of 1.0 moved about with the benchmark. A beta of 1.5 means a 1% benchmark move lined up, on average in that sample, with about a 1.5% move in the investment, in the same direction. A beta of 0.6 moved less. A negative beta moved the other way.

On a classroom pair of standard deviations, beta is correlation times the investment’s standard deviation divided by the benchmark’s. Correlation 0.75, investment 20%, benchmark 15% gives 0.75 × 20 / 15 = 1.00. Delta is a different sensitivity: the option against the stock, not the stock against the index.

The number depends on the benchmark, the window, the frequency, and the return calculation. It is an estimate for that sample. It is not a permanent property of the ticker.

R-squared is how much of the investment’s return variation lined up with the benchmark in a straight-line fit. It runs from 0 to 1. With correlation 0.75, R-squared is 0.56.

Beta and R-squared answer different questions. A beta of 1.3 with an R-squared of 0.85 says the sensitivity was high and the benchmark accounted for most of the variation. The same beta of 1.3 with an R-squared of 0.20 says the slope is still 1.3, and the benchmark accounts for little of the path. R-squared describes that fit. It does not say the market caused the returns.

Sharpe is excess return per unit of standard deviation. Excess return is the investment return minus the risk-free rate.

(return − risk-free rate) / standard deviation

A return of 12%, a risk-free rate of 4%, and a standard deviation of 16% give (12% − 4%) / 16% = 0.50. The series paid 0.50 of excess return per unit of volatility in that window. A higher Sharpe paid more for the variability. A negative Sharpe means the return sat below the risk-free rate you used. The ratio changes with the window, the frequency, and the risk-free rate.

One classroom stock, read as a set:

The 15% annualized return and 30% standard deviation produce a Sharpe of 0.40 when the risk-free rate in the sample is 3%. The beta of 1.4 says the stock moved more than the benchmark. The R-squared of 0.80 says most of the variation sat with that relationship.

A beta of 1.25 describes the sample. It does not set tomorrow’s beta. A standard deviation of 28% does not fix next year’s range. A Sharpe of 0.8 does not promise that payoff again. An R-squared of 0.75 does not assign three quarters of future returns to the market. The page describes the series you supplied.

Before an option, the four numbers are context for the underlying: a wide standard deviation and a high beta are a different stock from a quiet one. They still are not the implied volatility in the option price. Position size and the structure come after that distinction.

Return risk prints standard deviation, beta, R-squared, and Sharpe from the return, the two standard deviations, the risk-free rate, the benchmark return, and the correlation you type.