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Sortino ratio

The Sortino ratio is a risk-adjusted performance metric that evaluates the return of an investment relative to its downside risk. Unlike the Sharpe ratio, which considers total volatility, the Sortino ratio focuses only on harmful volatility by using the downside deviation instead of the total standard deviation. This distinction leads to different applications and interpretations of the two ratios. Its formula is:

Sortino Ratio = \frac{R_{p} - T}{σ_{d} (T)}

Here $R_{p}$ is the portfolio return, $T$ is the minimum acceptable return (MAR) — the threshold below which an outcome counts as a shortfall — and $σ_{d} (T)$ is the target downside deviation measured against that same $T$ . The same $T$ must appear in both places. Rollinger and Hoffman are emphatic on this point,⁵⁸ and mixing them — excess return over the risk-free rate divided by downside deviation about some other target — produces a number with no coherent interpretation, which is nonetheless what a good deal of software reports.

Setting $T = R_{f}$ recovers a direct analogue of the Sharpe ratio and is the sensible default. Setting $T$ to your actual required return — the 4% real you need for the plan to work, say — is the version that earns the extra complexity, because it measures shortfall against the thing you actually care about, not Treasury bills.

The downside risk $σ_{d}$ is calculated using a continuous formula:

σ_{d} = \sqrt{\int_{- \infty}^{T} {(T - r)}^{2} f (r) dr}

where $T$ is the target or minimum acceptable return (MAR), $r$ is the random variable representing the return, and $f (r)$ is the probability distribution of returns, e.g., the log-normal distribution. This continuous approach allows for the use of annual returns directly, aligning more naturally with how investors typically set their investment goals. In contrast, the discrete method requires conversion of annual targets into monthly targets to accumulate enough data points for meaningful analysis, potentially misrepresenting the actual risk involved.

What you and every spreadsheet will actually compute is the discrete estimator:

σ_{d} = \sqrt{\frac{1}{n} \sum_{t = 1}^{n} min {(r_{t} - T, 0)}^{2}}

and the detail that matters is the $min$ : returns above the target enter the sum as zeros — they are not dropped from it. Work five annual returns — 16%, $- 4$ %, 10%, $- 10$ %, 18% — against $T = 4 %$ . The three good years contribute zero; the two shortfalls contribute ${(- 8 %)}^{2}$ and ${(- 14 %)}^{2}$ ; divide by $n = 5$ , not the two shortfall years: $σ_{d} = \sqrt{(64 + 196) ∕ 5} \approx 7.2 %$ . With a 6% mean return the Sortino ratio is $(6 - 4) ∕ 7.2 \approx 0.28$ . Divide by the number of shortfall years instead — the classic implementation error — and you get 11.4%, a materially different number that will not match any correctly built report.

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