Sharpe vs Sortino: Why the Same Average Return Can Produce Opposite Rankings

A higher Sharpe ratio does not necessarily mean fewer losing months. A higher Sortino ratio does not establish protection against a crash. Sharpe puts variability of excess returns in the denominator; Sortino focuses on shortfalls below a specified target. The first decision is which question the measure should answer.

Consider two hypothetical six-month records with identical arithmetic average returns. Their rankings reverse without changing a single observation—only the definition of risk changes. The short sample below is a calculation exercise, not a claim about any fund or evidence of persistent investment skill.

Define the return series before choosing the ratio

Use returns measured in the same currency, at the same frequency, with distributions and relevant costs treated consistently. Remove the effect of external deposits and withdrawals before interpreting account growth as performance. Returns on total account equity are not directly comparable with returns quoted only against futures margin.

Let d be each period’s strategy return minus that period’s risk-free return. The sample Sharpe ratio is the arithmetic mean of d divided by its sample standard deviation, using n−1 in the variance denominator. If the risk-free return varies, subtract it observation by observation before estimating volatility; subtracting an annual rate from a monthly average is not a substitute.

For this example the monthly risk-free return and the Sortino target T are both zero. Define downside deviation as √[Σmin(r−T,0)²/n]. Sortino is the mean monthly return minus T, divided by that downside deviation. This convention divides by all n observations, including months with no shortfall. Document any different convention before comparing published figures.

One sample, two defensible but different rankings

Strategy A returns −2%, −1%, 1%, 2%, 3% and 3%. Strategy B returns −1%, −1%, −1%, −1%, 2% and 8%. Both average 1% a month. The following measures are monthly, not annualized.

For A, squared deviations from the 1% mean sum to 22 in percentage-point-squared units: √(22/5)=2.0976%. B’s corresponding sum is 66, giving √(66/5)=3.6332%. Dividing the common 1% mean by each standard deviation produces Sharpe ratios of 0.4767 and 0.2752. A wins that comparison.

Below the zero target, A contributes 4+1=5 squared percentage points. Its downside deviation is √(5/6)=0.9129%. B contributes four observations of 1, giving √(4/6)=0.8165%. Sortino therefore favors B, at 1.2247 versus A’s 1.0954. B loses more often, but its observed shortfalls are smaller; the 8% gain raises its total volatility.

Neither ranking says everything an investor needs. Compounded six-month returns are about 6.0370% for A and 5.8193% for B. Equal arithmetic averages do not imply equal wealth growth. Replace B’s 8% month with 2% and its mean becomes zero, as do both ratios here. Its attractive score depends heavily on one observation.

The target and the denominator are substantive choices

At a 1% monthly target, a positive 0.5% month is still a shortfall. At a zero target, it is not. Set the target before inspecting results: lowering the hurdle until a strategy looks good does not improve its ability to meet the original objective.

If A’s squared shortfalls are divided by its two losing months instead of all six, downside deviation becomes √(5/2)=1.5811% and Sortino becomes 0.6325. This is a change of measurement, not a deterioration in the strategy. A comparison that silently mixes these conventions is invalid.

An effective 5% annual target corresponds to (1.05)^(1/12)−1, or approximately 0.4074% a month. Dividing 5% by 12 gives 0.4167%, a simple-rate allocation rather than an exactly equivalent compounded target. Monthly shortfall and failure to meet a full-year objective remain different events even after conversion.

When every observation meets the target, downside deviation is zero. The ordinary ratio then has no finite value; it is not proof of infinitely good risk management. An uneventful sample may simply contain too little information about adverse outcomes, especially when valuations update infrequently.

Annualization, negative ratios and leverage need separate treatment

Multiplying a monthly Sharpe by √12 is a conventional scaling under assumptions about aggregation and serial dependence, not an unconditional identity. With autocorrelated excess returns, annual sum variance includes cross-month covariance terms. Stale prices and overlapping positions can make apparently smooth monthly returns misleading.

Do not automatically apply the same rule to Sortino. Selecting monthly shortfalls and then aggregating is not the same operation as first measuring annual returns and then selecting annual shortfalls. State the horizon, target conversion and estimator. A six-month sample does not become reliable annual evidence because a scaling factor is applied.

Negative Sharpe ratios are particularly easy to misrank. A mean excess return of −1% divided by volatility of 2% gives −0.50; the same mean divided by 4% gives −0.25. The larger number arises from more volatility, not a better mean outcome. Show the shortfall explicitly before drawing a preference from the ratio.

Proportionally scaling excess returns leaves Sharpe unchanged in an ideal frictionless setting. Borrowing spreads, margin calls, forced sales and nonlinear costs can defeat that condition. Sortino also depends on how the target is specified under scaling. Neither statistic determines a suitable position size.

A high score can still conceal tail exposure

A strategy that collects small gains and occasionally suffers a large loss may look excellent before the large loss enters the sample. Both measures are estimated from observed returns; neither invents a missing liquidity shock, overnight gap or nonlinear short-option loss.

Inspect the worst observations, loss concentration, drawdowns and recovery times alongside the ratios. Use explicit stress scenarios for exposures absent from the record. Rolling windows can reveal dependence on a few favorable months, but choose the window rule in advance rather than searching for the most flattering interval.

For an addition to an existing portfolio, examine correlation, covariance and component risk contributions. A lower standalone ratio may coexist with a useful portfolio interaction. That is a reason to analyze the combined holdings, not a blanket argument for buying low-scoring assets.

Six checks before accepting a performance ranking

1. Obtain the complete matched return series; reconcile currency, distributions, fees, cash flows, missing values and valuation times.
2. Write down the risk-free return and the target separately, each consistent with the observation frequency.
3. Recalculate excess-return standard deviation and full-sample downside deviation; identify exactly where n−1 and n are used.
4. Separate arithmetic average, compounded holding-period return and ratios. Flag zero denominators and negative numerators.
5. Test extreme months, tail exposures, autocorrelation and rolling-window stability; disclose the annualization method.
6. Evaluate the actual combined portfolio, including dependence, financing, liquidity and drawdown constraints.

A ranking is more informative when it survives reasonable changes in targets, windows and cost assumptions without changing the question being asked. If one month, a denominator convention or an annualization shortcut overturns the conclusion, that fragility belongs in the assessment—not hidden behind the best-looking headline number.