Should upside volatility count as risk?
Sharpe's denominator counts every deviation from the average, in both directions. So a portfolio that surprises you with a huge gain is penalised exactly as much as one that surprises you with a huge loss. Almost nobody actually experiences it that way. The Sortino ratio takes the complaint seriously.
The formula
Sortino ratio = (portfolio return − target return) ÷ downside deviation
The target is whatever you consider a non-event: usually zero, sometimes the risk-free rate, occasionally a required return. The denominator counts only the periods that fell short of it:
Downside deviation = √( (1/N) × Σ min(rᵢ − target, 0)² )
Read the divisor carefully. It is N — every period in the sample, not only the losing ones. Dividing by the count of negative periods instead is a common error, and it cuts two ways. Arithmetically it spreads the same sum over fewer periods, so the downside deviation comes out larger and the ratio comes out lower — on the six months below, dividing by 2 instead of 6 lifts the deviation from 1.47% to 2.55%. Conceptually it also changes the question being asked: you are then measuring the typical size of a loss when one happens, ignoring how often losses happen at all, which flatters a portfolio that loses small and often and punishes one that loses rarely but deeply.
Worked example
Six monthly returns: +5%, −3%, +4%, +6%, −2%, +2%. Target = 0%.
Mean return: (5 − 3 + 4 + 6 − 2 + 2) ÷ 6 = 12 ÷ 6 = 2.0%
Downside terms — only the negative months contribute, the rest enter as zero:
- (−3)² = 9
- (−2)² = 4
- Sum = 13, divided by N = 6 → 2.167
- Downside deviation = √2.167 = 1.47%
Sortino = 2.0 ÷ 1.47 = 1.36
Now the same series through Sharpe's lens. Deviations from the 2% mean are +3, −5, +2, +4, −4, 0; their squares sum to 70; 70 ÷ 6 = 11.67; standard deviation = 3.42%.
Sharpe-form ratio = 2.0 ÷ 3.42 = 0.59
The gap between 1.36 and 0.59 comes from two changes at once, and only one of them is the famous one. The Sortino denominator ignores the three months that beat the average (+5, +4, +6) — but it also measures each loss against the 0% target rather than against the 2% mean, so the −3% month contributes 3² = 9 where the Sharpe form charged it 5² = 25, and the −2% month contributes 4 where the Sharpe form charged 16.
Separate the two effects. Keep the 2% mean as the reference point but count only the months that fell below it — their deviations are −5 and −4, so 25 + 16 = 41, and the denominator drops from 3.42% to √(41 ÷ 6) = 2.61%, lifting the ratio to 0.77. Now move the reference point from the mean down to the 0% target and the denominator falls the rest of the way to 1.47%, lifting the ratio to 1.36.
So of the 0.77-point gap, excluding the upside is worth about 0.18 — roughly a quarter, and re-anchoring the losses to zero is worth about 0.59 — roughly three quarters. Sortino is usually sold as the ratio that stops punishing gains, but whenever the mean sits well above the target — as it does here — most of its advantage comes from the target rather than from the excluded upside. Reverse that, with a mean barely above the target and a few enormous up months, and the split reverses too. Which is right depends on what you were worried about — and that is the honest answer, not a dodge.
Where Sortino gets fragile
Fewer observations feed the denominator, so it is estimated from less data and is correspondingly noisier than Sharpe. Push it far enough and it breaks: a series with no periods below target has a downside deviation of zero and an undefined Sortino. Software often prints a spectacular number instead. That is a statement about the shortness of the sample, not about the strategy.
The same warning as always applies with extra force here: a favourable Sortino describes months that already happened, under a target you chose. It ranks nothing about the future and identifies nothing worth owning.
Try it now
- An index and a concentrated fund are below. Switch both to Monthly — you want roughly sixty observations, not twelve hundred — and read the monthly return series off each.
- For each series compute the mean, the full standard deviation, and the downside deviation using a 0% target — remembering to divide by the total number of months, not just the negative ones.
- Compute both ratios for both series. Which asset gains most from being judged on downside risk only? Measure its largest up-move and its largest down-move on the chart, and write one sentence linking the answer to the shape of its distribution: a series punished by full volatility for rising is a series Sortino will treat kindly.