Sortino Ratio Formula
The Sortino ratio divides a strategy's excess return over a target by its downside deviation, which measures only the volatility of returns below that target. Upside volatility does not count against the score, so a Sortino ratio rewards strategies whose dispersion is concentrated in good outcomes.
Formula
Copy the exact expression or work through it step by step below.
Sortino = (E[R_p] - T) / DD
DD = sqrt( (1/n) x sum_over_R<T ( min(0, R_i - T) )^2 )
Annualized: Sortino_annual = Sortino_period x sqrt(N) Variables
E[R_p]
Mean portfolio return
Arithmetic mean of the strategy's per-period return over the sample, expressed in the same frequency as the target.
T
Target return (MAR)
The minimum acceptable return per period below which a return is treated as a shortfall. Common choices are zero or the per-period risk-free rate. Returns at or above T contribute nothing to the denominator.
DD
Downside deviation
Root mean square of the negative deviations from the target. Importantly, the sum of squared shortfalls is divided by n, the total number of observations, not just the count of below-target periods. This keeps the measure comparable across samples with different shortfall frequencies.
n
Number of observations
Total count of return periods in the sample. The full count is used as the divisor inside the downside deviation, by Sortino and Price's original definition.
N
Periods per year
Annualization factor: 252 for daily, 52 for weekly, 12 for monthly returns. The ratio scales by the square root of N under the i.i.d. assumption, exactly as the Sharpe ratio does.
Step By Step
- 1
Choose a target return T. Use zero for absolute downside risk, or the per-period risk-free rate to match a Sharpe comparison.
Set T = 0 to penalize only periods where the strategy lost money.
- 2
For each period, compute the shortfall: the return minus the target, but replace any non-negative result with zero so only below-target periods contribute.
Returns of +3%, -2%, +1%, -4% against T = 0 give shortfalls of 0, -2%, 0, -4%.
- 3
Square each shortfall, sum them, divide by the total number of observations n, then take the square root to get downside deviation.
Squared shortfalls 0, 0.0004, 0, 0.0016 sum to 0.0020; divided by 4 is 0.0005; the square root is 0.02236, or 2.24%.
- 4
Subtract the target from the mean return and divide by the downside deviation to get the per-period Sortino ratio.
Mean return -0.5% over T = 0 divided by 2.24% gives -0.224 per period.
- 5
Annualize by multiplying by the square root of the number of periods per year.
A monthly Sortino of 0.30 annualizes to 0.30 x sqrt(12) = 1.04.
Worked Example
Trend strategy, four monthly returns, target T = 0
Monthly returns
+4%, -3%, +5%, -2%
Target T
0%
Periods per year
12
Mean return = (4 - 3 + 5 - 2)/4 = 1.0% = 0.01. Shortfalls below 0 are -0.03 and -0.02; squared they are 0.0009 and 0.0004, summing to 0.0013. Divide by n = 4: 0.000325. Downside deviation = sqrt(0.000325) = 0.01803 = 1.803%. Monthly Sortino = 0.01 / 0.01803 = 0.555. Annualized = 0.555 x sqrt(12) = 0.555 x 3.464 = 1.922.
Annualized Sortino of about 1.92. Because the two up months (+4%, +5%) are larger than the two down months, the downside deviation stays small relative to the mean, lifting the Sortino well above what a Sharpe ratio would report on the same series.
Common Variations
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Sources & References
- Performance Measurement in a Downside Risk Framework — Frank A. Sortino and Lee N. Price, Journal of Investing (1994)
- Downside Risk — Frank A. Sortino and Robert van der Meer, Journal of Portfolio Management (1991)
Related Content
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Sortino Ratio
Sortino ratio: same numerator as Sharpe, denominator only counts downside volatility. When it's the right number to look at.
Sharpe vs Sortino
Sharpe vs Sortino: when the gap between the two tells you something real about a strategy's tail behaviour — and when it's just noise from a small sample.
Sharpe Ratio Formula
The Sharpe ratio formula: excess return over the risk-free rate divided by return volatility, then annualized. Every variable defined, with a worked example.
Downside Deviation Formula
The downside deviation formula: root mean square of returns below a target, over total observations. The risk term inside the Sortino ratio.