Two funds with the same average return, worked month by month, and the denominator error that halves the ratio.
The Sortino ratio is the average return in excess of a minimum acceptable return, divided by the downside deviation: the square root of the average squared shortfall below that minimum, taken over every period, not only the bad ones. On twelve illustrative monthly returns averaging 0.55 per cent against a 3 per cent annual minimum, one fund scores an annualised Sortino of 1.34 and another with the same average return scores 2.12, even though the second has the lower Sharpe ratio.
Worked in full in Asset Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The Sharpe ratio penalises all volatility. A manager whose good months are very good is marked down for them, as if a 3.9 per cent month were a risk. The Sortino ratio counts only the months that fell short of the target, which is closer to what a client means by risk. The price of that precision is a denominator that is easy to compute wrongly.
| Month | Fund A | Fund B |
|---|---|---|
| 1 | 1.8 | 0.4 |
| 2 | −0.9 | 0.2 |
| 3 | 2.1 | 3.9 |
| 4 | 0.6 | 0.3 |
| 5 | −1.2 | −0.4 |
| 6 | 1.5 | 0.1 |
| 7 | 0.9 | 0.5 |
| 8 | −0.6 | 3.2 |
| 9 | 2.4 | 0.2 |
| 10 | 0.3 | −0.3 |
| 11 | −1.5 | −1.0 |
| 12 | 1.2 | −0.5 |
| Average | 0.550 | 0.550 |
The minimum acceptable return (MAR) is 3.00 per cent a year, 0.25 per cent a month, set equal to the cash rate so that Sharpe and Sortino share a numerator. Both funds earn 6.60 per cent a year on a simple sum, 3.60 points above the MAR. Twelve months is far too short a record to judge a manager; the length is chosen so that every step can be checked by hand.
Step 1. Excess return over the MAR
Fund A: 0.550 − 0.250 = 0.300 per cent a month.
Step 2. Shortfalls
For each month, min(0, return − MAR). Fund A falls short in four months: −1.15, −1.45, −0.85 and −1.75. The other eight months count as zero.
Step 3. Downside deviation
Square the shortfalls and sum them: 1.3225 + 2.1025 + 0.7225 + 3.0625 = 7.2100. Divide by all twelve months, 0.6008, and take the root: 0.7751 per cent a month. In Excel: =SQRT(SUMPRODUCT((B2:B13<MAR)*(B2:B13-MAR)^2)/COUNT(B2:B13)).
Step 4. Ratio and annualisation
Monthly Sortino = 0.300 / 0.7751 = 0.387. Annualise by multiplying by the square root of 12, 3.4641: 1.34. Equivalently, 3.60 per cent of annual excess over an annual downside deviation of 2.69.
Fund B goes through the same steps. It falls short of 0.25 per cent in seven months, but never by much: the shortfalls run from −0.05 to −1.25 and their squares sum to only 2.8775. Its downside deviation is 0.4897 a month, 1.70 a year, and its Sortino is 2.12.
| Fund A | Fund B | |
|---|---|---|
| Excess return over MAR | 3.60 | 3.60 |
| Standard deviation | 4.62 | 5.11 |
| Downside deviation | 2.69 | 1.70 |
| Sharpe ratio | 0.78 | 0.70 |
| Sortino ratio | 1.34 | 2.12 |
On Sharpe, Fund A wins: Fund B is 0.91 times as good. On Sortino the ranking reverses and Fund B is 1.58 times as good. Both statements are arithmetically right. Fund B's extra volatility comes from two strong months, 3.9 and 3.2 per cent, and its worst month is −1.0 against −1.5 for Fund A. If the client's concern is losing money relative to cash, Fund B has been the safer manager. If the concern is tracking a smooth path, it has not. The Sharpe ratio worked on a full mandate shows how that measure is built.
Annualising by the square root of 12 assumes the monthly returns are independent. Funds that hold illiquid or appraisal-based assets report smoothed returns, which understates both the standard deviation and the downside deviation and flatters both ratios. For those funds, compute the ratios on quarterly or annual data as well and report the lower figure. The ranking between Fund A and Fund B should survive the change of frequency before anyone acts on it.
| MAR, annual | Monthly | Sortino A | Sortino B | Sharpe A | Sharpe B |
|---|---|---|---|---|---|
| 0.00% | 0.000 | 2.99 | 5.39 | 1.43 | 1.29 |
| 3.00% | 0.250 | 1.34 | 2.12 | 0.78 | 0.70 |
| 5.00% | 0.417 | 0.53 | 0.78 | 0.35 | 0.31 |
The MAR moves the Sortino ratio far more than the Sharpe, because it changes both the numerator and which months count as shortfalls. At a zero MAR Fund B scores 5.39; at 5.00 per cent it scores 0.78. A Sortino ratio quoted without its MAR is not a number, and two managers' ratios can only be compared at the same one.
Dividing the sum of squared shortfalls by the number of shortfall months instead of by all months. For Fund A that gives a downside deviation of 1.3426 a month instead of 0.7751 and a Sortino of 0.77 instead of 1.34. The error is not even consistent: Fund B, with seven small shortfalls rather than four large ones, drops only to 1.62. A second version of the same mistake takes the standard deviation of the negative returns, which measures how spread out the losses are rather than how large they are. The downside deviation is a root mean square of shortfalls with zeros included; the zeros are the reward for the months that met the target.
Sortino = (average return − MAR) / downside deviation, with every period in the denominator's count. On these figures it reverses a Sharpe ranking, 2.12 against 1.34, because it stops charging a manager for good months. State the MAR beside every ratio, and use at least several years of data before reading anything into the difference. The free workbook for this case builds tracking error, information ratios and Sharpe ratios on a five-year mandate, gross and net of fees.
There is no universal threshold, because the ratio depends on the MAR and the period. On the same twelve illustrative months Fund B scores 5.39 at a zero MAR and 0.78 at 5.00 per cent. Compare managers only at the same MAR, over the same dates, and treat any figure from fewer than several years of data as noise.
Both share the numerator, return above a hurdle. Sharpe divides by the standard deviation of all returns; Sortino divides by the downside deviation, which only counts shortfalls below the MAR. A fund with strong up months scores better on Sortino: here Fund B's Sharpe is 0.70 against 0.78 for Fund A, while its Sortino is 2.12 against 1.34.
By all periods. Shortfalls are set to zero in months that meet the MAR and those zeros stay in the count. Dividing by the four shortfall months only gives Fund A a downside deviation of 1.3426 a month instead of 0.7751, and a Sortino of 0.77 instead of 1.34.
This article is one calculation from Asset Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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