Two reports show positive returns. One advertises a higher Sharpe ratio; the other emphasizes its Sortino ratio. Can Lin tell which record is stronger by choosing whichever number is larger?
She first needs to know what each numerator and denominator measure. A ratio compresses a return history under particular assumptions. It cannot supply the missing currency, dates, costs, or observations, and it does not prove that a strategy will remain useful.
What you will be able to do
- Reconstruct a historical Sharpe ratio from period-by-period differential returns.
- Explain how Sortino and Calmar use different definitions of an unfavorable outcome.
- Save an auditable calculation card with frequency, reference return, target, window, and missing-data policies.
Start with complete observations
This independent synthetic C6 supplement has two accounts starting at $100 and four complete years, Y1 through Y4. It does not replace G14's one-year case. Returns include distributions and have already deducted the same assumed costs. Personal taxes and external cash flows are excluded. Only the starting value and year-end values are supplied; the examples contain no within-year path.
| Year | Account A return | Account B return | Model reference return |
|---|---|---|---|
| Y1 | −10% | −15% | 2% |
| Y2 | 20% | −15% | 2% |
| Y3 | 10% | 15% | 2% |
| Y4 | 0% | 50% | 2% |
The 2% annual rate is a same-currency model risk-free reference, supplied as a dollar-denominated teaching input. It is not a current Treasury yield or a promise of preserved purchasing power. A real analysis must identify an appropriate risk-free reference and its source. The fact that another case, such as C2, assumes zero interest on account cash does not automatically make its performance reference zero.
Keep returns as decimals internally. A return of −10% is −0.10; a difference between 5% and 2% is 3 percentage points, or 0.03. Using percentages consistently can produce the same dimensionless ratio, but mixing decimals with percentage units cannot.
Sharpe: differential return relative to its variability
A historical Sharpe ratio divides the arithmetic mean of period differential returns by the standard deviation of that same sequence.1 Here, subtract the reference return each year. We use the sample standard deviation, with denominator n − 1.
For A, the differential returns are −12%, 18%, 8%, and −2%. Their sum is 12 percentage points and their mean is 3%. Their deviations from that mean are −15, 15, 5, and −5 percentage points.
In decimal units, the squared deviations sum to 0.05. Divide by 3, then take the square root: the sample standard deviation is about 12.9099%. The historical Sharpe ratio is 3% / 12.9099% = 0.2324. That is a ratio, not a 23.24% investment return.
For B, the mean differential return is 6.75%, the sample standard deviation is about 30.9233%, and the ratio is 0.2183. B has a larger average differential return, but also a much larger denominator. A's slightly higher ratio does not establish a statistically meaningful difference in skill from four constructed observations.
Because the reference is constant in this example, subtracting it changes the mean but not the standard deviation. If the reference varies by period, construct the actual differential-return sequence and calculate its variability. Do not assume the two standard deviations remain equal.
Match the frequency before annualizing
These inputs are annual returns, so the displayed ratios already use an annual observation frequency. Four annual observations do not justify multiplying the result by 4 or by the square root of 4. The number of observations and the number of periods in a year are different quantities.
With monthly observations, use monthly account and reference returns. Do not divide an annual return by daily volatility and call it the same calculation. Converting an annual reference into a compound-equivalent monthly rate requires taking the twelfth root of one plus the annual rate, then subtracting one. Dividing by 12 is not a generally exact conversion.
Multiplying a monthly Sharpe ratio by the square root of 12 requires assumptions about aggregation, including the absence of serial correlation in the simplified summed-return treatment. Compounding makes the relationship more complex.1 Serial dependence, changing variability, and a short history do not disappear when a report applies a familiar multiplier. Record the convention; do not confuse annualization with additional evidence.
Costs, selection of favorable dates, and stale or retrospectively revised prices also matter. A small reported denominator can reflect smoothed observations rather than a position that could actually be traded with little risk.
Sortino: shortfalls below a stated target
The Sortino ratio compares the average return relative to a target with downside deviation. The target, often called a minimum acceptable return, must be stated and selected for a reason. It need not equal the reference used for Sharpe.2
For this first comparison, set the annual target to 2%. For each year, subtract 2% from the return and keep only a negative result; replace a positive result with zero. Square those four values, average them across all four years, and take the square root. This is our specified downside-deviation convention.3
A's shortfall sequence is −12%, 0%, 0%, and −2%. The sum of squared decimal shortfalls is 0.0148. Dividing by 4 and taking the square root gives 6.0828%. A's arithmetic mean return is 5%, so its numerator is 5% − 2% = 3%. The Sortino ratio is 0.4932.
Notice two different denominators in the calculations: the sample variance for Sharpe divides squared deviations by 3; this downside calculation divides squared shortfalls by 4. Downside deviation is not the standard deviation of the two negative shortfalls. Dropping the zero entries and averaging only unfavorable years would change the measure.
B's shortfalls are −17%, −17%, 0%, and 0%. Its downside deviation is 12.0208%. Divide the 6.75% mean excess over the target by that denominator for a Sortino ratio of 0.5615. B ranks higher by this ratio even though A ranks slightly higher by Sharpe. The result is consistent: they penalize different features of the return histories.
Specify the target and formula when comparing software output. Different downside conventions can generate different numbers under the same “Sortino” label. A higher value under one convention is not proof that it captures every relevant risk, especially rare losses absent from the sample.
Calmar: compound growth relative to an observed drawdown
Our Calmar calculation divides the compound annual growth rate, or CAGR, by the positive magnitude of maximum drawdown.4 Both come from the same four-year window. We are not silently imposing a default 36-month window or using daily drawdowns when only annual endpoints are available.
| Observation | A value | B value |
|---|---|---|
| Start | 100 | 100 |
| End Y1 | 90 | 85 |
| End Y2 | 108 | 72.25 |
| End Y3 | 118.8 | 83.0875 |
| End Y4 | 118.8 | 124.63125 |
A finishes at 1.188 times its initial value. Its CAGR is the fourth root of 1.188 minus one, about 4.4009%. Its largest observed peak-to-trough decline is 10%, from 100 to 90. Calmar is therefore 0.4401.
B's CAGR is about 5.6591%. Its largest observed decline runs from the initial 100 to 72.25, or 27.75%. Its Calmar ratio is 0.2039. B's higher ending value does not erase its deeper intervening decline.
The supplied endpoints can miss a much deeper decline during a year. Call these year-end-observed drawdowns, rather than complete daily risk histories. Maximum drawdown also does not tell you everything about recovery time, liquidity, or whether a household could stay invested.
| Measure, under the conventions above | A | B |
|---|---|---|
| Sharpe, annual observations | 0.2324 | 0.2183 |
| Sortino, annual target 2%, all-period denominator | 0.4932 | 0.5615 |
| Calmar, four years and year-end observations | 0.4401 | 0.2039 |
Compare within a row using matching definitions. A Sortino value larger than a Sharpe value is not an improvement in the account: those are different scales with different denominators. A benchmark-relative information ratio is another measure, discussed in G51; do not substitute it for Sharpe without changing the question.
Know when the number is undefined or misleading
If an account returns exactly 4% in every supplied year while the reference stays at 2%, its differential-return standard deviation is zero. Ordinary division does not produce a finite Sharpe ratio. An “infinite” value or a blank cell does not establish that the account is safe.
Similarly, a series that never falls below the target has zero downside deviation under this convention. An observed path with no drawdown gives Calmar a zero denominator. One observation is insufficient for a sample standard deviation. Missing inputs, insufficient observations, zero denominators, and a valid zero ratio are distinct situations; preserve the reason instead of forcing them into one number.
When mean differential return is negative, increasing volatility can move a negative Sharpe ratio closer to zero. A mechanical “higher is better” ranking can then be economically misleading. Read the numerator, denominator, and return path, not just their quotient.
None of these measures independently resolves tail losses, financing demands, execution costs, changing market conditions, or selection from many attempted strategies. Those questions still require evidence outside the ratio.
Explain the ranking and test two changes
Explanation. B finishes with more money. Why can A still have a higher Sharpe ratio?
Reference answers · reveal
Answer. Terminal wealth depends on compounded returns; Sharpe here uses mean annual differential return divided by its sample variability. B has both a larger numerator and a much larger denominator. The ratio is not a ranking of ending wealth.
Repair a calculation. A report divides A's CAGR of 4.4009% by 12.9099% and labels the answer “Sharpe.”
Answer. It used geometric growth in place of mean period differential return and did not subtract the 2% reference in that numerator. The specified calculation is 3% / 12.9099% ≈ 0.2324.
Transfer 1. Keep A's returns and Sharpe reference unchanged, but change only the Sortino target to zero. What changes?
Answer. Mean return relative to the target is now 5%. Only −10% contributes a squared shortfall; downside deviation is the square root of 0.01 / 4, or 5%. Sortino becomes 1.0000. Sharpe remains 0.2324 and Calmar remains 0.4401. The evaluation target changed; the account did not earn a new return.
Transfer 2. Reorder B's annual returns to −15%, 50%, −15%, and 15%. Rebuild the path before recalculating the ratios.
Answer. The path is 100 → 85 → 127.5 → 108.375 → 124.63125. Terminal wealth and CAGR are unchanged. The same annual observations also preserve mean differential return, sample standard deviation, and downside deviation, so Sharpe and Sortino are unchanged. The largest observed drawdown is now 15%, making Calmar about 0.3773. This is an illustration of order sensitivity, not evidence that an investor could choose the favorable ordering in advance.
Save a calculation card, not three isolated numbers
Start T13, your performance-metric calculation card, with A's complete differential-return sequence, its mean, deviations, sample denominator, and Sharpe result. If adding Sortino, retain the target and all four shortfall entries, including zeros. If adding Calmar, retain the complete observed equity path, running peaks, window, and CAGR calculation.
Record the question, data version, currency, costs, frequency, dates, reference and its source, target, annualization convention, and policies for missing values and zero denominators. For real data, state the observation frequency's drawdown blind spots. Keep enough precision to reproduce the displayed result, and round only for presentation.
Ask another person to reconstruct one result from the saved inputs. If no one has done so, label the record as a self-check or AI-assisted check rather than an independent human verification. The practical test is whether a reader can recover what the ratio measures and what it leaves unknown.
Sharpe describes overall variation in differential returns; Sortino focuses on shortfalls below a target; Calmar connects compound growth with the observed path's drawdown over the same window. Align the definitions before reading a ranking. None of the three replaces an account ledger or a judgment about the money you can commit.
References
- William F. Sharpe: The Sharpe Ratio, 1994, historical differential returns and “Time Dependence.”
- Deborah Kidd: The Sortino Ratio, CFA Institute, 2012. Targets and interpretation depend on the stated convention.
- Rollinger and Hoffman: Sortino—A “Sharper” Ratio, calculation discussion. The authors explain the full-series denominator; this is not a claim that CME mandates one universal formula.
- Morningstar: Calmar Ratio. Compound annual return relative to maximum drawdown.
Editorial status
AI-assisted draft synchronized to the accepted second-round Chinese source. All numerical accounts are synthetic. See the author review. A separate AI reviewer has completed an independent full-text review. Human financial review and target-reader testing have not been performed and are not scheduled for this edition.