Worked examples, in Risk & Portfolio Construction
Risk-Adjusted Returns: Worked Examples
A single bad day is invisible in the mean return but dominates every downside metric. That is the lesson the third scenario is built to demonstrate. All scenarios use a ten-day series at zero risk-free rate; they are short and stylized so you can trace every number. The annualized ratios run large because the square-root-of-252 factor amplifies a small daily edge, so read the metrics against each other rather than against a real-world benchmark.
3 examples. Published May 26, 2026. By AI Fin Hub Research.
Worked examples
See the inputs and outcome together
Each scenario keeps the starting point, the outcome, and the actual lesson in one place so the page reads like a decision notebook, not a data dump.
- 1
Symmetric chop
A series that alternates small gains and losses around a near-zero mean. Wins and losses are balanced, so downside deviation is well below total standard deviation.
Annualized Sharpe 0.30, Sortino 0.46, Omega 1.04, max drawdown 1.1%, Calmar 3.5.
Returns
0.01, -0.01, 0.01, -0.01, 0.01, -0.01, 0.012, -0.008, 0.009, -0.011
Risk-free rate
0%
Sortino sits above Sharpe because downside deviation (0.70%) is smaller than total standard deviation (1.06%). Omega just above 1.0 confirms gains barely outweigh losses. This is the balanced baseline where every ratio tells the same modest story.
- 2
Pure uptrend with no losing days
Ten consecutive small gains. There are no negative returns at all, which breaks any metric that divides by downside risk.
Annualized Sharpe 49.9, Sortino 0 (no downside), Omega infinite, max drawdown 0%, Calmar undefined.
Returns
0.005, 0.004, 0.006, 0.003, 0.007, 0.002, 0.005, 0.004, 0.006, 0.005
Risk-free rate
0%
With zero losing days the downside deviation is zero, so Sortino returns 0 and Omega is infinite, and a zero drawdown leaves Calmar undefined. The absurd Sharpe is the lesson: a flawless short series produces meaningless ratios. Always check the sample is long enough to contain real drawdowns.
- 3
One catastrophic day
Nine quiet positive days and a single minus-5 percent shock. The mean stays positive, but the loss day reshapes the whole risk profile.
Annualized Sharpe 2.15, Sortino 2.51, max drawdown 5.0%, Calmar 16.1, skew minus 2.65, excess kurtosis 5.07.
Returns
0.008, 0.009, 0.007, 0.01, 0.008, 0.009, 0.007, 0.008, -0.05, 0.009
Risk-free rate
0%
The single shock drives skew to minus 2.65 and excess kurtosis to 5.07, the unmistakable fingerprint of tail risk that the Sharpe of 2.15 hides. Max drawdown jumps to the full size of that one day. When skew is sharply negative, trust the drawdown and Calmar over the Sharpe.
Patterns
Run the numbers next
- Calculators Sharpe Ratio Calculator
Paste daily returns; get Sharpe, Sortino, Calmar, and Omega side-by-side with a recommendation on which ratio fits your distribution.
- Calculators Returns Distribution Analyzer
Paste a returns CSV. Histogram, normal QQ plot, skewness, excess kurtosis, Jarque-Bera test, tail-weight index. See why Sharpe alone misleads.
Sources
- The Sharpe Ratio , Sharpe, W. F., Journal of Portfolio Management (1994)
- Performance Measurement in a Downside Risk Framework , Sortino, F. A. and van der Meer, R., Journal of Investing (1991)
Keep the topic connected
- Learn Sharpe Ratio
Sharpe ratio defined, when it lies (skew, fat tails, autocorrelation), and how to read a Sharpe number you didn't compute yourself.
- Learn Sortino Ratio
Sortino ratio: same numerator as Sharpe, denominator only counts downside volatility. When it's the right number to look at.
- Learn Drawdown
Drawdown explained: peak-to-trough decline, why max drawdown alone is misleading, and the recovery math that actually matters.