Sharpe vs Sortino in 5 lines of Python

Every fund factsheet shows a Sharpe ratio. The better ones also show Sortino - and when the two tell different stories, that gap is information.

Sharpe divides excess return by total volatility: upside surprises are punished exactly like losses. Sortino divides by downside deviation only - volatility you would happily take (big up days) costs you nothing.

Here is both, in five lines:

import numpy as np
r = np.random.default_rng(14).normal(0.0004, 0.01, 252) - 0.02 / 252  # daily excess returns
sharpe = r.mean() / r.std(ddof=1) * np.sqrt(252)
downside = np.sqrt(np.mean(np.minimum(r, 0.0) ** 2)) * np.sqrt(252)
sortino = r.mean() * 252 / downside
print(f"Sharpe {sharpe:.2f}   Sortino {sortino:.2f}")

Output: Sharpe 1.04 Sortino 1.55.

Three things worth internalising:

1. For symmetric (normal-ish) returns, Sortino ≈ √2 × Sharpe. For a zero-mean normal distribution, the expected squared downside is exactly half the variance - so the ratio of the two ratios is mechanically √2 ≈ 1.41. Our simulated series lands at 1.55/1.04 ≈ 1.49: close to √2, because the draw is roughly symmetric. A Sortino/Sharpe ratio near 1.4 tells you nothing about skill - it is what randomness produces.

2. The interesting funds are the ones far from √2. A trend-following CTA with a long right tail (many small losses, a few huge wins) shows Sortino ≫ √2 × Sharpe. A short-volatility strategy - steady small gains, rare disasters - shows the opposite until the disaster is in the sample. If Sortino/Sharpe ≈ 2.5, ask about skew before you ask about alpha.

3. Downside deviation uses min(r, 0), not a filter. A common bug is r[r < 0].std() - dropping the non-negative days changes the denominator's sample size and inflates the ratio. The convention divides the squared shortfalls by all n observations, which is what np.mean does above.

Both ratios annualise with √252 - keep the numerator and denominator on the same clock, or you will manufacture performance out of unit errors.