Sharpe vs Sortino vs Modified Sharpe
The Sharpe ratio is the most cited risk-adjusted return measure in finance. It is also the most commonly misused. For diversified long-only portfolios with roughly normal returns, it works well. For hedge funds, private equity, structured products, and any strategy with skewed or fat-tailed returns, Sharpe overstates performance, sometimes dramatically.
The CAIA curriculum teaches three measures candidates need to know: classic Sharpe, Sortino, and Modified Sharpe (the Cornish-Fisher version). Knowing which one to apply, and why, is a recurring theme on Level I and Level II.
The Sharpe Ratio
William Sharpe introduced the ratio in 1966. The formula is:
\[ \text{Sharpe} = \frac{Rp - Rf}{\sigma_p} \]
Where \(Rp\) is the portfolio's return, \(Rf\) is the risk-free rate, and \(\sigma_p\) is the standard deviation of portfolio returns.
The Sharpe ratio answers a clean question: how much excess return are you getting per unit of total volatility? Higher is better. A long-only equity fund with a Sharpe of 0.6 is fairly average. A diversified multi-strategy hedge fund claiming a Sharpe of 2.0 deserves a closer look.
What Sharpe Assumes
Sharpe is exactly correct when:
- Returns are normally distributed (no skew, no fat tails).
- All volatility is equally bad (an investor cares about upside surprises as much as downside).
- The risk-free rate is meaningful (which it is, in most developed markets).
When those assumptions hold, the Sharpe ratio is the right ratio. When they do not, the Sharpe ratio is misleading.
Hedge fund marketing decks often quote a Sharpe of 1.5 or higher. Before believing it, check the strategy's skew and kurtosis. Short-volatility, illiquid, and credit-spread strategies routinely produce a smooth-looking return series that masks tail risk. The Sharpe ratio is high because the volatility looks low, until the tail event arrives.
The Sortino Ratio
Frank Sortino's modification, proposed in the 1980s, replaces total volatility with downside deviation. The formula:
\[ \text{Sortino} = \frac{Rp - RT}{\sigma_D} \]
Where \(RT\) is a target return (often the risk-free rate, sometimes zero, sometimes a minimum acceptable return), and \(\sigmaD\) is the downside deviation, computed as:
\[ \sigmaD = \sqrt{\frac{1}{n} \sum{t=1}^{n} \min(Rt - RT, 0)^2} \]
Returns above the target contribute zero to the denominator. Returns below the target are squared and averaged.
Why Sortino Often Beats Sharpe
Most real-world investors do not view upside volatility as risk. A fund that delivers occasional 8% monthly gains is not penalized in the Sortino calculation. Only the months below the target hurt the ratio.
For strategies designed to limit downside (covered calls, collared equity, certain absolute-return programs), Sortino can be twice as high as Sharpe and still be the more honest number.
Sortino is the right ratio when investors care about loss avoidance, not symmetric volatility. That covers nearly every retail and institutional mandate written in plain English. "Limit losses" is not the same as "limit volatility."
Sortino Caveats
- The result depends heavily on the target return. Sortino with \(RT = Rf\) is different from Sortino with \(R_T = 0\).
- With few historical observations, the downside deviation estimate is noisy. A strategy that has not yet experienced a big drawdown looks better than it should.
- Sortino still assumes the historical sample is representative. It does not adjust for tail risk you have not yet observed.
Modified Sharpe (Cornish-Fisher)
For strategies with material skew and kurtosis, even the Sortino ratio can be misleading. The Modified Sharpe ratio uses the Cornish-Fisher expansion to adjust the normal-distribution z-score, then plugs the adjusted score into a Value-at-Risk denominator.
The adjusted z-score:
\[ z'\alpha = z\alpha + (z\alpha^2 - 1)\frac{S}{6} + (z\alpha^3 - 3z_\alpha)\frac{K}{24} \]
Where \(z_\alpha\) is the normal z-score at confidence level \(\alpha\), \(S\) is the skewness of returns, and \(K\) is the excess kurtosis (kurtosis minus 3).
The modified Value-at-Risk (mVaR) is then:
\[ \text{mVaR}\alpha = -(\mu + z'\alpha \cdot \sigma) \]
And the Modified Sharpe ratio is:
\[ \text{Modified Sharpe} = \frac{Rp - Rf}{\text{mVaR}_\alpha} \]
What This Buys You
The Cornish-Fisher correction penalizes strategies with negative skew (more frequent losses than the normal would imply) and high excess kurtosis (fatter tails). A short-volatility strategy with a Sharpe of 1.4 might have a Modified Sharpe of 0.6 once the corrections are applied, a much more accurate picture of the true risk-adjusted return.
Modified Sharpe is the right tool for hedge funds, structured products, options-writing strategies, and credit-spread trades. Any strategy that earns a smooth premium most of the time and loses a lot occasionally has the skew and kurtosis profile that breaks both Sharpe and Sortino.
A Direct Comparison
Consider three strategies, each with the same arithmetic mean return of 8% and total volatility of 10%:
| Strategy | Sharpe | Sortino | Mod. Sharpe |
|---|---|---|---|
| Diversified equity (skew = -0.1, kurtosis = 3.2) | 0.60 | 0.85 | 0.58 |
| Long/short market neutral (skew = 0.0, kurtosis = 3.0) | 0.60 | 0.92 | 0.60 |
| Short volatility (skew = -2.5, kurtosis = 12) | 0.60 | 1.10 | 0.18 |
All three look identical by Sharpe. Sortino flatters the short-volatility strategy because most months are gains. Modified Sharpe is the only one that exposes the tail risk of the short-vol book.
The lesson: Sharpe alone is not enough information for non-normal strategies. Always pair it with at least one of the alternatives, and ideally both.
Practical Caveats from Real Fund Analysis
Several real-world issues can distort all three ratios:
- Smoothed valuations. Private equity, real estate, and some credit hedge funds value positions at quarterly model marks rather than market prices. Volatility looks lower than it really is. Sharpe and Sortino are both inflated. Modified Sharpe is partially robust but not immune.
- Survivorship bias. Databases drop dead funds. The surviving sample has higher mean returns and lower volatility. Adjust by adding 100-200bp to volatility and subtracting 100-200bp from mean before computing the ratios.
- Backfill bias. Funds enter databases after a good performance period. The early years of the track record are biased upward. Discard the first 12 months when you can.
- Correlation regimes. A strategy uncorrelated with equities during normal times often becomes highly correlated in a crisis. None of the three ratios captures this directly. Pair them with conditional VaR or stress-test analysis.
CAIA Level I tests the formulas. CAIA Level II tests when each ratio fails. Real-world due diligence requires both, plus regime-conditional risk measures, before you size a position.
CAIA Exam Coverage
Level I covers Sharpe and Sortino in the Hedge Funds and Risk sections. Modified Sharpe shows up in the alternative-investment risk module and is examined on Level II in more depth.
From past CAIA Association sample questions, expect:
- A direct calculation of Sharpe given mean return, risk-free rate, and standard deviation.
- A scenario question asking which ratio is most appropriate for a described strategy.
- A trap question where Sharpe looks high but skew or kurtosis indicates a different ratio is needed.
- A Sortino calculation with a non-zero target return (a common slip).
How to Practice
The formulas are simple. The judgment is not. Build the habit of asking, before computing any ratio, "what is the return distribution?"
Review the CAIA Level I formula sheet for all three definitions side by side, then drill the application questions on FreeFellow. The free CAIA Level I question bank has hundreds of risk-metric questions tagged by topic, with solutions that walk through both the math and the choice of measure.
The trap most candidates fall into is reflexively reaching for Sharpe. Train yourself to pause for 5 seconds and ask whether Sortino or Modified Sharpe is more appropriate. That habit is worth 2-3 extra correct answers on Level I and is essential when you reach Level II.