Week 17: Performance Metrics — Sharpe, Sortino, Calmar, IR, Treynor, Alpha and Beta

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1. Why This Is Important

A return number on its own is almost useless. "I made 18% last year" is a sentence; it is not an evaluation. It tells you nothing about how much risk was taken to earn it, how often the strategy lost money, or whether the same dollar in an index fund would have done better. Professional investors, allocators, due-diligence teams, and honest self-assessing retail investors all live in the world of risk-adjusted return — return per unit of something painful.

You need this material for four reasons.

Evaluating funds and managers. Every fund tear-sheet on the

planet quotes Sharpe, max drawdown, and tracking error. If you cannot read those numbers and intuit what they mean — what a Sharpe of 0.4 versus 1.2 actually feels like in client account statements — you will overpay for mediocre managers and miss genuinely good ones.

Honestly grading yourself. A 30% year is impressive only if

you took less than 30% in vol to get it, and only if you can show it was not pure beta to a 30% market. Without Sharpe, Sortino, alpha, and beta, your year-end review is just storytelling.

Picking the right metric for the right job. Sharpe is the

default but it punishes upside vol the same as downside. Calmar focuses on the worst pain. IR tells you whether your active bets actually paid. Treynor looks only at the systematic risk you could not diversify away. Each metric answers a different question; using the wrong one gives you the wrong answer.

The vol tail wags the dog. Standard deviation

assumes returns are normally distributed. They are not. Tails are fat. So Sharpe consistently under-penalises strategies that quietly accumulate left-tail risk. Sortino and Calmar partially fix that. Knowing which metric flatters which strategy is most of the alpha in due diligence.

This lesson works through the whole taxonomy, runs every metric on the Damodaran 1928-2024 dataset for four model portfolios, and shows you how the rank order across metrics can reorder your preferences.

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2. What You Need to Know

2.1 The Sharpe Ratio — Excess Return per Unit of Total Volatility

The Sharpe ratio is the foundation. Bill Sharpe (Nobel 1990) wrote it down in 1966. The formula is simple:

$$ \text{Sharpe} = \frac{R_p - R_f}{\sigma_p} $$

Numerator: the excess return — your portfolio's return minus the risk-free rate (3-month T-Bill). Denominator: the **total standard deviation** of your portfolio's returns.

The Sharpe ratio answers: *how much return did this portfolio earn per unit of total volatility?* Higher is better. A few rough benchmarks for annualised Sharpe over a long horizon:

Sharpe (annualised)	Interpretation
< 0	Lost to the risk-free rate. Negative compensation for risk.
0.0 - 0.3	Mediocre. The S&P 500 averages about 0.4 over a century.
0.3 - 0.6	Decent. Most balanced portfolios live here.
0.6 - 1.0	Genuinely good — if real and persistent.
1.0 - 2.0	Excellent. Top quartile hedge funds, well-run risk parity.
> 2.0	Suspicious. Either short data window, hidden tail risk, or fraud.

Two important practical points.

The frequency rescaling trap. Sharpe is normally quoted annualised. If you compute it from monthly returns, you must multiply by $\sqrt{12}$, not 12. From daily, by $\sqrt{252}$. This follows from the assumption that monthly returns are independent — which they are not, exactly, but the convention has stuck. A strategy with a monthly Sharpe of 0.30 has an annualised Sharpe of $0.30 \times \sqrt{12} \approx 1.04$, not 3.6.

The vol-tail problem. Sharpe uses $\sigma$, which assumes returns are roughly symmetric around the mean. They are not. 1987's -22% Black Monday was a 20-sigma event under a normal model — meaning it should not have happened in the lifetime of the universe. Yet there it was. So Sharpe systematically rewards strategies that look smooth most of the time but blow up rarely (short volatility, illiquid credit, leveraged carry). The volatility tail wags the dog — do not pick managers on Sharpe alone.

2.2 Sortino — Downside Deviation Only

The Sortino ratio is Sharpe's first repair. It replaces total volatility with downside deviation only: the standard deviation of returns below a target, usually zero or the risk-free rate.

$$ \text{Sortino} = \frac{R_p - R_f}{\sigma_d} \quad \text{where} \quad \sigma_d = \sqrt{\frac{1}{N}\sum_{r_i < t}(r_i - t)^2} $$

By design, upside volatility no longer hurts your score — only losses do. For a positively-skewed strategy (lots of small losses, occasional big gains; trend-following is the classic example) the Sortino ratio is materially higher than Sharpe. For a negatively- skewed strategy (selling vol, picking up nickels in front of a steam-roller) Sortino is closer to or worse than Sharpe.

A useful rule of thumb: when Sortino is at least 1.4× Sharpe, the strategy has positive skew and the manager is genuinely managing downside. When Sortino is barely higher than Sharpe, the return distribution is symmetric or worse — left tail and right tail look similar, and the manager is just running risk.

2.3 Calmar — Return per Unit of Worst-Case Pain

Calmar is the cleanest answer to: *what is my return divided by my worst draw-down?*

$$ \text{Calmar} = \frac{\text{Annualised return}}{|\text{Max drawdown}|} $$

If a strategy returns 12% annualised and the worst peak-to-trough draw was 30%, Calmar is 0.40. Calmar values above 0.5 are good; above 1.0 is rare and excellent over a long sample. The S&P 500 since 1928 has a Calmar of roughly 0.10 (≈10% return / ≈86% 1929-32 drawdown).

The strength of Calmar: it focuses on a single number every investor actually cares about — the worst loss they would have sat through. The weakness: it depends on the sample window. A strategy launched in 2010 that never met its 2008 has a flattering Calmar; a strategy whose track record happens to start right before a crisis looks worse than it is. Always ask: *does this Calmar include the worst available regime?*

2.4 Information Ratio — Active Return per Unit of Tracking Error

For an active manager — one who claims to beat a specific benchmark — Sharpe alone is insufficient. The right question is: *per unit of how much you deviate from the benchmark, how much extra return do you produce?*

The Information Ratio:

$$ \text{IR} = \frac{R_p - R_b}{\sigma(R_p - R_b)} = \frac{\text{Active return}}{\text{Tracking error}} $$

Numerator: the active return (your return minus the benchmark's return). Denominator: the tracking error (standard deviation of the difference). A closet-indexer with low tracking error and a small positive active return can have a high IR; a wild stock- picker with both large active return and large tracking error can have a mediocre one.

Industry shorthand:

• IR < 0.0: Worse than the benchmark. Fire the manager.
• IR = 0.5: Top quartile institutional active manager.
• IR = 1.0: Top decile. Genuinely great.
• IR > 2.0: Vanishingly rare; demand to see the strategy in detail.

IR is the metric every pension consultant uses to grade active managers. If you ever pay a fee above passive, you should know yours.

2.5 Treynor — Excess Return per Unit of Beta

Sharpe divides by total risk. Treynor divides only by systematic risk — the part you cannot diversify away:

$$ \text{Treynor} = \frac{R_p - R_f}{\beta_p} $$

where $\beta_p$ is the slope of your portfolio's returns regressed on the market's. A diversified equity sub-portfolio with $\beta = 1$ and 12% excess return has Treynor 0.12. A market-neutral fund with $\beta \approx 0$ produces a divide-by-near-zero — Treynor is undefined or infinite, which is a clue Treynor is the wrong metric for hedged strategies.

Use Treynor when you are evaluating a sub-portfolio inside a larger book — for example, deciding whether the 40% of your equity sleeve that is in tech is earning enough for its market exposure, ignoring its idiosyncratic noise that diversifies away at the parent level. Use Sharpe when you are evaluating a stand-alone investment.

2.6 Jensen's Alpha and Beta — From the CAPM Regression

Beta and alpha both come out of the same equation: the CAPM regression of excess returns:

$$ R_p - R_f = \alpha + \beta (R_m - R_f) + \varepsilon $$

Run that regression on your monthly returns versus the S&P 500's, and:

• $\beta$ is the slope. It tells you how much your portfolio

moves for each unit of market move. $\beta = 1.2$ means a 1% market drop drags you down ~1.2%.

• $\alpha$ is the intercept. It is what you earned above the

return CAPM said you should have earned for taking that beta exposure. Annualised, alpha is the holy grail.

A few honest realities:

Most retail strategies have alpha statistically indistinguishable

from zero. Alpha is rare. Treat any alpha estimate from fewer than 60 monthly observations as extremely noisy.

Beta is often more useful than alpha for risk decomposition.

If $\beta = 1.4$, your "stock pick" is really 1.4× S&P leverage plus some noise; the size of that lever explains most of what is happening to the equity curve.

Alpha can be persistent for short windows by luck. It can also

be persistent because of an undisclosed factor exposure (small-cap, value, low-vol, momentum). Modern attribution strips those out before declaring "alpha".

The interactive at the end of this lesson lets you draw the CAPM scatter live: portfolio monthly excess returns vs S&P 500 excess returns, with $\alpha$ as the intercept and $\beta$ as the slope.

2.7 The Metrics Disagree, On Purpose

The whole reason there are five-plus ratios is that **they emphasise different parts of the return distribution**. The chart below runs Sharpe, Sortino, Calmar, and Treynor on four canonical portfolios using Damodaran 1928-2024:

Notice: by Sharpe, the 60/40 mix scores highest because the correlation discount on volatility (Week 4) lifts the denominator's denominator. By Calmar, all-bonds wins because the bond series' worst drawdown over the sample is shallower than the 1929-32 stock loss. By Treynor, the all-stock portfolio looks fine because beta =1 by construction — but Treynor on the bond portfolio is calculated against equity beta and looks unreliable. Same data, four different rankings.

This is not a bug. It is the entire point. A defensible portfolio review reports at least three metrics and explains where they agree and where they disagree.

2.8 Sharpe Through Time — The Regime Story

Even for a single asset, the Sharpe ratio is not a constant. Below is the rolling 10-year Sharpe of the S&P 500 (excess over 3-month T-Bills) since 1937:

The line is wild. In the 1970s, a 10-year window earning roughly zero excess over T-Bills produced a Sharpe near 0.0. In the 1980s-1990s tailwind, Sharpe climbed past 1.0. The 2000s double-decade-of-disappointment (dot-com + GFC) cratered it back toward zero. The 2010s rebuilt it to ~1.5 on the back of QE-driven multiple expansion.

The takeaway: be very careful when someone quotes "the long-run Sharpe of the S&P 500 is X." It is X over the chosen window. The 1980-2020 regime was anomalous — and so was its Sharpe.

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3. Common Misconceptions

"Sharpe of 2 is great." It is too great. Real,

long-horizon, capacity-bearing Sharpes above ~1.2 are rare. A short-window Sharpe of 2 usually decomposes into either selection bias (the survivors), short data (overfit), or hidden tail risk (about to blow up).

"Higher Sharpe = lower risk." Higher Sharpe = better

excess return per unit of measured risk. If the risk measure (standard deviation) misses fat tails, the apparent Sharpe lies — the volatility tail wags the dog.

**"Calmar is fairer than Sharpe because it uses real

drawdowns."** Calmar is sample-dependent. A strategy with no crisis in its track record has artificially high Calmar. The right comparison is Calmar over a common window including stress periods.

"Sortino rewards skill." Sortino rewards return shape, not

skill. A leveraged long-only equity book with no risk management has a high Sortino in any sustained bull market.

"Beta = risk." Beta = systematic risk *to the market you

regressed on*. A "low-beta" stock with $\beta=0.4$ to the S&P 500 may have $\beta=2$ to oil prices. The beta you compute depends entirely on the chosen market index.

"Alpha proves skill." Alpha proves *unexplained excess

return given the chosen factor model*. If your model omits small-cap, value, momentum, or quality, you will read off alpha that is really just a known factor premium. Most "alpha" in academic backtests pre-1995 has since been shown to be omitted-factor exposure.

"Information ratio measures the same thing as Sharpe." No

— Sharpe is total excess over total vol. IR is active excess over tracking error. A manager with low IR but high Sharpe is just running market beta.

"Treynor is better than Sharpe because it uses beta."

Treynor only makes sense when the portfolio is part of a larger diversified book where idiosyncratic risk genuinely diversifies away. For a stand-alone retirement account, Sharpe is the correct metric.

"You annualise Sharpe by multiplying by 12." No. By

$\sqrt{12}$. Multiplying by 12 inflates Sharpe by a factor of 3.46 and is a classic résumé fraud signal.

"Risk-adjusted return is the only thing that matters."

Risk-adjusted return matters for picking among strategies of similar absolute return. But a 0.6 Sharpe at 12% return grows your wealth far more than a 1.2 Sharpe at 4% return. Real wealth is built on the absolute return; Sharpe just helps you choose between two paths to the same end.

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4. Q&A Section

Q1. Which metric should I quote first when describing a fund? A1. Sharpe is still the default and the easiest for a literate audience to interpret. Pair it with max drawdown and one of {Sortino, Calmar} so the reader can spot fat-tail strategies. Quoting only Sharpe in 2026 is a yellow flag.

Q2. What risk-free rate do I use? A2. For US-dollar-denominated portfolios, the 3-month T-Bill yield matched in time to your return series. Damodaran's annual table includes this column. Annualised series use the year-end T-Bill; monthly series use the prevailing T-Bill divided by 12.

Q3. My monthly Sharpe is 0.4, my annualised is 1.4. Why the gap? A3. Annualised Sharpe = monthly Sharpe × √12 = 0.4 × 3.46 = 1.39. That is correct.

Q4. How long a window do I need before Sharpe is meaningful? A4. At minimum 3 years (36 monthly observations). Below that, the standard error of Sharpe is so large the number is essentially random. Five to ten years is typical for institutional allocation decisions.

Q5. What if my portfolio has no market beta — is Treynor useful? A5. No. Treynor with $\beta \approx 0$ is mathematically unstable (divide by near-zero). Use Sharpe and Calmar for market-neutral strategies. Treynor is for sub-portfolios with clear directional market exposure.

Q6. Sortino > Sharpe. Should I always prefer high Sortino strategies? A6. Probably yes — but only if you confirm the sample includes a real drawdown event. A trend-following strategy looks fantastic on Sortino during a sustained trend; the test is how it does in a choppy, mean-reverting regime.

Q7. Why do hedge funds quote IR more than Sharpe? A7. Because hedge fund LPs typically benchmark them against an index (long-short equity vs. S&P 500, market-neutral vs. T-Bills, etc.). IR is the natural metric for "did your active risk pay?" Sharpe answers a different question: "did your absolute return compensate me for absolute volatility?"

Q8. Can a portfolio have positive alpha but a negative Sharpe? A8. Yes, in stress periods. Alpha just measures unexplained excess over CAPM. If the market and your portfolio both lose money, but yours loses *less than CAPM said you should have lost given your beta*, alpha is positive while raw Sharpe is negative. 2008 Treasury managers had this experience.

Q9. How do I compute beta for a long-short portfolio? A9. Same regression: monthly excess returns of the portfolio versus monthly excess returns of the S&P 500. The slope is your net beta. Gross beta (sum of long beta + short beta absolute values) is a separate exposure measure for risk attribution.

**Q10. The risk-free rate has been near zero for a decade. Does that distort Sharpe?** A10. It inflates it. With $R_f \approx 0$, raw return equals excess return, so Sharpe rises mechanically when rates fall. To compare across regimes, always use the contemporaneous T-Bill rate, not a constant assumption.

Q11. Why does the rolling Sharpe of the S&P 500 swing so wildly? A11. Because both numerator and denominator move with the regime. In low-vol bull markets (1990s, 2010s) the numerator is high and denominator low — Sharpe explodes. In stagflation (1970s) or crisis decades (2000s) numerator collapses and denominator rises — Sharpe craters. Regime drives almost everything in quoted single-number stats.

Q12. What single number would Horace use to grade his own year? A12. Two numbers, not one: realised CAGR and max intra-year drawdown. Then a sanity check: Sharpe over 3-year and 10-year rolling windows to see whether the year was on-trend or a fluke. "Risk-adjusted return" in a single ratio is always partial.