Someone shows you a track record: up 40% last year. Impressive — until you ask the only question that matters. Compared to what? If the market was up 45%, that "great" strategy lost on a risk-adjusted basis: the returns weren't skill, they were beta — market exposure you could have bought for the price of an index fund. Real skill is what's left after you subtract the free ride, and most track records never do the subtraction. Four numbers do it for you.
1. Jensen's alpha — the return the market can't explain
Regress your excess returns on the benchmark's excess returns. The slope is your beta — how much of your movement is simply the market, levered up or down. The intercept is Jensen's alpha (Jensen 1968): the average return left over after paying for the market risk you took.
α = mean(r − rf) − β · mean(m − rf)
Positive alpha is the closest thing to a skill signature there is. Zero alpha with a beta of 1.4 means you did not beat the market — you levered it, and called the leverage talent.
2. Treynor — reward per unit of systematic risk
The Sharpe ratio divides excess return by total volatility. The Treynor ratio (Treynor 1965) divides it by beta — reward per unit of the systematic risk you cannot diversify away. For a strategy held inside a broader portfolio, where idiosyncratic risk washes out and only market exposure survives, Treynor is the more honest denominator: it prices the risk that actually remains.
3. Tracking error — how far you stray
Tracking error is the standard deviation of the active return — you minus the benchmark, period by period. It answers a plain question: how tightly does the strategy hug the thing it is measured against? A "market-neutral" fund with a tracking error near zero is just an index fund wearing a higher fee. A high tracking error is not bad in itself — it simply means you are making real, differentiated bets, for better or worse.
4. Information ratio — the honest scorecard
Put active return over active risk and you get the information ratio (Grinold & Kahn): mean active return divided by tracking error. This is the number allocators actually rank managers by, because it measures whether you add value consistently rather than just when the tide comes in. A strategy that matches its benchmark exactly has an information ratio of precisely nothing — no active risk, no active reward, no signal. We made the case for a general audience in "Is It Alpha, or Just Borrowed Beta?" and the accompanying note.
5. Computing them
Our open-source orderflow-metrics library ships all four, dependency-free, in TypeScript and Python — the beta regression included:
import { jensensAlpha, treynorRatio, trackingError, informationRatio } from "orderflow-metrics";
const returns = [0.012,-0.005,0.008,-0.014,0.011,0.003,-0.009,0.015,-0.006,0.010];
const benchmark = [0.008,-0.004,0.006,-0.010,0.009,0.002,-0.007,0.011,-0.005,0.007];
jensensAlpha(returns, benchmark); // 0.00021 — return the benchmark can't explain
treynorRatio(returns, benchmark); // 0.00186 — excess return per unit of beta
trackingError(returns, benchmark); // 0.0027 — stdev of the active return
informationRatio(returns, benchmark); // 0.2963 — active reward per unit of active risk
Here the strategy earns a small positive alpha and a positive information ratio — modest, genuine, benchmark-beating edge rather than borrowed beta. The Python distribution exposes the same functions (jensens_alpha, treynor_ratio, tracking_error, information_ratio). They pair with the standalone Sharpe/Sortino ratios and the wider market-microstructure toolkit — install from our open-source page (npm and PyPI, MIT-licensed).
6. Conclusion
A return without a benchmark is a story; a return with an information ratio is evidence. Jensen's alpha isolates the part your benchmark cannot explain, Treynor prices the systematic risk you kept, tracking error measures how far you dared to stray, and the information ratio asks whether the edge is repeatable. Where did the return come from — skill, or exposure? The regression answers what the headline can't. Explore the rest of the toolkit in our quantitative research library, or read the implementation in our open-source metrics.