Sharpe vs Treynor vs Sortino vs Jensen's alpha: one portfolio, worked example
Search for any single risk-adjusted return metric and you'll find a formula and a plugged-in example: return, standard deviation, beta, done. What's harder to find is one continuous walkthrough that starts from a single raw monthly return series, derives every input by hand, and then feeds that same data into five different metrics side by side, so you can see exactly where they agree, where they diverge, and why. That's what this page does, using one hypothetical portfolio and one benchmark index over twelve months.
The data: one portfolio, one benchmark, twelve months
Fund P is a concentrated growth portfolio. Index B is the broad benchmark it's measured against. Both are hypothetical, built only to make the arithmetic transparent. Assume an annual risk-free rate of 3%, so a monthly risk-free rate of 0.25%.
| Month | Fund P return | Index B return | Active return (P − B) |
|---|---|---|---|
| 1 | 4% | 2% | 2% |
| 2 | −2% | −1% | −1% |
| 3 | 5% | 3% | 2% |
| 4 | 1% | 1% | 0% |
| 5 | −3% | −2% | −1% |
| 6 | 6% | 3% | 3% |
| 7 | 2% | 1% | 1% |
| 8 | −1% | 0% | −1% |
| 9 | 4% | 2% | 2% |
| 10 | −5% | −2% | −3% |
| 11 | 7% | 3% | 4% |
| 12 | 6% | 2% | 4% |
Fund P's twelve returns sum to 24%, for a mean of 2% a month. Index B's sum to 12%, a mean of 1% a month. Every metric below is built from these same 24 numbers plus the 0.25% monthly risk-free rate — nothing else changes between them.
Step 1 — mean, standard deviation, and the Sharpe ratio
Subtract the 2% mean from each of Fund P's returns, square the result, and average the twelve squared deviations: (4,−16…) arithmetic works out to a sum of squared deviations of 174, so variance = 174 ÷ 12 = 14.5, and standard deviation σ_P = √14.5 ≈ 3.81%.
The Sharpe ratio divides excess return by total risk (all of it, systematic and unsystematic):
Sharpe = (R_P − R_f) ÷ σ_P = (2% − 0.25%) ÷ 3.81% ≈ 0.46 per month.
Annualizing a ratio built on standard deviation means multiplying by √12 (≈3.46), since volatility scales with the square root of time: annualized Sharpe ≈ 0.46 × 3.46 ≈ 1.59.
Step 2 — downside deviation and the Sortino ratio
Frank Sortino's critique of the Sharpe ratio was that it penalizes upside swings exactly as harshly as downside ones. The Sortino ratio fixes that by only counting returns that fall below a minimum acceptable return (MAR), here set to 0%. Fund P's four negative months are −2%, −3%, −1%, and −5%; squaring each (4, 9, 1, 25) and dividing by all twelve months (Sortino's own convention, not just the four downside ones) gives a downside variance of 39 ÷ 12 = 3.25, so downside deviation ≈ 1.80%.
Sortino = (2% − 0.25%) ÷ 1.80% ≈ 0.97 per month, annualized ≈ 0.97 × 3.46 ≈ 3.36.
Sortino's ratio comes out roughly twice as large as Sharpe's here for one mechanical reason: eight of Fund P's twelve months were gains, so most of its volatility is upside volatility that Sharpe counts against it and Sortino ignores.
Step 3 — beta, for the Treynor ratio and Jensen's alpha
Beta needs the covariance between Fund P and Index B, and Index B's own variance. Multiply each month's Fund-P deviation (from its 2% mean) by that month's Index-B deviation (from its 1% mean), sum the twelve products (79), and divide by 12: covariance ≈ 6.58. Index B's own squared deviations sum to 38, so its variance = 38 ÷ 12 ≈ 3.17.
β = covariance ÷ variance_B = 6.58 ÷ 3.17 ≈ 2.08
A beta near 2 means Fund P moves roughly twice as far as Index B in either direction — consistent with a concentrated growth fund. The correlation between the two series here works out to about 0.97, meaning almost all of Fund P's risk is systematic (market-driven) rather than fund-specific; that matters below, because Treynor only prices in the systematic slice.
Step 4 — the Treynor ratio
Jack Treynor's 1965 measure swaps standard deviation for beta in the denominator, on the reasoning that a well-diversified investor has already diversified away unsystematic risk, so only market risk should be priced:
Treynor = (R_P − R_f) ÷ β = (2% − 0.25%) ÷ 2.08 ≈ 0.84% per month per unit of beta.
Unlike Sharpe and Sortino, Treynor's output is a return figure, not a dimensionless ratio, so annualizing it by simple multiplication (0.84% × 12 ≈ 10.10%) is a teaching approximation; a real desk would compound the monthly returns properly rather than multiply.
Step 5 — Jensen's alpha
Michael Jensen's 1968 measure asks a sharper question than either ratio above: given the market risk Fund P actually took (its beta of 2.08) and what Index B actually returned (1% this month, on average), what return should the Capital Asset Pricing Model (CAPM) have predicted — and did Fund P beat or miss that number?
Expected return = R_f + β(R_B − R_f) = 0.25% + 2.08(1% − 0.25%) ≈ 1.81%
α = R_P − Expected return = 2% − 1.81% ≈ 0.19% per month, or roughly 2.29% a year at simple annualization — the slice of Fund P's return that beta alone doesn't explain.
Step 6 — tracking error and the information ratio
The last column of the table above, active return, is Fund P's return minus Index B's in each of the same twelve months. Its mean is 1% (2% − 1%, exactly as it should be), and its standard deviation — the tracking error — comes out to ≈ 2.12%. Richard Grinold's information ratio (later formalized with Ronald Kahn) divides one by the other:
Information ratio = active return ÷ tracking error = 1% ÷ 2.12% ≈ 0.47 per month, annualized ≈ 0.47 × 3.46 ≈ 1.63.
Unlike Treynor and Jensen's alpha, which strip out unsystematic risk on the assumption it's diversified away, the information ratio charges for all of the variability in how Fund P deviates from its specific benchmark — it's the metric built for judging a manager's consistency against the exact index they're paid to beat.
One portfolio, five verdicts
| Metric | Divides excess/active return by | Fund P, annualized | Use it when… |
|---|---|---|---|
| Sharpe ratio | Total standard deviation | ≈ 1.59 | Fund P is judged as a standalone holding, upside and downside volatility both count against it |
| Sortino ratio | Downside deviation only | ≈ 3.36 | Only losses matter to the investor (e.g. near retirement), and penalizing upside swings would be misleading |
| Treynor ratio | Beta (systematic risk) | ≈ 10.10% per unit of β | Fund P is one holding in an already-diversified book, so only market risk should be priced |
| Jensen's alpha | CAPM-expected return | ≈ 2.29% a year | You want a single percent figure for skill: return earned above what beta alone would predict |
| Information ratio | Tracking error vs. Index B | ≈ 1.63 | Judging a manager's consistency at beating one specific named benchmark, not risk-adjusted return in general |
Why the numbers don't agree
All five metrics are built from the same 24 monthly return figures, yet they don't rank the same way because each one is answering a different question about risk. Sharpe and Sortino diverge because Fund P's volatility is mostly upside (Sortino ignores that, Sharpe doesn't). Treynor and Jensen's alpha diverge from Sharpe because they price only the systematic slice of risk (beta), which here happens to capture nearly all of it, since Fund P and Index B are correlated at about 0.97 — for a less-correlated fund, Sharpe and Treynor would tell noticeably different stories. The information ratio is the odd one out entirely: it isn't measuring risk-adjusted return against a risk-free rate at all, but consistency of outperformance against one specific benchmark.
To practice applying these formulas against new scenarios, try PassDrill's risk management practice questions, which cover the Sharpe, Treynor, and Sortino ratios, Jensen's alpha, and Value at Risk individually.
This page is educational material to help you understand and practice these formulas; it is not investment, financial, or professional advice, and no specific fund, benchmark, or allocation is recommended.
Source: Jack L. Treynor, "How to Rate Management of Investment Funds," Harvard Business Review 43 (1965); Michael C. Jensen, "The Performance of Mutual Funds in the Period 1945–1964," The Journal of Finance 23, no. 2 (1968); Frank A. Sortino and Robert van der Meer, "Downside Risk," Journal of Portfolio Management 17, no. 4 (1991); Richard C. Grinold and Ronald N. Kahn, Active Portfolio Management (McGraw-Hill, 1999), for the information ratio's standard formulation; CFA Institute curriculum, Portfolio Management topic area, for the standard Sharpe-ratio treatment.