Jensen's Alpha Calculator — realised return minus CAPM-required

This Jensen's alpha calculator takes a portfolio's realised return and subtracts the return the capital asset pricing model would have required for that portfolio's beta. What remains is Jensen's alpha: the part of the outcome that the market exposure does not account for. It is measured in percentage points, it can be negative, and it is the classic performance-attribution measure introduced in the study of mutual fund performance.

Arb Digital builds free tools that keep a number's assumptions in view. Every return, rate, beta and premium here is one you type in. The page publishes no market data, names no fund, index or security, and recommends nothing. It computes a published model from your own figures and is explicit about how much of the answer depends on the beta and the benchmark you chose.

What This Jensen's Alpha Calculator Does

The capital asset pricing model says that an asset's expected return should equal the risk-free rate plus its beta multiplied by the equity risk premium. That is a statement about what an investor bearing a given amount of non-diversifiable risk would require. It is not a forecast, and on its own it says nothing about performance.

Jensen's alpha turns it into a performance measure by comparing the required return against what actually happened. If a portfolio with a beta of 1.15 returned 11.4 per cent while the model required 9.72 per cent, the alpha is 1.68 percentage points. The interpretation is narrow but useful: the portfolio delivered 1.68 points more than its market exposure alone would explain, over that window, against that benchmark, with that estimate of beta.

The calculator also reports the same alpha net of the fees and costs you enter, because the gross and net figures are frequently quoted as though they were interchangeable and are not. And it reports the appraisal ratio — the alpha divided by the residual volatility — which is the closest thing on the page to a risk-adjusted version of the same idea.

How to Use It

1. Enter the realised return for a stated window. One year, three years annualised, whatever you have — but every other input must describe the same window. Our annualized return calculator puts a holding-period figure on an annual basis.
2. Use a beta measured against the same benchmark as the market return you enter. The stock beta calculator derives one from paired return data, and the portfolio beta calculator produces a value-weighted beta from holdings.
3. Choose how to supply the market leg. Enter a market return and the premium is derived as market minus risk-free, or enter the premium directly if that is what you hold.
4. Say whether the return is gross or net. If it is gross, enter the fee and cost drag so the net alpha figure is meaningful. If it is already net, enter zero.
5. Read the required return in the grid, not just the alpha. Seeing 9.72 next to 11.4 is what makes the alpha legible; the single headline number on its own hides where it came from.

The Formula: How Jensen's Alpha Is Calculated

Write R_p for the realised portfolio return, R_f for the risk-free rate, β for the portfolio's beta and R_m for the market return. The CAPM-required return is R_f + β(R_m − R_f), and Jensen's alpha is α = R_p − [R_f + β(R_m − R_f)]. The bracketed term is the security market line evaluated at that beta, so alpha is the vertical distance of the portfolio from that line.

Work the defaults. The risk-free rate is 4.2 per cent and the market return 9.0 per cent, so the equity risk premium is 4.8 points. A beta of 1.15 requires 1.15 × 4.8 = 5.52 points of risk premium, and the required return is 4.2 + 5.52 = 9.72 per cent. Against a realised 11.4 per cent, the alpha is 11.4 − 9.72 = 1.68 percentage points.

The excess over the risk-free rate is a different and larger number: 11.4 − 4.2 = 7.2 points. Most of that 7.2 was payment for bearing market risk, and only the residual 1.68 is unexplained. Confusing the two is the most common misreading of the measure. Net of 0.85 points of fees, the alpha falls to 0.83 points, and the appraisal ratio is 1.68 ÷ 3.6 = 0.4667.

The Boundary With the CAPM Calculator

Our CAPM calculator does one half of this arithmetic and stops there. It takes a risk-free rate, a beta and either a market return or a premium, and outputs the required return — the cost of equity, in corporate finance language. It computes no alpha and it takes no realised return, because a required return is a statement about what an investor should demand, not a measurement of what a portfolio produced.

This page starts where that one finishes. It takes the same three or four inputs, produces the same required return, and then subtracts it from a realised return that the CAPM calculator never asks for. That extra input is the entire boundary between the two tools. If you want the discount rate or the cost of equity for a valuation, the CAPM calculator is the right page; if you want to assess a track record against that rate, this one is.

The distinction matters in interpretation as well as in inputs. A CAPM required return of 9.72 per cent is not a prediction that the portfolio will return 9.72 per cent. It is a hurdle implied by a model. Alpha therefore measures performance against a model's hurdle, and inherits every one of that model's assumptions along the way. The MIT OpenCourseWare lectures on the CAPM and APT set out the security market line and the evaluation of securities against it, including the alternative factor models that dispute the single-beta version.

Why a Benchmark-Relative Alpha Is Not a Risk-Adjusted Ratio

Alpha and the risk-adjusted ratios answer different questions, and the difference is structural rather than a matter of emphasis. Alpha subtracts a benchmark return and produces percentage points. The ratios divide by a measure of risk and produce a dimensionless number. You cannot rank two portfolios on alpha and expect the ranking to match a ratio, and neither is wrong when they disagree.

• The Sharpe ratio calculator divides excess return over the risk-free rate by total volatility. It asks how efficiently a portfolio converted total risk into return, and it has no benchmark and no beta. A portfolio can post a large alpha while taking so much idiosyncratic risk that its Sharpe ratio is poor.
• The Sortino ratio calculator divides the same numerator by downside deviation against a minimum acceptable return, so it only penalises variability that fell below the threshold. Alpha treats an upside and a downside deviation identically.
• The Treynor ratio calculator divides excess return by beta. It is the closest cousin, because it uses the same beta, but it produces return per unit of systematic risk rather than a return in points. Alpha and Treynor always agree on the sign for a positive-excess portfolio, and often disagree on the ranking.
• The information ratio calculator divides active return over a stated benchmark index by tracking error. It is benchmark-relative like alpha, but it divides rather than subtracts, and its benchmark is an index rather than a model's required return.

The practical consequence: a large alpha earned with wild residual volatility is a weaker result than the same alpha earned steadily, and nothing in the alpha figure shows that. The appraisal ratio in the grid is the standard repair — alpha divided by residual volatility — and it is the measure that tells you how much of the alpha is likely to be signal.

Alpha Is Only as Good as the Beta and the Benchmark

Two inputs carry almost all the fragility. The first is beta, which is an estimate from a past window and moves with the window's length, its periodicity and the index chosen. A beta of 1.15 rather than 1.05 changes the required return by 0.48 points on these inputs, which is nearly a third of the alpha. An alpha reported without the beta that produced it is not checkable.

The second is the benchmark itself. If the index does not represent the portfolio's investable universe, the alpha is measuring a style difference rather than a decision. A small-cap portfolio measured against a large-cap index will show alpha whenever small companies outperform, and that alpha belongs to the size exposure, not to the manager. This is the criticism that produced multi-factor models, in which several betas replace the single market beta and much of the apparent alpha is reclassified as exposure to a known factor.

Estimation noise is the third problem. Both the realised return and the beta come from the same limited sample, and over a short window a positive alpha is entirely achievable by chance. Professional practice tests whether an alpha is statistically distinguishable from zero rather than reading the point estimate, which usually needs many years of data. The CFA Institute's professional learning resources cover performance attribution and its statistical limits for practitioners.

Fees, Costs, and the Appraisal Ratio

An alpha quoted gross of fees is a description of a strategy; an alpha quoted net of fees is a description of what an investor received. Both are legitimate figures and they answer different questions, which is why this page shows them side by side rather than picking one. On the defaults, a gross alpha of 1.68 points becomes 0.83 net, meaning roughly half the excess went to costs.

The appraisal ratio, alpha divided by residual volatility, is what turns the point estimate into something comparable across portfolios. Two managers each producing 1.68 points of alpha are not equivalent if one did it with 3.6 per cent of residual volatility and the other with 12 per cent. The second is far more likely to have been lucky, and the ratio makes that visible. Where residual volatility is zero the ratio is undefined, and the calculator says so rather than printing a figure. For the underlying dispersion arithmetic, the standard deviation calculator handles the residual series.

Common Mistakes to Avoid

• Confusing alpha with the excess return over the risk-free rate — on the defaults those are 1.68 and 7.2 points, and almost all of the difference is payment for market risk.
• Using a beta measured against a different index from the market return entered — the two must describe the same benchmark or the required return is incoherent.
• Comparing a gross alpha with a net one — costs took roughly half the alpha in the default example, and the two figures are quoted interchangeably far too often.
• Reading a positive alpha over one year as skill — both the return and the beta are noisy estimates from a short sample, and a point estimate is not a test.
• Ignoring what the benchmark excludes — alpha against a mismatched index mostly measures style exposure, which is why multi-factor models reclassify so much of it.

Related Free Tools From Arb Digital

For the required-return half of this calculation on its own, use the CAPM calculator, which takes no realised return and computes no alpha. For the beta that feeds either page, the stock beta calculator works from paired returns and the portfolio beta calculator from holdings. The expected return calculator handles probability-weighted expectations rather than model-implied ones.

For the risk-adjusted ratios that answer neighbouring questions, see the Sharpe ratio calculator, the Sortino ratio calculator, the Treynor ratio calculator and the information ratio calculator. For the return inputs, the annualized return calculator and standard deviation calculator do the groundwork. Everything else is on the free tools hub.

Frequently Asked Questions

This tool is provided for educational and estimating use only. It is not investment advice and does not recommend any fund, manager, security or strategy. Past performance does not indicate future results, and anyone making an investment decision should take advice from a qualified financial adviser.