Alpha Calculator

Work out the return left over once the return explained by market exposure has been accounted for.

Jensen's alpha is the distance between what a portfolio actually returned and what the market exposure it carried was expected to pay, and this alpha calculator measures that distance in percentage points. The capital asset pricing model sets the bar — the risk-free rate plus beta of whatever the benchmark earned above that rate — and alpha is the gap above or below it over whatever period the figures cover. Because the bar comes from CAPM, the same figure is written as CAPM alpha as often as it is written as Jensen's alpha.

Formula: alpha = actual return - (risk-free + beta x (benchmark return - risk-free))

Your figures

Portfolio return

What the portfolio actually returned over the period you are measuring.

Benchmark return

What the series you are measured against returned over that same period.

Risk-free rate

Over the same period as the returns beside it, from whatever you treat as risk-free — a deposit rate, a short bill yield, or zero to measure the raw return. Annual returns need an annual rate.

Beta

The portfolio's beta against that same benchmark, from the beta calculator or your own estimate. A beta of 1 means the expectation is simply the benchmark's return.

Results

Alpha

+0.20%

beyond the +11.800% the market exposure carried was expected to pay

Expected return for this betaThe risk-free rate plus beta multiplied by the market risk premium. This is the bar alpha is measured from, and nothing else.: +11.800%
Return explained by market exposureBeta multiplied by the premium. The part of the expectation that came from being in the market at all rather than from anything the portfolio did.: +7.800%
Excess over the benchmarkPortfolio less benchmark. This is the figure alpha is routinely confused with, and they only agree when beta is exactly 1.: +2.000%
Market risk premiumBenchmark less the risk-free rate: what the market paid over the period for being in it.: +6.000%
Beta usedAs entered. Every figure above scales with it, so it is the input worth checking first.: 1.300

What this means

The bar was +11.800%: the 4.000% risk-free rate plus 1.300 of the +6.000% the benchmark paid above it. The portfolio returned +12.000%, so alpha is +0.200% — the part of the return the market exposure carried does not account for. Beating the benchmark is a different test, and here the two disagree: the gap over the benchmark is +2.000%, while alpha is +0.200%. The difference is the +7.800% that carrying 1.300 of the market's risk was expected to pay on its own. Rearranged, alpha is the gap over the benchmark less (beta − 1) multiplied by the premium, which is why a beta above 1 eats outperformance in a rising market and credits it back in a falling one.

Alpha and beating the benchmark are different questions. A portfolio with a beta of 1.5 returning 12% against a benchmark's 10%, with 2% risk-free, beat the benchmark by two points and has an alpha of −2, because the exposure it carried was expected to return 14%. The same beta returning −12% against −10% trails by two points and has an alpha of +4.
Beta is an input, so everything on this page inherits its accuracy. A beta estimated from twelve monthly returns carries real uncertainty, and quoting a result to two decimals from it is false precision — which is why the beta calculator reports R² beside the slope.
Every figure entered has to cover the same period. An annual return beside a monthly risk-free rate produces a number, and that number means nothing.
The expectation comes from the capital asset pricing model, which prices one thing: exposure to the benchmark. A record whose return came from small companies, from value, from momentum or from carrying credit risk will show that as alpha here, because there is nothing else in the model for it to land in.
Nothing here is fetched. The returns, the benchmark, the rate and the beta are all as entered, so the same arithmetic works on a backtest, a factsheet and a live account. The risk-free rate a field opens with is a round illustration, not a current market rate.
This describes a record that has already happened. It summarises the figures entered and says nothing about the ones that follow them, and no particular level of it carries a meaning beyond the arithmetic that produced it.

The link reopens this calculator with the same numbers. Nothing you enter is sent to a server.

Continue with these figures

About the Alpha calculation

Four numbers drive this Jensen's alpha calculator: the portfolio's return, the benchmark's return, the risk-free rate over that same period, and the beta between the portfolio and that benchmark. Beta comes from the beta calculator or from your own estimate; nothing is fetched, and no index or rate is assumed on your behalf.

The reason the page exists is that alpha and beating the benchmark are constantly treated as the same measurement, and they coincide at exactly one value of beta. Everywhere else they can disagree in sign as well as size: a portfolio can finish ahead of its benchmark with negative alpha, and finish behind it with positive alpha, and both results are the arithmetic working correctly.

Alpha formula

market risk premium = benchmark return − risk-free rate

expected return     = risk-free rate + beta × market risk premium

alpha               = actual return − expected return

rearranged:
alpha = (actual − benchmark) − (beta − 1) × market risk premium

actual return: What the portfolio returned over the period, as a percentage, on whatever basis you are measuring it.
benchmark return: What the series being measured against returned over exactly the same period.
risk-free rate: The rate over that same period from whatever you treat as carrying no market exposure — a deposit rate, a short bill yield, or zero to measure the raw figure.
beta: The portfolio's sensitivity to that benchmark. A beta of 1 makes the expectation simply the benchmark's return.
market risk premium: What the benchmark paid over the risk-free rate. Alpha's whole sensitivity to beta runs through this term.

The middle line is the alpha formula finance textbooks print, and it does one job and only one: setting the bar. It says a portfolio holding beta of the market's risk should on average be paid beta of the market's premium, on top of the rate available without holding any market risk at all. Alpha is whatever the record produced beyond that, so the entire measurement rests on the claim that market exposure is the thing worth pricing.

The rearranged line makes the gap with plain outperformance explicit. Alpha equals the margin over the benchmark less (beta − 1) multiplied by the premium, so above a beta of 1 in a rising market the second term is positive and eats into the margin, while in a falling market it reverses and credits it back. A portfolio with a beta of 0.7 that returned 7% against a benchmark's 10%, with 2% risk-free, trailed by three points but was only expected to make 7.6%, so its alpha is −0.6 rather than −3. The same beta returning −5% against a benchmark's −10% trails nothing and earns +1.4, because carrying seven tenths of a 12-point fall was expected to cost 6.4%.

Everything on the page therefore scales with beta, and it scales through the premium. With a market risk premium of six points, a beta wrong by 0.25 moves alpha by a point and a half. That is why an alpha quoted to two decimal places from a beta estimated over twelve monthly returns is false precision rather than accuracy, and why the beta calculator reports R² next to the slope.

One factor is being priced and no others. A record whose return came from small companies, from cheap ones, from momentum, from carrying credit risk, from illiquidity, or from writing options will show every one of those as alpha here, because the model contains nothing else for them to land in. Alpha is what the single-factor bar failed to explain, which is a narrower and more honest claim than skill.

Alpha: a worked example

A portfolio that returned 12% over a year while its benchmark returned 10%, with a 4% risk-free rate over that same year and a beta of 1.3 measured against that benchmark.

Market risk premium10 − 4 = 6.00%
Return explained by market exposure1.3 × 6 = 7.80%
Expected return for this beta4 + 7.80 = 11.80%
Alpha12 − 11.80 = +0.20%
Excess over the benchmark12 − 10 = +2.00%
Cross-check with the rearrangement2.00 − (0.3 × 6) = +0.20%

Two full points ahead of the benchmark, and two tenths of a point ahead of the bar. Nine tenths of the apparent outperformance was the reward for carrying 1.3 times the market's risk into a year the market rose, and only the remaining 0.20 points is alpha. Had the beta been 1.4 rather than 1.3, the same 12% return would have produced an alpha of −0.40 and the portfolio would have failed the test while still beating the benchmark by two points.

How to use the Alpha calculator

Put all four figures on the same period. An annual return beside a monthly risk-free rate produces a number, and that number means nothing.
Measure beta against the same series you enter in the benchmark box. A beta taken against a broad index and a benchmark return taken from a sector index are two different comparisons stapled together.
Enter the risk-free rate as the rate for that period, not an annual headline. For a monthly measurement that is roughly the annual rate divided by twelve; for a raw comparison, enter zero and read the result as return over the benchmark scaled by beta.
Read the expected return row before the alpha. It is the bar the whole page is measured from, and if it looks wrong the input to check is beta rather than the returns.
Compare the alpha against the excess-over-benchmark row, which is the figure a plain excess return calculator would have given you on its own. Where the two differ, the difference is entirely the (beta − 1) term, and that is the thing worth understanding about the record.

Limitations of the Alpha calculator

Every calculation makes assumptions. These are the ones that matter here.

Alpha inherits every bit of beta's estimation error. Beta is an input here, its own standard error is often 0.3 or more over a short record, and each 0.1 of beta moves alpha by a tenth of the market risk premium.
The bar comes from a single-factor model. Exposure to size, value, momentum, quality, credit or illiquidity is not in it, so returns earned from any of those are reported as alpha rather than identified as what they are.
One period gives one number and no measure of how reliable that number is. A regression over a series would produce a t-statistic alongside the alpha; a single-period calculation cannot, and a small positive alpha over a short record is indistinguishable from luck.
The model assumes the beta entered describes the whole period. A portfolio that ran at 0.7 for six months and 1.6 for the next six has no single sensitivity, and averaging them into one produces an expectation that was never the bar at any point in the record.
Nothing here is annualised or compounded. The alpha reported covers exactly the period the inputs cover, and alphas from different period lengths cannot be compared or added.
The benchmark return you enter is usually an index figure that no one could have held for nothing. If the portfolio return is net of fees and dealing costs and the benchmark is not, part of what appears as negative alpha is the cost of implementation rather than the manager.
The risk-free rate is whatever you decide it is. Bill yields, deposit rates and overnight rates differ enough over a year to move alpha by several tenths of a point through the premium term.

Frequently asked questions about Alpha

Is alpha the same thing as beating the benchmark?

Only when beta is exactly 1. At any other beta the two measure different things: outperformance is the raw gap between the portfolio and the benchmark, while alpha is the gap against a bar that has been scaled up or down for the market exposure carried. They can disagree in sign, which is the single most useful thing this page shows.

What level of alpha counts as good?

This page does not offer a threshold, and any figure quoted from one period should be read with the sampling error in mind rather than against a benchmark of goodness. A positive alpha describes a record in which the return exceeded the single-factor expectation; whether it would persist is a separate question that one number over one period cannot answer.

Where should the beta come from?

From the beta calculator on this site, fitted from your own paired returns, or from any estimate you trust — the field is an input precisely so that no assumption is made about it. Whichever source you use, check the period and the benchmark match the ones in the other three boxes, because a mismatched beta silently changes the bar.

Can alpha be positive in a year the portfolio lost money?

Yes, and that is the case where the model earns its keep. If the benchmark fell 10%, the risk-free rate was 2% and the portfolio ran at a beta of 1.2, the expectation was a loss of 12.4%. A portfolio that lost 9% over that period beat the bar by 3.4 points, and alpha is positive while the account is smaller than it started.

Why is my alpha negative when the portfolio beat its benchmark?

Because beta was above 1 in a period the benchmark rose. The extra exposure was expected to pay (beta − 1) multiplied by the market risk premium on its own, and if the margin over the benchmark came in below that, the portfolio was paid less than the risk it took should have earned. The expected-return row shows exactly what the bar was.

Is this the same alpha as the intercept in a factor regression?

It is the same idea measured a cruder way. A regression fits the intercept and the slope together across a whole series and reports a standard error for both; this page takes beta as given and evaluates one period against it. The two agree in structure and will differ in value, and the regression version is the one that can tell you whether the figure is distinguishable from zero.

Should the risk-free rate be zero if I only care about beating the index?

Entering zero is legitimate and changes what is measured: the bar becomes beta multiplied by the benchmark return, and alpha becomes the return beyond a beta-scaled version of the index. That answers a real question, but it is no longer Jensen's alpha, and it will differ from the standard figure by (1 − beta) times the rate you left out.

Last reviewed 29 August 2026. See our methodology and editorial policy.

More performance calculators