How to use the alpha calculator
The calculator has two tools.
Jensen’s alpha is the quick version, for when you already know the annual figures:
- Enter the portfolio’s or stock’s return for the period, including dividends.
- Enter the market’s return for the same period: for US stocks, usually the S&P 500 total return.
- Enter the beta. Our beta calculator estimates it, and stock pages and fund factsheets list it.
- Enter the risk-free rate, usually the average 3-month US Treasury bill yield over the period.
From return series estimates everything from raw data:
- Paste the portfolio’s returns and the benchmark’s returns for the same periods, in percent, oldest first.
- Choose the frequency and enter the annual risk-free rate. The calculator converts it to a per-period rate, subtracts it from both series and runs a regression.
The alpha formulas
Jensen’s alpha compares the actual return with the return the capital asset pricing model (CAPM) predicts:
Jensen's alpha α = Rp − Expected returnRp is the portfolio's return, Rm the market's return, Rf the risk-free rate and β the portfolio's beta, all for the same period. Rm − Rf is the market risk premium.
From return series, alpha and beta come out of one regression of excess returns:
Annualized alpha = α × N
t-statistic = α ÷ SE(α)rf is the risk-free rate per period, (1 + annual rate)1/N − 1, and N the number of periods in a year (12 for monthly returns). SE(α) is the standard error of the intercept. Simple outperformance is the compound annual return of the portfolio minus that of the benchmark.
A worked example
Jensen’s alpha. A portfolio returned 14% in a year when the market returned 10% and Treasury bills paid 4%. Its beta is 1.2. CAPM expects 4% plus 1.2 times the 6.00% market risk premium, which is 11.20%. The portfolio beat that by 2.80%: that is its alpha. The simple outperformance, +4.00%, overstates the skill, because part of the extra return was payment for the extra market risk.
From return series. The example contains 36 monthly returns of an actively managed stock portfolio and the S&P 500. The portfolio compounded at 16.09% a year against 14.02% for the index: a simple outperformance of +2.07% a year. $10,000 grew to $15,646 instead of $14,824.
The regression tells a different story. The portfolio’s beta is 1.33, so in a rising market it should beat the index just by amplifying it. After that adjustment the annualized alpha is -0.67%, with a t-statistic of -0.26: the manager added nothing beyond leveraged market exposure.
Alpha versus simple outperformance
Beating the benchmark and generating alpha are different things, and the difference is beta. With the market at 10% and a 4% risk-free rate:
| Portfolio | Return | Beta | Outperformance | CAPM expected | Alpha |
|---|---|---|---|---|---|
| The example | 14% | 1.2 | +4.00% | 11.20% | +2.80% |
| High beta | 14% | 1.8 | +4.00% | 14.80% | -0.80% |
| Low beta | 9% | 0.7 | -1.00% | 8.20% | +0.80% |
| Market-like | 12% | 1 | +2.00% | 10.00% | +2.00% |
The high-beta portfolio beat the market by the same margin as the example, yet its alpha is negative: with that much market exposure it should have done even better. The low-beta portfolio trailed the market but has positive alpha, because it took much less risk. Only when beta is 1 do outperformance and alpha match.
This matters most in strong markets. In a year when stocks rise 25%, almost any aggressive portfolio beats the index; in the following sell-off the same portfolios fall harder. Alpha separates that market exposure, which an index fund with leverage could provide cheaply, from genuine stock-picking skill.
Is the alpha real? The t-statistic
An alpha estimated from a few years of returns is noisy. The return series tool reports a t-statistic: the alpha divided by its standard error. As a rule of thumb, an absolute value of 2 or more suggests the alpha is unlikely to be pure chance; below that, the data cannot tell skill from luck. In the example, the t-statistic of -0.26 means the negative alpha is not significant. The honest conclusion is “no evidence of skill in either direction”.
Two related numbers help judge an active portfolio. Tracking error, 6.43% here, is the volatility of the return difference against the benchmark: how far the portfolio strays. The information ratio, 0.42, divides the average active return by the tracking error. Note that it is based on simple outperformance, not on alpha, so it rewards the high beta as well. Our tracking error calculator covers both in more detail.
What this calculator does not include
- Other risk factors. CAPM adjusts only for market risk. A portfolio tilted toward small companies, cheap value stocks or momentum can show alpha that is really exposure to those factors. Our factor exposure calculator tests for them.
- The right benchmark. Alpha against the S&P 500 means little for a portfolio of small caps or foreign stocks. Use the index that matches what the portfolio holds.
- Fees and taxes. Use returns after fees to see what an investor actually received.
- Changing beta. The regression assumes one beta for the whole period. A manager who changes market exposure over time can distort both beta and alpha.
- Annualization. Alpha is annualized by multiplying the per-period figure by N, the usual convention; compounding it instead gives a slightly different number for large alphas.