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CAPM / beta estimation

Regresses an asset’s return on the market’s to recover its beta and alpha, splits its variance into the part the market explains and the part it does not, and tracks how the beta moves over a rolling window.

What it needs

Asset return
Market / benchmark return

Frequently asked questions

What is beta in CAPM?

Beta measures how much an asset's return moves with the market. A beta of 1 moves one for one with the market, above 1 amplifies market moves, below 1 dampens them. In CAPM, beta is the only risk that earns a premium, because diversification removes the rest.

How do you calculate beta?

Beta is the covariance of the asset's excess return with the market's excess return, divided by the variance of the market's excess return. Equivalently, it is the slope of a regression of the asset's excess returns on the market's, which is how this calculator estimates it.

What is the CAPM formula?

Expected return = risk-free rate + beta × (expected market return − risk-free rate). The term in brackets is the equity risk premium, and beta scales it to the asset's exposure to market risk.

What is Jensen's alpha?

Jensen's alpha is the intercept of the CAPM regression: the average return an asset earned above what its beta predicted. A positive, statistically significant alpha suggests outperformance after adjusting for market risk; an insignificant one means the returns are explained by beta alone.

What does a beta of 1.5 mean?

It means the asset has tended to move 1.5 per cent for every 1 per cent move in the market, in either direction. It carries more market risk than the average stock, so CAPM requires a higher expected return for holding it.

The intuition

A fund's return moves for two reasons: the whole market moved, and something specific to the fund happened. Regressing the fund's excess return on the market's excess return splits those apart. The slope is beta, how much market risk the fund carries. The intercept is alpha, the part the market does not explain - which is the only part a manager can claim credit for.

When to use it

Estimating a stock or fund's systematic risk, computing a cost of equity, separating skill (alpha) from market exposure (beta), or checking whether a manager's returns are just leverage on the index.

Where the estimator comes from

The Sharpe-Lintner CAPM says expected excess return is proportional to covariance with the market: E[R_i]-R_f=\beta_i\,(E[R_m]-R_f) with \beta_i=\mathrm{Cov}(R_i,R_m)/\mathrm{Var}(R_m). That is a statement about EXPECTATIONS and has no error term. The regression actually estimated is Jensen\'s time-series form, R_{it}-R_{ft}=\alpha_i+\beta_i(R_{mt}-R_{ft})+\varepsilon_{it}, and OLS on it returns exactly the ratio above because the slope of a simple regression IS \mathrm{Cov}(x,y)/\mathrm{Var}(x). The theory\'s prediction is the testable restriction \alpha_i=0: under CAPM there is nothing left over.

The model

R_{i,t} - R_{f,t} = \alpha_i + \beta_i\,(R_{m,t} - R_{f,t}) + \varepsilon_{i,t}

What it assumes

The relationship is linear and beta is constant over the estimation window - the rolling-beta chart is there to check the second part.
The market return is a good proxy for the true market portfolio.
Residuals are roughly homoskedastic and not autocorrelated; with monthly data over a few years this is usually acceptable.
The estimation window contains no change in the fund's mandate or leverage. Beta is estimated as a constant, so a strategy change mid-window produces an average of two betas that describes neither.

Statistical properties

Beta is a ratio of second moments, so it is estimated far more precisely than alpha. Five years of monthly data pins beta to roughly ±0.15 and leaves alpha with a standard error of several percent a year.
Alpha is the intercept, so its units are the return interval. A monthly alpha of 0.3% is about 3.7% a year, not 0.3%.
R-squared is the share of the fund's variance explained by the market. For a diversified equity fund it is typically 0.85 to 0.95; a low value means the fund is doing something the market index does not capture, not that beta is wrong.
Returns are heteroskedastic and mildly autocorrelated in calendar time, so plain OLS standard errors understate uncertainty. Use HAC (Newey-West) if you intend to test alpha seriously.
Beta estimated on daily data is biased downward for illiquid assets because prices adjust with a lag. Dimson (1979) corrects this by adding lagged market returns and summing the coefficients.
The CAPM has been rejected repeatedly as a description of the cross-section: Fama and French (1992) found beta had almost no explanatory power for average returns once size and value were included. It remains useful as a risk decomposition, not as a prediction.

How to read the output

Beta above 1 means the asset amplifies market moves; below 1 means it dampens them; the reported t-statistic tests β = 1, not β = 0.
Alpha is the average return the market does not explain. Annualise it before quoting it. On a short sample a "significant" alpha is very often luck.
R² is the fraction of the asset's variance that is market risk. The rest is idiosyncratic and, in a diversified portfolio, diversifiable - CAPM says it earns no premium.
A rolling beta that drifts is a warning: one number for the whole period is hiding a changing exposure.

A worked reading

Fifteen years of monthly returns. Beta = 1.1955 (SE 0.0407), so the fund carries about 20% more market risk than the index, and the t of 4.80 reported beside it tests β = 1 rather than β = 0 - the interesting null for a fund is "as risky as the market", not "unrelated to the market". R² = 0.8287, so the market explains 83% of month-to-month variation and the risk decomposition splits the variance 82.9% systematic against 17.1% idiosyncratic. Alpha is 0.1786% a month, annualising to 2.1438%, with a monthly standard error of 0.1970% and a t of 0.91. That is the number to be careful with: two standard errors on the annualised figure is roughly ±4.7%, so the honest statement is that alpha is somewhere between about -2.5% and +6.9% a year. Consistent with real skill, and equally consistent with none. Fifteen years of monthly data is not enough to separate them, and saying so is more useful than quoting 2.1% as though it were established.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Alpha large, standard error larger	Normal. Alpha is hard to estimate	Do not report significance from a short sample
Beta near zero on an equity fund	Wrong market series, or levels used instead of returns	Check both series are returns over the same interval
R-squared below 0.3	The benchmark does not match the mandate	Use the index the fund is actually run against
Beta above 2	Leverage, or a single sector	Not an error, but say which it is
Residuals cluster in calm and wild periods	Volatility clustering, as always in returns	Alpha is unaffected; its standard error is not. Use HAC
Alpha vanishes when size and value are added	It was factor exposure, not skill	Report the factor model instead. This is the usual outcome

Common mistakes

Estimating beta on price levels or total returns instead of excess returns - subtract the risk-free rate, or use returns already in excess form.
Reading a positive alpha as skill without checking it against a multi-factor model; much "alpha" is exposure to size, value or momentum in disguise.
Using too short a window. Beta estimates are noisy; three to five years of monthly data is a common minimum.
Comparing betas estimated against different market proxies or over different windows. Beta is a property of a regression, not of a stock, and two defensible choices can produce noticeably different numbers.

Where to read this properly

Sharpe (1964), "Capital Asset Prices", Journal of Finance 19(3) - the original.
Jensen (1968), "The Performance of Mutual Funds in the Period 1945-1964" - where alpha comes from.
Fama and French (1992), "The Cross-Section of Expected Stock Returns" - the rejection.
Campbell, Lo and MacKinlay, The Econometrics of Financial Markets (1997), ch. 5.

Related Finance & risk

Black–Scholes option pricing · Bond price, duration & convexity · Merton distance to default · EMI / home loan planner · Multi-factor model · Portfolio optimization · Value at Risk & Expected Shortfall · Performance & drawdown · Time value of money & NPV/IRR · Equity valuation (DDM, FCF, residual income) · Cost of capital (WACC) · Binomial option pricing · Option strategies (payoff & profit) · Forwards, futures & swaps · Credit risk & Basel capital · Extreme value theory (tail risk)

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CAPM / beta estimation

What it needs

Frequently asked questions

What is beta in CAPM?

How do you calculate beta?

What is the CAPM formula?

What is Jensen's alpha?

What does a beta of 1.5 mean?

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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