# What is beta? Market sensitivity, formula and a worked example Canonical URL: https://marketheist.io/portfolio/learn/beta By Jeffrey Lin · Published 2026-11-06 · Updated 2026-11-06 · Data through Sep 2026 Category: Risk ## Key takeaways - Beta measures how much a fund tends to move when the market moves: a beta of 1 moves with the market, below 1 moves less, above 1 moves more. - It comes from the capital asset pricing model (Sharpe 1964), where beta is the only risk the market pays for, because the rest can be diversified away. - Beta measures co-movement, not total risk: a fund can have a low beta and high volatility if it swings for its own reasons, as gold and long bonds do. - R-squared says how much of a fund's month-to-month movement the market explains; when it is low, beta is a weak summary of the fund. - MarketHeist estimates beta from monthly total returns over the months the fund and the benchmark share, without subtracting the interest earned on cash. ## Definition Beta is the sensitivity of an investment's returns to the returns of a market benchmark: the slope of a line fitted through their paired returns. A beta of 1 means the investment has tended to move one-for-one with the market, above 1 more than the market, below 1 less, and a negative beta in the opposite direction. Also known as: beta coefficient, market beta, stock beta, systematic risk, CAPM beta, equity beta, upside and downside capture. ## How it works Put each month's return of the market on a horizontal axis and the same month's return of a fund on the vertical axis, one dot per month. Draw the straight line that fits the cloud of dots most closely. Its slope is the fund's beta: how far the fund's return has moved, on average, for each percentage point the market moved. The slope combines two things. One is how closely the two move together, their correlation. The other is how large the fund's swings are relative to the market's. Written out, beta is the covariance of the two return series divided by the variance (uncertainty) of the market's, which is the same as the correlation times the ratio of the two volatilities. A fund with the market's volatility and a correlation of one half has a beta of one half; so does a fund twice as volatile as the market with a correlation of one quarter. Beta became central through the capital asset pricing model of William Sharpe (1964), with John Lintner and Jan Mossin publishing the same result. In that model investors hold the market portfolio and lend or borrow at a risk-free rate, so any risk a holding adds that is not tied to the market can be diversified away and earns no reward. The expected excess return of an asset is then its beta times the market's expected excess return. Decades of tests found the line between beta and average return flatter than the model predicts, but beta survived as the standard measure of how much market risk a holding carries. Beta is like the gear ratio between two wheels: it says how far one turns when the other does, but not whether the second wheel also spins on its own. That second part is what R-squared and the regression's residuals measure. A single beta averages over rising and falling markets. Upside and downside capture split it: the fund's average return in months the market rose, divided by the market's average in those months, and the same for months the market fell. A defensive fund with a capture of less than 100% on the way down and close to 100% on the way up is what many investors hope for; most funds show similar figures in both directions. ## Formula **Beta (Sharpe 1964)** ``` \beta_i = \frac{Cov (R_i,R_m)}{Var (R_m)} = \rho_{i,m} \cdot \frac{\sigma_i}{\sigma_m} ``` where: - βᵢ: beta of investment i against the market m - Rᵢ, Rₘ: returns of the investment and of the market benchmark in the same periods - Cov(Rᵢ, Rₘ): covariance of the two return series - Var(Rₘ): variance of the market's returns: its standard deviation squared - ρᵢ,ₘ: correlation of the two return series - σᵢ, σₘ: standard deviations (volatilities) of the investment and of the market **The capital asset pricing model** ``` E[R_i] - R_f = \beta_i (E[R_m] - R_f) ``` where: - E[·]: expected (long-run average) return - R_f: risk-free rate, such as the Treasury-bill rate - E[Rₘ] − R_f: the market risk premium The model's claim: only beta is rewarded. Risk that is not shared with the market can be diversified away, so in equilibrium it earns no premium. **As MarketHeist estimates it: a regression on monthly returns** ``` r_{i,t} = a + \beta r_{m,t} + \epsilon_t, R^2 = 1 - \frac{\sum \epsilon_t^2}{\sum (r_{i,t} - \bar{r}_i)^2} ``` where: - r_i,t, r_m,t: monthly total returns of the fund and the benchmark in month t, nothing subtracted (risk-free rate 0) - a: intercept: the fund's average monthly return not explained by the benchmark (×12 is the annualized alpha) - β: slope: the beta estimate - εₜ: residual: the part of month t's return the line does not explain - R²: share of the fund's month-to-month movement that the line explains; with one benchmark it equals the correlation squared - r̄ᵢ: the fund's average monthly return With a single benchmark the least-squares slope equals the covariance-over-variance formula above exactly. The annualized standard deviation of εₜ is the fund's tracking error against the regression line. **Upside and downside capture** ``` C_{up} = \frac{avg (r_i∣r_m > 0)}{avg (r_m∣r_m > 0)}, C_{down} = \frac{avg (r_i∣r_m < 0)}{avg (r_m∣r_m < 0)} ``` where: - C_up, C_down: upside and downside capture, as a percentage of the benchmark's average move - avg(rᵢ | rₘ above 0): the fund's average monthly return over the months the benchmark rose; the down-month version averages over the months it fell Fund companies often compute capture from compounded returns over the up and down months; MarketHeist uses simple averages of monthly returns. ## Worked example The technology sector fund (XLK, the S&P 500's technology stocks) against the S&P 500 (SPY), 333 monthly total returns from Jan 1999 to Sep 2026. XLK's beta was 1.31: in an average month, a 1% move in the S&P 500 went with a move of about 1.31% in XLK, in the same direction. Beta is correlation times relative volatility, and the inputs show where it comes from: a correlation of 0.85 and a volatility of 23.2% a year, against 15.0% for the S&P 500. The sensitivity cut both ways. In months when the S&P 500 rose, XLK's average return was 133.4% of the index's average; in months when the index fell, XLK's average loss was 132.1% of the index's. A beta above 1 amplifies the market's falls as much as its rises. Beta describes only the part of XLK's movement that it shared with the market. The R-squared was 72.6%, and the rest showed up as a tracking error of 12.1% a year around the regression line: sector-specific swings, such as the collapse of technology stocks after 2000, that the S&P 500 shared only in part. The intercept of the regression, the annualized alpha, was 0.4% with a t-statistic of 0.15, far from the usual threshold of 2. Over this window XLK's results are described by its market exposure; nothing in the data separates any extra return from zero. The alpha and R-squared guide covers how to read that number. Data through Sep 2026. Reproduce id: 40578a056b87caff (this guide, its lab setup and the data month). ## How MarketHeist computes it - **Data**: Monthly total returns (dividends reinvested). Funds younger than the period shown are extended backward with older funds, spliced on returns, not prices. - **Window**: The worked example and the lab use the common window of the subject and the benchmark, Jan 1999 to Sep 2026 (333 months), ending at the data month of this page; every month is one point of the regression. - **Portfolio streams**: A fund is its proxy-extended monthly return stream. A built-in portfolio is read as the MarketHeist MCP reads it: its composed stream with its own rebalancing rule and overlay. A custom mix is rebalanced yearly. - **Regression**: Ordinary least squares of the subject's monthly returns on the benchmark's, with an intercept and a risk-free rate of 0. Beta is the slope; alpha is the intercept times 12; R-squared is the share of the ups and downs of the subject's returns that the line explains; the t-statistics use the classical standard errors; the residual standard error is the root of the residual sum of squares over n - 2, times the square root of 12. Up (down) capture is the subject's average return in the benchmark's up (down) months over the benchmark's average in those months; months with a benchmark return of exactly 0 are in neither. - **Method**: Ordinary least squares of the fund's monthly returns on the benchmark's monthly returns, with an intercept. Beta is the slope. With one benchmark this equals the covariance of the two series divided by the benchmark's sample variance. (`ols` in packages/shared/engine/factors.ts) - **Returns and window**: Monthly total returns (dividends reinvested) of both picks over their common window, ending in the latest data month. A template is first simulated with its own rebalancing rule. Funds younger than the window are extended backward with older funds or indexes, as everywhere on MarketHeist. - **Risk-free rate**: 0. Nothing is subtracted from either series, so the slope is the beta of total returns, not of excess returns. Beta changes little; the intercept changes more (see the alpha guide). - **Standard errors**: Classical least-squares standard errors with n − 2 degrees of freedom (residuals assumed independent, with constant spread). The t-statistic of beta is the slope divided by its standard error. (`ols` in packages/shared/engine/factors.ts) - **R-squared and tracking error**: R-squared is 1 minus the residual sum of squares over the total sum of squares of the fund's returns. The tracking error shown is the standard error of the regression, the square root of the residual sum of squares over n − 2, times √12: the spread around the fitted line, not the spread of the plain return difference. (`betaSide` in content-engine/lab/compute4.ts) - **Upside and downside capture**: The fund's simple average monthly return over the months the benchmark's return was above (below) zero, divided by the benchmark's simple average over the same months; not compounded. The lab refuses a pair with fewer than 12 up or 12 down months. (`betaSide` in content-engine/lab/compute4.ts) - **Engine functions**: `ols` (packages/shared/engine/factors.ts), `pearson` (packages/portfolio/engine/correlation.ts) ## Beta (market sensitivity) compared with its neighbours Beta and the statistics beside it | Statistic | Question it answers | Scale | Blind to | | --- | --- | --- | --- | | Beta | How much does the fund move per unit move of the market? | 1 = moves with the market | Movement the market does not share | | Correlation | Do the two move in the same direction? | −1 to 1 | Size of the moves | | R-squared | How much of the fund's movement does the market explain? | 0% to 100% | Direction and size of the sensitivity | | Volatility | How large are the fund's swings overall? | Percent a year | Whether the swings come from the market | | Alpha | What average return is left after the market exposure? | Percent a year | How reliable the estimate is (see its t-statistic) | | Upside / downside capture | How much of the market's rises and falls did the fund take? | Percent of the market's move | Months the market was flat; compounding | ## Limitations - Beta is estimated from one historical window and changes over time. A sector's beta can drift by a large amount from one decade to the next, and estimates over a few years are noisy; Blume (1971) found that betas measured in one period move toward 1 in the next. - The answer depends on the benchmark. A fund's beta against the S&P 500 and against a world stock index can differ, and a beta against a stock index says little about a bond fund. - Monthly data hide what happens within a month, and a fund whose prices lag (thinly traded or smoothed holdings) shows a beta that is too low on short intervals. - A single straight line assumes the sensitivity is the same in rising and falling markets. When it is not, as with options-like strategies, the up and down capture figures say more than beta. - Low beta is not low risk. Gold, long Treasuries and commodities have had betas near zero against stocks and still fell by large amounts on their own. ## Frequently asked questions **What does a beta of 1.5 mean?** That over the period measured, the investment moved on average about one and a half times as much as the benchmark in the same direction: a market rise of one percentage point went with a rise of about one and a half points, and a fall with a fall of about one and a half. It is an average relationship; individual months scatter around it, and how much they scatter is what R-squared measures. **Can beta be negative?** Yes. A negative beta means the investment has tended to rise when the market fell and the reverse. Long-term Treasuries in some decades, and strategies that buy protection against crashes, have had negative betas against stocks. A beta near zero means no consistent straight-line relationship; it does not mean the two cannot fall together in a crash. **Is beta the same as volatility?** No. Volatility measures the size of a fund's swings from any cause; beta measures only the part tied to the market. A gold fund can be as volatile as the stock market with a beta near zero, because its swings are mostly unrelated to stocks. **How is beta calculated in Excel?** Put the fund's and the benchmark's monthly returns over the same months in two columns and use SLOPE(fund returns, benchmark returns). COVARIANCE.S(fund, benchmark) / VAR.S(benchmark) gives the same number, and RSQ gives R-squared. Use total returns with dividends for both. **Why does my broker show a different beta for the same fund?** Data providers differ in the benchmark, the frequency (daily, weekly or monthly), the length of the window (often three or five years), whether they subtract a risk-free rate and whether they adjust the estimate toward 1. Each choice moves the number; compare betas only when they are computed the same way. ## References 1. Sharpe, W. F. (1964) Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk . Journal of Finance 19(3). The capital asset pricing model. 2. Lintner, J. (1965) The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. Review of Economics and Statistics 47(1). 3. Blume, M. E. (1971) On the Assessment of Risk. Journal of Finance 26(1). Betas estimated in one period regress toward 1 in the next. 4. Fama, E. F. and French, K. R. (2004) The Capital Asset Pricing Model: Theory and Evidence . Journal of Economic Perspectives 18(3). A survey of the model's empirical record. ## Try it App: https://marketheist.io/portfolio/p/classic-60-40 MCP prompt (MarketHeist MCP server, https://marketheist.io/mcp): ``` Using MarketHeist.io, what is the beta of XLK against SPY on monthly returns, and how much of XLK's movement does the S&P 500 explain (R-squared)? Compare with the utilities fund XLU. ``` Historical simulation, not advice.