# Alpha A Statistics concept (Relationships) in the LuxAlgo Library, with 1 indicator implementation. ## What is Alpha? Alpha is the part of an asset's or strategy's return that its market exposure does not explain. In Jensen's classic formulation, regress the asset's returns in excess of the risk-free rate on the benchmark's excess returns using [linear regression](https://www.luxalgo.com/library/concept/linear-regression/): the slope of that fit is beta, and the intercept is alpha. Positive alpha means the asset earned more over the window than its benchmark sensitivity alone would have predicted; negative alpha means it earned less. The formulation traces to Michael Jensen, whose 1968 study of mutual fund performance used the intercept of a CAPM regression to test whether managers beat the market after adjusting for risk; most did not. Under the capital asset pricing model, expected return is the risk-free rate plus beta times the market premium, so anything persistently above that line reads as skill or mispricing. Factor models later complicated the story: once size, value, and momentum factors join the regression, much of what looked like index-relative alpha is reclassified as factor exposure. Alpha is always relative to the model you specified. Alpha is an estimate, not a property. It changes with the benchmark chosen, the lookback window, and the return frequency, and short-window readings are noisy. Chart and screener implementations usually compute it as a rolling regression intercept against a reference index, which makes it a relative-performance lens: it shows who is outrunning their market exposure right now, with no promise the outrunning continues. For chart use the read is comparative. A rolling alpha series shows whether outperformance is expanding or fading, and its stability matters as much as its level, since an intercept that flips sign each window is mostly noise. Fit quality matters too: when the asset's [correlation](https://www.luxalgo.com/library/concept/correlation/) with the benchmark is weak, slope and intercept are both estimated poorly and the reading deserves little weight. ## How to calculate alpha Alpha is the intercept of a returns-on-returns regression; rolling implementations repeat these steps per window. 1. Pick a benchmark the asset genuinely tracks and a window, commonly 60 to 250 daily or 36 to 60 monthly returns. 2. Compute per-period returns for both series, subtracting the per-period risk-free rate for the textbook Jensen version; at daily frequency this barely moves the result. 3. Regress the asset's excess returns on the benchmark's: the slope is beta, and the intercept, the return left when the benchmark contributes nothing, is alpha. 4. Annualize for readability if desired, noting platforms differ in whether they scale or compound the per-period intercept. 5. Judge reliability: check the fit's R-squared and recompute on shifted windows; an alpha that survives window changes deserves more attention than a single print. ## How it's calculated The portion of a strategy's or portfolio's return not explained by its benchmark exposure, per the CAPM (Jensen's alpha). ``` α = R_p - (R_f + β × (R_m - R_f)) β = Cov(R_p, R_m) / Var(R_m) R_p: portfolio or strategy return over the measurement period R_f: risk-free rate over the same period R_m: benchmark (market) return over the same period β: portfolio beta versus the benchmark Cov(x, y): covariance of the two return series Var(x): variance of the return series α: alpha, the risk-adjusted excess return ``` All returns must cover the same period and frequency, and alpha is usually annualized. A common shortcut skips beta and the risk-free rate and reports the plain excess return R_p - R_m. Alpha is a backward-looking estimate and sensitive to the benchmark chosen. ## How traders use it - Screening: ranking symbols by rolling alpha against an index surfaces names outperforming beyond what their beta explains, a shortlist for further work rather than a buy list. - Strategy evaluation: computing alpha on backtest returns separates timing or selection skill from returns that are simply market exposure in disguise; a system with high raw returns and near-zero alpha is mostly repackaged beta. - Portfolio reads alongside beta: near-zero beta with positive alpha describes a return stream largely independent of the index, very different from a high-beta book with the same headline return. - Persistence checks: before acting on a rolling-alpha ranking, see whether the series persists or reshuffles, in effect its [autocorrelation](https://www.luxalgo.com/library/concept/autocorrelation/); a leaderboard that reorders weekly is mostly noise. - Source-of-return analysis: pairing alpha with [distribution-of-returns profiling](https://www.luxalgo.com/library/concept/distribution-of-returns-profiling/) shows whether the excess came from steady accrual or a few outlier days, which changes how repeatable it is. ## Alpha vs related statistics - **Correlation** (https://www.luxalgo.com/library/concept/correlation/): Correlation measures how tightly two return series move together and says nothing about which earned more. Alpha asks the follow-up: given that relationship, what return remains once the benchmark's contribution is removed? A stock can track its index tightly and still carry positive or negative alpha. - **Z-score** (https://www.luxalgo.com/library/concept/z-score/): A z-score normalizes a value against its own history, measuring unusualness in standard deviations. Alpha normalizes performance against a contemporaneous benchmark through regression. One is a self-referential yardstick, the other market-relative; screeners often show both. - **Percentile Rank** (https://www.luxalgo.com/library/concept/percentile-rank/): Percentile rank orders values within a lookback or across a universe with no model of why returns differ. Alpha is model-based, crediting only what beta cannot explain: ranking raw returns rewards high-beta names in an up market, while ranking by alpha is designed not to. ## FAQ ### How is alpha calculated? The Jensen method regresses the asset's excess returns (returns minus the risk-free rate) on the benchmark's excess returns. The regression intercept is alpha: the average return left over after the beta-scaled market contribution is removed. Simplified chart versions often skip the risk-free adjustment and report the raw regression intercept over a rolling window; at daily or intraday return frequencies the difference is usually small because the per-period risk-free rate is tiny. ### Does positive alpha mean a real edge? Not by itself. Alpha estimates move with benchmark choice, window length, and plain luck, and a strategy can print positive alpha in one sample and lose it in the next. It is evidence worth weighing, strongest when it persists across windows and survives realistic costs, but past alpha is not a guarantee of future alpha. ### What is the difference between alpha and beta? They are the two coefficients of one regression. Beta is the slope: how much the asset moves per unit of benchmark move, a measure of exposure. Alpha is the intercept: the average return remaining after that exposure is accounted for. A fund can have high beta and zero alpha (an expensive index proxy) or low beta and positive alpha (returns largely independent of the market). ### Does the choice of benchmark change alpha? Substantially. Against a broad index, a semiconductor stock may show large alpha that mostly vanishes against a semiconductor ETF, because sector strength was doing the work. The benchmark defines what counts as explained, so quote alpha with its benchmark and window attached; screening mixed sectors against one index flatters whatever the strong sector happens to be. ### How do I know whether an alpha estimate is statistically meaningful? Formally through the intercept's standard error and t-statistic, which most chart tools omit; informally through robustness. Recompute across windows, frequencies, and adjacent benchmarks, or resample in the spirit of [Monte Carlo price paths](https://www.luxalgo.com/library/concept/monte-carlo-price-paths/) to see how often skill-free series print comparable intercepts. Even genuine edges need long samples to separate from luck. ### Why do two platforms report different alpha for the same stock? Because alpha is convention-heavy: return frequency, window length, risk-free handling, annualization, and dividend treatment all move the number, and adjusted versus unadjusted prices alone can flip a small alpha's sign. Match settings before concluding anything changed in the market. ## Implementations in the Library - Alpha (LuxAlgo): https://www.luxalgo.com/library/indicator/alpha/ ## Related concepts - Correlation: https://www.luxalgo.com/library/concept/correlation/ - Beta: https://www.luxalgo.com/library/concept/beta/ - Autocorrelation: https://www.luxalgo.com/library/concept/autocorrelation/ - Lead-lag Detection: https://www.luxalgo.com/library/concept/lead-lag-detection/ - Cointegration: https://www.luxalgo.com/library/concept/cointegration/ - Pairs Trading Stack: https://www.luxalgo.com/library/concept/pairs-trading-stack/ - Stationarity & Efficiency Tests: https://www.luxalgo.com/library/concept/stationarity-and-efficiency-tests/ --- Source: https://www.luxalgo.com/library/concept/alpha/ (LuxAlgo Library, the encyclopedia of trading & technical analysis). Free to use with attribution: https://www.luxalgo.com/library/license/