📐 Beta & Active Return
Beta measures how strongly a portfolio tends to follow its benchmark, while active return isolates the part of the result the benchmark does not explain. Both are relative figures: they describe a portfolio only in the company of the series it was compared against, and they change when that series changes.
🔢 Formula
Four quantities are computed from the two aligned return series — the portfolio, \(p\), and the comparison series, \(b\).
Beta is the sample covariance of the two series divided by the variance of the comparison:
Both moments use the unbiased \((N-1)\) estimators, so at least two common observations are required.
Active return is the difference between the two compounded returns over the whole window:
Tracking error is the dispersion of the per-period difference \(a_t = r_{p,t} - r_{b,t}\), annualized with the measured factor:
Information ratio relates the average of that difference to its dispersion, annualized on the same basis:
The factor \(f\) is the same observed annualization factor used everywhere else in the analysis — counted from the data rather than assumed. See Observed Annualization.
The two series must be aligned
Both series are compared point by point, so they must cover the same observations: a comparison is computed only when the two series have the same length and at least two observations. The alignment is performed before the metrics are reached, which is why a result that used a shortened window says so through its observation count rather than through a silently shifted comparison.
🚫 When Beta Has No Value
Beta is not always defined, and when it is not, no number is published.
A comparison series with zero variance yields no beta. Beta expresses how the portfolio responds to movements of the comparison series; a series that does not move offers nothing to respond to. Mathematically the denominator vanishes, and any value returned in its place would be an artefact of the arithmetic rather than a measurement. The result reports the absence instead.
The same discipline applies to the information ratio: if the per-period difference between the two series never varies, its dispersion is zero, and the ratio is left unpublished rather than forced.
An absent value is a result
A missing beta is not a failure of the calculation — it is what the data supports. Reading it as "no relationship" would be the wrong conclusion: it means the comparison series provided no variation against which a relationship could be measured at all.
➗ Active Return Is a Difference of Compounded Returns
The order of operations matters, and it is the source of the most common misreading of this number. Active return compounds each series over the window first and subtracts afterwards. It is not the compounding of the per-period differences.
Three consequences follow.
It does not add up across time. The active return of a year is not the sum — nor the compounding — of the active returns of its months. Each figure belongs to the window it was computed on, and windows cannot be chained.
It is not the return of a strategy. It is not what an investor would have earned holding the portfolio and shorting the benchmark: that position would compound the difference, and would also carry financing and rebalancing effects that no subtraction of two compounded returns can represent.
It does not decompose into skill. Active return states that the portfolio ended the window ahead of or behind its comparison, by that amount. It attributes nothing: the gap may come from different exposures, different timing, or from a comparison that was never an appropriate yardstick to begin with.
💡 Interpretation
Beta describes sensitivity, not quality. A beta of \(1\) means the portfolio has tended to move one-for-one with the comparison series; below \(1\) it has moved less than the series, above \(1\) more. The sign matters more than any threshold: a negative beta means the portfolio has tended to move in the opposite direction.
It is worth separating beta from correlation, because they are easily conflated. From the definitions above, \(\beta = \rho_{p,b} \times \dfrac{\sigma_p}{\sigma_b}\): correlation captures only the direction of the relationship, while beta also carries the ratio of the two volatilities. A portfolio can track its comparison closely and still have a beta far from \(1\) simply because it fluctuates more, or less, than the series it follows. See Correlation.
Active return is the distance between the two finish lines, measured over the window.
Tracking error is how consistently that distance was travelled. It is the volatility of the difference, so a large value says the portfolio departed from its comparison frequently or violently — in either direction, since it is a dispersion and carries no sign.
Information ratio puts the two together: it expresses the average per-period difference in units of its own dispersion, annualized. A higher ratio means the difference was more regular relative to how much it varied; a ratio near zero means the difference, whatever its sign, is small compared with the variation around it. No value is good or bad in itself — the figure depends on the length of the window and on the comparison chosen, and the same number measured over a different window is not the same statement.
⚠️ Limitations
Every figure here inherits its benchmark
Beta, active return, tracking error and information ratio are all measured against a comparison series. Change the series and every one of them changes, without anything about the portfolio having changed. A relative figure quoted without naming what it was measured against is incomplete. See Benchmark Selection.
Beta sees a straight line, over one window
Beta is the slope of a linear relationship estimated over the observed period. A portfolio whose behaviour differs between calm and stressed conditions is summarised by a single slope that describes neither. And because it is estimated, a short window produces a figure that reflects the particular sequence it happened to contain.
Sensitivity is not explanation
A beta near \(1\) says the portfolio moved with the comparison series; it does not say the series drives the portfolio, nor that the remainder is skill. Two series can move together through a common cause with no relationship to one another — the same caution that applies to correlation applies here, with the ratio of volatilities added on top.
🔗 Related
- 🎯 Benchmark Selection — the choice that every figure on this page inherits
- 🔗 Correlation — direction of the relationship, with the volatility ratio removed
- 📊 Volatility — the dispersion that turns a correlation into a beta
- 📏 Observed Annualization — the measured factor used to annualize tracking error and information ratio