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Benchmark-relative

Tools · Benchmark-relative statistics

The arithmetic an allocator runs on a manager.

Paste a portfolio and the benchmark it should be measured against. You get alpha, beta, tracking error, the information ratio, upside and downside capture and R² — and, for each, the convention used to compute it. That last part matters more than it sounds: two people can compute an information ratio from the same two series, both correctly, and get different numbers.

Calculator

This page is showing two example series. Over the 36 example months the portfolio has a beta of 1.20 to the benchmark, annualised alpha of +3.8%, tracking error of 8.4% and an information ratio of 0.55. Enable JavaScript to paste your own; the table, chart and method on this page do not need it.

Portfolio monthly returns, one per line or comma separated

Benchmark the same months, in the same order

Risk-free rate per cent a year

Affects alpha and beta only through the excess returns. Tracking error, the information ratio and the capture ratios do not use it.

Beta: 1.20; to the benchmark
Alpha: +3.8%; annualised, arithmetic
Tracking error: 8.4%; active return, annualised
Information ratio: 0.55; active over tracking error
Upside capture: 1.36; over 19 up months
Downside capture: 1.03; over 17 down months
Appraisal ratio: 0.48; alpha over residual volatility
R²: 0.73; variance the benchmark explains

Growth of 100, portfolio and benchmark36 months

Solid is the portfolio, dashed the benchmark. The gap at the right-hand end is the cumulative difference; everything else on this page is an attempt to say how much of that gap was exposure and how much was something else.

Benchmark-relative statistics for the example series, with the convention used for each
Statistic	Example	Convention used
Beta	1.20	Slope of a least-squares regression of the portfolio’s excess return on the benchmark’s excess return.
Alpha, annualised	+3.8%	Jensen’s alpha: the regression intercept, multiplied by 12. Arithmetic annualisation, not compounded.
Tracking error	8.4%	Sample standard deviation of the active return, portfolio minus benchmark, times √12.
Residual volatility	8.1%	Standard deviation of the regression residual, times √12. Lower than tracking error whenever beta is not 1.
Active return, annualised	+4.6%	Mean of the monthly differences, times 12.
Information ratio	0.55	Active return divided by tracking error, both annualised. The common convention.
Appraisal ratio	0.48	Alpha divided by residual volatility. Often also called the information ratio, which is why the two disagree.
Upside capture	1.36	Compound portfolio return over the 19 months the benchmark rose, divided by the benchmark’s compound return over the same months.
Downside capture	1.03	The same over the 17 months the benchmark fell. Below 1 means the portfolio fell less.
Capture ratio	1.32	Upside capture divided by downside capture.
R²	0.73	Proportion of the portfolio’s variance explained by the benchmark. Equal to the square of the correlation.
Correlation	0.85	Pearson correlation of the two return series.

Method

The conventions used here.

Each of these statistics has more than one definition in common use. These are the ones this page applies, stated so that a figure from it can be reproduced or disagreed with.

Beta and alpha

r_p − r_f = α + β(r_b − r_f) + ε

One least-squares regression of the portfolio’s excess return on the benchmark’s excess return, on the monthly observations. β is the slope, α the intercept. Because the risk-free rate is constant across the series, the slope is identical whether excess or raw returns are used; the intercept is not. Alpha is annualised by multiplying the monthly intercept by 12 — arithmetic, not compounded.

Tracking error, and the other one

TE = stdev(r_p − r_b) × √12

The sample standard deviation, with n − 1 in the denominator, of the arithmetic difference between the two returns each month. The competing convention measures the standard deviation of the regression residual instead, which is a different quantity whenever β is not exactly 1. Both are reported above: 8.4% for the active return and 8.1% for the residual.

Information ratio — the ambiguity

IR = active return ÷ tracking error

Both annualised, giving 0.55 on the example series. The alternative divides alpha by residual volatility, which gives 0.48 on the same two series. Both are called the information ratio in practice. The second is more precisely the appraisal ratio, and where a manager quotes one figure it is worth asking which. The difference is not rounding: it is the difference between measuring departure from the benchmark and measuring departure from the benchmark’s explained part.

Capture ratios

upside = compound(r_p where r_b > 0) ÷ compound(r_b where r_b > 0)

The months are split by the sign of the benchmark, each side is compounded, and the two compound figures are divided. This is the convention in general use. Taking the ratio of arithmetic means instead gives a different answer. Both conventions become unstable when the benchmark’s compound return over the qualifying months is near zero, and neither is meaningful when there are very few such months — 19 up and 17 down in the example.

R²

The proportion of the portfolio’s variance the benchmark accounts for. For a single-benchmark regression this is exactly the square of the correlation between the two series. A low R² is not a criticism of the portfolio; it is a warning that alpha and beta measured against that benchmark describe very little of what the portfolio did.

Annualisation, throughout

Returns are annualised by compounding, means by multiplying by 12, and standard deviations by √12. The √12 step assumes the monthly observations are independent; where they are not, tracking error is understated and the information ratio overstated. The ratio interpreter quantifies that effect.

Limits

What these numbers cannot tell you.

The benchmark choice decides the answer. Alpha and beta are defined relative to one series. Change it and every figure here changes, including the sign of alpha. A benchmark chosen after the fact, or one with a different currency basis or investable universe, produces arithmetic that is correct and meaningless.
Alpha is an estimate with a wide error. The regression intercept has a standard error of its own, and on a few dozen monthly observations it is usually large relative to the intercept. An annualised alpha of a few per cent over three years is rarely distinguishable from zero.
Beta assumes one linear relationship throughout. A portfolio that reduces exposure in falling markets has a different beta up and down, and a single regression averages the two into a figure that describes neither.
Capture ratios are sensitive to how many months qualify. With few up or down months, one month moves the figure sharply, and the ratio of the two capture figures compounds that sensitivity.
None of this is a forecast. Every figure describes the overlap between two series that have already happened. Beta estimated on one period is a poor predictor of beta in the next, particularly across a change in market conditions.
The series have to be comparable. Same months, same order, same currency, and both net or both gross. A portfolio net of fees measured against a gross index produces a negative alpha that is an artefact of the comparison.

General information

This calculator is provided for general information only and is directed to wholesale and professional investors. It is not personal advice: it does not take into account the objectives, financial situation or needs of any person, and it is not an offer, invitation or recommendation to acquire any financial product. Investing involves risk, including the possible loss of capital. Past performance is not a reliable indicator of future performance.

Definitions

Terms on this page.

Each links to the glossary entry, which states the convention the term assumes as well as what it means.

Alpha — the return a benchmark does not account for.
Beta — how much of the benchmark’s movement was taken on.
Tracking error — how much the return has varied from the benchmark’s.
Information ratio — active return per unit of tracking error.
Upside and downside capture — the share of the benchmark’s gains and losses taken.
R² — the proportion of variance the benchmark explains.
Benchmark — the alternative the portfolio is measured against.

Questions

Common questions.

How is tracking error calculated?

Take the difference between the portfolio return and the benchmark return each month, take the sample standard deviation of those differences, and multiply by the square root of 12 to annualise. That is the active-return convention, which this page uses. The other convention in use takes the standard deviation of the residual from a regression of the portfolio on the benchmark, which differs whenever beta is not exactly 1. Both figures are shown above so the gap between them is visible.

What is a good information ratio?

The question has the same problem as asking what a good Sharpe ratio is: an information ratio is a Sharpe ratio computed on the active return, so it carries the same margin of error and that error is wide on short records. Before comparing one to a threshold, put it through the ratio interpreter with the length of the record behind it.

Why do I get a different information ratio elsewhere?

Almost always because of the denominator. Active return over tracking error and alpha over residual volatility are both called the information ratio, and they agree only when beta is exactly 1. On the example series here they are 0.55 and 0.48. Other sources of difference are the annualisation convention, whether the series are both net of fees, and whether a risk-free rate was subtracted.

What does an upside capture above 1 and a downside capture below 1 mean?

That over the months the benchmark rose the portfolio captured more than all of the rise, and over the months it fell the portfolio fell by less. Read the two together with the number of months in each: over 19 up months and 17 down months, a favourable pair of capture figures can come from one or two large months rather than from a consistent pattern.

Can I use daily or quarterly returns?

The calculator annualises on the assumption of monthly observations, so paste monthly data. For a different frequency, convert the inputs first with the return frequency converter, or read the annualised figures as scaled by the wrong constant.

Is anything I paste sent to KyperX Capital?

No. Both series are parsed and every statistic computed in your browser. Nothing is transmitted, recorded or stored, and this website receives none of it.