Sharpe, Sortino, Treynor and Information Ratio Compared

Four ratios divide a reward by a risk, and each divides by a different risk, so each answers a different question. The Sharpe ratio uses total volatility. The Treynor ratio uses beta. The information ratio uses active risk against a benchmark. The Sortino ratio uses downside deviation, and where that is not recorded it cannot be computed at all.

A return figure on its own is half a sentence. Somebody earned 14.2 per cent and somebody else earned 9.0 per cent, and until what each of them sat through to get there is known, almost nothing has been learned about either. The four ratios in this guide finish that sentence differently, and they disagree.

The running example is the Anantara Multi-Asset Portfolio, an invented discretionary mandate of Rs 500 crore run by Faiz Ahmad Ansari for an invented charitable endowment whose investment committee is chaired by Rukmini Deshpande. Its policy weights are the shape the holder chose in advance, and the actual weights drift away from them between rebalancings. The weights state what the mandate intended, and every ratio below is computed from the record the mandate actually produced.

The record covers one stated twelve month period and nothing else, and every number here belongs to it. The composite benchmark is described and never named, and the assumed risk-free rateThe return an investor is treated as earning without taking on the risk of the thing being measured. The analyst chooses and states it; it is not a property of the portfolio. of 6.5 per cent is a figure somebody selected rather than a property of either entity.

The 1.6 point gap between 14.2 and 12.6 is a gross figure, measured before the cost of the dealing that produced it. The record separately notes turnover of 34 per cent over the same year, and none of the return figures carries that cost, so the 1.6 point gap stays gross wherever it appears.

What does a risk adjusted return ratio actually do?

A risk adjusted return ratio puts a reward on top of a line and a risk underneath it, and divides. There are only two things that can sensibly go on top and a whole shelf of things that can go underneath, so the denominator is where all the disagreement lives.

Two households each clear Rs 12,000/- a month from a spare room. One rents to a salaried tenant on a two year agreement; the other rents by the night, some months full and some empty. Same rupees, very different amount of not knowing what next month looks like. A risk adjusted returnA return figure divided by a measure of the risk carried to produce it, so that two results earned under different conditions can be laid side by side. is that instinct written down as a fraction, and the argument between the four ratios is entirely an argument about how to measure the not knowing.

Across all four ratios the numerator takes only two forms. Either it is the return above the risk-free rate, or it is the return above a benchmark. The first asks whether taking any risk at all was worth it. The second asks whether departing from that benchmark was worth it. The denominator, by contrast, changes every single time. Total volatility, beta, active risk, downside deviation: four different answers to the question "risk of what, exactly".

Choosing a ratio is choosing which risk the manager is willing to be judged on, and a manager who selects the ratio before the result has already selected what will be asked about it. None of the four is more honest than the others, and a reader who does not know which one is in front of them cannot tell what has been claimed.

What is Active Return, and what is it measured against?

Active return is the portfolio return less the benchmark return, over the same period: 14.2 less 12.6, or plus 1.6 percentage points gross. On Rs 500 crore that is Rs 8,00,00,000/-.

Two words in that definition are doing all the work, and they are "same period". Active returnThe difference between what a portfolio returned and what its stated comparison returned, over one and the same stretch of time. is a difference of two figures drawn from one window. Pull them from different windows and the subtraction still produces something that looks like a percentage, with no error message anywhere.

A portfolio figure covering April to March, set against a benchmark figure covering January to December, gives a difference belonging to neither period, and it can easily be larger than the active return being measured. Pairing two windows destroys active return silently. The silence is what makes the mistake dangerous rather than merely wrong.

What is Active Risk, and why is it not the same number as volatility?

Taken as a series rather than as one figure for the year, the active return is, in each sub-period, the portfolio return less the benchmark return. The series of differences has a mean, the active return already stated, and a spread. Active riskThe standard deviation of the run of differences between a portfolio and its comparison. Active risk says how much that difference jumps about, not how much the portfolio does., also called tracking errorThe same quantity as active risk under a second name. The two are used interchangeably in performance reporting., is that spread, and for the Anantara portfolio it is 3.7 per cent.

The 3.7 per cent is not free. The two volatilities and the beta already fix it at 3.7114 per cent, and the figure below works the terms. The step by step working of the same line, with every input named, is covered separately.

Three of those four figures are free and the fourth is determined, so a report quoting all four supplies a consistency test whether it meant to or not. Put 3.2 per cent in place of 3.7114 and no sample produces that set.

Active risk is not the portfolio's own volatility, and the two are confused more often than any other pair here. The Anantara portfolio ran 11.8 per cent of total volatility and 3.7 per cent of active risk in the same year. One says how much the portfolio moved; the other says how much the gap between portfolio and benchmark moved. Total volatility and active risk come apart in either direction: a very volatile portfolio can hug a benchmark that moves in step with it, and a steady one can wander a long way from a benchmark that is steadier still.

A household's monthly grocery bill swings a lot, and so does everybody else's. The same prices move for everyone. The bill is volatile, and the bill relative to a neighbour's is steady. Asked how unpredictable the budget is, the first figure answers; asked whether the household shops differently from its neighbours, only the second does.

Active Return vs Active Risk: how do two statistics of one series combine?

Both describe the same object: the series of period by period differences between portfolio and benchmark. Active return is its mean and active risk is its dispersion, the first and second things computed about any run of numbers, and neither implies the other.

The independence of the mean from the dispersion is the point. A mean of plus 1.6 points can sit on a dispersion of 1 per cent, quietly and steadily ahead, or on a dispersion of 8 per cent, ahead at the end of a year that swung wildly either side of the benchmark. Same headline, different evidence about whether it was repeatable.

Dividing the mean by the spread asks how much active return was produced per unit of active risk taken. The mean over the spread is the information ratioActive return divided by active risk. The ratio asks how much the departure from a benchmark paid per unit of how far the departure wandered., and for the Anantara portfolio it is 1.6 over 3.7, or 0.43.

Two denominators are in circulation, and both land on 0.43. Working that has to reconcile back to exactly 1.6 points uses the unrounded 3.7114 rather than the printed 3.7.

Because the ratio is a quotient of two statistics, quoting it alone throws away both, and no reader can put them back. An information ratio of 0.43 is consistent with plus 1.6 points over 3.7 per cent and equally with plus 4.3 points over 10 per cent, two entirely different mandates. Report the pair, then the ratio.

What does the Sharpe ratio measure, and what can it not see?

The Sharpe ratioReturn above the risk-free rate divided by total volatility. The ratio asks what the whole of the risk taken returned, without separating the sources of that risk. divides the return above the risk-free rate by total volatility. William Sharpe set it out in 1966 under the name reward to variability ratio.

For the portfolio: 14.2 less 6.5 is 7.7, and 7.7 over 11.8 is 0.653. For the benchmark: 12.6 less 6.5 is 6.1, and 6.1 over 10.4 is 0.587. The portfolio comes out ahead. Volatility squares the deviations, so a rise and an equal fall count the same and the measure is symmetric by construction.

The limitation is structural rather than a flaw. Total volatility contains both the part of the portfolio's movement that goes up and down with the benchmark and the part that does not, and the Sharpe ratio divides by the sum of the two. Two portfolios with identical Sharpe ratios can be built one entirely from market exposure and the other almost entirely from decisions unrelated to the market, and the ratio will not distinguish them.

Three figures already in the record give the size of that split: the beta of 1.08 and the two volatilities. Beta squared times the benchmark variance is the part that moves with the benchmark, and the remainder is 9.4 per cent of the portfolio's total variance.

When the portfolio is the whole of what somebody holds, total risk really is the risk they carry, and the Sharpe ratio is the right measure. For one sleeve of something larger the Sharpe ratio is the wrong measure. The part of a sleeve's risk that duplicates the rest of the holding is not the same burden as the part that does not.

What does the Treynor ratio measure, and when does it disagree?

The Treynor ratioReturn above the risk-free rate divided by beta rather than by total volatility. The ratio measures reward per unit of exposure to the benchmark alone. keeps the same numerator and swaps the denominator for betaA measure of how much a portfolio has historically moved for a given move in its benchmark. Beta as an exposure is covered separately, under systematic risk.. Jack Treynor set out the ratio of excess return to beta in 1965.

For the portfolio: 7.7 over 1.08 is 7.13. For the benchmark, whose beta against itself is 1.00: 6.1 over 1.00 is 6.10. The denominators sit on completely different scales, so the Treynor figures do too. A Treynor ratio of 7.13 is not "better" than a Sharpe ratio of 0.653; only Treynor ratios compare with Treynor ratios.

The swap buys a denominator that ignores everything except exposure to the benchmark. Ignoring everything but the shared exposure is a strength for one sleeve inside a larger holding, where only that shared exposure matters to the whole. The same blindness is a weakness when the portfolio is the entire holding. The risk the denominator drops is then risk somebody is genuinely carrying. Which applies is a property of the holder, and the ratio cannot say which.

What does the Sortino ratio need that this record does not carry?

The Sortino ratioA ratio that replaces total volatility with a measure built only from deviations below a stated reference point, so upside movement is not counted as risk. exists because of a real complaint about the Sharpe ratio's denominator. Volatility squares deviations, so a good surprise and an equally sized bad one contribute identically, and an investor who does not lose sleep over the good one can reasonably object. The Sortino ratio answers by swapping in downside deviationA dispersion measure built only from the returns that fell below a stated reference point, ignoring the ones above it.. Frank Sortino is the measure's namesake.

And now the honest position. The Anantara record carries total volatility of 11.8 per cent and a worst drawdownThe fall from a high point to the following low point within a stated window, measured as a single distance rather than as a spread of many observations. of 9.7 per cent for the stated year, and it carries no downside deviation at all, so the Sortino ratio is not computable here. A stated refusal is the correct output here, not a gap somebody should fill.

The tempting substitution is the drawdown, and it is wrong for a reason that generalises. A drawdown is one distance, the fall from a high point to the following low point. A downside deviation is a dispersion, needing the whole run of returns below the reference point rather than the worst stretch. The second cannot be got from the first, and a drawdown in a Sortino denominator produces a figure that looks like a Sortino ratio and is not one.

A drawdown is measured within a stated window, and a different window gives a different figure for the same portfolio. The window travels with the number for that reason. The 9.7 per cent belongs to the stated twelve months and to nothing else.

Sharpe Ratio vs Information Ratio: which question does each one answer?

People spot one difference and stop, so the Sharpe ratio and the information ratio get treated as interchangeable more often than any other pair here. There are two: both the numerator and the denominator change, and the two changes are independent.

The numerators use different baselines. The Sharpe ratio measures the return above the risk-free rate of 6.5 per cent, and asks whether taking risk at all was worthwhile. The information ratio measures the return above the composite benchmark's 12.6 per cent, and asks whether departing from that benchmark was worthwhile. The denominators use different risks. Total volatility of 11.8 per cent is everything the portfolio did, and the Sharpe ratio divides by that. Active risk of 3.7 per cent is only how far the portfolio wandered from the benchmark, and the information ratio divides by that.

The Sharpe ratio asks whether the total risk was worth running. The information ratio asks whether the departure from the benchmark was worth running. A manager can score well on one and badly on the other. A mandate that hugs its benchmark and adds a small steady sliver reads a modest Sharpe ratio and a strong information ratio. The mandate's total risk is whatever the benchmark's risk is, and the sliver was earned with almost no wandering. A mandate that swung far from its benchmark all year and finished barely ahead reads the other way round.

Why do two ratios rank the same pair by different margins?

On this record the Sharpe and Treynor ratios agree about direction and disagree about size, and the disagreement is not noise. The size of the disagreement is information about where the portfolio's extra risk came from.

Work it in the ratios of the denominators. Volatility of 11.8 against 10.4 is 1.135, so the portfolio ran 13.5 per cent above the benchmark's volatility. Beta of 1.08 against 1.00 ran only 8 per cent above. The Sharpe ratio divides by the first and the Treynor by the second, so the heavier penalty lands on the Sharpe margin: 11.2 per cent against 16.9 per cent.

The difference between the two margins is precisely the risk the Treynor ratio cannot see, priced. Pricing it costs one extra division. If the two agree closely, the portfolio's risk is nearly all shared with the benchmark; if they diverge sharply, a large slice of its movement is unrelated to the benchmark, and whoever is judging needs to know before choosing which measure to trust.

What happens to the whole set when the risk-free rate moves?

The two returns are pinned and the two volatilities are pinned, and only the assumed risk-free rate changes. Commit to an answer before touching the control.

The margin runs from about 4.5 per cent at the bottom of the range to 18.8 per cent at the top while nothing about either entity changes. The Sharpe ratio is not a property of a portfolio; it is a property of a portfolio and an assumed rate together, so quoting one without the other is quoting half a measurement. Subtracting one rate from two returns and dividing by two volatilities does not preserve the ratio between them.

How does 1.6 points of gross excess return split apart?

Before the arithmetic, the question is what share of the 1.6 points belongs to the manager, knowing only that the portfolio ran a beta of 1.08.

The benchmark earned 6.1 points above the risk-free rate. A portfolio carrying a beta of 1.08 against it would be expected to return 6.5 plus 1.08 times 6.1, or 13.088 per cent. The portfolio returned 14.2 per cent, so the difference is plus 1.112 percentage points, still gross. The residual 1.112 points is what the term alphaThe part of a return left over once the return expected from the exposure carried has been subtracted. Being a leftover, it changes if the expectation changes. names, after Michael Jensen, 1968.

So the 1.6 points splits in two. The extra 0.08 of beta, applied to the benchmark's 6.1 points, accounts for 0.488 points, and 0.488 plus 1.112 returns 1.600 exactly. On Rs 500 crore that is Rs 2,44,00,000/- of extra market exposure and Rs 5,56,00,000/- of everything else, adding to Rs 8,00,00,000/-.

A headline excess return says nothing until it has been split, and roughly three tenths of this one was bought rather than earned. Every measure above is downstream of what the excess return consists of.

A second split asks where the excess came from rather than how much was exposure, dividing into an allocation effect and a selection effect. Both sum to 1.60 on different bases. The two splits are not alternative estimates of one quantity. Each answers a different question, and taking a term from one and pairing it with a term from the other produces a figure that means nothing. Only the beta decomposition is worked through here.

Why must every ratio carry its period, its rate and its benchmark?

Because all three are inputs, and changing any one changes the output while the portfolio sits perfectly still. The rate does exactly that, as shown above. The period does it too. A twelve month figure and a thirty six month figure are different measurements wearing the same units. So does the benchmark. An information ratio of 0.43 against one comparison is a different statement from 0.43 against another.

A ratio quoted without its period, its assumed risk-free rate and its benchmark is not a weak figure; it is an uninterpretable one, and treating it as a weak figure is how it ends up in a comparison it cannot survive. A weak figure can be used carefully. An uninterpretable one cannot be used at all.

Who actually uses these ratios, and on what day?

An investment committee like the one Rukmini Deshpande chairs reads these ratios to decide which question to ask next, not whether the manager is clever. A wide gap between the Sharpe and Treynor margins says a slice of the year's risk was unrelated to the benchmark, so the next question is where it came from. A respectable information ratio on a large active risk says the mandate wandered a long way to arrive somewhere reasonable, so the next question is whether it was meant to.

An analyst preparing that pack refuses to publish a bare ratio. Every figure leaves the desk with its three labels attached, and any figure arriving without them goes back to its source first. Almost all of the errors above are labelling errors dressed as arithmetic, so the labelling habit removes most of them.

A lender wants to know how far the value of the collateral can travel between two reporting dates. So a lender reading a borrower's investment portfolio cares less about the Sharpe ratio and more about the drawdown and the active risk.

The household version is done with a pen. Write down what the savings returned last year, what a plain comparison returned over exactly the same months, and how much the difference jumped around. The difference is an active return, and how far it jumped around is an active risk. The discipline is the same at every size: name the window, the comparison and the assumed rate, and only then divide.

When does the choice between these four ratios stop mattering?

Three situations. The first is a portfolio whose correlation with its benchmark is 1.00. Beta is then the portfolio volatility divided by the benchmark volatility, so the Treynor ratio becomes the Sharpe ratio multiplied by the benchmark's volatility, and everything measured against that benchmark ranks in the same order on both. Only the scale differs. The Anantara record is not that case, and its correlation of 0.9519 is why the two margins read 11.2 and 16.9 per cent.

The second is a mandate taking no active positions: active return and active risk are both zero, the information ratio has nothing to report, and the Sharpe ratio is the only one left with anything to say. The third is the ordinary one, where two candidates sit closer together than the precision of the inputs and a volatility printed to one decimal place cannot separate them. The choice stops mattering exactly when the risks in the two denominators stop differing, and every other time it is the whole of the answer.