Risk Adjusted Return Ratios: The Working, Step by Step

Given a return, a risk-free rate, a benchmark return and a risk measure, this calculator works the Sharpe, Treynor and information ratios and prints every line of the arithmetic. Each output travels with its window, its risk-free rate and the benchmark it was measured against. A ratio stripped of those three cannot be set beside any other ratio.

Every default in that panel comes from one place. The Anantara Multi-Asset Portfolio is an invented discretionary mandate of Rs 5,00,00,00,000/- run for a single institutional holder, an invented charitable endowment. Over one stated twelve month period its return was 14.2 per cent at a volatility of 11.8 per cent. Against it stands a composite benchmark built from a broad equity index at 60 per cent with a broad bond index at 40 per cent, neither of them named anywhere here, whose return across those same twelve months was 12.6 per cent at a volatility of 10.4 per cent. The beta against that benchmark was 1.08, the tracking error 3.7 per cent, and the risk-free rate for the period is taken as 6.5 per cent.

The defaults work out in full as follows. The 14.2 per cent less the 6.5 per cent risk-free rate leaves 7.7 percentage points. Over a total volatility of 11.8 per cent that is a Sharpe ratio of 0.6525, and over a beta of 1.08 it is a Treynor ratio of 7.1296. The same 14.2 per cent less the benchmark's 12.6 per cent leaves an active return of plus 1.6 percentage points gross. Over an active risk of 3.7 per cent that is an information ratio of 0.4324. The benchmark's own Sharpe ratio is 0.5865, the active risk derived from the two volatilities and the beta is 3.7114, and the Sortino ratio is refused because no downside deviation exists in this record. On the mandate base the gross active return is Rs 8,00,00,000/- and the active risk is Rs 18,50,00,000/- of spread.

What each ratio measures, and what each one is blind to, is set out under the introduction to risk-adjusted return ratios. The working itself runs in a fixed order, carries three labels on every output, and stops with a named refusal when a figure it needs is not in the record.

One habit before that starts. A computed figure and the figure a record carries are not always the same thing. The exact value is worked to four decimal places, and the record's rounding is named beside it. A surface that computes exactly should show the exact figure and then name the rounding the record holds, rather than quietly publishing one and reconciling against the other.

Which figures must be in hand before any of this arithmetic runs?

Start with the shopping list. The commonest defect on a ratio is not a slip in the division. The defect is a figure that was fetched from somewhere else. The Sharpe ratio needs a return, a risk-free rate and a total volatility. The Treynor ratio needs a return, a risk-free rate and a beta, plus the name of the benchmark that beta was measured against. A beta without its benchmark is not an input at all. The information ratio needs a portfolio return, a benchmark return, and the standard deviation of the difference between the two.

Three of those categories are period specific, and the period is the part that bites. A return belongs to a windowThe exact span of time a figure was measured over, with a start and an end. Two figures measured over spans of different lengths are not two readings of the same thing., a volatility belongs to a window, an active risk belongs to a window, and only the benchmark's identity is timeless. Every input to this calculator has to be fixed to one window before any of them is entered. A mismatched window produces an output that looks completely ordinary and carries no warning of any kind.

The same problem appears when two shops are compared. One quotes rice at Rs 62/- a kilo, the other Rs 310/- a bag. Neither number is wrong. Until the bag is known to hold five kilos the two cannot be put beside each other, and an assumption that it holds two picks the wrong shop with complete confidence. The window on a return figure does exactly the work the kilo does there, and it is invisible in the same way.

How is the Sharpe ratio worked, line by line?

Two operations, in a fixed order. Take the portfolio return for the stated year, 14.2 per cent, subtract the risk-free rate for the same twelve months, 6.5 per cent, and 7.7 percentage points is left. The surviving 7.7 points are the numeratorThe figure on the top of a division, the thing being shared out. Change it and the answer moves in the same direction.. Then divide by the portfolio's total volatility for the same window, 11.8 per cent. The result is 0.6525. The record carries the same figure rounded to 0.653.

The subtraction runs first. The order sounds too obvious to state until the two common departures are worked through. Skipping the subtraction altogether gives 14.2 divided by 11.8, or 1.2034. The reading 1.2034 has the shape of a Sharpe ratio, sits in a range Sharpe ratios genuinely occupy, and is nearly double the correct one. Reversing the order instead gives 14.2 divided by 11.8 with 6.5 then subtracted. The answer is minus 5.2966, absurd on its face and caught the moment anyone looks at it. The dangerous error is the one that produces a plausible number, so skipping the subtraction costs far more than reversing the order, even though reversing the order looks like the worse mistake.

The risk-free rate of 6.5 per cent takes away 45.8 per cent of the 14.2 per cent return, leaving 54.2 per cent of it standing as the 7.7 point numerator, and that surviving slice is the only part of the return either the Sharpe or the Treynor arithmetic ever divides. Leaving the subtraction out therefore does not shave a little off the answer; it changes what is being measured.

The denominatorThe figure underneath a division, the thing being divided by. The denominator sets the scale of the answer. Two answers built on different denominators are not on one scale. here is built from squared deviations, so it counts a rise and an equal fall alike. Squaring removes the sign, and there is nothing more to that line.

How is the Treynor ratio worked, and what changes underneath?

One line changes and nothing else. The numerator is the same 7.7 percentage points. Underneath, the total volatility of 11.8 per cent is replaced by the beta of 1.08, measured against the composite benchmark. The division gives 7.1296. The record carries the result as 7.13.

Notice what just happened to the units. The Sharpe denominator was a percentage; the Treynor denominator has no unit at all. So the two outputs are not two readings on one dial, they are readings on two dials whose faces happen to be printed in the same typeface. A Sharpe ratio and a Treynor ratio computed on the same portfolio over the same window can never be ranked against each other, and either can only be set beside the same ratio computed for something else over the same window.

The household version is the vegetable seller who quotes one price per kilo and another per bundle. Nobody calls the bundle dearer because 40 is a bigger number than 22; the first question is what a bundle weighs. The beta plays the part of the bundle here.

The same two lines run on the benchmark side, and the calculator prints both sides because a single-sided ratio has nothing to be compared with. The benchmark returned 12.6 per cent, so its numerator is 6.1 percentage points. Divided by its own volatility of 10.4 per cent that is 0.5865, carried in the record as 0.587. Divided by its beta against itself, 1.00 by construction, it is 6.1000. William Sharpe set out the first of these two measures in 1966 and Jack Treynor the second in 1965.

How is Active Return computed from two return figures?

Subtract one from the other, and stop. The portfolio returned 14.2 per cent for the stated twelve months and the benchmark 12.6 per cent over exactly the same twelve months, so active returnA portfolio return with its benchmark return taken away. The active return is what is left after the benchmark has been accounted for, and it can be negative. is 14.2 less 12.6, which is plus 1.6, and the unit is percentage pointsThe unit for a gap between two figures that are themselves percentages. Going from 12.6 per cent to 14.2 per cent is a rise of 1.6 points, not a rise of 1.6 per cent., gross. Both return figures are stated before any cost, so the 1.6 is a gross figure; a net one would need a cost input, and no cost figure is among the inputs here.

The unit is not decoration. Dividing 14.2 by 12.6 gives 1.1270. The quotient is a ratio of the two returns and a different quantity altogether. An active return is a difference of two returns and never a ratio of them, so it is quoted in percentage points, and a report quoting it in per cent has published a different number under the right name.

The gap has a rupee size too. On a base of Rs 5,00,00,00,000/- one percentage point of active return is Rs 5,00,00,000/-, so the 1.6 points gross is Rs 8,00,00,000/- and the 3.7 per cent of active risk is Rs 18,50,00,000/- of spread. Both are worth printing. A committee reads a rupee figure faster than it reads a ratio.

How is Active Risk computed, and how is the answer checked?

Active riskThe standard deviation of the series of differences between a portfolio and its benchmark. Also written as tracking error. Active risk is not the standard deviation of either one on its own. has two routes into the same box, and a record worth trusting produces the same figure down both. The direct route measures it: take the series of period-by-period differences between the portfolio and the benchmark and compute the standard deviation of that series over the stated window. On this record that is 3.7000 per cent. The record also carries the same figure under the name tracking error.

The second route derives it from three figures already in the pack. Square the portfolio volatility to reach 139.24, add the squared benchmark volatility of 108.16 to reach 247.40, then subtract twice the beta times the benchmark variance, 2 times 1.08 times 108.16, or 233.6256. The subtraction leaves 13.7744, whose square root is 3.7114 per cent.

Agreement between the two routes is the point rather than a coincidence, and the agreement says something about the record itself. Portfolio volatility, benchmark volatility, beta and active risk are not four figures anyone is free to set independently. Fix any three and the fourth is already decided. A record quoting all four is quoting one figure twice and inviting a contradiction. An earlier version of this record carried 3.2 per cent for the tracking error, which no single sample can produce alongside 11.8, 10.4 and 1.08, and it was corrected to 3.7 on exactly this identity. The last row of the identity table in the panel above runs that check on whatever is entered.

The identity has a blunt consequence for anyone filling in the box marked active risk. The portfolio's own volatility of 11.8 per cent is sitting right there in the same pack, it is also a standard deviation, and it is the wrong figure by a factor of 3.19. Substituting it produces an information ratio of 0.1356 rather than 0.4324, and nothing anywhere reports an error. The arithmetic was carried out perfectly on the wrong number. The third control above the panel does exactly that substitution, so the failure can be watched rather than only read about.

Active Return vs Active Risk: how does one divide into the other?

Active return and active risk are not two ideas that happen to be paired. The two are statistics of one and the same series. Build the period-by-period series of portfolio return less benchmark return: its mean over the window is the active return, and its standard deviation over the same window is the active risk. The information ratio is the first divided by the second, 1.6 gross over 3.7. The quotient is 0.4324, carried in the record as 0.43.

Two properties fall out of that division, and both matter when someone reconciles the working. The first is a rounding trap. The record's printed 0.43 multiplied back by 3.7 gives 1.591, and the record states 1.6 points gross, so the two do not tie. The unrounded 0.4324 multiplied by the same 3.7 gives 1.5999, and that ties. Any working that has to reconcile exactly uses the unrounded figure, and the related case where a leftover is named rather than shown is set out under rounding and residual reconciliation.

The second property is that the ratio is blind to size. Double the active return to 3.2 points gross and double the active risk to 7.4 per cent and the ratio is still 0.4324; quarter both, to 0.8 points gross over 1.85 per cent, and it is 0.4324 again. The information ratio says nothing whatever about how large the departures from the benchmark were, so a portfolio taking cautious positions and one taking very large ones can print the identical figure. The second control above the panel drives that case at fourfold scale on the live figures.

Sharpe Ratio vs Information Ratio: what does each one read on this record?

Run both on the same portfolio over the same stated twelve months and print them side by side. The Sharpe ratio reads 0.6525, carried in the record as 0.653; the information ratio reads 0.4324, carried as 0.43. Two figures, one portfolio, one window, and they are not two attempts at the same measurement.

Both lines differ between the two ratios, not one. The Sharpe ratio subtracts the risk-free rate of 6.5 per cent and divides by the total volatility of 11.8 per cent; the information ratio subtracts the benchmark return of 12.6 per cent and divides by the active risk of 3.7 per cent. Different baseline, different risk measure. Because both the thing subtracted and the thing divided by are different, the two readings are answers to two separate questions and can point in opposite directions on the same record in the same year.

Everyday version: a shop can be doing well against last year and badly against the shop across the road, in the same month, and both statements can be true because they measure against different things. Neither reading corrects the other.

What has to be printed beside every output?

Every figure this calculator produces carries three labels. The window here is one stated twelve month period. The risk-free rate here is 6.5 per cent. The benchmark here is a composite of a broad equity index at 60 per cent with a broad bond index at 40 per cent, described and never named.

Naming the three is not tidiness. Move the window and the whole set of inputs changes. Move the risk-free rate to 5.5 per cent and the Sharpe ratio becomes 0.7373 on figures that did not move. Change the benchmark and both the active return and the active risk are different quantities. Each of the three silently changes the value. A ratio printed without its window, its rate and its benchmark is not an under-documented figure but an uninterpretable one.

The three do not damage equally either. The risk-free rate touches only the Sharpe and Treynor readings. The information ratio subtracts the benchmark return instead. The benchmark touches everything except the Sharpe ratio, and it reaches the Treynor ratio through the beta rather than directly. The window touches all five outputs. Every input is drawn from it. An unstated window is the costliest of the three omissions, and the only one that puts every line of the pack in doubt at once.

The same discipline exists in an ordinary household bill. An electricity reading of 240 units means nothing until the meter start date, the end date and the tariff slab sit beside it. Nobody would accept a bill that printed the units alone. A ratio quoted bare asks for exactly that.

What happens when an input is not in the record?

The calculator reports a refusal and names the figure that is missing. The refusal is the whole behaviour, and it is worth stating outright, because the alternative is so tempting. Take the Sortino ratio on this record. Its numerator is available: the same 7.7 percentage points. Its denominator is a downside deviationA spread figure computed only from the readings that fell below a chosen level, ignoring the ones above it. A downside deviation needs the underlying series, not a summary of it., and no downside deviation exists anywhere in this record.

The record offers the worst drawdownThe fall from the highest point reached to the lowest point after it, inside a stated window. A different window gives a different figure, so the window is quoted every time., 9.7 per cent for the portfolio against 8.1 per cent for the benchmark, both measured peak to trough inside the same stated year. The drawdown is a fall over the same window and it looks like it might serve. A drawdown cannot serve: a drawdown is one distance between two points, and a downside deviation is a spread across many readings. Substituting one for the other returns a figure with the shape of a Sortino ratio and none of its content, and because the shape is right nobody downstream will question it.

Where is each of these seven figures actually read off?

Each of these seven figures is read off one particular line of one reporting pack, and reading a figure off the wrong line is the failure that starts most of the others. The seven places are collected together here.

Who runs these lines, and on which day?

Faiz Ahmad Ansari runs the Anantara Multi-Asset Portfolio, and the investment committee of the endowment holding it is chaired by Rukmini Deshpande. Neither computes these ratios for the pleasure of it. The ratios exist because a committee meeting has an agenda and somebody has to say, in about ninety seconds, whether the year's departures from the benchmark were on a scale that matches what came back.

The order in that room runs backwards from the order set out here. The first thing anyone reads is the active return, plus 1.6 points gross or Rs 8,00,00,000/- on the mandate base. The second is the active risk, 3.7 per cent or Rs 18,50,00,000/- of spread. Only then does the information ratio of 0.4324 mean anything. The ratio is the first divided by the second, and neither figure alone would have told the committee what it wanted to know.

A credit officer at a lender does the same arithmetic on a borrower without calling it this. A trader whose income averaged Rs 90,000/- a month over the year has a good income, and a trader whose income averaged Rs 90,000/- a month while swinging between Rs 20,000/- and Rs 2,10,000/- has the same average and a completely different file. The officer is dividing an average by a spread, exactly as the information ratio does. Every user of these ratios is doing the same one thing. Each of them refuses to read a headline figure until the spread that produced it is sitting beside it.

The analyst on the other side of the table has a narrower job and it is mostly checking. Does every figure in the pack come from one period column. Do the printed and the derived active risk agree, 3.7000 against 3.7114 on this record. Does the ratio, multiplied back by its denominator, return the active return the pack states. Three checks, none of which needs a model, and the second of them is what caught the tracking error of 3.2 per cent that could not stand beside 11.8, 10.4 and 1.08.

The Anantara Multi-Asset Portfolio, the charitable endowment that holds it, the composite benchmark, Rukmini Deshpande and Faiz Ahmad Ansari are invented.
Educational material. Not advice on any investment, tax, budget or market position.