How to Set a Portfolio Risk Budget, and What Limits It

A risk budget states how much variability a portfolio may run in total and how much of that may come from decisions differing from the benchmark. Setting one means fixing both positions from the record, converting the active limit into the return it could earn at a stated ratio, then allocating that limit across decisions so the combined total stays inside it.

Think about the way a household sets a monthly spending limit before it thinks about any single purchase. The limit is not a wish. The limit is worked out from what actually comes in, it is written down before anybody goes shopping, and it is then split across rent, school fees, food and the rest. Two things about that household limit matter here. The household limit starts from a measured position rather than an ambition, and the individual allowances have to be reconciled against the total rather than simply listed.

Setting a risk budgetA written statement of how much variability a portfolio may run, set before the decisions that will use it are taken. works the same way, and it goes wrong in the same two places. A budget that was not started from the current measured position is a number rather than a constraint, and a budget whose parts were added together has been reconciled by the wrong arithmetic. Everything below is the order, one step at a time, worked on the Anantara Multi-Asset Portfolio, an invented discretionary mandate of Rs 500 crore run for an invented charitable endowment, whose investment committee is chaired by Rukmini Deshpande and whose mandate is run by Faiz Ahmad Ansari.

What exactly is being budgeted, and how many budgets are there?

Step one names the quantity. Naming the quantity sounds like a formality, and it is the step most often skipped. Everybody in the room believes they already agree on what risk means. The people in the room usually do not.

There are two budgets here, not one. Total riskHow much the portfolio's own value moves, measured on the portfolio's return series on its own, without reference to any comparison. is how much the portfolio itself moves. Active riskHow much the difference between the portfolio and its comparison moves, measured on the series of differences rather than on the portfolio. is how much the difference between the portfolio and its benchmark moves. Total risk and active risk are measured on different series and they answer different questions. Both were settled in the guide on risk-adjusted ratios that opens this sequence. Step one names them and moves on.

A manager can sit comfortably inside one budget while running far outside the other, so a document that names only one has left the other unlimited by omission. Naming both budgets is the whole reason step one exists. The difference shows in a household saying it will spend no more than Rs 60,000/- a month and a household saying it will spend no more than Rs 60,000/- a month and no more than Rs 5,000/- of that differently from last month. The second sentence constrains something the first one does not touch.

Where does the budget start, if not from an aspiration?

Step two fixes the total risk position, and it takes it from the record rather than from what anybody would like it to be. For the Anantara Multi-Asset Portfolio the stated twelve month period shows a portfolio volatility of 11.8 per cent against the composite benchmark's 10.4 per cent, both measured over the same window, with the risk-free rate of 6.5 per cent stated beside every risk-adjusted figure on this platform.

One sentence about what that number is, and the measure itself is covered in the statistics and equities layers. Volatility squares the deviations, so a rise and an equal fall count exactly the same: it measures spread, not loss. A limit set without knowing the current position is a number rather than a constraint, so a budget starts from where the portfolio actually is.

The policy design carries its own volatility figure, and the two are worth holding apart. On the mandate's own stated assumptions the policy portfolio at 60, 30 and 10 has a variance of 125.3725 and therefore a volatility of 11.196986 per cent. The record prints that volatility as 11.20 per cent. The stated year delivered 11.8 per cent instead. One year is one draw and the assumptions were never a forecast, so neither number corrects the other, and saying so is the finding rather than a hedge.

How does the active position get stated, and what checks it?

Step three fixes the active risk position, separately, and then checks it a second way before anybody budgets against it. The record gives 3.7 per cent for the stated year, measured on the series of differences between the portfolio and the composite benchmark.

The second route uses three figures the record already carries: the portfolio volatility of 11.8 per cent, the benchmark volatility of 10.4 per cent and the beta against that benchmark of 1.08. Squaring the first two gives 139.24 and 108.16, and twice 1.08 times 108.16 is 233.6256. Adding the first two and subtracting the third gives 13.7744. The square root of 13.7744 is 3.7113878 per cent, rounding to 3.7114, and the record prints 3.7. The four risk figures are not four free inputs: given any three of them the fourth is determined, so a record where the measured figure and the identity disagree contains an error rather than two valid numbers.

Agreeing to two decimal places is not the same as agreeing, so the measured figure and the derived figure have to be looked at side by side. The record prints 3.7 and the identity returns 3.7114. The two figures are the same number reported to different precision, and both are kept below: the printed 3.7 wherever the record's own figure is being quoted, and the unrounded 3.7114 wherever something has to reconcile exactly. An earlier version of this record carried 3.2 per cent. No single sample can produce 3.2 alongside the other three figures, and the entry was corrected.

One thing this step is not doing. Step three is not splitting the year's gross excess return into where it came from. The record carries two separate decompositions of the same gross 1.6 points, one asking how much was market exposure and one asking whether allocation or selection produced it, and both sit in the monitoring sequence rather than here. Step three decomposes risk and not return, so no term from either return decomposition is borrowed anywhere below. Where the total risk of the portfolio is split by asset class, the figure comes from the allocation sequence's treatment of risk contribution, and the split was computed there.

What does an active risk limit convert into?

Step four is the step that turns a budget from a formality into a decision. Step four converts the limit into the return that limit could earn. Active returnThe difference between what the portfolio returned and what its comparison returned, over the same stated window. is the information ratioActive return divided by active risk, so it says how much departure from the benchmark was earned for each unit of departure taken. multiplied by the active risk, and the ratio itself was built in the guide on risk-adjusted ratios that opens this sequence. Step four only uses it.

Run it on the record. The gross excess return for the stated year was plus 1.6 percentage points, and the active risk was 3.7 per cent, so the ratio is 1.6 divided by 3.7. The division gives 0.432432, and the record prints 0.43. Now the rounding rule, and a reconciliation quietly fails at exactly this point. Rounded, 0.43 times 3.7 returns 1.591 rather than 1.6. Unrounded, 0.4324 times 3.7 returns 1.6 to the precision the record carries. A budget that fails to reproduce the record it was built from will not be trusted by anybody who checks it, so every conversion below uses the unrounded ratio and reports to two places only at the end.

What would a larger budget have to earn to be worth it?

Step five runs the conversion forward at limits the portfolio did not actually run. Running it forward is the only way to see what a bigger budget would buy. Hold the ratio at 0.4324 and read three limits off it. An active risk budget of 1.0 per cent implies 0.43 of a gross percentage point of active return. The recorded 3.7 per cent implies the gross 1.6 points. Double that to 7.4 per cent and the implied figure is 3.2 gross points.

Look at what moved and what did not. The implied return moved exactly in step with the limit. Multiplying by a fixed ratio does nothing else. The ratio itself sat still. The ratio is the only thing that would have to improve for a larger budget to be worth more for each unit of risk spent, and nothing about spending more risk improves it. A street vendor who doubles the stock on the cart doubles the money at stake and doubles the takings on a good day, and has done nothing whatever to the margin on each item sold. The margin on each item sold is worth arguing about.

Why do the separate decision budgets not add up?

Step six allocates the limit across the decisions that will use it, and this is where the arithmetic bites hardest. The instinct is to divide: a total of 3.7 per cent across two decisions must be 1.85 per cent each. The even split is correct exactly once, in the case nobody actually has. The property that defeats the instinct has a name, non-additivityThe property that two risks combined are not in general the sum of the two risks taken separately., and it is why this step needs arithmetic of its own rather than a division.

The correlation between decisionsHow closely two separate decisions inside a portfolio tend to move in the same direction at the same time, which governs how their risks combine. is what settles it, and the reason for it is covered under systematic risk. If two decisions move in perfect lockstep they do simply add, and 1.85 per cent each is right. If the two decisions are independent, the combined figure is the square root of the sum of the two squares. Each may then run 3.7 divided by the square root of two, or 2.6163 per cent. At a correlation of 0.50 the combination runs through the square root of three, so each may run 3.7 divided by 1.7320, or 2.1362 per cent.

Check the middle one rather than trusting it. Two decisions of 2.6163 per cent, independent of each other, combine as 2.6163 squared twice, giving 13.7744. The square root of 13.7744 is 3.7114 per cent. The 3.7114 is the budget, to the precision the identity carries. The same 3.7 per cent budget permits individual positions 41 per cent larger under independence than under lockstep, and a committee that allocated by adding has assumed the worst case without ever saying so out loud.

There is a mirror image of the same point, and it is the one that catches committees out. Take the permitted sizes and add them, the way a spreadsheet naturally would. In lockstep, two lots of 1.85 per cent add to exactly 3.70, the budget itself. At a correlation of 0.50 two lots of 2.1362 add to 4.2724. Under independence two lots of 2.6163 add to 5.2326. Simple addition matches the budget in one case only and overstates it everywhere else. An added-up allocation is therefore a reconciliation done with the wrong arithmetic rather than a conservative one.

What gets checked first, capacity or comfort?

Step seven runs the check in one particular order, and the order is the whole content of the step. Capacity, meaning what the situation can absorb, is computed first. Comfort, meaning the level the committee is willing to live with, is applied inside capacity and never around it. The ordering rule is settled under risk tolerance against risk capacity; this step only applies it.

Run it on the record and see what is actually there to check against. The Anantara mandate carries an equity band of 50 to 70 per cent. On a Rs 500 crore portfolio that band runs from Rs 2,50,00,00,000/- at the lower edge to Rs 3,50,00,00,000/- at the upper one, or Rs 250 crore to Rs 350 crore. The mandate also caps any single holding at 5 per cent of the portfolio, or Rs 25,00,00,000/-. The mandate bans unlisted holdings, and it states a minimum credit standing on the fixed income sleeve as a policy rather than as a rating symbol.

Now check the positions against the bands. Checking the positions is the arithmetic this step exists for. Equity sits at 60.0 per cent, or Rs 3,00,00,00,000/-, so it is inside the band with Rs 50 crore of room in each direction. The largest holding is 4.6 per cent of the portfolio, or Rs 23,00,00,000/-, so it is inside the cap with Rs 2 crore of room. A band forces its own edge, so a step whose output fell outside one would not be an aggressive answer but an inadmissible one. Every stated position on this record sits inside every written band. A step that proposed 45 per cent equity, for instance, would be proposing Rs 225 crore against a floor of Rs 250 crore, and the band would force the number back to 50.0 per cent with the Rs 25 crore difference named rather than absorbed.

The capacity check also needs to know where the total risk actually sits, and that split was computed in the allocation sequence's treatment of risk contribution rather than here. On the mandate's own stated assumptions the money splits 60.0, 30.0 and 10.0 per cent while the risk splits 95.6 per cent to equity, 4.4 per cent to fixed income and effectively nothing to cash. Capacity is therefore tested where the risk is rather than where the money is, and on this record those are two very different places.

What does a risk budget never do?

Step eight closes the sequence by writing down two things: what would trigger a review of the budget, and which claims a budget can never support. A review triggerA written condition which, if it occurs, requires the budget to be looked at again rather than left to stand by default. is a condition written in advance, and it makes reopening the budget a scheduled event rather than an argument.

Then the refusals, and they matter more than they look. A budget is a limit and not a target. Spending a limit in full is not an achievement, and whether it was worth spending is answered by the ratio and by nothing else. A household that spends its entire monthly allowance has not thereby done well; it has done exactly what the allowance permitted, and whether that was money well spent is a completely separate question about what it bought.

The last refusal names the limits the record never wrote down. The Anantara mandate has a written equity band and a written single holding cap. The mandate has no written active risk limit and no written total risk limit. The budget is therefore shown sitting inside the band the mandate does have, with the other two recorded as not supplied rather than filled in with a figure that would make the picture look complete. An invented limit would read exactly like a real one to anybody reading the record later, and that is why the gap is drawn rather than filled.

What has the sequence produced by the end?

Eight steps, eight figures, in one order. Written out as a table the sequence stops looking like a method and starts looking like a chain, where each row needs the row above it to have produced something. A step that leaves something unresolved has to say so rather than let the next step assume it was settled, so the last column matters most.