Risk Contribution: Where a Portfolio's Risk Actually Sits

Risk contribution is the share of total portfolio volatility that each holding actually supplies, and it is almost never the same as the share of the money. In the Anantara Multi-Asset Portfolio, equity holds 60 per cent of the money and supplies 95.6 per cent of the risk. A risk budget sets limits in those terms rather than in money terms.

A portfolio return is the weighted average of its parts. A portfolio volatility is not. So what exactly is each part supplying?

The Anantara Multi-Asset Portfolio, an invented discretionary mandate of Rs 5,00,00,00,000/-, is run by Faiz Ahmad Ansari for an invented charitable endowment whose committee Rukmini Deshpande chairs. Its policy weightsThe share each asset class is meant to carry, decided in advance by the holder. Actual weights drift away from it as prices move. are equity 60.0 per cent at Rs 3,00,00,00,000/-, fixed income 30.0 at Rs 1,50,00,00,000/- and cash 10.0 at Rs 50,00,00,000/-, summing to the whole exactly.

The mandate states its own risk assumptions, invented for teaching: equity volatility 18.0 per cent, fixed income 5.0, cash 0.5, a correlation of 0.20 between equity and fixed income, and cash uncorrelated with both. Each of those figures is an assumption the holder wrote down rather than a forecast, and a different set produces different risk contributions from the same portfolio.

If equity holds 60 per cent of the money, does it carry 60 per cent of the risk?

No, and the gap is not a rounding matter. Equity holds three fifths of the capital and supplies nineteen twentieths of the outcome, or 95.6 per cent; fixed income holds nearly a third of the money and supplies 4.4 per cent; cash holds a tenth and supplies two thousandths of one per cent. Almost every conversation about how a portfolio is positioned is conducted in money units while the thing the room cares about lives in risk units, and the two rows look nothing like each other.

The everyday version. A household takes in Rs 60,000/- a month from a salary, Rs 30,000/- from a small shop and Rs 10,000/- of interest on a deposit: three sources in money terms. Ask where the surprise comes from when the month's income lands higher or lower than expected. The deposit never surprises anybody, the salary once a year, the shop every month. The household holds three sources of income and runs one source of uncertainty, and the money split will never show that.

A portfolio does the same thing: on the Anantara assumptions the money row and the risk row disagree by 35.6 percentage points on the equity line.

Volatility does not add, so the volatility ranking gives the direction of the answer but not its size. The size takes arithmetic.

What is risk weighting, and how does it differ from money weighting?

Risk weighting is the set of shares of total portfolio risk, one per class, summing to one. Money weighting is the set of shares of total portfolio capital, one per class, also summing to one. The difference is hard to hold on to because both rows are three numbers adding to a hundred.

Money weightingThe share of the total capital sitting in each part of a portfolio. Almost every report prints that row. says where the capital is and risk weighting says where the outcome will come from, so a portfolio can be spread on one row while being concentrated on the other. Anantara is the ordinary case: three classes, a mainstream split, no leverage and no derivative anywhere.

The money row is an observation: read it off a custodian statement and two people will agree. The risk row is a computation, and two people using different volatility and correlation assumptions get different risk rows from the identical portfolio. The assumptions therefore have to be written down beside the answer, and a risk row printed without them is not a measurement.

What is risk contribution, and how is it actually computed?

Risk contribution comes in two forms and confusing them is the standard error on this subject, so both are built here.

The portfolio variance is built from a covarianceA number describing how two things move together: the correlation between them multiplied by both of their standard deviations. grid, three by three for three classes, every entry of it from the stated assumptions. The diagonal holds the variances: 18.0 squared is 324.0, 5.0 squared is 25.0, 0.5 squared is 0.25. The off-diagonal entry between equity and fixed income is the correlation times both volatilities, 0.20 times 18.0 times 5.0, or 18.0, and cash is assumed uncorrelated with both, so its entries are zero.

From that grid take one intermediate quantity per class, the row product: for each class, multiply every entry in that class's row by the weight of the class in that column, and add.

Run the same three lines for the other two classes. Fixed income: 0.60 times 18.0 is 10.80, plus 0.30 times 25.0 is 7.50, giving 18.30. Cash: everything is zero except 0.10 times 0.25, giving 0.025.

One step more before the two forms. Multiply each row product by its own weight and add the three: 0.60 times 199.80 is 119.880, plus 0.30 times 18.30 is 5.490, plus 0.10 times 0.025 is 0.0025. The three products sum to 125.3725, the portfolio variance in squared percentage points. The square root of 125.3725 is 11.1970 per cent, and the record carries that as 11.20. The volatility of the whole is not the weighted average of the parts, and the 1.15 point difference between them is the diversification. The difference exists only because the correlation is 0.20 rather than 1.00.

The correlation does two jobs at once and they pull in different directions. At 0.00 the volatility falls to 10.9038 per cent while equity's risk share rises to 98.1; at 1.00 the volatility rises to 12.3001 while the risk share falls to 87.8. A higher correlation makes the portfolio move more in total while making the risk look better spread. Never read a risk share without the volatility beside it.

The marginal form

Marginal contribution to riskA rate of change, not a quantity currently held: how much the portfolio's volatility moves for a very small increase in one weight. is how much the portfolio volatility changes for a small increase in one weight, and it is that class's row product divided by the portfolio volatility. Equity: 199.80 divided by 11.1970 is 17.8441. The same division gives fixed income 1.6344 and cash 0.0022.

Set those three beside the standalone volatilities they came from. Equity keeps 99.1 per cent of its own 18.0 at the margin. Fixed income keeps only 32.7 per cent of its own 5.0: at a correlation of 0.20 most of what fixed income does is cancel against equity. A marginal contribution is not a class's own volatility, it is that volatility after the rest of the portfolio has had its say, and the discount is largest for the classes doing the diversifying.

So 17.8441 is a rate: at these weights a small shift of money into equity raises the portfolio volatility at about 17.84 points per unit of weight. Watch that work once on a real move.

Take one percentage point of money out of fixed income and put it into equity, so the weights become 61, 29 and 10. The marginal figures predict a rise of 0.01 times the difference between 17.8441 and 1.6344, or 0.1621 points. Recomputed from scratch the volatility is 11.3593 per cent against 11.1970, a rise of 0.1623. The match is what a marginal figure is for, and also where it stops: a marginal figure describes a small step, and the further the step runs the more the prediction drifts.

The total form

Total contribution to riskThe amount of the portfolio's volatility one part currently supplies, being its marginal contribution multiplied by its own weight. is the marginal figure multiplied by the weight itself. Equity: 17.8441 times 0.60 is 10.7065. Fixed income: 1.6344 times 0.30 is 0.4903. Cash: 0.0022 times 0.10 is 0.0002. Total contributions are quantities, measured in the same percentage points as the portfolio volatility.

The marginal form answers what happens if I add a little, and the total form answers how much of what I already have comes from here, and reaching for one where the other is meant is the most common error on this subject. A marginal figure does not sum to anything meaningful. A total contribution is a share of a whole.

Why do the total contributions add up to the portfolio volatility exactly?

Add them. 10.7065 plus 0.4903 plus 0.0002 is 11.1970, exactly the portfolio volatility. AdditivityThe property that a set of parts sums exactly to the whole, which is what lets a quantity be divided up and handed out in shares. is the name of that property, and it is not a convention anybody agreed on. The total contributions were built as each weight times its own row product divided by the volatility. The sum of the weights times their row products is the variance, and the variance divided by the volatility is the volatility, so additivity falls straight out of the arithmetic.

A quantity that does not add up cannot be allocated, and additivity is therefore the only reason risk can be budgeted at all. A household budget works because rent, food, school fees and savings add to the month's income; if they summed differently depending on the order in which they were written down, no share could be handed to anybody.

Volatility itself does not add: 18.0 and 5.0 in no combination produce 11.1970 without the covariance machinery. Total contributions do add. Risk budgeting is possible because volatility can be sliced into shares that sum to it, and every risk budget rests on that slice.

What do the two rows look like once they are turned into shares?

Dividing each total contribution by the portfolio volatility gives the risk weights. Equity: 10.7065 divided by 11.1970 is 95.6 per cent. The same division gives fixed income 4.4 per cent and cash 0.002. The totals summed to the volatility, so the three risk weights sum to 100 per cent.

Read the cash line of that table against its money line. On these assumptions cash is not managing risk, it is holding value still, and those are two different jobs that get described with the same word.

Say the same thing in money. One standard deviation of the portfolio, at 11.1970 per cent of Rs 5,00,00,00,000/-, is Rs 55,98,49,310/-. Split by the total contributions, equity supplies Rs 53,53,22,621/- of it, fixed income Rs 2,45,15,525/-, and cash Rs 11,164/-. Additivity holds in rupees just as it holds in percentage points, so the three add back to Rs 55,98,49,310/- exactly.

Fifty crore of capital, eleven thousand rupees of risk. Anybody calling that a risk management decision has to say which risk. The cash may be doing something else worth doing, such as meeting a spending commitment without selling anything, and the honest sentence names that job rather than borrowing the language of risk.

How stubborn is that risk share when the money moves?

A committee asks that next. If equity's risk share is uncomfortably high at a 60 per cent money weight, move some money out: how much does the risk share fall?

Very little, and this is the part people find hardest to believe. Take equity from 60 down to 50 per cent of the money, the bottom of the mandate band, with the released money going into fixed income and cash pinned at 10 per cent. Recompute from scratch: the row products become 169.20, 19.00 and 0.025, the variance 92.2025, the volatility 9.6022 per cent, and equity's risk share falls from 95.6 to 91.8 per cent. A sixth of the equity money moved and the risk share moved by 3.8 percentage points.

Push harder. At 30 per cent equity, half the policy weight, with fixed income at 60 and cash still at 10, the volatility falls to 6.6815 per cent and equity's risk share is still 72.6 per cent.

Across the permitted band, from 50 to 70 per cent, the risk line is nearly flat: it moves 6.1 points while the money line moves 20. Inside the mandate the manager has a lever that changes the money a great deal and the risk share hardly at all.

The curve has an exchange rate in it, and it halves at every step. Ten more points of equity money weight bought from 30 buys 12.33 points of risk share; from 40 it buys 6.84, from 50 it buys 3.86, from 60 it buys 2.24. The closer a portfolio already sits to having its risk all in one place, the less the money lever does.

The total risk does change: the portfolio volatility falls from 12.8375 per cent at a 70 per cent equity weight to 9.6022 at 50. The band exists for exactly that fall. The source of the risk barely changes. The volume can be turned down; which instrument is playing cannot be changed.

How risk budgeting works: which of the five steps is the awkward one?

A risk budget is a set of intended shares of total risk, written down in advance, against which the actual shares are checked. Running one is five ordered steps, and the difficulty lives in the process rather than in the definition.

Steps one, two and five cost nothing. Step three is the arithmetic run in reverse: the intended risk share is known and the money weight that produces it is wanted, and because the relationship is not linear it is solved numerically rather than by dividing something.

Step four is where budgets die. A risk budget is an internal decision and a mandate is a commitment made to the holder, so when the money weights implied by the budget fall outside the mandate the mandate is what binds and the budget has to be rewritten.

Suppose Rukmini Deshpande's committee decided equity should supply half the risk rather than 95.6 per cent. Solve step three for that share, cash still pinned at 10 per cent, and the equity money weight that produces it is about 19.6 per cent. The mandate requires equity between 50 and 70 per cent, so the budget is asking for a money weight thirty points below the floor, and no amount of goodwill closes that gap.

So the committee writes down the share it can reach, around 91.8 per cent at the floor, records that the share it wanted is not attainable, and states why. A year later, when somebody asks why the portfolio is so equity-driven, the answer is already written down with its arithmetic.

What does a risk budget rule out that a money budget does not?

A risk budget rules out calling a portfolio balanced because the money is spread. Ruling that sentence out is a reporting job before it is a construction job.

Once the risk row is printed next to the money row, a whole class of comfortable sentences becomes impossible to write. A shift from 60 to 55 per cent equity is not a meaningful de-risking when the risk share moves from 95.6 to 94.0 per cent. The budget's value is that it removes the vocabulary, not that it changes the portfolio.

The risk row works in the other direction too. A class held at 3 per cent of the money at four times the volatility of everything else is a rounding line on the money row and a number that gets discussed on the risk row, and a money budget can never surface that.

The same gap sits in a household asset list. A deposit, a small equity plan, a plot of land, gold in a locker: nothing in that list says which of the four moves the value of the whole from year to year.

Risk budget vs tracking error: are these the same limit?

No, and collapsing them is the most common confusion in this area. The two limits are stated in different currencies.

A risk budget divides total risk among the classes. Total risk is the dispersion of the portfolio's own returns, measured against nothing but itself, and for the Anantara portfolio over the stated twelve month period the realised volatility was 11.8 per cent.

A tracking errorThe standard deviation of the difference between a portfolio's return and its benchmark's return over a stated period. The figure says how far the two drift apart, not how much either moves. limit constrains something else: active riskThe risk that comes from being positioned differently to a benchmark rather than from the market itself. Holding the benchmark exactly leaves none of it., the dispersion of the difference between the portfolio and its benchmark. For the same stated twelve month period the Anantara portfolio's tracking error against its composite benchmark was 3.7 per cent.

A portfolio holding its benchmark exactly, position for position, would have a tracking error of zero while carrying the benchmark's whole volatility; a low-volatility set of positions very different from the benchmark runs the reverse. Neither figure can be read off the other. A report quoting one of them has said nothing about the other.

With two volatilities and a beta known, is the tracking error still free?

No, and this is where the two currencies do connect. Given a portfolio volatility, a benchmark volatility and a betaHow much a portfolio has tended to move for a given move in its benchmark over a stated period. A beta of 1.08 means it moved about 1.08 times as much. for the same period, the tracking error is determined rather than chosen: it is the square root of the portfolio variance plus the benchmark variance less twice the beta times the benchmark variance.

Run it on the stated twelve month period. The portfolio volatility was 11.8 per cent, so its variance is 139.24. The benchmark volatility was 10.4 per cent, so its variance is 108.16. The beta was 1.08. Twice the beta times the benchmark variance is 2.16 times 108.16, or 233.6256. Then 139.24 plus 108.16 is 247.40, less 233.6256 leaves 13.7744. The square root of 13.7744 is 3.7114 per cent, and the record carries it rounded as 3.7.

Only three of those four figures can be chosen independently, so a report quoting a portfolio volatility, a benchmark volatility, a beta and a tracking error for one period has quoted one figure twice. That makes the identity a working check on any report printing all four.

One rounding note. The record also carries an information ratio for the period: 1.6 percentage points of gross excess return divided by the tracking error. Dividing by the printed 3.7 gives 0.4324; dividing by the unrounded 3.7114 gives 0.4311. Both round to the 0.43 the record carries. Show the division in full and name the denominator. A reader who recomputes will otherwise think one of the two figures is wrong.

How does anybody use this in a room, on a Tuesday?

Three habits, none of them needing software. The first: every statement about how a portfolio is positioned gets made in both rows, printed side by side, so nobody has to convert in their head and nobody can quietly use whichever row supports the sentence they wanted.

The second: the assumption set travels with the risk row wherever it goes. Anybody wanting to argue with the 95.6 per cent has to argue about 18.0, 5.0, 0.5 and 0.20. Arguing about four numbers beats arguing about whether the portfolio feels balanced.

The third is a reversal test. Ask what the risk row would look like if the equity assumption fell from 18.0 to 12.0 per cent. A conclusion that survives a wide range of the input is informative. A conclusion that flips is a fact about the assumption rather than the portfolio, and a committee is better off knowing that before it acts.

An analyst at a lender runs the same steps for a different reason: they want to know how far the value of the collateral can move, and the money split will not tell them. An endowment secretary preparing spending plans wants to know which single assumption, if wrong, would move the whole picture. In both settings the question is where the movement will come from, and the money row does not answer it.

At the mandate floor the money outside equity is 50 per cent and supplies 8.2 per cent of the risk, so ten more points of money outside equity bought 3.8 points of risk share. The exchange rate between the two rows is a poor one.