Risk Parity: Equalising Risk Contribution Across Assets

Risk parity chooses weights so that every class contributes the same share of the portfolio's volatility. Risk parity needs volatilities and correlations and no expected returns at all. The missing returns are the whole of what it buys and the whole of what it costs. On the Anantara Multi-Asset Portfolio's own stated assumptions the equal risk contribution weights come out at 2.27 per cent equity, 8.18 per cent fixed income and 89.55 per cent cash.

The Anantara Multi-Asset Portfolio holds 60 per cent of its money in equity, and equity supplies 95.6 per cent of the portfolio's risk. In the terms that produce the outcome, a three class portfolio is a one class portfolio with two decorations. Risk parityA rule for choosing weights so that each holding or class supplies an identical share of the portfolio's total volatility. The rule uses volatilities and correlations only. takes that observation literally and asks what weights would make the three shares identical.

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 committee is chaired by Rukmini Deshpande. Its policy mix is equity 60.0 per cent at Rs 300 crore, fixed income 30.0 per cent at Rs 150 crore and cash 10.0 per cent at Rs 50 crore, and equity must sit between 50 and 70 per cent.

Every number here is computed from one short assumption set. The assumptions are the holder's own, not forecasts, market expectations or anybody's published estimates. A different set produces a different portfolio, and every figure computed from them would move.

The three classes do not move together, and the size of that effect can be measured rather than asserted.

The 95.6, 4.4 and 0.002 shares come from risk contribution and the risk budget, and the mean-variance problem is settled under portfolio optimisation.

What is risk parity actually solving for?

Find the set of weights at which each class's total contribution to the portfolio's volatility is the same number. Not the same money. Not the same expected return. The same contribution to risk. With three classes each supplies a third of the volatility. A portfolio built that way is also called an equal risk contributionThe condition that every class in a portfolio supplies the same amount of total risk. With three classes it means each supplies exactly one third. portfolio.

A class's total risk contribution is its weight multiplied by the covariance between that class and the whole portfolio, divided by the portfolio's volatility, and the three contributions add up to the portfolio volatility exactly. The identity was established under risk contribution and the risk budget. Risk parity turns it around: instead of computing the three contributions from a set of weights somebody chose, it searches for the weights that make the three contributions equal.

Run that search on this assumption set and it converges on a portfolio whose volatility is 0.7756 per cent, with each class supplying 0.2585 volatility points. Three lots of 0.2585 is 0.7756, and that sum is the check that the answer is an answer rather than something close to one.

Nothing in the objective refers to what anything is expected to earn. Risk parity asks a question with no return in it at all. The answer it gives belongs to a different question, not to a better version of the one mean-variance construction asks.

Written out on the mix the portfolio actually holds, the identity looks like this.

Why does equalising the contributions look attractive here?

Because the starting point is genuinely lopsided, and the lopsidedness is invisible in the document a committee usually reads.

A household puts a third of its savings into a deposit, a third into a monthly equity plan and a third into the flat it lives in. Then the local employer that pays the salary, sets the rents and holds up flat prices has a difficult year, and all three move together. The money was split three ways; the exposure never was.

A sound motivation is one thing. Whether the method produces a portfolio this mandate could actually hold is a separate question, and the gap between the two is wide.

Risk Parity vs Equal Weighting: does equalising the money equalise the risk?

Equal weightingA rule that gives every holding or class the same share of the money, without reference to its size, its volatility or anything else. gives every class the same share of the money and stops. Risk parity gives every class the same share of the risk and lets the money land wherever that requires. Equal weighting is set against the other money based rules under weighting schemes.

Run equal thirds on this assumption set: 33.33 per cent of Rs 500 crore in each class, Rs 1,66,66,66,667/- apiece to the nearest rupee, with the three contributions recomputed from scratch.

The risk shareThe percentage of a portfolio's total volatility that one class supplies. The three shares add to one hundred per cent by construction. figures come out at 88.77, 11.16 and 0.06 per cent. A third of the portfolio changed hands, and equity's share of the risk fell from 95.6 to 88.8 per cent, under seven points.

Equal weighting is keyed to a count: three classes, therefore a third each, with no look at how variable any of the three is. When one carries thirty six times the volatility of another, a count is the wrong key. Equal weighting needs no assumption set at all; risk parity needs most of one.

Mean-Variance Optimization vs Risk Parity: what separates the two?

Mean-variance constructionChoosing weights by solving for the highest expected return at a stated volatility, or the lowest volatility at a stated expected return, from expected returns, volatilities and correlations. is the frontier machinery Harry Markowitz set out in Portfolio Selection in 1952, and what it is handed and what it returns are worked through under portfolio optimisation. On inputs alone, the difference between the two is a matter of counting.

Nine numbers against six, and the three that vanish are the three expected returns. Portfolio optimisation and resampled efficiency both found expected returns to be the worst estimated inputs in the problem, with a mean-variance answer moving a long way when one of them moves a little. Risk parity does not improve those estimates. The method refuses to use them.

Change equity's expected return assumption from 12.0 to 13.0 per cent, a full point, and re-solve for equal risk contribution.

No weight was ever a function of a return, so every weight is identical to four decimal places. Only the expected return of the resulting portfolio moves. The move is 0.0227 points, or 1.00 point multiplied by equity's 2.2709 per cent weight. Removing the return inputs removes the estimation problem attached to them, and with it every statement of the portfolio's expected return.

The portfolio still has an expected return. The figure has to be computed afterwards, from outside the method.

What weights does equal risk contribution produce on these assumptions?

The condition is that each class's weight multiplied by its covariance with the whole portfolio comes out the same for all three, and exactly one set of positive weights satisfies it: 2.2709 per cent equity, 8.1751 per cent fixed income and 89.5540 per cent cash.

Each class's contribution works out at 0.200498 in variance units, three of which is 0.601494, and the square root of that is 0.7756 per cent, the portfolio volatility. Each class therefore supplies 33.3333 per cent of the risk, which is what was asked for. In money that is Rs 11,35,43,402/- in equity, Rs 40,87,56,248/- in fixed income and Rs 4,47,77,00,350/- in cash, summing to Rs 500 crore to the rupee.

Reaching equal risk contribution from the policy mix means selling roughly nineteen twentieths of the equity holding and three quarters of the fixed income holding, and putting almost the whole portfolio into the class with 0.5 per cent volatility. The result is not a tilt but a different portfolio wearing the same mandate's name.

Why does nearly all of the money land in the least volatile class?

Because the objective forces the arithmetic there. To make a class with 0.5 per cent volatility contribute as much risk as one with 18.0 per cent volatility, it has to carry enough weight to close a gap of thirty six times. The weights therefore move roughly the other way round from the volatilities, and roughly the other way round from 60 per cent is a very small number.

A tea seller and a steel wholesaler put money into a common pot and want the pot's month to month swings to come equally from each of them. The steel business swings thirty six times as hard, so the tea seller has to put in almost all the money. The equality of swing forced the sizes; nobody decided the tea business was better.

Multiplying each weight by its own volatility at the answer shows the mechanism directly.

The first two products are identical to six decimal places, at 0.408756, and cash differs at 0.447770. With every correlation zero, equal risk contribution would collapse into inverse volatilityA weighting rule that sets each weight proportional to one divided by that class's volatility, ignoring correlations entirely. weighting. Equity and fixed income are correlated at 0.20 while cash is taken as uncorrelated, so equal risk contribution does not collapse into inverse volatility weighting here.

Inverse volatility weighting lands close to the answer here and does not reach it: its risk shares come out at 35.29, 35.29 and 29.41 per cent rather than 33.33 each, so a portfolio built that way and described as risk parity is describing itself as something it is not. The gap is small here only because one of the three correlations is non-zero.

Across exactly two assets, equal risk contribution is inverse volatility weighting exactly, and the correlation between them drops out of the algebra altogether. The two asset case is worked below.

What does the equal risk contribution portfolio expect to earn?

6.2589 per cent, carried in the register as 6.26: 2.2709 per cent of 12.0, plus 8.1751 per cent of 7.5, plus 89.5540 per cent of 6.0. Almost the whole of the portfolio is cash, so almost the whole of that number is the cash assumption.

The policy mix expects 10.05 per cent on the same assumptions. Against it the equal risk contribution portfolio gives up 3.79 percentage points of expected return, and against the cash assumption of 6.0 per cent it adds 0.26 points. Risk parity did not fail here: it did precisely what it was asked, and what it was asked never included earning anything.

The instinct on seeing 6.26 per cent is that the method is broken. It is not. A rule that equalises risk contribution and refuses to look at returns puts the money where the volatility is lowest, and on most assumption sets a holder would write down that is also where the expected return is lowest.

Placed on expected return against volatility, the equal risk contribution mix sits alone in the bottom left corner. The position is a statement about where the mix sits, not about whether it suits anybody.

Each destination here removes some of equity's risk share and charges for it in expected return, and dividing one by the other prices them against each other.

Can this mandate hold the equal risk contribution portfolio?

No, and the test takes one comparison. The mandate requires equity between 50 and 70 per cent; equal risk contribution puts it at 2.2709 per cent. AdmissibilityWhether a proposed set of weights is permitted by the constraints the mandate already carries. A mix that breaks a stated limit is inadmissible however good its properties are. is a check against limits written down before this arithmetic was run.

The miss is 47.73 percentage points of equity weight. The miss is not a criticism of risk parity. A method chosen for one property was run inside constraints written for another, and the constraints were written first.

What does the mandate band force instead?

If the answer is out of reach, how close can the mandate legally get? Pin equity at 50 per cent, the floor, and hunt across every permitted split of the remaining 50 per cent for the mix leaving the largest risk share as small as possible. The result is the least unequal portfolio the band allows.

The mix is 50 per cent equity, 50 per cent fixed income and no cash at all, and it leaves equity supplying 88.83 per cent of the risk, at an expected return of 9.75 per cent and a volatility of 9.8107 per cent. Pinning cash at its policy 10 per cent instead, giving 50, 40 and 10, leaves equity at 91.75 per cent.

The record states an equity band, a single holding cap, a listing rule and a minimum credit standing on the fixed income sleeve. Nothing in the record sets a minimum cash holding, so whether the 50, 50 and nothing mix is available is NOT SUPPLIED, and both versions are computed. Either way, inside the band equity's risk share cannot be pushed below about 88.8 per cent, so the band rules out getting anywhere near the answer.

Drawn across the whole range of equity weights, the relationship has a shape worth reading carefully.

The curve is steep exactly where the mandate forbids the portfolio to stand and flat everywhere it is allowed to stand. Across the permitted band the money weight moves a full twenty points and the equity risk share runs from 88.83 to 96.42 per cent, under eight. Equity stops carrying more than a third of the risk only at an equity weight of 15.47 per cent, less than a third of the mandate's floor.

A weighting rule decides how much money sits in each class and stops there.

What if cash is dropped and the two risky classes are equalised?

The two class version is the one usually presented, and it is cleaner because the awkward class is gone. Equalise risk contribution between equity and fixed income only, with no cash at all. With exactly two assets the correlation cancels: the weights go inversely with the volatilities, so equity takes 5 parts and fixed income 18, or 21.7391 and 78.2609 per cent.

Equity's risk share is 50.00 per cent and fixed income's is 50.00 per cent, exactly, at a volatility of 6.0621 per cent and an expected return of 8.4783 per cent, giving up 1.57 points against the policy mix rather than 3.79. And it is still inadmissible: 21.74 per cent equity sits 28.26 percentage points below the mandate's 50 per cent floor.

Where does the risk share actually start to move?

Almost nowhere, until the very end. Draw a straight line in weight space from the policy mix to the equal risk contribution mix and walk along it, calling the position along that line the blendA setting between nought and one that mixes two sets of weights: nought gives the first set, one gives the second, and a half gives the average of the two. setting, nought at the policy mix and one at the answer. Risk shares do not blend. Recompute them at every stop rather than interpolating.

Weights blend because a straight line between two sets of weights is a straight line by construction. Anything computed from those weights does not, and averaging the two ends gets the answer badly wrong.

At nought the weights are 60.0, 30.0 and 10.0 per cent and the risk shares are 95.6, 4.4 and 0.002 per cent. Three quarters of the way the weights are 16.7, 13.6 and 69.7 per cent, a portfolio no longer recognisable as the policy mix, and equity still supplies 90.5 per cent of the risk. Only at the far end do the shares become 33.3 each.

Three quarters of the journey has moved 43.3 percentage points of equity weight and bought 5.15 points of risk share; the last quarter moves 14.4 points of weight and buys 57.13 points of share. The curve does nothing for most of its length and then falls off a shelf.

An exchange rate hides in that curve. Early on, a percentage point of equity weight buys about 0.12 points of risk share; in the last stretch it buys about 3.96 points. A proposal that moves toward risk parity but stops short of the far end is buying at the first price and describing the second.

The finding here is not the one reached under risk contribution and the risk budget. There, equity's money weight moved along the mandate band with cash pinned at 10 per cent, and the risk share fell only to 91.8 per cent at the floor. Moving toward cash instead finds the share equally unresponsive. Two levers, the same stubbornness, and it belongs to the arithmetic rather than to either path.

Reaching the equal risk contribution mix in one step would move Rs 3,97,77,00,350/- into cash, 79.6 per cent of the portfolio in one way trade value, against turnover of 34 per cent in the stated twelve month period. Getting to the answer would move more than twice as much money as the whole of that year moved, and turnover carries a cost no return figure here shows.

What would have to change for a risk parity portfolio to meet a return objective?

Three things could change. None of them is a weighting decision, and none of them is put forward.

The first is that the objective drops the return requirement: if the mandate wants risk contributions equalised and does not want a return target, 6.26 per cent is the answer rather than a failure. The second is that the equity band is rewritten; the 50 per cent floor is the holder's own choice rather than a rule imposed from outside, so it can be moved, but that is a change to the mandate rather than to the portfolio.

The third route is the one people reach for. A risk parity portfolio can be scaled up with borrowed money, raising the risk of the low volatility classes until the whole portfolio carries the risk the holder wanted. Whether this mandate permits borrowing at all is NOT SUPPLIED by the record, and borrowing changes the portfolio itself rather than only its weights. It introduces a lender, a cost of borrowing and a path to losses no unlevered mix has, and it is covered separately.

How does a committee actually use any of this on a Tuesday?

Not by adopting risk parity, and not by rejecting it: by using its arithmetic as a measuring instrument on whatever is already in front of them. The number risk parity is built around costs two lines to produce, and wanting it is not the same as wanting the portfolio.

Any paper proposing a change in weights arrives with the weights before and after. The committee asks for two more rows: the risk shares before and after, computed from the volatilities and the one correlation the holder has already written down. Neither a forecast nor a view is needed for them. If the first number did not move, the paper has not proposed a change in risk.

A lender asks a narrower question. If this holder had to raise cash in a difficult month, which class is driving the value, and is that the class that will be hardest to sell? On these assumptions the answer is equity on both counts, and the money weight of 60 per cent understates it. An analyst asks a third version: at a 95.6 per cent risk share, a year's swing cannot be attributed to anything other than equity without a specific argument.

The household version is the same instrument at a different scale. Write down each savings pot, beside it how much that pot swings in a bad year, and multiply. The pot with the biggest product is where the outcome comes from, and it is rarely the pot with the most money in it. The instrument is worth more than the method: a holder can decline to build a risk parity portfolio and still use its question whenever somebody says a portfolio is balanced.

What is risk parity not?

Risk parity is not a better optimiser: it solves a different problem and returns an answer a mean-variance problem never would. Nor is it inverse volatility weighting. The two coincide across exactly two assets and sit close together where correlations are mostly zero, and nowhere else. Nor is it equal weighting with better manners; equal weighting is keyed to a count and risk parity to a covariance.

Risk parity is also not a claim about performance. How a risk parity portfolio has actually done over some period is a question about a record, and no set of weights computed from assumptions can answer it. The claim is arithmetical and checkable: on these stated assumptions, equal risk contribution needs 2.27 per cent equity, expects 6.26 per cent, and sits far outside a band the holder wrote before any of this was computed.

And equal risk contribution is not a mix put forward for anybody. The three candidate portfolios here show what each construction rule produces and what each one costs. Which of them, if any, belongs in a portfolio depends on a specific holder's purpose, obligations and constraints.