Case 041Portfolio construction and optimisationHard

Formulate and solve a three-asset mean-variance problem with a budget constraint, no shorting and a 50% position limit, with and without the limit, and explain where the capped weight goes.

BlackRockNew York · 2025

The situation

Aranyak Multi-Asset Fund, an Indian scheme, chooses weights in three assets. Equities are expected to return 12% with 18% volatility, corporate bonds 8% with 5%, and gold 9% with 15%. The equity-bond correlation is 0.2, equity-gold 0 and bond-gold 0.1.

The committee uses a mean-variance objective with a risk aversion of 2, written in the usual form: expected return minus half of 2 times variance, which is expected return minus variance. Weights must sum to one, nothing may be shorted, and the risk policy says no asset may exceed 50% of the fund.

Your task

Write the problem down, solve it without and with the 50% cap, and explain where the weight removed by the cap ends up and why.

Quick check

The cap cuts equities by about 16 points. Where does most of that weight go?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Without the cap the fund holds about 66% equities, 5% bonds and 29% gold; with it, 50%, 21% and 29%. Almost every point cut from equities goes to bonds, because bonds add the least variance per point of weight and cutting equities lowers their risk further. Spreading the freed weight pro rata would cost about 5 basis points of certainty-equivalent return against the true capped optimum.

Step 1How do you write the problem down before solving it?

Think of packing for a flight. You want the most useful things, each item has a weight, the bag has a limit, you cannot pack a negative shirt, and the airline caps any one bag. A portfolio problem has the same four parts: an objective, a budget, a sign restriction and position limits. The objective rewards expected return and charges for variance at a rate set by the risk aversionA number that sets how much expected return the investor gives up to remove one unit of variance. Higher means more cautious. coefficient. Interviewers asking you to formulate the problem want all four spoken before any number.

The relationship
max⁡w  w⊤μ−λ2 w⊤Σws.t.∑iwi=1,    wi≥0,    wi≤0.5\max_{w}\; w^{\top}\mu - \frac{\lambda}{2}\, w^{\top}\Sigma w \quad \text{s.t.} \quad \sum_i w_i = 1,\;\; w_i \ge 0,\;\; w_i \le 0.5
wthe three weights
muexpected returns: 12%, 8%, 9%
Sigmathe covariance matrix built from the volatilities and correlations
lambdarisk aversion, 2 here
What it says in wordsChoose the weights that give the best return after a charge for variance, spending exactly the whole fund, shorting nothing and putting no more than half in any asset.
CovarianceEquitiesBondsGold
Equities0.032400.001800.00000
Corporate bonds0.001800.002500.00075
Gold0.000000.000750.02250
Variances sit on the diagonal: equities 0.0324, bonds 0.0025 and gold 0.0225, so a point of gold weight carries nine times the own-variance of a point of bonds, while equities and gold share no covariance at all.
Step 2What does the optimiser hold without the cap?

At the optimum every asset held must earn the same marginal utility: its expected return less the risk it adds at the margin, lambda times its covariance with the whole portfolio. If one asset's marginal utility were higher, moving a point into it would help. Solving that condition gives 66.2% equities, 5.0% bonds and 28.9% gold, an expected return of 10.94% with 12.73% volatility, and all three marginal utilities equal at 7.69%. Gold earns its place as the diversifier because it has no correlation with equities; bonds barely make it in, because at this risk aversion their 8% return looks thin next to equities.

Where the 16 points cut from equities go: almost all to bonds10%20%30%40%60%70%0%EquitiesCorporate bondsGold66.25.028.9Without the cap50.021.228.8With the 50% cap50.07.342.7Cap, refilled pro ratacap 50%Weights in per cent; each group sums to 100
Without the cap the fund holds 66.2% equities; the 50% cap moves the 16.2 freed points almost entirely into bonds, which rise to 21.2% while gold stays near 28.8%, whereas a pro rata refill would wrongly push gold to 42.7%.
Step 3Where does the capped weight go, and why not pro rata?

Fix equities at 50% and re-solve for the other two. Bonds take 16.2 points and gold changes by -0.03, so bonds absorb more than all of the freed weight. Two forces do it. First, a point of bonds adds one ninth of the own-variance of a point of gold, so bonds are the cheaper place to put weight. Second, cutting equities lowers the risk each bond adds, because the two are correlated at 0.2, while gold's marginal risk is untouched, because its correlation with equities is zero. Bonds become more attractive and gold does not, so the next-best diversifier takes the weight.

Pro rata looks fair but is not an optimum. It would put 7.3% in bonds and 42.7% in gold, giving 11.13% volatility against 10.27% for the true capped answer, and a certainty-equivalent return of 9.19% against 9.23%. The cap itself costs 8 basis points of certainty-equivalent return; refilling it pro rata throws away another 5.

At the capped optimum, bonds and gold tie; equities sit above the lineEquities, held at the cap8.68%Corporate bonds, free7.67%Gold, free7.67%common level 7.67%shadow price 1.01 ptsExpected return minus risk cost of the next point of weight, per cent a year
At the capped optimum bonds and gold have the same marginal utility of 7.67%, while equities sit 1.01 points higher, which is the shadow price telling the committee what the 50% limit costs at the margin.
SolutionEquitiesBondsGoldReturnVolatilityCertainty equivalent
No cap66.2%5.0%28.9%10.94%12.73%9.31%
Cap, solved50.0%21.2%28.8%10.29%10.27%9.23%
Cap, pro rata50.0%7.3%42.7%10.43%11.13%9.19%
The capped optimum gives up 8 basis points of certainty-equivalent return against no cap, and the pro rata refill gives up 5 more while carrying higher volatility than the properly solved capped portfolio.
Step 4What would you tell the committee about the cap?

That it is cheap insurance against bad inputs. If the equity return estimate were 11% instead of 12%, the uncapped equity weight would fall from 66.2% to 50.2%: one point of forecast error moves the answer as far as the cap does. Optimisers magnify errors in expected returns, so position limits are how a committee stops a confident forecast from becoming a concentrated fund. Two further constraints belong in the real formulation: a turnover limit, and the category rules for Indian multi-asset funds, which set a minimum holding in each asset class. Confirm the current minimum; if it is above the 5.0% the uncapped answer puts in bonds, that lower bound binds too.

Where candidates lose it

The common loss is assuming the freed weight is spread in proportion to the existing holdings. That is what a spreadsheet does when you clip a weight and rescale, and it gives a portfolio that is not optimal: gold balloons and volatility rises.

The second is writing the objective as return minus 2 times variance when the committee meant a risk aversion of 2 in the usual half-lambda form. The two differ by a factor of two in risk aversion; with the larger charge the cap never binds, so state your convention before solving.

What the interviewer asks next

  • What changes if the equity-gold correlation is 0.3 instead of zero?
  • Add a 10% minimum in each asset. Which constraint binds now?
  • How would you build a turnover penalty into the objective?
  • Why do practitioners shrink expected returns before optimising?

Asked at BlackRock, Quantitative Research, New York, 2025 (Wall Street Oasis): asked how I would formulate a portfolio optimization problem, including the objective function and practical constraints

Company names and figures are illustrative.