Risk Management puzzles, solved step by step
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040A portfolio holds Rs 60 crore of asset A with 20% volatility and Rs 40 crore of asset B with 30% volatility, and their correlation is 0.5. Compute the portfolio's volatility and each asset's contribution to it.Bank market riskQuant risk
Try it first
Asset A is 60% of the capital. What share of the portfolio's risk does it contribute?
Show the worked solution
Portfolio volatility is about Rs 20.78 crore, 20.8% of the Rs 100 crore, and each asset contributes exactly half of it. A carries 60 x 20% = Rs 12 crore of standalone volatility and B carries 40 x 30% = Rs 12 crore. The variance is 144 + 144 + 2 x 0.5 x 12 x 12 = 432, so volatility is 20.78. With equal standalone risk, each contributes Rs 10.39 crore: 60% of the capital, 50% of the risk.
Why convert to rupee volatility first?
A household spends 60% of its budget on rent and 40% on a car loan. If the rent is fixed and the car loan's rate floats, most of the budget's uncertainty comes from the smaller item. Risk depends on size times volatility, so the first step is to express each position's volatility in rupees: that puts the two assets on one scale. Here the smaller, more volatile position carries exactly as much standalone risk as the larger, calmer one: Rs 12 crore each.
The relationship12 rupee volatility of each position, Rs crore 0.5 correlation between A and B What it says in wordsPortfolio variance is each position's variance plus twice the shared part.How do you split the portfolio's risk between the two assets?
Use the Euler allocationSplitting total risk so each position gets its own variance plus its share of every covariance, divided by total volatility; the pieces add up exactly to the total.: each asset's contribution is its own variance plus its covariance with the other, divided by total volatility. For A that is (144 + 0.5 x 12 x 12) over 20.78, which is 216 over 20.78, Rs 10.39 crore. B's contribution is the same Rs 10.39 crore, and the two add back to exactly Rs 20.78 crore, so the split is 50/50 while the capital split is 60/40.
Asset A holds 60% of the capital but contributes only 50% of the risk, Rs 10.39 crore of the Rs 20.78 crore portfolio volatility, because asset B's higher volatility makes its Rs 40 crore as risky as A's Rs 60 crore. Two points to add. First, the diversification benefitThe amount by which portfolio risk falls short of the sum of the standalone risks, because the positions do not move in perfect lockstep.: standalone risks sum to Rs 24 crore, the portfolio carries Rs 20.78 crore, and Rs 3.22 crore disappears because correlation is below 1. Euler allocation spreads that benefit across both assets rather than handing it to whoever joined last. Second, the limit: contributions depend on a correlation estimate, and correlations rise in stress. At a correlation of 1 there is no benefit left to share.
Where candidates lose it
The common error is assuming risk shares match capital shares, 60/40, because A is the bigger position. The whole point of the puzzle is that the smaller, more volatile asset can carry as much risk as the larger one.
The second error is adding standalone volatilities, Rs 24 crore, as the portfolio risk. That ignores correlation and overstates risk by more than Rs 3 crore; always go through variance.
What the interviewer asks next
- How much of B would you sell to make the risk split 60/40?
- What happens to both contributions if the correlation rises to 0.9?
- Why do risk-parity funds size positions by risk contribution instead of capital?
066A long-short pair holds equal rupee amounts of two stocks, long one and short the other, each with 2% daily volatility. What happens to the pair's daily volatility when the correlation between the stocks falls from 0.9 to 0.6?Bank market riskAsset manager risk
Try it first
The correlation drops by a third. What does the pair's volatility do?
Show the worked solution
It doubles, from about 0.89% to 1.79% a day, as a share of one leg. For equal and opposite legs with the same volatility, pair variance is 2 x sigma squared x (1 - rho). The unhedged part, 1 - rho, goes from 0.1 to 0.4, four times as large, so volatility doubles. On Rs 100 crore a leg, daily volatility goes from about Rs 0.89 crore to Rs 1.79 crore.
Why does a modest fall in correlation hit a hedged book so hard?
Two boats tied close together on a river rise and fall almost as one; the rope between them barely strains. Loosen the tie and every wave that hits one boat and not the other shows up as strain on the rope. A long-short pair is the rope. It only carries the part of the two stocks that does not move together, and that part is 1 minus rho, which grows fast as correlation slips from a high level.
With equal legs of 2% daily volatility, the pair's volatility is 0.89% at a correlation of 0.9 and 1.79% at 0.6, double, because the uncorrelated part 1 minus rho quadruples from 0.1 to 0.4. What does this mean for the desk's risk numbers?
VaR scales with volatility, so the pair's VaR doubles with no change in position. On Rs 100 crore a leg the 99% one-day VaR moves from about Rs 2.1 crore to Rs 4.2 crore, entirely from a correlation the desk does not control. Correlations between related stocks do fall in exactly the markets where one company's news breaks from its peer's, which is when the pair was supposed to be safe.
The relationship\sigma each leg's daily volatility, 2% \rho the correlation between the two stocks, 0.9 then 0.6 What it says in wordsThe pair's volatility is each leg's volatility times the square root of twice the uncorrelated share.Say the limitation. The formula assumes equal volatilities and equal notionals; a real pair is often beta-weighted, which changes the coefficients but not the lesson. A risk limit that assumes the historical correlation of 0.9 is a limit on a number that can quietly double.
Where candidates lose it
The trap is reasoning linearly: correlation fell by a third, so risk rises by a third. The risk depends on 1 minus rho, which went up fourfold, and the square root turns that into a doubling.
The second trap is treating the pair as market neutral and therefore low risk. Neutral to the market is not neutral to correlation, and a good answer names correlation as the position's real exposure.
What the interviewer asks next
- At what correlation does the pair's volatility equal a single leg's 2%?
- How would you hedge the correlation risk itself?
- If the legs are beta-weighted instead of equal, how does the formula change?
091Asset A has 10% volatility. Asset B has 30% volatility and a correlation of minus 0.2 with A. What weight in B minimises the portfolio's volatility, and what is that minimum?Asset manager riskQuant risk
Try it first
Can adding some of the 30% volatility asset make the portfolio less risky than A alone?
Show the worked solution
About 14.3% in B, giving a minimum volatility of about 8.78%. The minimum variance weight in B is A's variance less the covariance, over the sum of the variances less twice the covariance: 160 over 1,120. The portfolio is less risky than A alone for any weight in B up to 28.6%, even though B is three times as volatile.
How can a riskier asset lower total risk?
An umbrella seller and an ice cream seller at the same beach each have bad days, but rarely the same bad days. A shopkeeper who sells mostly ice cream and a few umbrellas has steadier takings than one who sells only ice cream, even if umbrella sales swing wildly. For a small weight, the risk an asset adds depends on how it moves with what you already hold, and a negatively correlated asset offsets part of the existing risk. B's own variance enters with the square of its weight, so at small weights it barely registers, while the offset enters in proportion to the weight.
The relationshipsigma_A, sigma_B volatilities 10% and 30%, so variances 100 and 900 in per cent squared rho correlation, minus 0.2 rho sigma_A sigma_B the covariance, minus 60 What it says in wordsThe minimum-risk weight in B is A's variance less the covariance, divided by the variance of the difference between the two assets.Portfolio volatility falls from A's 10% to a minimum of 8.78% at a 14.3% weight in the 30% volatility asset, stays below 10% until B's weight reaches 28.6%, and only then climbs toward B's own 30%. How do you check the minimum volatility?
Plug the weight back in: 0.857 squared times 100, plus 0.143 squared times 900, plus twice 0.857 times 0.143 times minus 60. That is 73.5 plus 18.4 less 14.7, about 77.1, whose square root is 8.78%. A shortcut confirms it: the minimum variance for two assets is the product of the variances times one minus rho squared, over the same denominator, 90,000 times 0.96 over 1,120, which is 77.1. Two routes agreeing is the check the interviewer is listening for.
Then give the limit. The answer rests on a correlation estimate, and correlations drift. If the correlation moved to zero, the best weight would fall to 10% and the minimum would rise to about 9.5%; if it turned positive in a stress, the benefit would shrink further. A risk manager treats the minimum variance weight as a range, not a point.
Where candidates lose it
The instinct that a 30% volatility asset must add risk is the trap. For small weights, correlation dominates and B's own volatility barely enters.
The mechanical slip is the sign of the covariance: with a negative correlation, minus rho sigma sigma becomes plus 60 in the numerator and minus two rho sigma sigma becomes plus 120 in the denominator. Write the covariance as a number, minus 60, before substituting.
What the interviewer asks next
- What is the minimum-variance weight if the correlation is zero?
- At what correlation does adding any B increase risk from the start?
- Why might a risk manager distrust a correlation estimated from calm years?