Risk Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 17
Topics: 13
Hard: 30

Showing 1–7 of 7 · filtered from 100Clear filters

003A portfolio holds 60% in a stock with a beta of 1.2 and 40% in cash. The index falls 10%. What move do you expect in the portfolio from market exposure alone?Correlation and diversificationWarm upAsset manager risk
Try it first
What is the expected move in the portfolio?
Show the worked solution
About 7.2% down. Portfolio beta is the weighted average of the holdings' betas: 0.6 times 1.2 for the stock plus 0.4 times zero for the cash, which is 0.72. Multiply by the index move of minus 10% and the expected move is minus 7.2%. The stock itself is expected to fall 12%, but the cash dilutes it.
Why does the cash count as zero?
Imagine a household where one earner's pay swings with the economy and the other's savings sit in a bank account. When the economy dips, only the first half of the income moves. Beta measures how much a holding moves for each 1% move in the index, and cash does not move with the index at all, so its beta is zero. The portfolio beta is then the weighted sum: 0.6 times 1.2, which is 0.72, plus 0.4 times zero.
The stock is 60% of the portfolio with a beta of 1.2 and contributes 0.72; the cash contributes nothing, so a 10% fall in the index maps to an expected 12% fall in the stock but only 7.2% in the portfolio.
The relationship
$\beta_p = \sum_i w_i \beta_i = 0.6 \times 1.2 + 0.4 \times 0 = 0.72 \qquad \Delta_p = 0.72 \times (-10\%) = -7.2\%$
w_i each holding's share of the portfolio
beta_i each holding's sensitivity to the index
What it says in wordsWeight each holding's beta by its share of the money, add them up, and scale the index move by the result.
What does beta leave out?
Everything that is not the market. Beta gives the expected move from market exposure; the stock's own news adds a separate, unpredictable move on top. On Rs 10 lakh the expected loss is Rs 0.72 lakh, about Rs 72,000, but the actual loss could be larger or smaller depending on what happens to that one company. A single stock position carries a lot of this idiosyncratic riskRisk specific to one company, such as a product failure or a management change, which does not move with the market as a whole., which is why a risk manager quotes beta as an expectation, not a forecast.
Also say that beta is estimated from past returns and drifts over time. A stock measured at 1.2 over the last three years can behave like 1.5 in a sell-off, because correlations tend to rise when markets fall. The 7.2% is the right answer to the question as posed; the conversation that follows is about how much to trust the 1.2.
Where candidates lose it
Candidates answer 12%, which is the stock's expected move, and forget that 40% of the money is in cash. The question is about the portfolio, and the weights are the point.
The second miss is presenting 7.2% as what will happen. Call it the expected move from market exposure, and name the stock-specific risk that sits on top.
What the interviewer asks next
How much of the stock would you sell to bring the portfolio beta to 0.5?
The cash is replaced with a bond fund with a beta of 0.1. What is the new portfolio beta?
How would you hedge the market exposure with index futures, and what risk would remain?
015A 60/40 portfolio holds equities with 15% volatility and bonds with 6% volatility, and the correlation between them is minus 0.2. What is the portfolio volatility?Correlation and diversificationCoreAsset manager risk
Try it first
Pick the portfolio volatility.
Show the worked solution
About 8.84%. The equity term is 0.6 times 15, squared, which is 81. The bond term is 0.4 times 6, squared, 5.76. The cross term is 2 times 0.6 times 0.4 times minus 0.2 times 15 times 6, which is minus 8.64. The variance is 78.12 and its square root is 8.84%, well below the 11.4% weighted average.
Why is the answer not the weighted average of 15% and 6%?
Think of two friends walking a dog on separate leads. If they always pull the same way, the dog is dragged as far as their combined pull. If one sometimes pulls left while the other pulls right, the pulls partly cancel. Volatilities only add in a straight line when the correlation is exactly one; any lower correlation lets the swings offset, and a negative one subtracts risk outright. The weighted average of 15% and 6% is 11.4%, the answer for a correlation of one.
The relationship
$\sigma_p^2 = (w_1\sigma_1)^2 + (w_2\sigma_2)^2 + 2\,w_1 w_2\,\rho\,\sigma_1\sigma_2 = 81 + 5.76 - 8.64 = 78.12$
w_1, w_2 the weights, 0.6 in equities and 0.4 in bonds
sigma_1, sigma_2 the volatilities, 15% and 6%
rho the correlation, minus 0.2
What it says in wordsAdd each asset's own variance contribution, then add the cross term, which is negative when the correlation is negative.
Equities contribute 81 and bonds 5.76 squared percentage points of variance, and the minus 0.2 correlation subtracts 8.64, leaving 78.12, a volatility of 8.84% against 9.31% at zero correlation and 11.4% at a correlation of one.
What do you add after the number?
Two things. First, equities carry almost all the risk: 81 of the 86.76 squared points before the cross term, so a 60/40 portfolio is mostly an equity risk portfolio with a bond cushion. A risk contributionThe share of a portfolio total variance that comes from one holding, including its share of the cross terms. breakdown makes that visible and is usually the next question. Second, the minus 0.2 is an estimate from history, and correlations between equities and bonds have changed sign across decades. If it turned positive at plus 0.3, the volatility would rise to about 10%.
Say the limit too. Volatility treats upside and downside swings alike and assumes the correlation holds in a crisis. In a sharp sell-off correlations can move together, so the diversification shown here is the benefit in normal conditions, not a promise for the worst month.
Where candidates lose it
The trap is answering 11.4%, the weighted average of the volatilities, which ignores diversification entirely. The second is getting the sign of the cross term wrong and adding 8.64 instead of subtracting it.
Say the formula before the numbers, and square the weighted volatilities first: 9 squared and 2.4 squared are easier out loud than 0.36 times 225.
What the interviewer asks next
What bond weight minimises the portfolio volatility?
What is the portfolio volatility if the correlation is plus 0.3?
What share of the portfolio's risk comes from equities once the cross term is split between the two?
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.Correlation and diversificationHardBank 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 relationship
$\sigma_P^2 = 12^2 + 12^2 + 2(0.5)(12)(12) = 432, \qquad \sigma_P = 20.78$
12 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?
053A hedge instrument has a correlation of 0.8 with your position. If you put on the best possible hedge, what share of the position's variance does it remove, and how much of the volatility is left?Correlation and diversificationCoreBank market riskQuant risk
Try it first
Your gut first: how much of the volatility does a 0.8 correlated hedge leave behind?
Show the worked solution
The best hedge removes 64% of the variance and leaves 60% of the volatility. With the minimum variance hedge ratio, the share of variance removed is the correlation squared, 0.8 squared or 0.64. That leaves 36% of the variance, and because volatility is its square root, 60% of the original volatility is still there.
Why does a correlation of 0.8 leave so much behind?
Two friends walk home along roughly the same road. Most of the time they are close, but each takes a detour now and then, and the gap between them on those days is what a hedge cannot touch. A hedge only cancels the part of your position that moves with the instrument. The part it cancels is rho squared of the variance, and whatever is left, called basis riskThe risk that the hedge and the position do not move together, so the hedge gains or loses a different amount from the position., is all yours. At 0.8 that residual is 36% of variance.
A hedge with a 0.8 correlation removes 64 of every 100 units of variance and leaves 36. Because volatility is the square root of variance, the same hedge leaves 60 of every 100 units of volatility, so most of the swing you feel day to day survives.
How do you get from variance left to volatility left?
Take the square root. Risk managers quote variance when they add risks and volatility when they talk about losses, and the square root between them is where candidates lose the answer. A hedge that removes 64% of variance sounds impressive; saying it leaves 60% of the volatility is the honest version, and it is the number that matters for a VaR limit, which scales with volatility.
The relationship
$\sigma_{\text{hedged}} = \sigma_P\sqrt{1-\rho^2} = \sigma_P\sqrt{1-0.64} = 0.6\,\sigma_P$
\sigma_P the volatility of the unhedged position
\rho the correlation between position and hedge, 0.8
What it says in wordsThe best hedge leaves the position's volatility times the square root of one minus the correlation squared.
The limitation: the 0.8 was measured on past data. In a stress the correlation can fall, and the residual grows just when you need the hedge most. That is why desks watch the stability of a hedge correlation as closely as its level.
Where candidates lose it
The fast wrong answer is 20% of the risk left, one minus the correlation. It treats correlation as the share of risk removed, which it is not. Squaring the correlation gives the share of variance explained.
The second trap is stopping at 36% and forgetting that volatility is the square root. Give both numbers, 64% of variance removed and 60% of volatility left, and say which one a VaR limit sees.
What the interviewer asks next
What correlation do you need to cut volatility in half?
What hedge ratio achieves this if the position has twice the hedge instrument's volatility?
Why might a proxy hedge's correlation drop in a crisis?
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?Correlation and diversificationHardBank 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
$\sigma_{\text{pair}} = \sqrt{\sigma^2 + \sigma^2 - 2\rho\sigma^2} = \sigma\sqrt{2(1-\rho)}$
\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?
078Each of 25 stocks has 20% volatility. What is the volatility of an equal-weighted portfolio if they are uncorrelated, and what floor does a pairwise correlation of 0.3 put under it?Correlation and diversificationCoreAsset manager riskQuant risk
Try it first
With a correlation of 0.3, roughly where does portfolio volatility settle however many stocks you add?
Show the worked solution
Uncorrelated, the portfolio's volatility is 4%; with a correlation of 0.3 it is 11.45%, above a floor of 10.95%. Uncorrelated risk shrinks with the square root of the count, 20% over 5. With correlation, the shared risk stays: variance is 20% squared times (1/25 plus 24/25 times 0.3). As the count grows the floor is 20% times the square root of 0.3.
Why does adding stocks stop helping?
Picture 25 shops in one town. Each has its own bad luck, a broken freezer or a rude cashier, and across 25 shops those mishaps average out. But if the town's main factory closes, every shop loses customers on the same day, and owning more shops in the same town does not help. Diversification removes each stock's own risk but cannot touch the risk the stocks share, and correlation is the measure of that shared part.
The relationship
$\sigma_p^2 = \sigma^2\left(\frac{1}{n} + \left(1 - \frac{1}{n}\right)\rho\right) = 0.04 \times (0.04 + 0.96 \times 0.3) = 0.01312$
sigma each stock's volatility, 20%
n number of stocks, 25
rho pairwise correlation, 0.3
What it says in wordsPortfolio variance is a shrinking own-risk term plus a shared term that stays; the square root of 0.01312 is 11.45%.
With no correlation, 25 stocks at 20% volatility give a 4.0% portfolio and the line keeps falling; with a correlation of 0.3 the same 25 stocks give 11.45%, and no number of stocks takes the portfolio below 10.95%.
How do you reach the numbers in your head?
Uncorrelated first: variance divides by n, so volatility divides by the square root of n, 20% over 5 is 4%. For the floor, let n run to infinity and the 1/n term vanishes, leaving variance of sigma squared times rho. The square root of 0.3 is about 0.55, so the floor is about 11%, and 25 stocks already capture almost all the diversification available. The 25-stock figure of 11.45% is only half a point above the 10.95% floor.
Then say the limitation. Correlations are estimated in normal markets and tend to rise in a sell-off, exactly when diversification is wanted. A portfolio sized at 11% volatility on a correlation of 0.3 can behave like one at 15% or more if the correlation jumps to 0.6.
Where candidates lose it
Candidates get the 4% and then apply the same square-root rule to the correlated case, which gives 4% again. The square-root rule is a special case that holds only when correlation is zero.
The second loss is saying diversification removes all risk given enough stocks. Name the floor, give the number, and say that correlation rises in a crisis.
What the interviewer asks next
How many stocks do you need to be within one point of the floor?
What happens to the floor if correlations jump to 0.6 in a crisis?
Why does a portfolio of index funds across countries not diversify as much as the correlation tables suggest?
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?Correlation and diversificationHardAsset 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 relationship
$w_B^* = \frac{\sigma_A^2 - \rho\sigma_A\sigma_B}{\sigma_A^2 + \sigma_B^2 - 2\rho\sigma_A\sigma_B} = \frac{100 + 60}{100 + 900 + 120} = 14.3\%$
sigma_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?

w_i	each holding's share of the portfolio
beta_i	each holding's sensitivity to the index

w_1, w_2	the weights, 0.6 in equities and 0.4 in bonds
sigma_1, sigma_2	the volatilities, 15% and 6%
rho	the correlation, minus 0.2

12	rupee volatility of each position, Rs crore
0.5	correlation between A and B

\sigma_P	the volatility of the unhedged position
\rho	the correlation between position and hedge, 0.8

Try it first

Why does the cash count as zero?

What does beta leave out?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer not the weighted average of 15% and 6%?

What do you add after the number?

Where candidates lose it

What the interviewer asks next

Try it first

Why convert to rupee volatility first?

How do you split the portfolio's risk between the two assets?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a correlation of 0.8 leave so much behind?

How do you get from variance left to volatility left?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a modest fall in correlation hit a hedged book so hard?

What does this mean for the desk's risk numbers?

Where candidates lose it

What the interviewer asks next

Try it first

Why does adding stocks stop helping?

How do you reach the numbers in your head?

Where candidates lose it

What the interviewer asks next

Try it first

How can a riskier asset lower total risk?

How do you check the minimum volatility?

Where candidates lose it

What the interviewer asks next