Risk Management puzzles, solved step by step
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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?Asset 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 relationshipw_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?
028A fund's annual volatility is 18%, its benchmark's is 16%, and the correlation between their returns is 0.95. What is the fund's tracking error?MSCIMonterrey · 2013
Try it first
Quick instinct: roughly how big is the tracking error?
Show the worked solution
About 5.73%. Tracking error is the volatility of the fund's return minus the benchmark's. Its variance is 18 squared plus 16 squared minus 2 x 0.95 x 18 x 16, which is 324 plus 256 minus 547.2, or 32.8. The square root is 5.73%, nearly three times the 2-point gap in volatilities.
What exactly is tracking error measuring?
Two friends walk to the same office. How far apart they are at any moment depends less on how fast each walks than on whether they take the same streets. Tracking error is the volatility of the return difference, fund minus benchmark, so it depends on how much the two move apart, not on how much each moves. The fund and index can both swing wildly and still track closely if they swing together. That is why the formula needs the correlationA number from minus 1 to 1 describing how closely two returns move together; 1 means perfect lockstep., not just the two volatilities.
The relationship\sigma_F, \sigma_B fund and benchmark volatility, 18% and 16% \rho correlation of their returns, 0.95 What it says in wordsThe variance of a difference is the two variances added, less twice the part they share.Drawing the two volatilities as sides 18 and 16 at the angle whose cosine is 0.95 makes tracking error the short third side, 5.73%; nudging correlation from 0.95 to 0.99 cuts it to 3.12%, and dropping it to 0.90 raises it to 7.85%. Why does the correlation matter more than the volatilities?
Look at the table in the figure. Keeping 18 and 16 fixed, moving correlation from 0.99 to 0.90 takes tracking error from 3.12% to 7.85%, more than doubling it. At high correlations each hundredth of correlation moves tracking error a lot, because the large shared term 2 x rho x 18 x 16 almost cancels the two variances. Now hold correlation at 0.95 and give both sides 16% volatility: tracking error is still 5.06%. The volatility gap contributes a little; the imperfect correlation contributes most.
Close with the limit. The formula uses a correlation estimated from history, and correlations drift, often falling in stressed markets. A fund reporting 5.7% tracking error in calm years can run well above it in a sell-off, so a risk team watches realised tracking error alongside the model figure.
Where candidates lose it
The fast wrong answer is 2%, subtracting the volatilities. It silently assumes correlation of exactly 1, which the question has just told you is false. Candidates who say it have treated volatility as if it were a return.
The second trap is fumbling the formula under pressure. Anchor it to one line you already know: the variance of A minus B is var A plus var B minus twice the covariance. Everything else follows.
What the interviewer asks next
- What correlation would give a tracking error of exactly 2%?
- The fund's beta to the benchmark is 1.07. Split the tracking error into a beta part and a residual part.
- Why might a fund with low tracking error still underperform its benchmark every year?
Asked at MSCI, Financial Tools, Monterrey, 2013 (Wall Street Oasis):
What's the tracking error formula?
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?Bank 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_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?
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?Asset 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 relationshipsigma 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?
