Portfolio Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 31
Topics: 13
Hard: 30

Showing 1–6 of 6 · filtered from 100Clear filters

015A Rs 50 crore equity portfolio has a beta of 1.2 to the index. How much index futures notional must you sell to bring the portfolio's beta down to 0.5?Portfolio risk mathsWarm upPortfolio implementationHedge funds
Try it first
How much notional do you sell?
Show the worked solution
Sell Rs 35 crore of index futures notional. At a beta of 1.2 the portfolio moves like Rs 60 crore of the index. At the target of 0.5 it should move like Rs 25 crore. The difference, (1.2 minus 0.5) x Rs 50 crore, is Rs 35 crore, assuming the futures move one for one with the index. At an assumed Rs 10 lakh a contract, that is about 350 contracts.
Why is the portfolio's market exposure not simply Rs 50 crore?
Think of a car that goes 1.2 km for every km a reference car goes. Holding Rs 50 crore of it is like holding Rs 60 crore of the reference. Beta converts a portfolio's value into index-equivalent exposure, so a Rs 50 crore book at beta 1.2 carries Rs 60 crore of market risk. Once you see the exposure in index rupees, the hedge is a subtraction: you want Rs 25 crore left, so you take away Rs 35 crore by selling futures, which carry a beta of one to the index.
The Rs 50 crore portfolio at beta 1.2 carries Rs 60 crore of market exposure; selling Rs 35 crore of index futures leaves Rs 25 crore, which is a beta of 0.5 on the portfolio.
The relationship
$N=(\beta_{target}-\beta_{now})\times V=(0.5-1.2)\times 50=-35$
N futures notional to trade, Rs crore; negative means sell
\beta_{now}, \beta_{target} the current beta 1.2 and the target 0.5
V the portfolio's value, Rs 50 crore
What it says in wordsThe futures notional equals the change in beta times the portfolio value, with a minus sign meaning a sale.
What does the hedge not do?
It removes market risk, not stock risk. After the hedge the portfolio still carries every stock-specific bet it had; only its sensitivity to the index has been cut. That is often the point: a manager who likes the stocks but not the market can keep the stock picks and trim the market bet. Say the limitations: beta is estimated from history and drifts, so the hedge is right only on average; futures need margin and must be rolled at expiry, and the futures price can move slightly differently from the index, which is called basis risk.
Where candidates lose it
The common answers are Rs 50 crore, hedging the whole value, and Rs 60 crore, hedging the whole beta-weighted value. Both take the beta to zero, not to 0.5. Hedge the change in beta.
Candidates also forget the direction. Lowering beta means selling futures; raising it means buying them. Say the sign with the number.
What the interviewer asks next
How much would you trade to raise the beta to 1.5 instead?
If the portfolio falls 10% in value, is the hedge still right?
Why might the hedged portfolio still lose money in a market fall?
025An asset has an expected return of 10% and volatility of 12%, and the risk-free rate is 6%. If you lever it 1.5 times, borrowing at the risk-free rate, what are the expected return, the volatility and the Sharpe ratio?Portfolio risk mathsCoreMulti-assetHedge funds
Try it first
What happens to the Sharpe ratio when you lever at the risk-free rate?
Show the worked solution
Expected return 12%, volatility 18% and the same Sharpe ratio of 0.33. With 1.5 in the asset and minus 0.5 in cash, the return is 6% plus 1.5 times the 4% excess, which is 12%, and the volatility is 1.5 x 12%, which is 18%. The Sharpe ratio is (12 minus 6) / 18, unchanged at 0.33. Leverage at the risk-free rate moves you along the line without changing its slope.
Why does leverage leave the Sharpe ratio alone?
Think of a recipe scaled up by half: every ingredient rises by 1.5 times, so the taste, which depends on the proportions, is the same. Borrowing at the risk-free rate multiplies both the excess return and the volatility by the leverage, so their ratio, the Sharpe ratio, does not change. The asset earns 4 points over cash with 12 points of volatility, 0.33 per unit of risk. At 1.5 times it earns 6 points over cash with 18 points of volatility: still 0.33. The borrowed cash itself has no volatility, so it adds none.
The asset at 12% volatility and 10% return and the 1.5 times levered portfolio at 18% and 12% sit on the same line from the 6% risk-free rate, so both have a Sharpe ratio of 0.33, while borrowing at 8% instead drops the levered point to 11% and a Sharpe ratio of 0.28.
The relationship
$E[r_L]=r_f+L(E[r]-r_f)=6+1.5(4)=12\% \qquad \sigma_L=L\sigma=18\% \qquad S=\frac{6}{18}=\frac{4}{12}\approx 0.33$
L the leverage, 1.5 times
r_f the risk-free rate, 6%, also the borrowing rate here
S the Sharpe ratio, excess return over volatility
What it says in wordsLeverage scales the excess return and the volatility by the same factor, so the Sharpe ratio is unchanged.
When does leverage lower the Sharpe ratio?
When borrowing costs more than the risk-free rate, which is the normal case for anyone but a government. At a borrowing rate of 8%, the borrowed half costs 2 points more than cash earns, so the levered return falls to 11% and the Sharpe ratio to 0.28. The line bends down to the right of the asset. Say the other limitations: volatility is not the only risk that scales, because a levered portfolio can be forced to sell after a large loss, and the higher volatility drags down compound growth even when the Sharpe ratio is unchanged. This is why a manager with a high Sharpe ratio, low-volatility strategy can lever it, while the same move on a volatile asset is far more dangerous.
Where candidates lose it
The common answer is that leverage raises the Sharpe ratio because it raises return, or lowers it because it raises risk. Both miss that the two rise together.
The arithmetic slip is levering the whole 10% return, answering 15%, instead of levering the 4% excess and paying 6% on the borrowed half. Write return as the risk-free rate plus leverage times the excess.
What the interviewer asks next
What leverage gives an expected return of 14%, and what is its volatility?
Why do investors who cannot borrow tend to hold riskier assets instead?
How does the answer change if the borrowing rate is 8%?
026A portfolio has an expected return of 10% a year and a volatility of 15%, and yearly returns are roughly normal and independent. What is the chance of losing money in any one year, and the chance that its average annual return over ten years is below zero?Portfolio risk mathsCoreWealth managementMulti-asset
Try it first
Before you calculate: roughly how likely is a losing ten-year average?
Show the worked solution
About 25% for one year and about 1.75% for the ten-year average. One year: zero is 10 over 15, or 0.67 standard deviations below the mean, and the normal table gives 25.2%. The ten-year average has a standard deviation of 15 over the square root of 10, which is 4.74, so zero is 2.11 standard deviations away and the chance falls to 1.75%.
Why is a losing year so common when the portfolio expects 10%?
Think of a bus that is due every 10 minutes but can be 15 minutes early or late on a normal day. Being late is not rare; it happens about one day in four. A 10% expected return with 15% volatility means zero sits only two thirds of a standard deviation below the average, and about a quarter of any normal distribution lies further out than that. So a client holding this portfolio should expect a losing year roughly one year in four, even if nothing is wrong.
Both curves are centred on 10%, but the one-year curve is wide and 25.2% of it lies below zero, while the ten-year average curve is narrow and only 1.75% of it lies below zero. The total ten-year return still spreads out, with a standard deviation of about 47 points.
What does the square root of ten do, and what does it not do?
Averaging independent years cancels part of the noise: good and bad years offset. The standard deviation of an average of n years is the one-year figure divided by the square root of n, so ten years takes 15 down to 4.74. The chance of a losing average falls sharply with time, but the spread of the total amount you end up with keeps growing. The total ten-year return has a standard deviation of 15 times the square root of 10, about 47 points, against 15 for one year. A longer horizon makes a loss less likely, not smaller when it comes.
The relationship
$z = \frac{0 - \mu}{\sigma / \sqrt{n}} = \frac{-10}{15/\sqrt{10}} = -2.11 \qquad P(\bar{R} < 0) = \Phi(-2.11) \approx 1.75\%$
\mu the expected annual return, 10%
\sigma the annual volatility, 15%
n the number of independent years averaged, 10
\Phi the standard normal cumulative probability
What it says in wordsMeasure how many standard deviations of the average zero sits below the mean, then read the tail off the normal table.
Name the assumptions when you give the number. Real yearly returns have fatter tails than a normal curve and are not fully independent; losing years tend to cluster. Both make the true ten-year figure higher than 1.75%, so treat it as a floor, not a promise.
Where candidates lose it
The common slip is to say the risk disappears with time, or to divide the 25% by ten. Neither is how averages behave: the spread of the average shrinks with the square root of the number of years, not with the number itself.
The subtler loss is stopping at 1.75% and calling long horizons safe. Say the second half: the total outcome still spreads out with time, so a patient investor faces fewer losing decades but not smaller losses when one arrives.
What the interviewer asks next
How many years until the chance of a losing average drops below 1%?
What happens to both answers if yearly returns have fat tails?
Is the chance of losing money over ten years the same as the chance of a negative average annual return?
039A 60/40 portfolio holds equities with 18% volatility and bonds with 6% volatility, and the two correlate at 0.1. What is the portfolio's volatility, and what share of its risk comes from equities?Portfolio risk mathsHardMulti-assetAsset allocation
Try it first
Roughly what share of the portfolio's risk comes from the 60% in equities?
Show the worked solution
Volatility is about 11.3%, and equities contribute about 93% of the risk. The variance is 0.36 x 324 plus 0.16 x 36 plus 2 x 0.6 x 0.4 x 0.1 x 18 x 6, which is 116.64 plus 5.76 plus 5.18, or 127.58. The square root is 11.3%. Equities' own term plus half the cross term is 119.2, which is 93% of the total.
Why does 60% of the money carry more than 90% of the risk?
Picture a household with two earners, one a salaried clerk and one a commission-only salesperson. The salesperson may bring in 60% of the money but almost all of the month-to-month swings. Risk contribution depends on weight times volatility, and variance squares it, so a sleeve three times as volatile with more capital swamps the other. Equities' variance term is 0.36 times 324, or 116.6; bonds' is 0.16 times 36, or 5.76. Twenty to one before the cross term.
Capital is split 60/40, but equities supply 93.5% of the portfolio's variance and bonds only 6.5%. The portfolio's volatility is 11.30%, so a 60/40 portfolio behaves almost entirely like an equity portfolio with the volume turned down.
How do you split the risk between the two sleeves?
Give each asset its own variance term plus half of the cross term. An asset's risk contribution is its weight times its covariance with the whole portfolio, and the contributions add to the total variance. For equities that is 0.6 x (0.6 x 324 + 0.4 x 10.8), which is 0.6 x 198.72, or 119.23. Divided by 127.58, that is 93.5%. Bonds take the remaining 6.5%.
The relationship
$\sigma_p^2 = w_e^2\sigma_e^2 + w_b^2\sigma_b^2 + 2 w_e w_b \rho\,\sigma_e\sigma_b \qquad RC_e = \frac{w_e\,(w_e\sigma_e^2 + w_b\,\rho\,\sigma_e\sigma_b)}{\sigma_p^2}$
w_e, w_b the capital weights, 60% and 40%
\sigma_e, \sigma_b the volatilities, 18% and 6%
\rho the correlation, 0.1
RC_e the equity share of portfolio variance
What it says in wordsPortfolio variance is each asset's own variance plus the cross term, and each asset's share is its weight times its covariance with the portfolio.
This is the arithmetic behind risk parity, which sizes each sleeve so that the risk contributions are equal rather than the capital. To give bonds half the risk here, the portfolio would hold roughly three times as much bond capital as equity, and often borrow to lift the return. The limitation: correlation is not stable. In some years stocks and bonds fall together, and the 7% can grow quickly.
Where candidates lose it
The first slip is saying the portfolio volatility is 0.6 x 18 plus 0.4 x 6, which is 13.2%. That ignores diversification and would be right only at a correlation of one. The second is reporting 60% as the equity risk share because that is the capital share.
Write the three variance terms, take the square root, then split. The whole point of the question is the gap between 60 and 93, so say it in a sentence.
What the interviewer asks next
What equity weight gives equal risk contributions from the two sleeves?
How does the equity risk share change if the correlation rises to 0.5?
Why might a pension fund still describe itself as 60/40 despite this?
050Asset A has a correlation of 0.9 with asset B and 0.9 with asset C. What is the lowest possible correlation between B and C?Portfolio risk mathsHardQuantitative asset managementRisk management
Try it first
Pick the lowest correlation B and C could have.
Show the worked solution
About 0.62. Think of each asset's returns as a vector and correlation as the cosine of the angle between two of them. A correlation of 0.9 is an angle of about 25.8 degrees. B and C are each within 25.8 degrees of A, so they are at most 51.7 degrees apart, and cos 51.7 degrees is 0.62. The formula gives the same: 0.81 minus the square root of 0.19 x 0.19.
Why can the third correlation not be anything you like?
If your office is 10 km from your home and the gym is also 10 km from your home, the office and the gym cannot be 50 km apart. Distances have to fit on a map. Correlations work the same way: they are cosines of angles between return vectors, and angles have to fit together in space, so two strong correlations force a third. A correlation matrix that breaks this rule does not describe any real set of assets, and it can make a risk model report a negative variance.
B and C each sit 25.8 degrees from A, because the cosine of that angle is 0.9. At most they are 51.7 degrees apart, so their correlation cannot fall below cos 51.7 degrees, which is 0.62.
How do you get 0.62 without drawing angles?
Split B and C into a part driven by A and a part independent of A. Each has 0.9 of A in it, and the independent parts carry the remaining variance, 1 minus 0.81, or 0.19. The correlation of B and C is 0.81 from the shared A part plus up to 0.19 either way from their independent parts, depending on whether those parts move together or against each other. So the range is 0.81 minus 0.19 to 0.81 plus 0.19: 0.62 to 1.00. The lowest value comes when the independent parts are perfectly opposed.
The relationship
$\rho_{BC} \ge \rho_{AB}\rho_{AC} - \sqrt{(1-\rho_{AB}^2)(1-\rho_{AC}^2)} = 0.81 - 0.19 = 0.62$
\rho_{AB}, \rho_{AC} the given correlations, 0.9 each
\rho_{BC} the correlation being bounded
1-\rho^2 the share of each asset's variance not explained by A
What it says in wordsThe shared link through A gives 0.81, and the parts unrelated to A can take away at most 0.19.
Why this matters on a risk desk: when analysts override individual correlations in a model, for a stress test or a view, they can create a matrix that no real market could produce. The fix is to check the matrix is positive semi-definite, and the lesson generalises. The bound is only strong when the given correlations are high; with two links of 0.7, the minimum for the third is -0.02, which forces almost nothing.
Where candidates lose it
The common answer is minus 1, from the idea that correlations are unrelated to each other, or 0.81, from multiplying the two links as if correlation were transitive. The first ignores the geometry; the second gives the answer only for one special case.
Say the angle picture in one sentence, give 0.62, and show the formula as a check. Then add that the same logic is why a hand-edited correlation matrix must be tested before it goes into a risk model.
What the interviewer asks next
What is the lowest possible correlation between B and C if both links are 0.5?
How would you check whether a 10 x 10 correlation matrix is valid?
Give a real-world example of three assets where this bound would bind.
071Over the last 100 trading days, portfolio A's five worst daily losses were Rs 5.2, 5.5, 6.0, 6.3 and 7.0 crore. Portfolio B's were Rs 5.2, 6, 9, 15 and 25 crore. Both report the same one-day 95% value at risk of Rs 5 crore. What is each portfolio's expected shortfall?Portfolio risk mathsCoreRisk managementInstitutional asset management
Try it first
How do the two portfolios compare once you look past the value at risk?
Show the worked solution
A's expected shortfall is Rs 6.0 crore; B's is about Rs 12.0 crore. With 100 days, the worst 5% are the five worst days, and expected shortfall is their average. A's five add to 30, an average of 6.0. B's add to 60.2, an average of 12.04. Both portfolios cross Rs 5 crore on the same number of days, but when B has a bad day it is, on average, twice as bad.
What does value at risk leave out?
Think of a river's flood mark. Saying the river tops the bank five days a year tells you nothing about whether it rises a hand's width or floods the town. Value at risk tells you the loss that is exceeded on the worst 5% of days, and nothing about how big those losses are. Both portfolios breach Rs 5 crore on five days in a hundred, so their VaR is identical. What happens beyond that line is where they differ, and it is exactly the part the measure does not describe.
Both portfolios have the same Rs 5 crore value at risk, but A's five worst days average Rs 6.0 crore while B's average Rs 12.0 crore, because B's tail stretches out to a Rs 25 crore day.
The relationship
$ES_{95} = \frac{1}{5}\sum_{i=1}^{5} L_{(i)} \qquad A: \frac{30}{5} = 6.0 \qquad B: \frac{60.2}{5} = 12.04$
L_(i) the i-th worst daily loss, Rs crore
5 the number of days in the worst 5% of a 100-day sample
What it says in wordsExpected shortfall is the average loss on the days worse than the value at risk.
Why do risk managers prefer expected shortfall?
Because it cannot be gamed as easily and it adds up sensibly. A trader can sell deep out-of-the-money options, which earn small premiums almost every day and lose heavily now and then. That book can show a low VaR while hiding a large tail. Expected shortfall looks inside the tail, so strategies that pile risk just past the cut-off show up in it. It also behaves well when books are combined: diversification never makes it worse, which VaR cannot promise.
Be honest about the limits. Five observations are a thin sample for estimating anything, and one Rs 25 crore day dominates B's figure. A longer history, or a model of the tail, would be needed before setting limits on it. The comparison is still the right first read: same VaR, and one portfolio's bad days are twice as painful.
Where candidates lose it
The trap is saying the two portfolios carry the same risk because the VaR is the same. The question is built to show that VaR stops at the edge of the tail.
The second miss is quoting B's worst day, Rs 25 crore, as its risk. That is a single observation; expected shortfall averages the whole tail, which is the fair comparison with A.
What the interviewer asks next
Why does VaR sometimes rise when two books are combined, and why can that not happen with expected shortfall?
What kind of strategy produces a tail like portfolio B's?
How would you estimate expected shortfall with only 100 days of data?

N	futures notional to trade, Rs crore; negative means sell
\beta_{now}, \beta_{target}	the current beta 1.2 and the target 0.5
V	the portfolio's value, Rs 50 crore

L	the leverage, 1.5 times
r_f	the risk-free rate, 6%, also the borrowing rate here
S	the Sharpe ratio, excess return over volatility

\mu	the expected annual return, 10%
\sigma	the annual volatility, 15%
n	the number of independent years averaged, 10
\Phi	the standard normal cumulative probability

w_e, w_b	the capital weights, 60% and 40%
\sigma_e, \sigma_b	the volatilities, 18% and 6%
\rho	the correlation, 0.1
RC_e	the equity share of portfolio variance

Try it first

Why is the portfolio's market exposure not simply Rs 50 crore?

What does the hedge not do?

Where candidates lose it

What the interviewer asks next

Try it first

Why does leverage leave the Sharpe ratio alone?

When does leverage lower the Sharpe ratio?

Where candidates lose it

What the interviewer asks next

Try it first

Why is a losing year so common when the portfolio expects 10%?

What does the square root of ten do, and what does it not do?

Where candidates lose it

What the interviewer asks next

Try it first

Why does 60% of the money carry more than 90% of the risk?

How do you split the risk between the two sleeves?

Where candidates lose it

What the interviewer asks next

Try it first

Why can the third correlation not be anything you like?

How do you get 0.62 without drawing angles?

Where candidates lose it

What the interviewer asks next

Try it first

What does value at risk leave out?

Why do risk managers prefer expected shortfall?

Where candidates lose it

What the interviewer asks next