Portfolio Management puzzles, solved step by step

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

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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
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What happens to the Sharpe ratio when you lever at the risk-free rate?
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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
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Before you calculate: roughly how likely is a losing ten-year average?
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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?
058Every stock in a market has 35% volatility and every pair of stocks has a correlation of 0.25. What is the volatility of an equally weighted portfolio of 1 stock, of 10 stocks, and of infinitely many?Portfolio risk mathsCoreNorthern TrustChicago · 2025Northern TrustChicago · 2025
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With infinitely many stocks, where does portfolio volatility settle?
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35% for one stock, about 20% for ten, and a floor of 17.5% for infinitely many. Portfolio variance is the stock variance times one over n, plus the correlation times what is left. With ten stocks that is 0.1225 times 0.325, a volatility of 20.0%. As n grows the one-over-n part vanishes and only the correlation term remains: 35% times the square root of 0.25, or 17.5%.
Why does adding stocks lower risk at all?
Ten shops in ten different towns do not all have a bad week at once; ten shops in one mall often do. Some of what moves a stock is its own news and some is the market everyone shares. Stock-specific shocks cancel out as you add names, because one company's bad quarter is offset by another's good one, but the shared market shock hits every name together and does not cancel. Correlation measures how much of each stock's movement is shared.
The relationship
$\sigma_p^2 = \sigma^2\left[\frac{1}{n} + \left(1-\frac{1}{n}\right)\rho\right] \;\to\; \sigma\sqrt{\rho} = 0.35 \times 0.5 = 17.5\%$
sigma each stock's volatility, 35%
rho the correlation between any two stocks, 0.25
n the number of stocks, equally weighted
What it says in wordsPortfolio variance is a shrinking stock-specific part plus a fixed shared part, and only the shared part survives as the portfolio grows.
Portfolio volatility falls from 35% with one stock to 20.0% with ten and then flattens towards a floor of 17.5%, because adding names removes stock-specific risk but cannot remove the risk all the stocks share.
How much of the benefit do the first ten stocks deliver?
Most of it. Going from one stock to ten cuts volatility from 35% to 20.0%, about 86% of the whole distance to the floor. Going from ten to thirty takes it only to 18.4%. Diversification pays off quickly and then almost stops, and the level where it stops is set by correlation, not by the number of holdings. That is why a manager worried about risk gains more from adding assets that are less correlated than from adding a fortieth stock of the same kind.
Say the limitation. Correlations are not fixed: in a sell-off they tend to rise together, which raises the floor exactly when diversification is needed. Real stocks also differ in volatility and correlation, so this uniform market is a teaching model; the shape of the curve survives, the exact numbers do not.
Where candidates lose it
The trap is saying diversification takes risk to zero, or reaching for the correlation without the square root and answering 8.75%. The floor is the square root of the shared variance, so it is volatility times the square root of the correlation.
The second miss is getting 20% for ten stocks by guesswork and being unable to show it. Write the variance formula first and plug in: 0.1225 times 0.1 plus 0.9 times 0.25.
What the interviewer asks next
What correlation would make a 10-stock portfolio half as risky as one stock?
Why do correlations tend to rise in a market sell-off, and what does that do to this floor?
How would you lower the floor itself rather than approach it?
Asked at Northern Trust, Asset Management, Chicago, 2025 (Wall Street Oasis): First one was more technical and asked about my understanding of AM, portfolio diversification and strategy
Asked at Northern Trust, Asset Management, Chicago, 2025 (Wall Street Oasis): Asked about my understanding of asset management, portfolio diversification and strategy
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
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How do the two portfolios compare once you look past the value at risk?
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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?
083Two trading desks each have a one-day 95% value at risk of Rs 10 lakh, and their daily P&Ls have a correlation of 0.3. Assuming normal returns, what is the combined value at risk, and how big is the diversification benefit?Portfolio risk mathsCoreBlackRockNew York · 2026
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Your first estimate of the combined value at risk?
Show the worked solution
About Rs 16.1 lakh, a diversification benefit of about Rs 3.9 lakh. Under normal returns value at risk is a fixed multiple of standard deviation, so it combines the same way: the square root of 10 squared plus 10 squared plus 2 x 0.3 x 10 x 10, which is the square root of 260. Adding the two desks' figures would overstate the risk by Rs 3.9 lakh.
Why can you not just add the two numbers?
Two friends each walk 10 minutes from a crossing, one north and one north-east. They do not end up 20 minutes apart; the angle between their paths matters. Under normal returns, a desk's value at risk is a fixed multiple of its standard deviation, and standard deviations combine like arrows: the angle between them is set by the correlation. Only at a correlation of 1 do the arrows point the same way and add to 20.
Placing the two Rs 10 lakh risks head to tail at the angle set by a 0.3 correlation gives a combined value at risk of Rs 16.1 lakh, against Rs 20 lakh if they moved together and Rs 14.1 lakh if they were uncorrelated.
The relationship
$\text{VaR}_{A+B} = \sqrt{10^2 + 10^2 + 2(0.3)(10)(10)} = \sqrt{260} \approx 16.1$
10 each desk's one-day 95% value at risk, Rs lakh
0.3 the correlation between the desks' daily P&Ls
What it says in wordsSquare each desk's figure, add twice the correlation times their product, and take the square root.
What assumption is doing the work, and when does it fail?
The square-root rule holds only when returns are jointly normal, or close to it, so that value at risk is a clean multiple of standard deviation. With fat tails or options in the book, value at risk need not be subadditive, and the combined figure can even exceed the sum. Correlations also rise in a crisis, so the Rs 3.9 lakh benefit is thinnest on exactly the days it is needed. Firms often report both the diversified total and the sum of the parts for that reason.
Where candidates lose it
The fast wrong answer is Rs 20 lakh, which quietly assumes a correlation of 1. The interviewer then asks why banks bother measuring correlation at all, and the candidate has nowhere to go.
The second loss is giving Rs 16.1 lakh without the normality condition. Say it: the square-root rule is a property of standard deviation, and value at risk inherits it only under normal returns.
What the interviewer asks next
At what correlation is the combined value at risk exactly Rs 15 lakh?
How much does each desk contribute to the combined figure?
Why is expected shortfall preferred over value at risk for limits?
Asked at BlackRock, Restructuring, New York, 2026 (Wall Street Oasis): Techincal and behavioral (VAR, market views, stock valuation) why blackrock, python experience?

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

sigma	each stock's volatility, 35%
rho	the correlation between any two stocks, 0.25
n	the number of stocks, equally weighted

L_(i)	the i-th worst daily loss, Rs crore
5	the number of days in the worst 5% of a 100-day sample

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 adding stocks lower risk at all?

How much of the benefit do the first ten stocks deliver?

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

Try it first

Why can you not just add the two numbers?

What assumption is doing the work, and when does it fail?

Where candidates lose it

What the interviewer asks next