Portfolio Management puzzles, solved step by step

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

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002A trading book's one-day 99% value at risk is Rs 2 crore. What are its 10-day and its one-month (21 trading day) value at risk under the usual scaling rule, and when does that rule fail?Portfolio risk mathsWarm upAQR Capital ManagementGreenwich · 2022
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Pick the 10-day value at risk before you calculate.
Show the worked solution
About Rs 6.3 crore over 10 days and Rs 9.2 crore over 21 days. With independent daily returns, variance adds across days, so volatility and value at risk scale with the square root of time: 2 x the root of 10 and 2 x the root of 21. The rule fails when returns trend or mean revert, when tails are fat, and when the book changes during the period.
Why the square root of time and not time itself?
Think of a person taking random steps left or right. After a hundred steps they are rarely a hundred steps from the start, because the left steps cancel the right ones; the typical distance is about ten, the square root of a hundred. Daily returns behave the same way when each day is independent. Variances add across independent days, so the spread of a ten-day return is the daily spread times the square root of ten, not times ten. Value at risk at a fixed confidence level is a multiple of that spread, so it scales the same way: Rs 2 crore becomes Rs 6.32 crore over ten days and Rs 9.17 crore over 21.
Scaled by the square root of time, a one-day value at risk of Rs 2 crore becomes about Rs 6.3 crore at ten days and Rs 9.2 crore at 21 days, far below the Rs 20 crore and Rs 42 crore that scaling by the number of days would give.
The relationship
$\text{VaR}_T=\text{VaR}_1\sqrt{T} \qquad 2\sqrt{10}\approx 6.32 \qquad 2\sqrt{21}\approx 9.17$
\text{VaR}_1 the one-day value at risk, Rs 2 crore
T the holding period in trading days
What it says in wordsMultiply the one-day figure by the square root of the number of days, which is valid only for independent, identically spread daily returns and an unchanged book.
When does the rule give the wrong answer, and in which direction?
The rule rests on three assumptions, and each one breaks in real markets. If returns trend, so a bad day tends to follow a bad day, the true ten-day loss is larger than Rs 6.3 crore; if they mean revert, it is smaller. Fat tails make the 99% point further out than a normal curve suggests, and the ratio between the tail and the spread need not hold across horizons. And a book is not frozen: over a month a desk cuts losing positions, which the scaling ignores. Say which way each one pushes the number and the interviewer knows you understand the rule rather than having memorised it.
One more thing worth saying: the scaling also assumes the expected daily return is zero. Over a day that is harmless. Over a year, drift matters and a simple square root rule starts to overstate the loss for a portfolio with a positive expected return.
Where candidates lose it
The fast wrong answer is Rs 20 crore, which treats ten independent days as ten worst days in a row. Candidates who know the square root rule sometimes lose the point anyway by stating it without its assumptions.
The follow-up is almost always when it fails. Have the three failures ready, trending returns, fat tails and a changing book, and say in which direction each pushes the number.
What the interviewer asks next
If daily returns have a positive autocorrelation of 0.2, is the true 10-day value at risk above or below Rs 6.3 crore?
Why do regulators ask for a 10-day horizon rather than one day?
What is the annual value at risk under the same rule, using 250 trading days?
Asked at AQR Capital Management, Quantitative Research, Greenwich, 2022 (Wall Street Oasis): Specific statistics questions on financial concepts. daily vs monthly return, VAR, more that i don't remember
003A car leaves town A for town B, 100 miles away, at 50 miles an hour. At the same moment a bird leaves B at 100 miles an hour, flies to meet the car, turns back to B, then turns again toward the car, and keeps shuttling until the car reaches B. How far does the bird fly?Logic brainteasersWarm upBlackRockNew York · 2025
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Answer inside ten seconds.
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200 miles. The car needs 100 miles at 50 miles an hour, which is two hours. The bird flies without stopping for those two hours at 100 miles an hour, so it covers 200 miles, however many times it turns. Summing the legs gives the same answer: 66.7 there and back, then 22.2 there and back, each pair a third of the one before, which adds to 200.
Why is summing the legs the slow way?
Picture a dog on a walk that runs ahead to the gate and back to you, over and over, until you reach the gate. Nobody counts the dog's sprints; you just ask how long the walk took and how fast the dog runs. When something moves at a constant speed for a known time, distance is speed times time, whatever path it traces. The bird's zig-zag looks like the hard part of the question. It is a distraction. The only thing that matters is when the flying stops, and that is when the car arrives.
The car takes two hours to cover 100 miles at 50 miles an hour, and the bird zig-zags between B and the car for exactly those two hours, so at 100 miles an hour it flies 200 miles; the shrinking legs of 66.7, 66.7, 22.2, 22.2 and so on add to the same total.
How do you check 200 the long way, in case the interviewer asks?
The bird and the car close at 150 miles an hour, so they first meet after 100 / 150 of an hour, 40 minutes, when the car is 33.3 miles from A and the bird has flown 66.7 miles. The bird flies 66.7 miles back to B. By then the car is 66.7 miles along, 33.3 miles from B, and the same geometry repeats on a gap a third as large. Each round trip is a third of the one before, so the legs form a geometric series: 133.3 times one over one minus a third, which is 200.
The relationship
$d=v_{bird}\times\frac{D}{v_{car}}=100\times\frac{100}{50}=200 \qquad \text{check: } 133.3\times\frac{1}{1-\tfrac{1}{3}}=200$
D the distance from A to B, 100 miles
v_{car}, v_{bird} the speeds, 50 and 100 miles an hour
What it says in wordsThe bird's distance is its speed times the car's travel time; the geometric sum of its legs confirms it.
Why ask this on a quantitative research desk? Because the same move, stepping back from the path to the total, is how you price anything path-dependent in your head: ask what is conserved or fixed before tracing every step.
Where candidates lose it
Candidates start computing the first meeting point, then the second, and lose the room in arithmetic. The interviewer is watching for the moment you ask how long the bird flies; some interviewers stop you once you begin summing legs.
The second trap is saying infinite because the bird turns infinitely often. A sum of infinitely many shrinking terms can be finite, and here it is.
What the interviewer asks next
What if the bird flew at 150 miles an hour?
If both towns sent a car toward each other at 50 miles an hour, how far does the bird fly?
How many times does the bird touch the car?
Asked at BlackRock, Quantitative Research, New York, 2025 (Wall Street Oasis): A car starts at point A going 50 miles an hour towards point B, and a bird starts at point B going towards point A at 100 miles per hour
031An office building has gross potential rent of Rs 10 crore a year. Vacancy runs at 8%, and operating costs are 30% of the rent actually collected. At an 8% cap rate, what is the building worth? Walk through it from gross potential rent.Private and real asset mathsWarm upInvescoNew York · 2025
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Which number does the cap rate divide?
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About Rs 80.5 crore. Vacancy of 8% takes Rs 10 crore of gross potential rent down to Rs 9.2 crore collected. Operating costs of 30% of that are Rs 2.76 crore, leaving net operating income of Rs 6.44 crore. Divide by the 8% cap rate: 6.44 over 0.08 is Rs 80.5 crore, or 12.5 times the income.
Why does the cap rate price net operating income and not the rent?
Think of buying a shop that a friend runs. You would not pay for the sales the shop could make if every shelf sold out; you pay for what is left after empty days and the electricity bill. A cap rate is net operating income divided by value, so the value is only ever as good as the income that actually reaches the owner after vacancy and running costs. Gross potential rent is the ceiling, not the income.
Rs 10 crore of gross potential rent loses Rs 0.8 crore to vacancy and Rs 2.76 crore to operating costs, leaving Rs 6.44 crore of net operating income. At an 8% cap rate that income is worth Rs 80.5 crore.
What are the lines between gross rent and value, in order?
Say them as a ladder. Gross potential rent is what a full building earns at the rent roll. Take off vacancy and bad debt to get effective gross incomeThe rent a building actually collects after vacancy and unpaid rent, before any running costs.. Take off operating expenses, such as maintenance, insurance, property tax and management, to get net operating income. Every line above NOI moves the price by 12.5 times its size at an 8% cap rate, so a small leak near the top is a large leak in value. One extra point of vacancy costs Rs 0.1 crore of rent, Rs 0.07 crore of NOI, and Rs 0.875 crore of value.
The relationship
$V = \frac{NOI}{c} = \frac{GPR\,(1-v)(1-o)}{c} = \frac{10 \times 0.92 \times 0.70}{0.08} = 80.5$
GPR gross potential rent, Rs 10 crore
v vacancy rate, 8%
o operating costs as a share of collected rent, 30%
c the cap rate, 8%
What it says in wordsValue is the income that survives vacancy and costs, divided by the yield buyers demand.
For an exit value, the same arithmetic runs on the NOI expected in the sale year and an exit cap rate, which analysts often set a little above the entry cap rate to allow for an older building. Say which cap rate you are using and why, and say that capital spending such as a new roof sits below NOI and is not captured by this shortcut.
Where candidates lose it
The costly slip is dividing gross potential rent by the cap rate, which gives Rs 125 crore and overpays by Rs 44.5 crore. The other is applying the 30% cost ratio to gross rent rather than collected rent, which gives NOI of Rs 6.2 crore and a value Rs 3 crore too low.
Walk the ladder aloud, line by line, and name what each deduction is. Interest never appears: it depends on how the buyer finances the building, not on the building.
What the interviewer asks next
The buyer expects NOI to grow 3% a year and plans to sell in five years at an 8.5% exit cap rate. What is the exit value?
Why might a buyer's cap rate for this building differ from the seller's?
What happens to value if operating costs rise to 35% of collected rent?
Asked at Invesco, Real Estate, New York, 2025 (Wall Street Oasis): Walk me through how to get the exit value of a property from Gross Potential rent, using Cap rate.
051A two-year bond pays an 8% annual coupon and repays 100 at maturity. Market yields for this bond are 7%. What is its price, and why is it above par?Bond mathsWarm upJ.P. Morgancloumbus · 2026
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Before you discount anything: where does the price land?
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About 101.81. Discount each cash flow at 7%: the year 1 coupon of 8 is worth 7.48 today and the year 2 payment of 108 is worth 94.33, which add to 101.81. The bond sits above par because it pays 8 when the market only asks for 7, and a buyer pays up for that extra coupon until the return on the price paid falls back to 7%.
What does pricing a bond actually mean?
Think of a friend who promises you Rs 8 next year and Rs 108 the year after. What would you hand over today? If you can earn 7% elsewhere, each promised rupee is worth less the further away it sits. A bond's price is every promised cash flow divided by one plus the yield, once for each year you wait, and then added up. Here that is 8 divided by 1.07, which is 7.48, plus 108 divided by 1.07 twice, which is 94.33. The total is 101.81.
The relationship
$P = \frac{8}{1.07} + \frac{108}{1.07^2} = 7.48 + 94.33 = 101.81$
8 the annual coupon on 100 of face value
108 the final coupon plus the principal
1.07 one plus the market yield
What it says in wordsDiscount each payment by the yield for as many years as you wait, then add.
Discounted at 7%, the year 1 coupon is worth 7.48 and the year 2 payment of 108 is worth 94.33, stacking to 101.81; the 1.81 above par is exactly the extra 1 a year of coupon over a 7% bond, valued today.
Why must a bond with a high coupon trade above par?
Suppose it traded at 100. A buyer would earn 8% on a bond when the market pays 7% for the same risk, so everyone would want it and the price would rise. The price climbs until the return on the price paid equals the market yield, and that happens at a premium over par. The premium is easy to see directly: compared with a 7% bond at 100, this one pays an extra rupee each year for two years, worth 0.93 plus 0.87, which is 1.81. That is a check on the long method, and it is quicker to say in the room.
The same logic runs the other way. A coupon below the market yield means a discount to par, and a coupon equal to the yield means exactly 100. If you are asked how you would price a bond in today's market, say that the yield comes from comparable bonds of the same credit and maturity; the arithmetic above is the easy part. A premium bondA bond whose price is above its face value, because its coupon is higher than the yield the market currently demands. also pulls back towards 100 as it nears maturity, so a buyer at 101.81 loses the premium slowly while collecting the fat coupon.
Where candidates lose it
The common slip is adding the extra coupons without discounting them and answering 102. The extra rupee in year 2 is worth only 0.87 today, and saying 102 tells the interviewer you know the direction but not the method.
The second loss is getting 101.81 without saying why it must be above 100. Give the one-line reason: the coupon beats the market yield, so buyers bid the price up until the return on the price paid is 7%.
What the interviewer asks next
What would the price be if yields were 9% instead?
Why does the premium on this bond shrink as it approaches maturity?
Where would you find the right yield to price a bond like this in practice?
Asked at J.P. Morgan, Generalist, cloumbus, 2026 (Wall Street Oasis): How would you price a bond in today's market
054Without a calculator: what IRR turns money into 2.5 times in four years, and what IRR turns it into 3 times in five years?Private and real asset mathsWarm upNeuberger BermanLondon · 2026
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Which pair is closest?
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About 26% and about 25%. For 2.5 times in four years, take the square root of 2.5 twice: 1.58 and then 1.257, so 25.7% a year. For 3 times in five years, anchor on the fact that 2 times in three years is 26%; 3 times in five sits just below it at 24.6%. The bigger multiple over the longer hold is actually the lower annual return.
Why is the simple average so far off?
A savings account that pays 26% a year does not add 26 rupees every year to your 100. It adds 26 in year one, then 26% of 126 in year two, and so on, so by year four you have about 250. The IRR is the steady annual rate that compounds to the multiple, so it is always well below the total gain divided by the years. The simple average for 2.5 times in four years would be 150% over 4, or 37.5%, which overstates the true 25.7% by more than ten points.
The relationship
$\text{IRR} = M^{1/n} - 1 \qquad 2.5^{1/4} - 1 = 25.7\% \qquad 3^{1/5} - 1 = 24.6\%$
M the money multiple, cash back over cash in
n the years held, with one cash flow in and one out
What it says in wordsThe IRR is the nth root of the multiple, less one.
How do you get there in your head?
Use one of two tricks. For four years, take the square root twice: the square root of 2.5 is about 1.58, and the square root of 1.58 is about 1.257. For odd holding periods, use logs: the natural log of 3 is about 1.10, divided by five is 0.22, and adding a little for compounding turns 22% into about 24.6%. Faster still is a small grid of anchors you know by heart, because interviewers ask the same handful of multiples and periods.
A grid of IRRs by multiple and holding period shows 2.5 times in four years at 25.7% and 3 times in five years at 24.6%, and a band of 2 times in three, 2.5 times in four and 3 times in five years all near 25%.
Notice the diagonal band. Two times in three years, two and a half in four and three in five all land within a point of 25%. That one pattern lets you place almost any private equity outcome in a second. Say the limitation too: this shortcut assumes a single cash flow in and a single cash flow out. Real funds call and return money in stages, and the IRR then depends on the timing, not just the multiple.
Where candidates lose it
The trap is dividing the gain by the years, 150% over four and 200% over five, and answering 37.5% and 40%. It sounds confident and is wrong by more than ten points, and a private markets interviewer hears it as not understanding compounding.
The quieter miss is thinking 3 times must be the better deal because the multiple is bigger. The extra year costs more than the extra half turn of money earns.
What the interviewer asks next
What multiple does a 20% IRR give over five years?
A fund returns 2 times in three years and another 2.5 times in six. Which would you rather have, and what else would you ask?
Why can two deals with the same multiple and the same holding period show different IRRs?
Asked at Neuberger Berman, Generalist, London, 2026 (Wall Street Oasis): The associate interview was quite technical, covered 5-7 questions on valuation and understanding of returns in private equity
055An open-ended fund holds Rs 500 crore of assets across 20 crore units, a NAV of Rs 25. Investors redeem Rs 50 crore. What happens to the NAV and to the number of units?Funds, ETFs and implementationWarm upMorningstarMumbai · 2025
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Straight after the redemption, what is the NAV?
Show the worked solution
The NAV stays at Rs 25; the units fall from 20 crore to 18 crore. Redemptions are paid at NAV, so Rs 50 crore buys back 2 crore units at Rs 25 each and those units are cancelled. The fund is left with Rs 450 crore of assets across 18 crore units, which is still Rs 25 a unit. A redemption shrinks the fund, not the value of each unit.
Why does money leaving not lower the NAV?
Think of a pizza cut into 20 equal slices. If two friends leave and take their slices with them, 18 slices remain, and each one is exactly as big as before. An open-ended fund cancels the units that are redeemed and pays out exactly what they were worth, so assets and units fall in the same proportion and the NAV per unit does not move. Rs 50 crore at Rs 25 a unit is 2 crore units cancelled, leaving Rs 450 crore across 18 crore units.
Rs 50 crore of redemptions cancels 2 of the fund's 20 crore units at Rs 25 each, so assets fall to Rs 450 crore and units to 18 crore, and the NAV is still Rs 25 a unit.
The relationship
$\text{NAV} = \frac{\text{assets} - \text{liabilities}}{\text{units}} = \frac{500 - 50}{20 - 2} = 25$
assets the market value of everything the fund holds, Rs crore
units units outstanding, crore
2 units cancelled, 50 divided by the NAV of 25
What it says in wordsTake the same amount off the top and the bottom in the same proportion, and the ratio does not change.
So what does a redemption change?
Two things, both real. First, the manager must raise the Rs 50 crore, usually by selling holdings, and the transaction costsBrokerage, taxes and the price impact of selling, paid out of the fund when it trades. of those sales are paid by the whole fund, including the investors who stayed. Large redemptions dilute the remaining investors through trading costs, not through the NAV arithmetic. Exit loads, where a scheme charges them and credits them back to the scheme, go the other way and cushion those who stay. Second, a fund that has to sell in a hurry may sell what is easiest to sell, which leaves the remaining portfolio less liquid than before.
The day's NAV is the one used, struck after the market closes, which is why a redeeming investor cannot know the exact price when placing the request. The precise cut-off times and load rules are set by the regulator and the scheme documents, so confirm the current ones rather than quoting them from memory.
Where candidates lose it
The trap is answering Rs 22.50: dividing the smaller Rs 450 crore by the old 20 crore units. Candidates who think of NAV as a share price picture money leaving as bad news for the price. In a fund, the leavers take their units with them.
The opposite miss, Rs 27.78, divides the old assets by the new unit count. Say it as one sentence: units are cancelled at NAV, so both halves of the ratio fall together.
What the interviewer asks next
Who bears the cost when a large redemption forces the fund to sell illiquid holdings?
What happens to the NAV when new money comes in instead?
How does an ETF handle outflows differently from an open-ended fund?
Asked at Morningstar, Private Markets, Mumbai, 2025 (Wall Street Oasis): They asked questions such as: What are derivatives? Can you explain NAV? What are ETFs?

\text{VaR}_1	the one-day value at risk, Rs 2 crore
T	the holding period in trading days

D	the distance from A to B, 100 miles
v_{car}, v_{bird}	the speeds, 50 and 100 miles an hour

GPR	gross potential rent, Rs 10 crore
v	vacancy rate, 8%
o	operating costs as a share of collected rent, 30%
c	the cap rate, 8%

8	the annual coupon on 100 of face value
108	the final coupon plus the principal
1.07	one plus the market yield

Try it first

Why the square root of time and not time itself?

When does the rule give the wrong answer, and in which direction?

Where candidates lose it

What the interviewer asks next

Try it first

Why is summing the legs the slow way?

How do you check 200 the long way, in case the interviewer asks?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the cap rate price net operating income and not the rent?

What are the lines between gross rent and value, in order?

Where candidates lose it

What the interviewer asks next

Try it first

What does pricing a bond actually mean?

Why must a bond with a high coupon trade above par?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the simple average so far off?

How do you get there in your head?

Where candidates lose it

What the interviewer asks next

Try it first

Why does money leaving not lower the NAV?

So what does a redemption change?

Where candidates lose it

What the interviewer asks next