Portfolio Management puzzles, solved step by step

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

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001A fund's returns have an R squared of 0.81 against its benchmark index. The fund's volatility is 20% a year and the index's is 18%. What are the correlation, the beta and the fund's residual volatility?Statistics and forecastingWarm upPerformance analysisAsset management

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Before you work it: how much of the fund's 20% volatility does the index fail to explain?

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Correlation 0.9, beta 1.0 and residual volatility of about 8.7%. Correlation is the square root of R squared, so 0.9. Beta is correlation times the ratio of volatilities, 0.9 x 20 / 18, which is exactly 1.0. The unexplained 19% of the fund's variance of 400 is 76, and the square root of 76 is 8.7%: the fund's own risk, on top of what the index explains.

Why does 81% explained still leave so much unexplained?

Think of a household's monthly spending. If rent explains most of how the bill moves, the groceries, travel and surprises that make up the rest can still swing it by a lot. The fund is the same. R squared splits variance, and variance is volatility squared, so a small share of variance becomes a much larger share once you take the square root back. The fund's variance is 20 squared, 400. The index explains 81% of it, which is 324. The other 76 belongs to the fund alone, and the square root of 76 is 8.72%.

The index explains 324 of the fund's variance of 400 and leaves 76 unexplained; in volatility terms that is 18 points from the index and 8.7 points of the fund's own, which combine to 20 only because volatilities add in squares.

How do you get the correlation and the beta from R squared?

In a regression on a single index, R squared is simply the correlation squared, so the correlation is the square root of 0.81, which is 0.9. Beta is the correlation scaled by how volatile the fund is relative to the index: 0.9 times 20 over 18 is exactly 1.0. So the fund moves one for one with the index on average, and carries about 8.7 points of volatility the index does not explain. Mention the sign: the root could be minus 0.9, but a long-only equity fund with a positive slope takes the positive root.

The relationship

\rho=\sqrt{R^2}=0.9 \qquad \beta=\rho\,\frac{\sigma_f}{\sigma_i}=0.9\times\frac{20}{18}=1.0 \qquad \sigma_\varepsilon=\sigma_f\sqrt{1-R^2}=20\sqrt{0.19}\approx 8.7\%

R^2	the share of the fund's variance the index explains, 0.81
\rho	the correlation between fund and index returns
\sigma_f, \sigma_i	the volatilities of the fund, 20%, and the index, 18%
\sigma_\varepsilon	the residual volatility, the part of the fund's risk the index does not explain

What it says in wordsCorrelation is the root of R squared, beta rescales it by the volatility ratio, and the residual volatility is the fund's volatility times the root of the unexplained share.

What does the residual number tell a portfolio manager?

With a beta of 1.0, the residual volatility is the fund's tracking errorThe volatility of the difference between a fund's return and its benchmark's return. against the index. An R squared of 0.81 sounds index-like, but 8.7 points of tracking error is a genuinely active book: almost half as volatile as the market itself. Say the limitation too. The split assumes the relationship is linear and stable over the sample; a fund whose beta drifted during the period shows a lower R squared for reasons that have nothing to do with stock picking.

Where candidates lose it

The common slip is treating R squared as a share of volatility and answering 19% of 20%, which is 3.8%. The interviewer is checking whether you know that variances add and volatilities do not, which is the same fact that sits under every portfolio risk calculation.

The second slip is computing beta as 0.9 and stopping, forgetting that beta needs the volatility ratio. Say the three formulas in order and the numbers follow.

What the interviewer asks next

If the fund's beta were 1.2 with the same volatilities, what R squared would that imply?
How would you tell whether the 8.7 points are skill or just unintended sector bets?
Why might R squared against a style index be much higher than against the broad market?

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.
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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
004A portfolio has a 25% chance of a down year, and each year is independent of the others. What is the chance of at least one down year over ten years?Behavioural and decision trapsWarm upWealth managementRetirement and pensions
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Quick instinct: how likely is at least one down year in ten?
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About 94%. The chance of avoiding a down year every single year is 0.75 multiplied by itself ten times, which is 5.6%. At least one down year is everything else: 1 minus 0.056, or 94.4%. Over a decade a falling year is close to certain, so a plan that treats one as a surprise is a plan built on the wrong base case.
Why work through the chance of it never happening?
Ask a commuter how likely they are to miss at least one train in a year of mornings, and the honest answer is: nearly certain, even if they miss one in a hundred. "At least one" questions have many routes to yes and a single route to no. Counting the one way it never happens and subtracting from 1 is always faster than counting every way it can happen. Here the only route to no down year is ten good years in a row, each with chance 0.75, so 5.6% of decades are clean and 94.4% are not.
With a one in four chance of a down year, the chance of at least one down year passes 50% by the third year and reaches 94.4% by the tenth, because the chance of dodging every one shrinks to 5.6%.
The relationship
$P(\text{at least one})=1-(1-p)^n=1-0.75^{10}\approx 0.944$
p the chance of a down year, 25%
n the number of years, 10
What it says in wordsThe chance of at least one down year is one minus the chance that every year is up.
Why does a wealth desk ask a probability question like this?
Because clients judge a portfolio year by year and plan over decades. A 25% chance in any one year sounds like a risk you might avoid; over ten years a down year is close to certain. The point of the number is to change the conversation from whether a down year comes to what the plan does when it does. Say the limitation as well: independence is an assumption. Real market years are not coin flips, and a regime of bad years clusters, which changes the count of down years without changing the lesson.
Where candidates lose it
The trap is adding: 25% times ten is 250%, which is obviously wrong, and candidates who spot that often retreat to 25%, which is just as wrong. The interviewer wants the complement said out loud.
The quieter trap is getting 94% and stopping. On a wealth or pensions desk, the follow-up is what that means for a client, so have one sentence ready on setting expectations before the first bad year arrives.
What the interviewer asks next
What is the chance of at least two down years in ten?
How many years before a down year is more likely than not?
If down years cluster, does the chance of at least one go up or down?
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
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How much notional do you sell?
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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?
027A trader is right on 70% of trades. Each winning trade makes 1% and each losing trade loses 3%. What does the average trade return, and what hit rate would the trader need just to break even?Probability and expected valueWarm upHedge fundsAsset management
Try it first
Gut call first: is this trader making money?
Show the worked solution
The average trade loses 0.2%, and break-even needs a 75% hit rate. Expected value is 0.7 x 1% minus 0.3 x 3%, which is 0.7% minus 0.9%, or minus 0.2%. To break even the winners must pay for the losers: p x 1 = (1 minus p) x 3, so p is 3 over 4, or 75%. The trader is right more often than wrong and still loses money.
Why does a 70% hit rate not settle the question?
A shopkeeper who makes a small profit on seven sales out of ten but sells the other three at a big loss can still close the month in the red. Counting the happy sales tells you nothing until you know how big each one was. Expected return is each outcome's probability times its size, added up, so a hit rate only means something next to the payoff ratio. Here the losers are three times the size of the winners, and that ratio is doing all the damage.
Over ten trades, seven winners of 1% add 7 points and three losers of 3% remove 9 points, so the trader nets minus 2 points, or minus 0.2% a trade. Expected return crosses zero only at a 75% hit rate, five points above what the trader achieves.
How do you find the break-even hit rate in one line?
Set expected value to zero and solve. The break-even hit rate is the loss size divided by the sum of the win and loss sizes: 3 over 1 plus 3, which is 75%. Every point of hit rate is worth 0.04% a trade here, because moving one trade in a hundred from loser to winner swings 4 points in total. So the trader is 5 points of hit rate, or 0.2% a trade, short of break-even.
The relationship
$E = pW - (1-p)L \qquad p^{*} = \frac{L}{W+L} = \frac{3}{1+3} = 75\%$
p the hit rate, 70%
W the average win, 1%
L the average loss, 3%
p^{*} the hit rate at which expected return is zero
What it says in wordsA strategy breaks even when the chance of losing, times the loss, equals the chance of winning, times the win.
Then say what you would change. The trader can cut losers sooner, let winners run further, or be more selective; lifting the average win to 1.5% with the same losses moves break-even to 3 over 4.5, about 67%. Averages also hide transaction costs, which push break-even higher still.
Where candidates lose it
Candidates hear 70% and say the trader is good, or multiply 70% by 1% and forget the losers entirely. The interviewer built the question so the hit rate looks impressive and the payoff ratio quietly wins.
The second loss is getting minus 0.2% and stopping. The follow-up is always the break-even hit rate or the break-even payoff ratio, so have the one-line formula ready.
What the interviewer asks next
Keep the 70% hit rate. How big must the average win be to break even?
A second trader is right 40% of the time, wins 3% and loses 1%. Who would you rather back?
How do trading costs of 0.05% a round trip change the break-even hit rate?
028Rs 10 a year forever, starting next year, is worth Rs 100 at a 10% discount rate. What is the same stream worth today if the first payment arrives only in year 4?Valuation riddlesWarm upAsset managementMutual funds
Try it first
Pick the answer before you work it.
Show the worked solution
About Rs 75.1. The formula C over r gives the value one year before the first payment. With the first payment in year 4, the stream is worth 10 over 0.10, or Rs 100, at year 3. Discount that back three years: 100 over 1.1 cubed is Rs 75.1. As a check, Rs 100 less the present value of the three missing payments, Rs 24.9, gives the same answer.
Where does the perpetuity formula put its answer in time?
Think of a pension that starts paying at retirement. Its value on the day before the first cheque is one number; its value to a 30-year-old is that number shrunk by three decades of waiting. C over r values a level perpetuity exactly one period before its first payment, so a delayed stream is the ordinary perpetuity discounted from that point. Here the first payment is in year 4, so C over r lands at year 3, where it reads Rs 100. One more discounting step gets it to today.
The stream of Rs 10 a year from year 4 is worth Rs 100 at year 3, and three years of discounting at 10% brings that to Rs 75.1 today. Subtracting the three missing payments, worth Rs 24.87 today, from an immediate perpetuity of Rs 100 gives the same Rs 75.1.
How do you check it without the formula?
Start from what you know: Rs 10 a year from year 1 is worth Rs 100. The delayed stream is that same stream with the first three payments cut out. Their present values are 9.09, 8.26 and 7.51, which add to 24.87, and Rs 100 less 24.87 is 75.13. Two methods agreeing to the paisa is what makes the answer safe to say out loud.
The relationship
$PV = \frac{C/r}{(1+r)^{3}} = \frac{100}{1.1^{3}} = \frac{100}{1.331} \approx 75.13$
C the yearly payment, Rs 10
r the discount rate, 10%
3 the years between today and one period before the first payment
What it says in wordsValue the perpetuity where it begins, then discount that single sum back to today.
This is the same step as a terminal value in a discounted cash flow. The Gordon growth value at the end of year 5 is a year 5 number and must be discounted five years, not six. Counting the periods wrong by one is the most frequent error in valuation models, and this puzzle is the cleanest place to catch it.
Where candidates lose it
The two wrong answers come from the timeline. Discounting by 1.1 to the fourth assumes the formula lands at the first payment date rather than one year before it, which gives about Rs 68.3. Subtracting Rs 30 at face value ignores that early money is worth more than later money.
Draw the timeline, even in the air with your finger. Say where C over r lands before you discount, and offer the subtraction check.
What the interviewer asks next
What is the stream worth if the payments grow at 3% a year from year 4?
How much is the delay costing, as a share of the undelayed value?
Where does the same off-by-one error show up in a DCF terminal value?
029A fund returned 14% with 18% volatility in a year when cash paid 6%. Its benchmark returned 12%, and its tracking error against that benchmark was 4%. What are its Sharpe ratio and its information ratio, and what does each one tell you?Performance measurementWarm upPerformance analysisAsset management
Try it first
Which number goes in the denominator of the information ratio?
Show the worked solution
The Sharpe ratio is about 0.44 and the information ratio is 0.5. Sharpe divides the excess over cash, 14 minus 6 or 8 points, by total volatility of 18: 0.44. The information ratio divides the excess over the benchmark, 14 minus 12 or 2 points, by the tracking error of 4: 0.5. Sharpe judges the whole portfolio's risk; the information ratio judges only the manager's active bet.
Why are there two ratios for one fund?
Picture judging a cook. One question is whether the whole meal was worth its price. Another is whether the chef's changes to the standard recipe made it better. The Sharpe ratio asks whether the fund's total return beat cash by enough to justify all its risk, and the information ratio asks whether the manager's departures from the benchmark earned enough to justify that active risk. An investor choosing an asset mix cares about the first; one who has already chosen the market and is picking a manager cares about the second.
The Sharpe ratio divides 8 points of return over cash by 18 points of total volatility and gives 0.44; the information ratio divides 2 points of return over the benchmark by 4 points of tracking error and gives 0.50. Each ratio pairs a reward with the risk that produced it.
How do you read 0.44 and 0.5 once you have them?
The tracking errorThe standard deviation of the difference between a fund's return and its benchmark's return, a measure of how far the fund strays. is small against total volatility because most of the fund's ups and downs are the market's, which the benchmark shares. A 0.5 information ratio from a single year is respectable on paper but statistically weak: one year of 2 points against a 4-point tracking error is half of one standard deviation. It would take many years at that rate before anyone could separate skill from luck with confidence. The Sharpe ratio is also best compared with the benchmark's own Sharpe over the same period, not read alone.
The relationship
$\text{Sharpe} = \frac{R_p - R_f}{\sigma_p} = \frac{8}{18} = 0.44 \qquad \text{IR} = \frac{R_p - R_b}{TE} = \frac{2}{4} = 0.5$
R_p the fund's return, 14%
R_f the cash rate, 6%
\sigma_p the fund's total volatility, 18%
R_b the benchmark return, 12%
TE the tracking error, 4%
What it says in wordsEach ratio is a reward divided by the risk taken to earn it; the two differ in which reward and which risk.
Say one limitation in the room: both ratios assume returns are roughly normal. A fund that sells insurance-like options can post a smooth, high Sharpe for years and then lose a large amount in one month, which neither ratio sees in advance.
Where candidates lose it
The frequent mix-up is putting total volatility under the information ratio, which gives 2 over 18, about 0.11, and makes a reasonable manager look poor. The other is measuring the Sharpe numerator against the benchmark instead of cash.
Say the pairing aloud before calculating: excess over cash with total risk, excess over benchmark with active risk. Then the numbers take ten seconds.
What the interviewer asks next
The benchmark had 16% volatility. What was its Sharpe ratio, and did the fund beat it on that measure?
How many years of a 0.5 information ratio before the excess return is statistically significant?
Why can a fund have a higher Sharpe ratio than its benchmark but a negative information ratio?
030You have two ropes and a lighter. Each rope takes exactly 60 minutes to burn from one end to the other, but it burns unevenly, so half the length does not mean half the time. How do you measure exactly 45 minutes?Logic brainteasersWarm upAsset managementReal assets
Try it first
What does lighting a rope at both ends give you, if it burns unevenly?
Show the worked solution
Light rope 1 at both ends and rope 2 at one end, together. When rope 1 burns out, light the other end of rope 2; when rope 2 burns out, 45 minutes have passed. Rope 1 lasts 30 minutes because two flames share its 60 minutes of burning. At that moment rope 2 has 30 minutes left, and lighting its other end halves that to 15. 30 plus 15 is 45.
Why does uneven burning not spoil the halving?
Two people eating one plate of food from opposite sides finish it in half the time one person would take, whether the food is piled high on one side or spread evenly. They simply meet off-centre. A rope holds 60 minutes of burning in total, and two flames consume it twice as fast, so it is gone in 30 minutes wherever the flames happen to meet. The unevenness decides the place, never the time. That is the only fact the puzzle needs.
Rope 1, lit at both ends, is gone after 30 minutes even though its flames meet away from the middle. Rope 2, lit at one end, has 30 minutes of burning left at that moment, and lighting its other end halves that to 15 minutes, ending at 45.
How do you know rope 2 has exactly 30 minutes left at the half-hour?
Rope 2 has been burning from one end for 30 minutes, so it has used 30 of its 60 minutes of material, whatever length that turned out to be. You never measure length; you only ever track time used and time left. Lighting the far end of what remains halves the 30 minutes left, and the rope finishes 15 minutes later. Say the timeline in that order and the answer is audible.
Interviewers often extend it: with the same two ropes you can also time 15, 30, 60 and 90 minutes, each by deciding which ends are burning at which moment. Answering one extension shows you own the principle rather than a memorised trick.
Why would a real asset desk ask this? It is a clean test of whether you separate the thing you can measure from the thing that is noisy. A rent roll is lumpy month to month; the annual total is what the valuation rests on. Saying that link in one sentence costs nothing.
Where candidates lose it
Candidates try to cut or fold the ropes, or to reason about lengths, which the uneven burning makes useless. The whole question is set up to see whether you let go of length and think only in minutes of burning.
The second slip is lighting rope 2's second end at the start. Rope 2 must be lit at one end at minute 0 so that exactly 30 minutes of it are used up when rope 1 finishes.
What the interviewer asks next
Using the same two ropes, how do you measure 15 minutes?
With three ropes, what is the longest time you can measure beyond 60 minutes?
Can you measure 20 minutes with two ropes? Why or why not?
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
Try it first
Which number does the cap rate divide?
Show the worked solution
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.

\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

p	the chance of a down year, 25%
n	the number of years, 10

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

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