Portfolio Management puzzles, solved step by step

Puzzles: 100
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043You have n cars, each fuelled to drive exactly 1,000 miles, and fuel can be moved from one car to another along the way. Tanks cannot be overfilled. How far can one car get, and how does that distance grow as n becomes very large?Logic brainteasersHardMillennium ManagementLondon · 2024
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With four cars, how far can one car get?
Show the worked solution
1,000 x (1 + 1/2 + 1/3 + ... + 1/n) miles, which grows without limit but only like the logarithm of n. Drive all n cars 1,000/n miles; together they have burned one full tank, so one car can refill the rest and be abandoned. Repeat with n minus 1 cars for 1,000/(n minus 1) miles, and so on. Four cars reach 2,083 miles, 100 cars about 5,187, and the distance tracks 1,000 x (ln n + 0.577).
When should a car drop out of the convoy?
Think of friends sharing water on a long walk, where every bottle is full at the start and nobody can carry more than one. The moment the group has drunk exactly one bottle's worth, one friend can pour the rest of theirs into everyone else's bottles and head home. A car should drop out the instant the convoy has burned exactly one tank in total, because that is the first moment its remaining fuel exactly fills the others. With n cars that happens after 1,000/n miles, since n cars burn fuel n times as fast as one.
Four cars travel 250, 333, 500 and 1,000 miles in successive legs as one car at a time tops up the rest and drops out, reaching 2,083 miles. The distance with n cars keeps growing but ever more slowly, from 2,929 miles with 10 cars to 5,187 with 100.
Why does the distance grow only like a logarithm?
The total is 1,000 times the harmonic sum 1 + 1/2 + ... + 1/n. Each extra car adds the shortest leg of the journey, 1,000/n miles, so the gains shrink as the convoy grows, and the harmonic sum rises like ln n plus about 0.577. It never stops growing, so any distance is reachable in principle, but slowly: 10 cars reach about 2,929 miles, and getting past 5,000 miles takes 83 cars. Doubling the fleet adds only about 1,000 x ln 2, roughly 693 miles.
The relationship
$D(n) = 1000\sum_{k=1}^{n}\frac{1}{k} \approx 1000\,(\ln n + 0.577)$
n the number of cars at the start
1000/k the leg driven while k cars remain
0.577 the Euler-Mascheroni constant
What it says in wordsEach leg is one tank shared among the cars still running, and the legs add up to a harmonic series.
Why a hedge fund asks it: the structure is the same as scaling a strategy. Each extra unit of capital or effort buys a smaller increment, and a candidate who sees the diminishing returns and names the rate of decay is showing the instinct the desk wants. Also be ready to argue optimality in one sentence: any plan that drops a car earlier wastes fuel it cannot hand over, and dropping later wastes the fuel spent carrying a car that is no longer needed.
Where candidates lose it
The quick wrong answers are n times 1,000, as if all the fuel could be pooled into one car, or a flat 1,000 because tanks cannot be overfilled. Both skip the key idea that the convoy itself consumes fuel while carrying the reserve.
Work n equals 2 out loud first: drive 500, pour the rest of car 2 into car 1, and drive 1,000 more, for 1,500. Then generalise. The interviewer wants the harmonic series and the words grows like log n.
What the interviewer asks next
Roughly how many cars do you need to travel 10,000 miles?
What changes if cars can come back to a depot and cache fuel along the road?
Where do you see diminishing returns of this shape in portfolio construction?
Asked at Millennium Management, Investments, London, 2024 (Wall Street Oasis): Suppose you have n cars, each fueled so that they can drive for 1000 miles.
045Two stock markets close at different times of day. Their same-day daily returns correlate at 0.30, and market B's return today correlates at 0.25 with market A's return yesterday. All other lagged correlations are zero, neither market's returns are autocorrelated, and both have the same daily variance. Roughly what is the correlation of their monthly returns?Statistics and forecastingHardAQR Capital ManagementGreenwich · 2022
Try it first
Is the monthly correlation higher, lower or the same as the daily 0.30?
Show the worked solution
About 0.54, close to 0.55. Add up 21 daily returns in each market. Each monthly variance is 21 times the daily variance. The monthly covariance collects 21 same-day terms of 0.30 and 20 lagged terms of 0.25, where A's day falls inside the same month as B's next day. So the correlation is (21 x 0.30 + 20 x 0.25) over 21, which is 0.538, tending to 0.55 for long windows.
Why would daily data understate how much two markets move together?
Two friends in different cities hear the same news; one hears it at lunch, the other the next morning. Compare their moods hour by hour on the same day and they look unrelated; compare their moods over the whole week and they look very similar. When two markets close at different times, news that arrives between the two closes shows up in one market's return today and the other's tomorrow, so same-day daily correlation misses part of the true co-movement. Longer windows add the delayed part back in.
News that arrives after market B has closed moves market A the same day and market B only the next day, which creates the 0.25 lagged correlation. Summing 21 days captures both the same-day 0.30 and 20 of the lagged pairs, so the monthly correlation is about 0.54.
How do you add up the covariance terms?
Write each month's return as the sum of its daily returns. The covariance of two sums is the sum of every pairwise covariance. With a unit daily variance, same-day pairs contribute 21 x 0.30, and the pairs of B's day t with A's day t minus 1 contribute 20 x 0.25, since only 20 such pairs sit inside a 21-day month. The variances are 21 each, with no autocorrelation to add. So the monthly correlation is (6.3 + 5.0) over 21, which is 0.538.
The relationship
$\rho_{M} = \frac{N\rho_0 + (N-1)\rho_1}{N} = 0.30 + 0.25 \times \frac{20}{21} \approx 0.54$
N trading days in the month, 21
\rho_0 the same-day daily correlation, 0.30
\rho_1 B today against A yesterday, 0.25
\rho_{M} the monthly correlation
What it says in wordsOver a month, the same-day and the one-day-lagged co-movement both count, less one lagged pair lost at the month's edge.
This matters for portfolio construction. A risk model built on daily same-day correlations between markets in different time zones will think they diversify each other far more than they do over the horizon an investor actually holds. The standard fixes are to use weekly or monthly returns, or to add lagged terms to the daily covariance, as this calculation does. The limitation: with only 12 monthly observations a year, the monthly estimate is itself noisy.
Where candidates lose it
Candidates often say correlation is a property of the two assets and does not change with the horizon, or that monthly data is noisier and so correlations must be lower. Both miss the timing effect, which is the entire question.
Draw the two closing times, say where the news lands, then add up the covariance terms. The interviewer wants to hear that you know lagged cross-correlation, and that you counted 20 lagged pairs rather than 21.
What the interviewer asks next
How would you estimate the lagged correlation from daily data?
If market B's returns were also positively autocorrelated, would the monthly correlation go up or down?
Why do daily correlations between Asian and US markets look lower than weekly ones?
Asked at AQR Capital Management, Quantitative Research, Greenwich, 2022 (Wall Street Oasis): what would be the difference between the correlation of daily vs monthly returns of a given year
047A Rs 100 crore bond portfolio holds Rs 40 crore of a bond with duration 2, Rs 35 crore with duration 5 and Rs 25 crore with duration 12. What is the portfolio's duration, and how much must move from the 2-year bond into the 12-year bond to lift it to 7?Bond mathsHardPIMCOLos Angeles · 2026
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How much must move from the duration-2 bond to the duration-12 bond to lift portfolio duration from 5.55 to 7?
Show the worked solution
Duration is 5.55, and Rs 14.5 crore must move from the 2-year bond to the 12-year bond. Portfolio duration is the value-weighted average: (40 x 2 + 35 x 5 + 25 x 12) over 100, which is 555 over 100, or 5.55. Each Rs 1 crore switched gains 10 years of duration on a hundredth of the portfolio, adding 0.1. Closing a gap of 1.45 needs Rs 14.5 crore.
Why is portfolio duration a simple weighted average?
Think of the average age of people in a room: each person counts in proportion to how many of them there are. Duration measures how much a bond's price moves for a one-point change in yield, and for a small parallel move the portfolio's rupee loss is just the sum of each bond's rupee loss, so its duration is the value-weighted average of the bonds' durations. The Rs 25 crore in the 12-year bond is only a quarter of the money but supplies more than half the duration: 3.00 of the 5.55.
Before the switch, the three bonds contribute 0.80, 1.75 and 3.00 years for a portfolio duration of 5.55. Moving Rs 14.5 crore from the 2-year to the 12-year bond changes the contributions to 0.51, 1.75 and 4.74, which total exactly 7.00.
What changes for the portfolio when duration goes from 5.55 to 7?
Solve for the switch with one line: the shift x changes duration by x times (12 minus 2) over 100, and that must equal 1.45. After the switch, a one-point parallel rise in yields costs about 7% of the portfolio, Rs 7 crore, instead of about 5.55%, Rs 5.55 crore. The portfolio gains more if yields fall and loses more if they rise. Its cash-flow profile also becomes a barbell, heavier at the long end, which gives it more convexity than a single bond with the same duration but more exposure to the long end of the curve if the curve steepens.
The relationship
$D_p = \sum_i w_i D_i = \frac{40(2) + 35(5) + 25(12)}{100} = 5.55 \qquad x = \frac{(7 - 5.55) \times 100}{12 - 2} = 14.5$
w_i each bond's share of portfolio value
D_i each bond's duration, in years
x the Rs crore switched from the 2-year to the 12-year bond
What it says in wordsDuration averages by value, so a switch moves it by the amount moved times the duration gap, over the portfolio's size.
Say the limits: a weighted average of durations describes small, parallel shifts in yields. If short and long yields move by different amounts, a portfolio at duration 7 built from a barbell behaves differently from one built from 7-year bonds, and the switch changes the portfolio's yield and credit mix as well. For large moves, convexity adds a second-order correction.
Where candidates lose it
The most common slip is taking a simple average of the three durations, 19 over 3, about 6.3, ignoring the amounts held. The other is solving for the switch but forgetting to divide by the portfolio size, which gives Rs 1.45 crore or some other scale error.
Say weighted by value first, give 5.55, then set up the switch as one equation. Close with what the higher duration means in rupees for a one-point move in yields; that sentence is the part the question actually asks about.
What the interviewer asks next
How would you reach duration 7 without selling any of the 2-year bond?
Why might the barbell at duration 7 behave differently from a bullet 7-year bond if the curve steepens?
What does a one-point parallel fall in yields do to the portfolio after the switch?
Asked at PIMCO, Generalist, Los Angeles, 2026 (Wall Street Oasis): Given a portfolio of these 3 bonds (I forgot exactly what they were) explain how the portfolio changes if duration increases.
049Fund rankings persist from one year to the next with a correlation of 0.2. A fund was in the top decile of its category last year. Where do you expect it to rank this year?Behavioural and decision trapsHardFund selectionPerformance analysis
Try it first
Where should you expect last year's top-decile fund to land this year?
Show the worked solution
Around the 64th percentile, still above average but most of the way back to the middle. On a normal scale, the top decile averages about 1.75 standard deviations above the median. With a correlation of 0.2, the expected position this year is a fifth of that, 0.35, which is about the 64th percentile. The quick linear version, 50 plus 0.2 x 45, gives the 59th.
Why should a top-decile fund be expected to fall back?
Think of a student who topped one exam. Part of that was knowing the subject and part was that the questions happened to suit them. Next time, the knowledge carries over and the luck does not. An extreme result is usually a mix of skill and a large dose of luck, and only the skill is expected to repeat, so the best forecast moves back towards the average by however much luck there was. A correlation of 0.2 says four fifths of last year's spread in rankings was the non-repeating kind.
With a year-to-year correlation of 0.2, the expected rank this year is a flat curve close to the 50th percentile. A top-decile fund, averaging about the 96th percentile last year, is expected around the 64th this year, far below the diagonal that perfect persistence would give.
How do you turn a rank correlation into an expected rank?
Work on a normal-score scale, where the regression is a straight line through the middle. The expected score this year is the correlation times last year's score, so a correlation of 0.2 keeps one fifth of the distance from average. The top decile's average score is the normal density at its cut-off, 1.28, divided by 0.1, which is about 1.75. A fifth of that is 0.35, and the normal table turns it into the 64th percentile. If you skip the normal scale and pull the 95th percentile a fifth of the way from 50, you get about the 59th, a fair first answer.
The relationship
$E[z_{t+1} \mid z_t] = \rho\, z_t \qquad \bar{z}_{top\,10\%} = \frac{\varphi(1.28)}{0.10} \approx 1.75 \qquad \Phi(0.2 \times 1.75) \approx 0.64$
\rho the year-to-year correlation of rankings, 0.2
z_t last year's normal score
\varphi, \Phi the standard normal density and cumulative probability
What it says in wordsNext year's expected score keeps only the correlation's share of this year's distance from the middle.
Then say what a fund selector does with it. Chasing last year's top decile buys mostly luck at a high price, often just as new money floods in. The answer is not to ignore performance but to weight longer records, look for a process that explains the result, and set expectations that even a genuinely good fund will usually rank well below where it did in its best year.
Where candidates lose it
The two extreme answers both lose the point. Expecting the fund to stay in the top decile ignores that a correlation of 0.2 is weak. Expecting it to fall below average overshoots: regression pulls results back towards the middle, not past it.
Say one fifth of the distance, give the 64th percentile on the normal scale or the 59th with the quick version, and explain that the difference comes from how extreme the top decile really is. Then draw the selection lesson.
What the interviewer asks next
What persistence correlation would keep a top-decile fund in the top quartile on average?
How does the answer change if you look at five-year rankings instead of one?
Why do flows into last year's best funds tend to make this problem worse?
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.
061Forecaster A predicts next year's index return with a bias of plus 1 point and an error standard deviation of 3 points. Forecaster B is unbiased, with an error standard deviation of 3.5 points. Whose mean squared error is lower, and what does an equal blend of the two give if their errors are independent?Statistics and forecastingHardBridgewater AssociatesNew York · 2024
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Which forecaster has the lower mean squared error?
Show the worked solution
A has the lower error, 10 against 12.25, and an equal blend cuts it to about 5.6. Mean squared error is bias squared plus variance: A is 1 plus 9, B is 0 plus 12.25. Averaging the two halves the bias to 0.5 and, because the errors are independent, quarters each variance: 0.25 plus 5.31, or 5.56. The blend beats both forecasters by a wide margin.
How can a biased forecaster beat an unbiased one?
Think of two watches. One always runs a minute fast but is otherwise steady; the other is right on average but wanders a few minutes either way. If you need to catch a train, the steady fast watch may serve you better. Mean squared error charges for two things, the average miss and the scatter around it, so a small steady bias can cost less than extra scatter. A's bias of 1 adds 1 to its error. B's standard deviation of 3.5 instead of 3 adds 3.25. On this measure, A is the better forecaster.
The relationship
$\text{MSE} = \text{bias}^2 + \sigma^2 \qquad A: 1 + 9 = 10 \qquad B: 0 + 12.25 = 12.25$
bias the forecaster's average error, in points of return
sigma the standard deviation of the error around that average
What it says in wordsThe average squared miss is the square of the average miss plus the scatter of the misses.
Why does averaging the two help so much?
Because independent errors partly cancel. When A is too high, B is as likely to be too low as too high, so the average of the two misses by less than either. An equal blend halves the bias and, with independent errors, cuts the variance to a quarter of the sum of the two variances. Here that is a quarter of 9 plus 12.25, which is 5.31, plus a bias term of 0.5 squared, 0.25. The total, 5.56, is little more than half of A's 10.
Forecaster A's error of 10 is 1 of bias squared plus 9 of variance, B's 12.25 is all variance, and an equal blend with independent errors cuts the total to 5.56 because averaging halves the bias and quarters each variance.
You can do slightly better by leaning towards A. The weight on A that minimises the error is B's variance over the sum of A's mean squared error and B's variance, about 55%, which gives 5.51. The gain over a plain 50/50 is tiny, which is the practical lesson: a simple average of decent, independent forecasts captures almost all of the benefit. The limitation is the word independent. Two economists reading the same data make correlated errors, and then averaging helps far less.
Where candidates lose it
The trap is choosing B on principle because unbiased sounds better. The question asks about mean squared error, and a candidate who does not split it into bias squared plus variance cannot compare the two.
The second loss is averaging the standard deviations for the blend, 3.25, and squaring it. Variances add, not standard deviations, and the blend divides each variance by four. Say independent out loud, because that assumption is doing the work.
What the interviewer asks next
What if the two forecasters' errors are correlated 0.8? What does the blend give then?
What weight on A minimises the blend's error, and why is it close to one half?
Why might a portfolio manager prefer the biased forecaster even though the blend is better?
Asked at Bridgewater Associates, Generalist, New York, 2024 (Wall Street Oasis): Was asked math questions about forecasting
066Five pirates, ranked A to E by seniority, must split 100 gold coins. The most senior pirate proposes a split and everyone votes. If at least half the votes, including his own, are in favour, the split stands; otherwise he is thrown overboard and the next pirate proposes. Pirates are perfectly rational, want to survive first and maximise coins second, and vote no when indifferent. What does A propose?Logic brainteasersHardHedge fundsQuantitative asset management
Try it first
How many coins does A keep?
Show the worked solution
A proposes 98 for himself, 0 for B, 1 for C, 0 for D and 1 for E. Solve from the end. With two pirates, D keeps all 100 because his own vote is half. With three, C buys E for 1 coin. With four, B buys D for 1. With five, A needs two votes and buys the two pirates who get nothing under B's plan, C and E, for one coin each, keeping 98.
Where do you start?
At the end, where there is no choice left. Think of planning a train journey with connections: you start from the time you must arrive and work back to when you must leave. A sequential game is solved backwards, because each pirate's vote depends only on what he would get if the current proposal failed. With two pirates left, D proposes 100 for himself; his own vote is half, which is enough. So E gets nothing if it ever comes to that, and E knows it.
Read the grid from the bottom up: each proposer keeps everything except one coin for each vote he needs, and he buys the pirates who would get nothing in the row below, so A ends with 98 and pays C and E one coin each.
How does each step follow from the one below?
With three pirates, C needs two votes, his own and one more. E gets 0 if C dies, so one coin buys E: C proposes [99, 0, 1]. With four, B needs two votes; under C's plan D gets 0, so B buys D for one coin: [99, 0, 1, 0]. A vote is worth exactly one coin more than that pirate's fallback, so a proposer always buys the cheapest voters, the ones left with nothing in the next round. With five, A needs three votes. Under B's plan C and E get nothing, so one coin each buys them, and A proposes [98, 0, 1, 0, 1].
State the assumptions, because the answer rests on them. If an indifferent pirate voted yes, A could buy votes for zero coins. If the rule needed a strict majority, the counts change. Interviewers often change one rule as a follow-up to see whether you rebuild the chain or reach for a memorised answer. The buy-side lesson is about incentives: what someone will accept depends on their alternative, not on fairness.
Where candidates lose it
The trap is reasoning forwards from fairness, proposing an even split or generous bribes to the next in line. Without the backward chain you cannot know who is cheap to buy, and B, the obvious ally, is in fact the most expensive vote because he inherits the power if A dies.
The second loss is skipping the stated assumptions. Say that indifferent pirates vote no and that exactly half passes; they decide whether the bribe is one coin or zero.
What the interviewer asks next
What if a proposal needs a strict majority rather than half?
With the same rules, what happens with 200 pirates and 100 coins?
Where do you see the same logic, what someone accepts depends on their outside option, in a debt restructuring?
067A sponsor buys a business at 8 times EBITDA of Rs 100 crore, funded 50% with debt. EBITDA grows 8% a year, and Rs 40 crore of free cash flow repays debt every year. What exit multiple after five years gives the sponsor 2.5 times its money?Private and real asset mathsHardNeuberger BermanNew York · 2022Neuberger BermanNew York · 2022
Try it first
Roughly what exit multiple does 2.5 times the money need?
Show the worked solution
About 8.2 times, barely above the 8 times paid. Entry value is Rs 800 crore, half debt, so equity is Rs 400 crore and the target is Rs 1,000 crore. Five years of Rs 40 crore repayments leave Rs 200 crore of debt, so the exit value must be Rs 1,200 crore. Year 5 EBITDA is 100 times 1.08 to the fifth, Rs 146.9 crore, and 1,200 over that is 8.17 times.
Why work backwards from the target?
Think of saving for a house: you start from the price you must reach and work out what monthly saving gets you there, instead of guessing savings and hoping. A paper LBO question that fixes the return is solved from the exit backwards, because the target return pins the equity value, and everything else follows from it. Equity in is half of Rs 800 crore, Rs 400 crore. 2.5 times that is Rs 1,000 crore of equity at exit.
The sponsor needs Rs 1,000 crore of equity and still owes Rs 200 crore of debt, so the business must sell for Rs 1,200 crore, which on year 5 EBITDA of Rs 146.9 crore is an exit multiple of 8.17 times.
Which step do candidates drop?
The debt still owed. The equity holders only get what is left after the lenders are repaid, so the enterprise value at exit is the equity target plus the remaining debt: Rs 1,000 crore plus Rs 200 crore. Enterprise value belongs to lenders and owners together, so you always add the debt back before dividing by EBITDA. Dividing Rs 1,000 crore alone by Rs 146.9 crore gives 6.8 times and a wrong story about a deal that works even with a lower multiple.
The relationship
$m_{exit} = \frac{2.5 \times 400 + (400 - 5 \times 40)}{100 \times 1.08^5} = \frac{1{,}200}{146.9} = 8.17\times$
2.5 x 400 the equity the sponsor needs back, Rs crore
400 - 5 x 40 debt still owed after five annual repayments
100 x 1.08^5 year 5 EBITDA, Rs crore
What it says in wordsThe exit multiple is the equity target plus remaining debt, divided by exit EBITDA.
Of the Rs 600 crore gain, EBITDA growth valued at the entry multiple supplies Rs 375 crore and debt paydown Rs 200 crore, so the multiple only needs to add Rs 25 crore, a rise from 8.0 to 8.17 times.
Then give the view. 2.5 times in five years is an IRR of about 20%, and this deal gets there almost entirely from growth and paydown. That is the comfortable kind of LBO: the return does not depend on a buyer paying more than the sponsor did. Say the simplifications as well: free cash flow is held flat at Rs 40 crore although EBITDA grows, and fees, interest on the debt and taxes are folded into that figure.
Where candidates lose it
The trap is forgetting the Rs 200 crore of debt still outstanding and dividing the equity target by EBITDA, which gives 6.8 times and makes the deal look safer than it is. The exit value must cover the lenders before the sponsor sees a rupee.
The second loss is stopping at the number. The interviewer wants to hear that 8.2 times against 8 times paid means the return is driven by growth and paydown, not by hoping for multiple expansion.
What the interviewer asks next
What IRR does the deal earn if the exit multiple falls to 7 times?
How does the answer change if the sponsor uses 60% debt and repays the same Rs 40 crore a year?
Why might free cash flow for debt repayment not stay flat at Rs 40 crore as EBITDA grows?
Asked at Neuberger Berman, Private Equity, New York, 2022 (Wall Street Oasis): The most difficult was the more advanced industry-specific technicals and paperback LBOs
Asked at Neuberger Berman, Private Equity, New York, 2022 (Wall Street Oasis): Interviews 4-5 were very technical again and also included multiple paperback LBOs and other, more advanced technicals.
072An equity index trades at 22 times forward earnings, pays out 40% of earnings as dividends, and its long-run earnings growth is expected to be 10% a year. The 10-year government bond yields 7%. Which asset is cheaper?Valuation riddlesHardPIMCOSan Diego · 2026
Try it first
The earnings yield is 4.5% and the bond yields 7%. What does that comparison tell you?
Show the worked solution
On expected return, equities offer about 11.8% against the bond's 7%, a premium of about 4.8 points. The earnings yield, 1 over 22 or 4.5%, looks worse than 7%, but it ignores growth. The dividend yield is 40% of 4.5%, or 1.82%, and adding 10% growth gives about 11.8%. Whether equities are cheaper depends on whether a 4.8-point premium pays enough for equity risk.
Why is 4.5% against 7% the wrong comparison?
Compare a fixed-rent lease with a shop whose profits grow each year. The lease might pay more in year one, but the shop's income keeps rising. A bond's yield is everything it will ever pay, while an earnings yield is only the starting point of a stream expected to grow, so the two cannot be compared directly. The earnings yield of 4.5% sits 2.5 points below the bond, and that gap says almost nothing about which is cheaper.
The earnings yield of 4.5% looks worse than the bond's 7%, but the index's expected return, a 1.8% dividend yield plus 10% growth, is about 11.8%, a premium of about 4.8 points over the bond.
How do you put them on the same footing?
Estimate the equity's expected return the way you would a bond's. For a stock or an index, that is roughly the dividend yield plus long-run growth, the logic of the Gordon growth model. The dividend yield is the payout ratio times the earnings yield, and the growth rate does the rest. Here 40% of 4.55% is 1.82%, and 10% growth takes the total to 11.82%. Against 7% on the bond, equities offer 4.82 extra points a year.
The relationship
$E[R] \approx \frac{D}{P} + g = 0.40 \times \frac{1}{22} + 10\% = 1.82\% + 10\% = 11.82\%$
D/P the dividend yield, the payout ratio times earnings over price
g long-run growth in earnings and dividends, 10%
What it says in wordsAn equity's expected return is roughly its dividend yield plus the rate at which its dividends grow.
Then test the growth number, because the answer rests on it. Growing earnings 10% while paying out 40% means reinvesting 60% at a return on equity of about 16.7%, which is demanding for a whole market. If growth were 8%, the premium would shrink to about 2.8 points. A view on which asset is cheaper is really a view on whether that premium, after testing growth, pays enough for the extra risk of equities. That is the judgement to state, with the numbers that drive it.
Where candidates lose it
The trap is comparing the earnings yield with the bond yield and declaring bonds cheaper. That comparison ignores growth and treats a rising income stream as if it were fixed.
The second loss is taking the 10% growth at face value. Check it against the payout ratio: growth needs reinvestment, and the implied return on equity tells you whether the number is plausible.
What the interviewer asks next
What equity risk premium would you need to call equities and bonds fairly valued here?
How does inflation change the comparison between an earnings yield and a nominal bond yield?
What growth rate makes the index's expected return exactly equal to 7%?
Asked at PIMCO, Debt Capital Markets, San Diego, 2026 (Wall Street Oasis): Which is cheaper us bonds or us equities
073Estimate how many individuals in India hold a direct equity portfolio worth more than Rs 50 lakh, the minimum ticket for a portfolio management service.Market sizing and estimationHardIndian wealth managementIndian asset management
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Which step decides the answer most?
Show the worked solution
Roughly 5 lakh people, within a range of about 3 to 10 lakh. Start from about 18 crore demat accounts, an assumption to confirm, and divide by 1.5 accounts per person: 12 crore individuals. Perhaps 40% hold a meaningful portfolio, 4.8 crore. Wealth is heavily skewed, so assume about 1 in 100 of those hold more than Rs 50 lakh: about 4.8 lakh. A Pareto-tail check gives the same order of magnitude.
Why is the headline account count a trap?
Think of a cricket academy with thousands of registered players. Asking how many could play first-class cricket is not a question about registrations; it is about the very top of the talent curve. Portfolio sizes are highly skewed, so the count above a high threshold is a small slice of the total, and the size of that slice, not the headline, decides the answer. Crores of accounts become a few lakh people above Rs 50 lakh.
About 18 crore demat accounts shrink to 12 crore individuals and 4.8 crore with meaningful portfolios, and only about 1 in 100 of those holds more than Rs 50 lakh, which is roughly 4.8 lakh people.
How do you defend the 1 in 100?
With a second route. Suppose 20% of the 4.8 crore holders have more than Rs 5 lakh. Wealth tails often thin out like a Pareto distributionA skewed distribution in which the share above any level falls by a fixed power as the level rises, often used for wealth and city sizes.: raise the threshold tenfold and the share falls by a factor of 10 to about 20. Two independent routes landing in the same range is what turns a guess into an estimate. From Rs 5 lakh to Rs 50 lakh is a tenfold rise, so between 4.8 and 9.6 lakh people sit above Rs 50 lakh, which brackets the first answer.
The relationship
$N = \frac{18}{1.5} \times 0.40 \times 0.01 \text{ crore} = 4.8 \text{ lakh}$
18 demat accounts, crore, a round assumption
1.5 accounts per individual
0.40 share holding a meaningful portfolio
0.01 share of those above Rs 50 lakh
What it says in wordsAccounts to people, people to active holders, then the thin top slice above the threshold.
Then say what the number is for, and what it misses. A portfolio management service can also be funded from bank deposits, mutual fund units or property sales, so the true addressable pool is wider than direct equity holders. The Rs 50 lakh minimum and every count used here are figures to confirm against current SEBI rules and depository data; the structure of the estimate is the part to trust.
Where candidates lose it
The trap is quoting the demat account count, or some large share of it, as the answer. It confuses accounts with people and the whole distribution with its top slice, and it lands a hundred times too high.
The second loss is presenting assumptions as facts. Say each number as a round assumption, show the skew check, and give a range, not a single figure said with false precision.
What the interviewer asks next
How would the answer change if the threshold rose to Rs 1 crore?
What data would you use to test the 1 in 100 assumption?
Why might the addressable market for this service be larger than the count of direct equity holders?

n	the number of cars at the start
1000/k	the leg driven while k cars remain
0.577	the Euler-Mascheroni constant

N	trading days in the month, 21
\rho_0	the same-day daily correlation, 0.30
\rho_1	B today against A yesterday, 0.25
\rho_{M}	the monthly correlation

w_i	each bond's share of portfolio value
D_i	each bond's duration, in years
x	the Rs crore switched from the 2-year to the 12-year bond

\rho	the year-to-year correlation of rankings, 0.2
z_t	last year's normal score
\varphi, \Phi	the standard normal density and cumulative probability

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