Portfolio Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 31
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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
Try it first
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.
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
Try it first
Before you discount anything: where does the price land?
Show the worked solution
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
Try it first
Which pair is closest?
Show the worked solution
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?
058Every stock in a market has 35% volatility and every pair of stocks has a correlation of 0.25. What is the volatility of an equally weighted portfolio of 1 stock, of 10 stocks, and of infinitely many?Portfolio risk mathsCoreNorthern TrustChicago · 2025Northern TrustChicago · 2025
Try it first
With infinitely many stocks, where does portfolio volatility settle?
Show the worked solution
35% for one stock, about 20% for ten, and a floor of 17.5% for infinitely many. Portfolio variance is the stock variance times one over n, plus the correlation times what is left. With ten stocks that is 0.1225 times 0.325, a volatility of 20.0%. As n grows the one-over-n part vanishes and only the correlation term remains: 35% times the square root of 0.25, or 17.5%.
Why does adding stocks lower risk at all?
Ten shops in ten different towns do not all have a bad week at once; ten shops in one mall often do. Some of what moves a stock is its own news and some is the market everyone shares. Stock-specific shocks cancel out as you add names, because one company's bad quarter is offset by another's good one, but the shared market shock hits every name together and does not cancel. Correlation measures how much of each stock's movement is shared.
The relationship
$\sigma_p^2 = \sigma^2\left[\frac{1}{n} + \left(1-\frac{1}{n}\right)\rho\right] \;\to\; \sigma\sqrt{\rho} = 0.35 \times 0.5 = 17.5\%$
sigma each stock's volatility, 35%
rho the correlation between any two stocks, 0.25
n the number of stocks, equally weighted
What it says in wordsPortfolio variance is a shrinking stock-specific part plus a fixed shared part, and only the shared part survives as the portfolio grows.
Portfolio volatility falls from 35% with one stock to 20.0% with ten and then flattens towards a floor of 17.5%, because adding names removes stock-specific risk but cannot remove the risk all the stocks share.
How much of the benefit do the first ten stocks deliver?
Most of it. Going from one stock to ten cuts volatility from 35% to 20.0%, about 86% of the whole distance to the floor. Going from ten to thirty takes it only to 18.4%. Diversification pays off quickly and then almost stops, and the level where it stops is set by correlation, not by the number of holdings. That is why a manager worried about risk gains more from adding assets that are less correlated than from adding a fortieth stock of the same kind.
Say the limitation. Correlations are not fixed: in a sell-off they tend to rise together, which raises the floor exactly when diversification is needed. Real stocks also differ in volatility and correlation, so this uniform market is a teaching model; the shape of the curve survives, the exact numbers do not.
Where candidates lose it
The trap is saying diversification takes risk to zero, or reaching for the correlation without the square root and answering 8.75%. The floor is the square root of the shared variance, so it is volatility times the square root of the correlation.
The second miss is getting 20% for ten stocks by guesswork and being unable to show it. Write the variance formula first and plug in: 0.1225 times 0.1 plus 0.9 times 0.25.
What the interviewer asks next
What correlation would make a 10-stock portfolio half as risky as one stock?
Why do correlations tend to rise in a market sell-off, and what does that do to this floor?
How would you lower the floor itself rather than approach it?
Asked at Northern Trust, Asset Management, Chicago, 2025 (Wall Street Oasis): First one was more technical and asked about my understanding of AM, portfolio diversification and strategy
Asked at Northern Trust, Asset Management, Chicago, 2025 (Wall Street Oasis): Asked about my understanding of asset management, portfolio diversification and strategy
059Why should I buy your college, and how much would you sell it for? Suppose it earns an operating surplus of Rs 40 crore this year, the surplus grows 5% a year for the foreseeable future, and a buyer wants a 12% return.Valuation riddlesCoreWellington ManagementBoston · 2024
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What price does the growing surplus support, before land?
Show the worked solution
About Rs 600 crore for the operating business, before any value in the land. Next year's surplus is Rs 40 crore grown 5%, or Rs 42 crore. A surplus that grows forever at 5% and is valued at 12% is worth next year's amount divided by the 7-point gap: 42 over 0.07 is Rs 600 crore. The reason to buy is the durability of that surplus: steady demand for seats and fees that can rise with costs.
What is the question really asking?
It is a stock pitch in disguise. The interviewer wants two things in order: why this asset produces reliable cash, and what that cash is worth. Answer the why with the quality of the surplus, and the how much with a valuation you can do out loud. For a college the why is simple to say: students keep applying every year, fees are paid in advance, and a college with a good name can raise fees roughly in line with its costs. Those are the reasons the surplus can be treated as growing and durable.
Why divide by 12% less 5%?
Think of a rented flat whose rent rises every year. A buyer asking for a 12% return on a rent that grows 5% needs only 7% from the current rent; the other 5% arrives through growth. A cash flow growing at g forever, valued at a required return r, is worth next year's cash flow divided by r minus g. Here that is Rs 42 crore over 0.07, which is Rs 600 crore. Dividing this year's Rs 40 crore instead gives Rs 571 crore, a common small slip: the buyer receives next year's surplus, not this year's.
The relationship
$V = \frac{S_1}{r - g} = \frac{40 \times 1.05}{0.12 - 0.05} = \frac{42}{0.07} = 600$
S_1 next year's surplus, Rs crore
r the buyer's required return
g the permanent growth rate of the surplus
What it says in wordsA growing perpetuity is worth next year's payment divided by the gap between the required return and the growth rate.
With growth fixed at 5%, the college is worth Rs 600 crore at a 12% required return, but Rs 700 crore at 11% and Rs 525 crore at 13%, because value depends on the gap between return and growth and that gap is small.
Now say what the number is sensitive to. One point on the required return moves the value by Rs 100 crore up or Rs 75 crore down, because a 7-point gap becoming 6 or 8 is a large change in proportion. The land and buildings may be worth more than the operating surplus, so a seller would also ask what the campus fetches as property. And check the structure before promising anyone the surplus: many colleges, in India among other places, are run by trusts or societies, and whether an owner can take surplus out at all is a legal question to confirm.
Where candidates lose it
The trap is diving into a formula without answering why. The question starts with why should I buy, and a candidate who opens with a number has skipped the half the interviewer cares about most.
The arithmetic trap is dividing Rs 40 crore by 12%, which treats a growing surplus as flat and values it at Rs 333 crore. Name the growth, use next year's surplus, and divide by the gap.
What the interviewer asks next
What growth rate is the seller implicitly assuming if he asks Rs 800 crore?
How would you value the land separately, and when would it exceed the value of the operating business?
What would make you use a higher required return for this college than for a listed education company?
Asked at Wellington Management, Investment Research, Boston, 2024 (Wall Street Oasis): Why should I buy your College and how much would you sell it for?
060Estimate how many narrow-body passenger jets, the single-aisle planes used on most short and medium routes, are delivered to airlines worldwide in a year.Market sizing and estimationCoreRothschild & CoParis · 2026
Try it first
What drives yearly deliveries of a plane that flies for decades?
Show the worked solution
About 1,200 a year, within a range of roughly 1,000 to 1,500. Size the fleet first: about 100,000 flights a day, three quarters on narrow-bodies, at five flights per jet a day, needs about 15,000 jets. Yearly demand is replacement plus growth: 15,000 over a 25-year life is 600, and 4% growth adds 600. Every input is an assumption to state and then check against published fleet data.
Where do you start with a plane that lasts 25 years?
Think of refrigerators in a city. Almost every home has one, but the shops sell only the ones that break down plus the ones new homes need. For a long-lived asset, yearly sales are a small flow out of a large stock: the stock divided by its life, plus the stock times its growth rate. So the job splits into two parts. Size the fleet from how the planes are used, then turn the fleet into a yearly flow.
About 75,000 narrow-body flights a day at five flights per jet needs a fleet of about 15,000, and replacing a twenty-fifth of that each year plus 4% growth gives roughly 1,200 deliveries a year.
How do you size the fleet without knowing it?
From usage, which is easier to guess. Take about 100,000 commercial flights a day worldwide, a round number to state as an assumption. Most flights are short and medium routes, so put three quarters on narrow-bodies: 75,000. A narrow-body on short routes flies several sectors a day; call it five. That needs 15,000 jets. Each assumption is a round number said out loud, so the interviewer can challenge one without losing the structure.
The relationship
$D = \frac{F}{L} + gF = \frac{15,000}{25} + 0.04 \times 15,000 = 1,200$
D deliveries a year
F the narrow-body fleet
L service life in years
g the fleet's yearly growth rate
What it says in wordsYearly deliveries are the jets that retire plus the jets that growth adds.
Now sanity-check. 1,200 deliveries is about 8% of the fleet, which sounds right for an asset that lasts decades. If the question asks about one maker, split by an assumed market share; two main makers sharing the market would each deliver around 600. For an asset manager the useful point is the sensitivity: deliveries swing with the growth assumption far more than the fleet does, which is why aircraft makers' order books are so cyclical. Treat every number here as an assumption to check against published fleet and delivery data before using it.
Where candidates lose it
The trap is sizing the whole fleet and calling that the yearly number, or dividing all passengers by seats per plane. Both answer how many jets exist, not how many are delivered, and they are wrong by a factor of ten or more.
The second loss is presenting assumptions as facts. Say each input as a round assumption, then show the answer moves predictably when one changes.
What the interviewer asks next
If fleet growth drops to zero for two years, what happens to deliveries?
How would you estimate the value of those deliveries in dollars a year?
What would you check first to test the 25-year life assumption?
Asked at Rothschild & Co, Asset Management, Paris, 2026 (Wall Street Oasis): Can You estimate number of flights solds by airbus
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
Try it first
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
064A corporate bond yields 180 basis points more than a government bond of the same maturity and has a spread duration of 5. How far can its spread widen over the next year before it earns no more than the government bond?Bond mathsCoreAQR Capital ManagementGreenwich · 2021
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How much widening does the 180 basis point spread absorb?
Show the worked solution
About 36 basis points. Over a year the bond earns 180 basis points more than the government bond. Every basis point the spread widens knocks about 5 basis points off its price, because its spread duration is 5. The extra yield is used up when widening times 5 equals 180, which is at 36 basis points. Beyond that the corporate bond does worse than the government bond.
What is the cushion, and what eats it?
Think of a shop that earns a steady margin on every sale but whose stock loses value when fashions change. The margin comes in slowly; a markdown hits all at once. A credit spread pays carry slowly over the year, while widening hits the price immediately, in proportion to spread duration. The carry is 180 basis points. A spread duration of 5 means a 1 basis point widening costs about 5 basis points of price.
The bond earns 1.80% more than the government bond if spreads do not move, loses 0.05% for every basis point of widening, and so falls behind once spreads widen more than 36 basis points.
The relationship
$\Delta s^* = \frac{\text{spread}}{\text{spread duration}} = \frac{180}{5} = 36 \text{ bp}$
Delta s* the widening at which the extra return is zero
spread the extra yield over the government bond, 180 basis points
spread duration the percentage price change for a 1 point change in spread, 5
What it says in wordsDivide the extra yield by the spread duration to find how much widening it can absorb.
What would you add to sound like a credit investor?
Two refinements, both worth a sentence. First, part of the spread pays for defaults, not risk. If expected default losses were 50 basis points a year, an illustration, only 130 is true cushion and the break-even falls to 26 basis points. The spread is not all profit, so the honest break-even uses the spread after expected losses. Second, the price loss is felt at the end of the year, when the bond is shorter; at a spread duration of about 4.2 then, the break-even is closer to 43. The 36 is the conservative, quick answer.
The ratio also compares bonds quickly. A short bond with a small spread can have a wider break-even than a long bond with a big one, because the long bond's duration magnifies every move. Credit portfolio managers often rank bonds by spread per unit of spread duration for exactly this reason.
Where candidates lose it
The trap is saying 180 basis points, as if the spread could widen by its own size before the bond loses out. Candidates forget that duration multiplies every basis point of widening into a larger price loss.
The second loss is treating the whole spread as profit. Mention expected default losses; it shows you know why the spread exists in the first place.
What the interviewer asks next
A 2-year bond at 90 bp and a 10-year at 220 bp with spread duration 8: which has the wider break-even?
How does roll-down along the credit curve change this answer?
Why might a portfolio manager hold the bond even if she expects 50 bp of widening?
Asked at AQR Capital Management, Investment Research, Greenwich, 2021 (Wall Street Oasis): Discussion on credit spreads on fixed income products and duration.

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

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

M	the money multiple, cash back over cash in
n	the years held, with one cash flow in and one out

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