Portfolio Management puzzles, solved step by step

Puzzles: 100
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041Two stocks both have an 11% cost of equity. The value stock's dividends grow 3% a year forever and the growth stock's grow 8%. Using a constant-growth model, what is each stock's equity duration, and roughly how much does each fall if the cost of equity rises by 50 basis points?Valuation riddlesHardBlackRockNew York · 2026BlackRockNew York · 2026
Try it first
Which stock is more sensitive to a rise in the discount rate, and by roughly how much?
Show the worked solution
Durations of 12.5 and 33.3 years; the value stock falls about 6% and the growth stock about 14%. With price equal to D over (r minus g), duration is 1 over (r minus g): 1 over 0.08 and 1 over 0.03. A 50 basis point rise takes the spreads to 8.5 and 3.5 points, so prices fall by 1 minus 8/8.5, which is 5.9%, and 1 minus 3/3.5, which is 14.3%.
Why does a stock have a duration at all?
Think of two people valuing a lottery ticket: one pays out a small sum every year starting now, the other pays little now and a lot decades from now. If the interest rate rises, the second ticket loses far more value, because its money is further away and gets discounted for longer. An equity is a stream of future cash flows, so like a bond it has a duration: the weighted distance to its cash, and the growth stock's cash sits much further out. In the constant-growth model that distance comes out as 1 over (r minus g).
At an 11% cost of equity, the value stock growing 3% has a duration of 12.5 years and loses 5.9% if the rate rises half a point, while the growth stock growing 8% has a duration of 33.3 years and loses 14.3%. The dashed lines show that the straight duration estimate overstates both falls.
How do you get the price changes exactly, and why is the duration estimate too big?
The price is proportional to 1 over (r minus g), so compare spreads before and after. For the value stock the spread moves from 8 to 8.5 points, a 5.9% price fall; for the growth stock it moves from 3 to 3.5, a 14.3% fall. Duration times the rate change gives 6.25% and 16.7%, a little larger, because price is a curved function of the rate: like a bond with convexity, the stock loses less than the straight-line estimate. The curvature matters more the longer the duration.
The relationship
$P = \frac{D_1}{r-g} \qquad D_{eq} = -\frac{1}{P}\frac{dP}{dr} = \frac{1}{r-g} \qquad \frac{P_{new}}{P_{old}} = \frac{r-g}{r + \Delta r - g}$
D_1 next year's dividend
r the cost of equity, 11%
g the perpetual growth rate, 3% or 8%
D_{eq} equity duration, the percentage price change per point of rate change
What it says in wordsIn a constant-growth model, sensitivity to the discount rate is one over the gap between the rate and the growth rate.
This is the arithmetic behind a familiar market pattern: when real yields rise sharply, long-duration growth stocks usually fall more than value stocks. Say the limitation too. The model assumes growth is fixed while the rate moves; in practice rates often rise because growth is strong, which can offset part of the fall. And no real company grows at 8% forever, so the growth stock's duration is a stylised upper figure.
Where candidates lose it
Candidates often say both stocks move the same because they share a cost of equity, or they compute the percentage change of r, about 4.5%, and apply it to both prices. Neither uses the spread between r and g, which is the whole mechanism.
State the formula, name the spread, and give the two durations first. Then offer the exact falls and say why they are smaller than the duration estimate.
What the interviewer asks next
What happens to the growth stock's duration if its growth rate rises to 10%?
Why might a rate rise driven by stronger growth hurt growth stocks less than this model suggests?
How would you hedge the rate sensitivity of a growth-heavy portfolio?
Asked at BlackRock, Restructuring, New York, 2026 (Wall Street Oasis): Which equities have duration ? multiple stocks vs value stocks
Asked at BlackRock, Risk and Quantitative Analysis, New York, 2026 (Wall Street Oasis): Which equities have duration? VaR, market views, stock valuation.
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
Try it first
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
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.
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
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
074An analyst takes ten years of monthly data, builds rolling 12-month fund and market returns sampled every month, regresses one on the other, and reports an alpha with a t-statistic of 3.6 from ordinary least squares. Why is that overstated, and roughly what is an honest t-statistic?Statistics and forecastingHardAQR Capital ManagementGreenwich · 2022AQR Capital ManagementGreenwich · 2022
Try it first
Roughly how large is the honest t-statistic?
Show the worked solution
Because the observations overlap, the honest t-statistic is close to 1, about 1.04. Ten years of monthly data give 109 rolling 12-month windows, but each shares 11 months with its neighbour. OLS assumes independent errors, so it treats 109 observations as 109 pieces of information. With the overlap, the variance of the estimate is understated about twelvefold, and 3.6 over the square root of 12 is about 1.0.
Which OLS assumption breaks?
Picture asking the same twelve people for their opinion, then swapping one person each month and asking again. You would not claim 109 independent surveys. OLS standard errors assume the regression errors are uncorrelated with each other, and overlapping windows make them strongly correlated, so the standard errors come out far too small. The alpha estimate itself is not biased; what is wrong is the claim about how precisely it is known, which is exactly what the t-statistic reports.
Two 12-month windows one month apart share 11 months, so their correlation is 11 over 12 and falls in equal steps to 1 over 12 at an 11-month gap, which inflates the true variance of the estimate about 12-fold.
How do you get to about 1?
Add up the overlap. If monthly returns are independent, two 12-month windows k months apart have a correlation of 12 minus k over 12. The variance of an average of such overlapping sums is inflated by one plus twice the sum of those correlations, which here is 1 plus 2 times 5.5, or 12. The t-statistic shrinks by the square root of the inflation factor, 3.6 divided by the square root of 12. That gives about 1.04: nowhere near significant, and consistent with having only about ten independent years.
The relationship
$\text{VIF} = 1 + 2\sum_{k=1}^{11}\frac{12-k}{12} = 12 \qquad t^* = \frac{3.6}{\sqrt{12}} = 1.04$
VIF variance inflation from the overlap
(12-k)/12 the correlation of two windows k months apart
t* the corrected t-statistic
What it says in wordsOverlap inflates the true variance about twelvefold, so divide the naive t-statistic by the square root of twelve.
How would you fix it? Three answers, in order of simplicity. Run the regression on non-overlapping monthly returns, which uses all the data with independent errors. Or keep the overlap and use standard errors that allow for autocorrelation, such as Newey-West or Hansen-Hodrick with 11 lags. Or sample the 12-month returns once a year, which leaves only ten points but honest ones. The interviewer wants to hear that the fix is about the standard errors, not the coefficient.
Where candidates lose it
The trap is defending 3.6 because the sample has over a hundred observations. The count of rows is not the count of independent pieces of information, and that is the whole point of the question.
The second loss is saying the alpha estimate itself is biased. It is not; the problem is its precision. Name the broken assumption, uncorrelated errors, and the fix, corrected standard errors or non-overlapping data.
What the interviewer asks next
What other OLS assumptions matter most for return regressions, and how would you check them?
Why do overlapping returns make long-horizon predictability look stronger than it is?
How many lags would you use in Newey-West standard errors here, and why?
Asked at AQR Capital Management, Investments, Greenwich, 2022 (Wall Street Oasis): they asked about how to fix ols assumptions
Asked at AQR Capital Management, Investments, Greenwich, 2022 (Wall Street Oasis): How would you fix violations of the OLS assumptions?
075A callable bond is priced at 100.0. If yields rise 50 basis points its price falls to 98.9; if yields fall 50 basis points it rises only to 100.6. What are its effective duration and effective convexity?Bond mathsHardAmundiLondon · 2018
Try it first
What sign does the convexity take?
Show the worked solution
Effective duration is about 1.7 and effective convexity about -200. Duration is the price difference across the two shocks over twice the price times the shock: 1.7 over 1.0, which is 1.7. Convexity is the sum of the shocked prices less twice the base, over the price times the shock squared: minus 0.5 over 0.0025, which is -200. The call caps the price, so it gains less than it loses.
Why effective duration rather than the usual formula?
Think of renting out a flat on a lease the tenant can cancel whenever cheaper flats appear. When rents fall, the tenant leaves and you do not keep the high rent; when rents rise, you are stuck. A callable bond's cash flows change with yields, because the issuer calls it when rates fall, so you measure its duration from how its price actually moves, not from a fixed schedule of coupons. That is effective duration: shock the yield both ways, reprice, and read the slope.
The relationship
$D_{eff} = \frac{P_- - P_+}{2P_0\Delta y} = \frac{100.6 - 98.9}{2 \times 100 \times 0.005} = 1.7 \qquad C_{eff} = \frac{P_- + P_+ - 2P_0}{P_0 \Delta y^2} = \frac{-0.5}{0.0025} = -200$
P_- price when yields fall 50 bp, 100.6
P_+ price when yields rise 50 bp, 98.9
P_0 the starting price, 100.0
Delta y the shock, 0.005
What it says in wordsDuration is the average slope across the two shocks; convexity is how much the two moved prices bend away from a straight line.
The callable bond rises only to 100.6 when yields fall 50 basis points, because the call caps it below 101, but falls to 98.9 when yields rise, so its effective duration is 1.7 and its convexity is negative, about -200.
What does negative convexity cost the holder?
It means the bond loses more when yields rise than it gains when they fall. The holder has sold the issuer an option to refinance, and the price of that option is the upside given up when rates fall. A straight bond with a duration of about 4 would gain about 2.0 points for a 50 basis point fall; this one gains 0.6. The holder is paid for this through a higher yield than an equivalent straight bond, and the question for a portfolio manager is whether that extra yield covers the option given away.
Two more things are worth saying. The duration of 1.7 is not fixed: as yields fall towards the level where a call becomes likely, duration shrinks further, and as they rise, it lengthens towards the bond's straight duration. That shifting is why callable bonds, and mortgage securities with the same feature, need effective measures rather than the textbook formulas. The curve in the figure is a stylised fit around the three prices given; a real one would come from an option pricing model.
Where candidates lose it
The trap is computing a positive convexity by habit, or dividing by the shock rather than the shock squared and getting minus 1. Write the formula, put the sign of 100.6 plus 98.9 minus 200 on the page, and the negative number is obvious.
The second loss is giving the numbers without the story. Say that the call caps the upside, which is why duration is short and convexity negative.
What the interviewer asks next
Estimate the price change for a 100 basis point fall using this duration and convexity. Why might it be wrong?
Why does a callable bond's effective duration lengthen when yields rise?
What would the price-yield curve of a putable bond look like?
Asked at Amundi, Rates, London, 2018 (Wall Street Oasis): What would your allocation be in today's market? What is effective duration?
095A fund returned 17% in a year when the risk-free rate was 6%, the market's excess return 8%, the size factor SMB 2% and the value factor HML 3%. Its loadings are 1.1 on the market, 0.4 on size and minus 0.3 on value. How much of the return do its factor exposures explain, and what is its alpha?Statistics and forecastingHardState StreetCambridge · 2019
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What is the fund's alpha?
Show the worked solution
The factors explain 14.7%, leaving alpha of 2.3%. Start from the 6% risk-free rate and add each loading times its factor return: 1.1 x 8 = 8.8 for the market, 0.4 x 2 = 0.8 for size, and minus 0.3 x 3 = minus 0.9 for value. The fund's 11 points over cash are mostly priced exposure; 2.3 points is left for skill.
Why not measure the fund against cash or the market alone?
A tutor whose students all score well may simply have been given the strongest students. To judge the teaching, you first adjust for who walked in. A fund with a beta above 1 and a small-cap tilt should earn more than the market in a year when the market and small caps did well, so its exposures must be priced before anything is called skill. The Fama and French three-factor model does exactly that with market, size and value.
From a 6% risk-free rate, market exposure adds 8.8, the size tilt 0.8 and the value tilt takes away 0.9, so the factors explain 14.7% and only 2.3 points of the fund's 17% remain as alpha.
The relationship
$R = R_f + \beta_M\,\text{MKT} + \beta_S\,\text{SMB} + \beta_V\,\text{HML} + \alpha = 6 + 8.8 + 0.8 - 0.9 + 2.3$
beta_M, beta_S, beta_V the fund's loadings on market, size and value: 1.1, 0.4, minus 0.3
MKT, SMB, HML the factor returns that year: 8, 2 and 3 points
alpha the part no factor explains
What it says in wordsEach exposure earns its factor's return; the leftover is alpha.
What does the split tell you about the manager?
Against the plain market's 14%, the fund beat by 3 points. Of those 3 points, 0.7 came from priced tilts in net terms, more beta and small caps less the growth tilt, and 2.3 is left over. The negative value loading matters too: a growth-leaning fund lost 0.9 points in a year value did well, so part of the manager's alpha was earned while swimming against a factor.
Then state the limits. One year of alpha is mostly noise; you would want a regression over many years with a standard error before calling 2.3 points skill. And the answer depends on the model: add momentum or quality factors and some of the 2.3 may turn out to be another priced exposure.
Where candidates lose it
The common slip is calling 11% or 3% the alpha, measuring against cash or the market without adjusting for beta and tilts. The interviewer is testing whether you price every exposure before crediting skill.
The second slip is sign handling on the value loading. A negative loading in a year the factor rose is a cost, minus 0.9, not a gain.
What the interviewer asks next
If HML had been minus 3% that year, what would the alpha be?
How would you tell whether 2.3% of alpha is statistically meaningful?
Why might adding a momentum factor change the answer?
Asked at State Street, Investment Banking, Cambridge, 2019 (Wall Street Oasis): some basic market knowledge, such as factor model (Fama French), portfolio optimization, risk analysis

D_1	next year's dividend
r	the cost of equity, 11%
g	the perpetual growth rate, 3% or 8%
D_{eq}	equity duration, the percentage price change per point of rate change

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

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Why does a stock have a duration at all?

How do you get the price changes exactly, and why is the duration estimate too big?

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When should a car drop out of the convoy?

Why does the distance grow only like a logarithm?

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Why would daily data understate how much two markets move together?

How do you add up the covariance terms?

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Why is portfolio duration a simple weighted average?

What changes for the portfolio when duration goes from 5.55 to 7?

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How can a biased forecaster beat an unbiased one?

Why does averaging the two help so much?

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Why work backwards from the target?

Which step do candidates drop?

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Why is 4.5% against 7% the wrong comparison?

How do you put them on the same footing?

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Which OLS assumption breaks?

How do you get to about 1?

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Why effective duration rather than the usual formula?

What does negative convexity cost the holder?

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Why not measure the fund against cash or the market alone?

What does the split tell you about the manager?

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