Portfolio Management puzzles, solved step by step

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

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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
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Roughly what exit multiple does 2.5 times the money need?
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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
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The earnings yield is 4.5% and the bond yields 7%. What does that comparison tell you?
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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
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Roughly how large is the honest t-statistic?
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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
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What sign does the convexity take?
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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?
083Two trading desks each have a one-day 95% value at risk of Rs 10 lakh, and their daily P&Ls have a correlation of 0.3. Assuming normal returns, what is the combined value at risk, and how big is the diversification benefit?Portfolio risk mathsCoreBlackRockNew York · 2026
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Your first estimate of the combined value at risk?
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About Rs 16.1 lakh, a diversification benefit of about Rs 3.9 lakh. Under normal returns value at risk is a fixed multiple of standard deviation, so it combines the same way: the square root of 10 squared plus 10 squared plus 2 x 0.3 x 10 x 10, which is the square root of 260. Adding the two desks' figures would overstate the risk by Rs 3.9 lakh.
Why can you not just add the two numbers?
Two friends each walk 10 minutes from a crossing, one north and one north-east. They do not end up 20 minutes apart; the angle between their paths matters. Under normal returns, a desk's value at risk is a fixed multiple of its standard deviation, and standard deviations combine like arrows: the angle between them is set by the correlation. Only at a correlation of 1 do the arrows point the same way and add to 20.
Placing the two Rs 10 lakh risks head to tail at the angle set by a 0.3 correlation gives a combined value at risk of Rs 16.1 lakh, against Rs 20 lakh if they moved together and Rs 14.1 lakh if they were uncorrelated.
The relationship
$\text{VaR}_{A+B} = \sqrt{10^2 + 10^2 + 2(0.3)(10)(10)} = \sqrt{260} \approx 16.1$
10 each desk's one-day 95% value at risk, Rs lakh
0.3 the correlation between the desks' daily P&Ls
What it says in wordsSquare each desk's figure, add twice the correlation times their product, and take the square root.
What assumption is doing the work, and when does it fail?
The square-root rule holds only when returns are jointly normal, or close to it, so that value at risk is a clean multiple of standard deviation. With fat tails or options in the book, value at risk need not be subadditive, and the combined figure can even exceed the sum. Correlations also rise in a crisis, so the Rs 3.9 lakh benefit is thinnest on exactly the days it is needed. Firms often report both the diversified total and the sum of the parts for that reason.
Where candidates lose it
The fast wrong answer is Rs 20 lakh, which quietly assumes a correlation of 1. The interviewer then asks why banks bother measuring correlation at all, and the candidate has nowhere to go.
The second loss is giving Rs 16.1 lakh without the normality condition. Say it: the square-root rule is a property of standard deviation, and value at risk inherits it only under normal returns.
What the interviewer asks next
At what correlation is the combined value at risk exactly Rs 15 lakh?
How much does each desk contribute to the combined figure?
Why is expected shortfall preferred over value at risk for limits?
Asked at BlackRock, Restructuring, New York, 2026 (Wall Street Oasis): Techincal and behavioral (VAR, market views, stock valuation) why blackrock, python experience?
086Two private deals each return 2.0 times the money invested, in a single payout at the end. One pays out after three years, the other after seven. What IRR does each earn?Private and real asset mathsCoreInvescoNew York · 2025
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Roughly, what IRR does the seven-year deal earn?
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About 26% a year for the three-year deal and 10.4% for the seven-year deal. The IRR of a single payout is the multiple to the power one over the years, less one: 2 to the one third and 2 to the one seventh. Same money back, very different speed. The multiple says how much, the IRR says how fast, and a deal needs both.
Why does the same multiple give such different returns?
Two friends each double their savings; one takes three years, the other seven. Nobody would call them equally good investors. A multiple of money ignores time entirely, while IRR is a rate per year, so the same 2.0x falls from 26% to 10.4% as the wait stretches from three years to seven. The curve is steep early: each extra year cuts the IRR most when the holding period is short.
A fixed 2.0 times multiple earns about 26% a year if it arrives in three years and only 10.4% if it takes seven; to match the three-year IRR, the seven-year deal would need about 5.0 times the money.
The relationship
$\text{IRR} = M^{1/T} - 1: \quad 2^{1/3} - 1 = 26.0\%, \quad 2^{1/7} - 1 = 10.4\%$
M multiple of money, 2.0
T years until the single payout
What it says in wordsFor one cash flow in and one out, the IRR is the yearly growth rate that turns the investment into the multiple.
So which number should an investor care about?
Both, because each can be gamed alone. A short, quick flip can post a high IRR on a small absolute gain, while a long hold can post a big multiple at a return below the cost of capital. Limited partners look at IRR for speed and at the multiple, often called the equity multiple or MOIC, for how much wealth was actually created. To match 26% over seven years, the second deal would need about 5.0 times the money, not 2.0.
One caution for the room: this clean formula holds only with one outflow and one inflow. With interim distributions, the IRR assumes those cash flows are reinvested at the IRR itself, which is rarely true, so a quoted IRR on a deal with early payouts flatters it.
Where candidates lose it
The common slip is linear: 100% gain over seven years is about 14% a year. That forgets compounding and overstates the return by more than three points.
The second loss is treating a higher IRR as automatically better. Say that a 26% IRR on a small cheque held briefly can create less wealth than 10% on a large one held for years.
What the interviewer asks next
What multiple does a 20% IRR produce over five years?
A deal returns 1.5x in one year. Is that better than 2.0x in three?
Why do sponsors sometimes use a credit line to delay capital calls, and what does it do to IRR?
Asked at Invesco, Real Estate, New York, 2025 (Wall Street Oasis): Lots of basic questions asked about IRR, EM, Cap Rates etc

087Estimate how many new passenger cars are sold in India in a year.Market sizing and estimationCoreAllianceBernsteinNew York · 2021

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Where should the estimate start?

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Roughly 40 lakh new cars a year, built from stated assumptions. About 30 crore households, 8% owning a car, gives about 2.4 crore cars. Ownership rising half a point a year adds 15 lakh first-time buyers; a 12-year life means 20 lakh replacements; fleets and second cars add about 5 lakh. Then compare with published industry sales data.

How do you turn a stock of cars into yearly sales?

A school's intake each year is not its total strength; it is the students leaving that year plus any growth in the roll. New car sales work the same way: they are the growth in the number of cars owned plus the cars being replaced, so the answer needs a stock and two flows. Say this structure first; the interviewer is scoring the tree more than the final number.

About 30 crore households with 8% owning a car gives a stock of about 2.4 crore cars, and new sales come from 15 lakh first-time buyers, 20 lakh replacements and about 5 lakh fleet and second cars, roughly 40 lakh a year.

Which assumption would you defend first, and which would you flag?

Take the population as roughly 140 crore and confirm the current figure; at about 4.7 people per household that is 30 crore households. The 8% ownership share is the number to flag, because ownership is concentrated in cities and in the top income bands. Replacement is the largest flow, so the assumed 12-year life matters most: at 10 years replacements rise to 24 lakh, at 15 they fall to 16.

Assumption	Base	If wrong	Effect on the total
Households owning a car	8%	10%	Replacement rises to 25 lakh
Yearly rise in ownership	0.5 point	0.3 point	First-time buyers fall to 9 lakh
Life of a car	12 years	10 years	Replacement rises to 24 lakh
Fleet and second cars	15% on top	25% on top	Adds about 3.5 lakh

The total is most sensitive to the replacement branch, because it is the largest flow and rests on two soft assumptions, ownership and car life.

Finish by naming what you would check. Industry bodies publish domestic sales every month, and an interviewer at a fund expects you to know the order of magnitude. If your estimate is far off, say which branch you would revisit rather than adjusting the total to fit.

Where candidates lose it

The common loss is starting from population times the share who want a car. That sizes the stock of potential owners, not the yearly flow, and the answer comes out ten times too big.

The second loss is mixing units: crore households and lakh sales in the same sentence without saying so. Label the stock in crore and the flows in lakh, and read the tree back once before giving the total.

What the interviewer asks next

How would the answer change if used cars absorbed most first-time buyers?
Size the market for two-wheelers the same way.
Which listed-sector metric would you track to see whether your replacement assumption holds?

Asked at AllianceBernstein, Investment Banking, New York, 2021 (Wall Street Oasis): Case study market sizing question

088An n by n by n cube, like a Rubik's cube, is built from small cubes. As a formula in n, how many of the small cubes show at least one face on the outside?Logic brainteasersCoreT. Rowe PriceBaltimore · 2020
Try it first
For n = 3, a standard Rubik's cube, how many small cubes show a face?
Show the worked solution
n cubed minus (n minus 2) cubed, which expands to 6n squared minus 12n plus 8. Everything not on the surface forms a hidden cube with one layer peeled from each side, so its side is n minus 2. For n = 3 that is 27 minus 1, or 26; for n = 10 it is 488. If the interviewer means the coloured squares instead, the answer is 6n squared.
Why count what you cannot see?
To count the tiles round the edge of a courtyard, it is easier to measure the whole yard and subtract the inner lawn than to walk the border without counting corners twice. The cubes not on the surface form one clean block of side n minus 2, so subtracting it from n cubed avoids every double-counting trap at the edges and corners.
In a 3 by 3 by 3 cube only the centre cube is hidden, so 26 of the 27 show a face; the same subtraction of a hidden core of side n minus 2 gives 8, 26, 56, 98 and 488 for n equal to 2, 3, 4, 5 and 10.
The relationship
$n^3 - (n-2)^3 = 6n^2 - 12n + 8$
n^3 all the small cubes
(n-2)^3 the hidden core after peeling one layer from every side
What it says in wordsSurface cubes are all cubes less the core you cannot see.
How do you check the formula a second way?
Count by type. There are always 8 corner cubes, 12 edges each carrying n minus 2 cubes between the corners, and 6 faces each with an (n minus 2) by (n minus 2) centre. That is 8 plus 12(n minus 2) plus 6(n minus 2) squared, which expands to 6n squared minus 12n plus 8. Two routes agreeing is the check the interviewer wants to hear.
Two edge cases show care. The formula needs n of at least 2; a single cube, n = 1, is all surface, and the formula would wrongly give 2. And the question is sometimes asked as surface area: the number of coloured squares is 6n squared, 54 for a standard cube. Ask which is meant before answering.
Where candidates lose it
The fast wrong answer is 6n squared, which counts the stickers, not the cubes: every edge cube is counted twice and every corner three times. On a standard cube that gives 54 against the true 26.
The second loss is giving the formula with no check. Say the corner, edge and face split once; it takes ten seconds and proves the algebra.
What the interviewer asks next
How many small cubes show exactly two faces, as a formula in n?
For what n are more than half the cubes hidden?
Now the cube is n by n by m. Generalise the count.
Asked at T. Rowe Price, Equities, Baltimore, 2020 (Wall Street Oasis): Give an equation that yields the surface area of an n by n by n Rubic's cube based on number of blocks per side.
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
096A manager holds the index but overweights stock A by 5 percentage points and underweights stock B by 5 points. A has 30% volatility, B has 25%, and they correlate at 0.6. What tracking error does this pair of bets create?Portfolio risk mathsHardMSCIMonterrey · 2013
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Before working it: is the tracking error above or below the 1.5% that the A bet alone would create?
Show the worked solution
A tracking error of about 1.25% a year. Tracking error is the volatility of the active weights. A's bet contributes (5% x 30%) squared, 2.25; B's contributes (5% x 25%) squared, 1.5625; and because the bets are opposite on correlated stocks, the covariance term is minus 2.25. The sum is 1.5625, whose square root is 1.25%.
Why does adding a second bet reduce the risk?
Buying an umbrella and selling a raincoat leaves you with little net exposure to rain, because both move with the weather. Overweighting A and underweighting B, when the two stocks tend to move together, is partly a hedge: when both rise, the gain on A is partly offset by the shortfall on B. Tracking error measures the risk left after that offset.
The variance of the active bets is A's own term 2.25 plus B's 1.5625 minus a covariance term of 2.25, which leaves 1.5625 and a tracking error of 1.25%, against 1.95% if the stocks were unrelated and 2.46% if both bets pointed the same way.
The relationship
$TE^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2\,w_A w_B\,\rho\,\sigma_A\sigma_B = 2.25 + 1.5625 - 2.25$
w_A, w_B active weights, +5% and -5%
sigma_A, sigma_B volatilities, 30% and 25%
rho correlation, 0.6
What it says in wordsTracking error is the portfolio volatility formula applied to the active weights instead of the holdings.
What is the neat coincidence, and what does it hide?
Here the covariance term exactly cancels A's own variance, so the answer equals B's bet alone, 5% x 25% = 1.25%. That is a coincidence of these numbers, not a rule: the covariance term is 2 x 0.6 x 30 x 25, which happens to equal 30 squared. Change the correlation to 0.5 and the answer moves. The general lesson holds, though: tracking error depends on the size of the bets and on how much they cancel.
Say the limits. Correlations are estimated and unstable, so a pair that looks like a hedge in calm markets can decouple when a stock-specific event hits. Real tracking error also includes every other small active weight, and many small bets can add up to more than one large one.
Where candidates lose it
The common slip is adding the two bets' risks, 1.5% plus 1.25%, as if they were independent and in the same direction. That ignores both the correlation and the opposite signs of the weights.
The quieter slip is getting the sign of the covariance term wrong. The weights have opposite signs, so the term is negative; say that out loud before plugging in numbers.
What the interviewer asks next
At what correlation would the tracking error be zero?
What tracking error would a 2% overweight in a stock with 40% volatility add on its own?
How would you decompose a portfolio's tracking error into contributions from each bet?
Asked at MSCI, Financial Tools, Monterrey, 2013 (Wall Street Oasis): What's the tracking error formula?

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

D/P	the dividend yield, the payout ratio times earnings over price
g	long-run growth in earnings and dividends, 10%

VIF	variance inflation from the overlap
(12-k)/12	the correlation of two windows k months apart
t*	the corrected t-statistic

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

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Why does adding a second bet reduce the risk?

What is the neat coincidence, and what does it hide?

Where candidates lose it

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