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  1. 028The correlation between X and Y is 0.2 and the correlation between Y and Z is 0.5. What range of values can the correlation between X and Z take?Data and statistics intuitionHardTower Research CapitalNew York · 2019

    Try it first

    Pick the range before any algebra.

    Show the worked solution

    Anywhere from about -0.75 to 0.95. Read each correlation as the cosine of an angle between two arrows. X is 78.5 degrees from Y, and Z is 60 degrees from Y. Z can swing to the same side as X, leaving them 18.5 degrees apart, or to the other side, 138.5 degrees apart. The cosines of those angles are 0.9485 and -0.7485.

    Why can a correlation be drawn as an angle?

    Think of three people walking away from the same lamp post. Knowing how far apart the first two point, and how far apart the second and third point, limits how far apart the first and third can point, but only loosely. If you standardise each variable, its correlation with another is the cosine of the angle between them, so correlations must behave like angles in space. A correlation of 0.2 is an angle of 78.5 degrees; 0.5 is 60 degrees.

    Correlations are cosines of angles, so the third angle can only swing so farYXZ, same sideZ, other sideX to Y: 78.5 deg (cos 0.2). Y to Z: 60 deg (cos 0.5)Known: 0.2 and 0.5. X and Z can be anywhere in-1-0.500.51-0.750.95If both known were 0.9: X and Z must be-1-0.500.510.621.00Centre 0.2 x 0.5 = 0.10Half-width sqrt(0.96 x 0.75) = 0.849
    X sits 78.5 degrees from Y and Z sits 60 degrees from Y, so the angle between X and Z runs from 18.5 to 138.5 degrees, which puts their correlation anywhere from -0.75 to 0.95; two correlations of 0.9 would pin it much tighter, from 0.62 to 1.
    The relationship
    ρXZ∈ρXYρYZ±(1−ρXY2)(1−ρYZ2)=0.10±0.849\rho_{XZ} \in \rho_{XY}\rho_{YZ} \pm \sqrt{(1-\rho_{XY}^2)(1-\rho_{YZ}^2)} = 0.10 \pm 0.849
    rho XY, rho YZthe two known correlations, 0.2 and 0.5
    square roothow much room the weak links leave, the product of the two sines
    What it says in wordsThe range is centred on the product of the known correlations and is as wide as the product of how far each is from perfect.

    Where does the formula come from, and when does it bite?

    The three-by-three correlation matrix must not imply a negative variance for any mix of X, Y and Z, which means its determinant cannot go below zero. Solving that condition gives the formula above, the cosine rule for the difference and sum of two angles. The constraint is loose when the known correlations are weak and tight when they are strong. With 0.2 and 0.5, almost the whole scale stays open. With 0.9 and 0.9, X and Z must correlate at least 0.62: two things each closely tied to Y cannot drift far from each other. Say the practical use: a risk model that fills in missing correlations by hand can break this rule and produce a matrix that is not valid.

    Where candidates lose it

    The fast wrong answer is 0.10, the product, as if correlation passed along a chain. The product is the centre of the range, not the answer. The other wrong answer is that nothing can be said, which ignores that angles must fit together.

    Candidates who know the formula often cannot say why it holds. Draw the arrows, say cosine of an angle, and the formula follows in one line.

    What the interviewer asks next

    • If X and Y have correlation 0.9 and Y and Z 0.9, what is the minimum correlation of X and Z?
    • Can three variables all have pairwise correlation of -0.6?
    • Why might a hand-edited correlation matrix in a risk model fail to invert?

    Asked at Tower Research Capital, Prop Trading, New York, 2019 (Wall Street Oasis): What if the correlation between X and Y is 0.2 and the correlation between Y and Z is 0.5.

  2. 061Two stocks have a correlation of minus 0.3 between their daily returns within each month, yet their monthly returns across a year have a correlation of plus 0.6. How can both be true?Data and statistics intuitionHardSquarepoint CapitalMontreal · 2024

    Try it first

    Which explanation fits?

    Show the worked solution

    Both hold when the two stocks share a slow driver that changes month to month, while their day-to-day moves push in opposite directions. On any one day the shared drift is a small part of each move and the opposing noise wins, so the daily correlation is negative. Over a month the drift adds up 21 times while the noise partly cancels, so the shared part dominates and the correlation turns positive.

    How can two stocks move apart by the day but together by the month?

    Take two neighbours' electricity meters. On a given day one runs high because guests are staying while the other family is away, so daily readings look opposed. But both bills climb every summer when the air conditioners come on. Correlation is not a fixed property of two assets; it belongs to a horizon, because different forces drive short moves and long moves. Daily returns are dominated by trading noise and stock-specific news; monthly returns by sector and macro trends.

    Daily wiggles pull apart, the monthly drift pulls togetherMonth 1: upMonth 2: downMonth 3: upABIllustration: 63 trading days. Same trend each month, opposite daily movesShare of variance from theshared driftOne day9.7%Correlation -0.3One month69.2%Correlation +0.6Drift variance grows with 21 x 21,noise only with 21
    Both stocks follow the same monthly drift while their daily wiggles run opposite, and because the shared drift's variance grows with the square of the horizon, it explains 9.7% of daily variance but 69.2% of monthly variance.

    Why does the shared driver win over a month?

    Write each daily return as a shared drift plus the stock's own noise. Over 21 days the drift adds up in a straight line, so a month's drift is 21 times a day's and its variance is 21 x 21 = 441 times larger. Noise does not line up day after day, so its variance grows only 21 times. Whatever the two stocks share pulls harder the longer you hold them, because shared moves stack while unshared moves wash out. With daily noise variance of 1, a noise correlation of minus 0.3 and a drift variance of 0.107, the monthly correlation comes out at exactly plus 0.6.

    The relationship
    ρmonth=212 σd2+21 ρe σe2212 σd2+21 σe2=441×0.107−21×0.3441×0.107+21=0.6\rho_{month} = \frac{21^2\,\sigma_d^2 + 21\,\rho_e\,\sigma_e^2}{21^2\,\sigma_d^2 + 21\,\sigma_e^2} = \frac{441 \times 0.107 - 21 \times 0.3}{441 \times 0.107 + 21} = 0.6
    sigma_d^2variance of the shared daily drift, 0.107
    sigma_e^2variance of each stock's own daily noise, set to 1
    rho_ecorrelation of the daily noise between the two stocks, minus 0.3
    21trading days in a month
    What it says in wordsOver a month the shared drift is weighted by 441 and the noise by only 21, so a drift that is a tenth of daily variance ends up driving the monthly correlation.

    The model also shows how sensitive the result is. Halve the drift variance and the monthly correlation drops from 0.6 to 0.39. Read across the whole year, daily returns would show a correlation of about -0.17: still negative, because on any single day the drift is too small to matter.

    What else could cause it, and what would you check?

    Two other mechanisms produce the same pattern. A lead-lag, where one stock reacts to shared news a day later, hides co-movement in daily data that monthly data captures. And short-term trading that pushes money from one stock into the other, such as a pairs or index rebalancing flow, creates opposing daily moves inside a shared trend. Then question the evidence. Twelve monthly points give a correlation of 0.6 a standard error of about 0.19, so the true figure could plausibly be anywhere from about 0.2 to near 1. Check other years before building a hedge on it, because a hedge ratio estimated on daily data can have the wrong sign for a position held for months.

    Where candidates lose it

    The common answer is that it is impossible, because a monthly return is just the sum of the daily ones. That treats correlation as if it were additive, when sums mix components that grow at different speeds with the horizon.

    The other weak answer is waving at noise: monthly data has only twelve points, so it is unreliable. That is a fair caveat, but it does not explain a positive sign; the interviewer wants the shared-driver mechanism first and the sample-size warning second.

    What the interviewer asks next

    • You hedge a three-month position using a hedge ratio from daily data. What goes wrong?
    • How would a one-day lead-lag between the stocks show up in daily and weekly correlations?
    • How many years of monthly data would you want before trusting a 0.6 correlation?

    Asked at Squarepoint Capital, Hedge Fund, Montreal, 2024 (Wall Street Oasis): correlation can be negative intra-month but positive across a year, how?

  3. 096Branch A has a lower default rate than branch B among salaried borrowers, and a lower rate among self-employed borrowers, yet a higher overall default rate. A lent to 100 salaried borrowers who defaulted at 2% and 900 self-employed who defaulted at 10%. B lent to 900 salaried at 3% and 100 self-employed at 12%. Show how this is possible, and say which branch underwrites better.Data and statistics intuitionHardBank creditRating agencies

    Try it first

    Before you compute: which branch has the higher overall default rate?

    Show the worked solution

    A's overall rate is 9.2%, B's is 3.9%, and A is still the better underwriter in each segment. A: 2 defaults among 100 salaried plus 90 among 900 self-employed, 92 in 1,000. B: 27 plus 12, 39 in 1,000. An overall rate is a weighted average of segment rates, and the weights are the mix of the book. A's book is 90% self-employed, where everyone's rate is high, so its average is pulled up by whom it lends to, not by how well it lends.

    How can a branch be better at everything and worse in total?

    Two surgeons. One takes the hardest cases in the hospital and loses fewer of them than anyone else would; the other takes mostly routine cases. Count deaths per patient and the first surgeon looks worse, because her patients were sicker to begin with. An overall rate is a weighted average of segment rates, and the weights are the mix of the book, so a branch that lends mostly to the riskier segment carries a higher average even when it is better in every segment. A's rate is 0.1 x 2% + 0.9 x 10% = 9.2%; B's is 0.9 x 3% + 0.1 x 12% = 3.9%.

    SegmentBranch ABranch BLower rate
    Salaried100 loans, 2%, 2 defaults900 loans, 3%, 27 defaultsA
    Self-employed900 loans, 10%, 90 defaults100 loans, 12%, 12 defaultsA
    Whole book1,000 loans, 9.2%, 92 defaults1,000 loans, 3.9%, 39 defaultsB
    A beats B in each segment and loses on the whole book because its mix is nine tenths self-employed.
    Width is the mix, height is the rate: A wins each segment and loses the totalBranch A: overall default rate 9.2%92 defaults among 1,000 loans2%Salaried, 100 loans2 defaults10%Self-employed, 900 loans90 defaultsoverall9.2%Branch B: overall default rate 3.9%39 defaults among 1,000 loans3%Salaried, 900 loans27 defaults12%Self-employed, 100 loans12 defaultsoverall3.9%Like with like: salaried 2% at A against 3% at B, self-employed 10% at A against 12% at B. A is lower in both.At B's mix, A would run at 2.8% against B's 3.9%. The overall gap is the mix, not the underwriting.
    Drawn with width as the number of loans and height as the default rate, A's book is one wide bar at 10% and a sliver at 2%, averaging 9.2%, while B's is one wide bar at 3% and a sliver at 12%, averaging 3.9%, so A is better in both segments and worse overall.

    How do you compare the two branches fairly?

    Give them the same customers. Apply each branch's segment rates to one common mix and the mix stops voting. At B's mix, 90% salaried, A's rates give 0.9 x 2% + 0.1 x 10% = 2.8% against B's actual 3.9%. At A's mix, B's rates give 0.1 x 3% + 0.9 x 12% = 11.1% against A's actual 9.2%. At a 50:50 mix, A is 6.0% and B is 7.5%. On any common mix A is lower, so A underwrites better; its higher total is the price of the book it chose to build. This reversal, where a comparison flips when groups are combined, is known as Simpson's paradox.

    The relationship
    overall rate=∑iwi riA:0.1(2%)+0.9(10%)=9.2%B:0.9(3%)+0.1(12%)=3.9%\text{overall rate} = \sum_i w_i\, r_i \qquad A: 0.1(2\%) + 0.9(10\%) = 9.2\% \qquad B: 0.9(3\%) + 0.1(12\%) = 3.9\%
    w_ithe share of the book in segment i, the mix
    r_ithe default rate within segment i
    What it says in wordsThe overall default rate is each segment's rate weighted by its share of the loans, so the mix can outweigh the rates.

    What should the analyst say beyond the arithmetic?

    Three things. First, both statements are true at once: A underwrites better, and A's book loses more money, because capital is charged on actual losses and A chose a riskier book. Whether that was a good choice depends on the pricing: if A earns enough extra yield on self-employed loans to cover the extra 8 points of defaults, its strategy can be sound. Second, ask why the mixes differ, since location, product and targets all shape who walks in. Before judging two rates, ask whether the two populations are the same; if they are not, standardise to one mix or compare segment by segment.

    Third, say the limitation. Each segment is itself a mix: of loan sizes, vintages and cities. A's salaried borrowers might be larger-ticket government employees and B's might be gig workers on salary slips, and a finer split could flip the comparison back. Standardising to the segments you can see removes the mix you know about, never the mix you do not. The honest close is that A looks the better underwriter on this cut, and you would want the segment rates by vintage before signing that view.

    Where candidates lose it

    The common failure is disbelief: candidates assume the overall rates must be wrong, or declare that B is the better underwriter because its total is lower. Do the two weighted averages aloud, 0.1 x 2 + 0.9 x 10 and 0.9 x 3 + 0.1 x 12, and the paradox dissolves into mix.

    The second loss is stopping at the explanation. The interviewer wants the fair comparison: put both branches on one mix and A is lower on every one (2.8% against 3.9% at B's mix). Then add that A's book still loses more rupees, which is a question about strategy and pricing, not underwriting skill.

    What the interviewer asks next

    • If A charges self-employed borrowers 4 points more interest than salaried ones, is its riskier book a sound strategy?
    • What single number would you report to the credit committee to compare the two branches' underwriting?
    • Give another finance example where combining groups reverses a comparison.