Financial Analysis puzzles, solved step by step

Puzzles
100
Traced to a firm
47
Topics
13
Hard
30
Topic
All topicsAccounting flow riddles10Valuation and multiples riddles10Ratio and margin riddles8Cost of capital, leverage and rates8Compounding and time value8Mental maths8Probability and expected value9Working capital and cash riddles6Percentages and averages7Estimation and market sizing7Logic and counting brainteasers7Pricing, costing and unit economics6Data and statistics intuition6
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–1 of 1 · filtered from 100Clear filters
  1. 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.