Debt Capital Markets puzzles, solved step by step

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100
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All topicsLeverage, coverage and cash flow9Mental maths and numeracy8Estimation and market sizing7Logic and brainteasers8Cost of capital and valuation riddles7Bond pricing and yield7Compounding, PIK and fees6Issuance and refinancing arithmetic8Credit spreads and default probability8Duration and convexity8Capital structure and recovery8Probability and expected value10Yield curve and forward rates6
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Showing 1–3 of 3 · filtered from 100Clear filters
  1. 016Numerical reasoning: an issuer's bond volumes grow 12%, then 15%, then fall 10% over three years. What is the total change over the period, and what is the average annual growth rate?Mental maths and numeracyCoreCorporate banking

    Try it first

    What is the total change over the three years?

    Show the worked solution

    A total rise of about 15.9%, or about 5.05% a year. Percentage changes multiply: 1.12 times 1.15 is 1.288, and a 10% fall from there leaves 1.1592. The steady yearly rate that gets to the same place is the cube root of 1.1592, about 1.0505. Adding the rates gives 17% and an average of 5.67%, both too high, because the fall hits a bigger base than the gains did.

    Why can you not add the percentages?

    A shop raises a price by 10% and later cuts it by 10%, and the tag ends below where it started, because the cut is taken on the higher price. Each percentage change is measured on the level it starts from, so a chain of changes multiplies rather than adds. Here the 10% fall comes after two good years, so it removes 12.88 points of the index, more than the 12 the first year added.

    Percentage changes multiply: each one is taken on the new base100Start+12.0Year 1 +12%+16.8Year 2 +15%-12.88Year 3 -10%115.92End15% of 11210% of 128.8total +15.92%5.05% a yearnot 17% and 5.67%
    Indexed to 100, volumes rise to 112, then by 16.8 to 128.8, then fall by 12.88 to 115.92, a total change of 15.92% rather than the 17% that adding the three rates suggests.

    How do you do it fast under test conditions?

    Break each multiplication into easy pieces. 1.12 times 1.15 is 1.12 plus 15% of 1.12, which is 1.12 plus 0.168, so 1.288; taking 10% off is 1.288 minus 0.1288, which is 1.1592. For the yearly rate, test a round guess: 1.05 cubed is 1.1025 times 1.05, which is 1.1576, just under 1.1592, so the answer is a shade above 5%. On a timed test, that is enough to pick the right option in seconds.

    The relationship
    1+G=1.12×1.15×0.90=1.1592g=(1+G)1/3−1≈5.05%1 + G = 1.12 \times 1.15 \times 0.90 = 1.1592 \qquad g = (1+G)^{1/3} - 1 \approx 5.05\%
    Gthe total change over three years
    gthe compound annual growth rate
    1/3one third, because there are three years
    What it says in wordsMultiply the yearly growth factors for the total, then take the cube root for the steady yearly rate.

    Why does the average of 5.67% overstate growth?

    The simple average of 12, 15 and minus 10 is 5.67%, but three years at 5.67% would give 18.0%, not 15.9%. Whenever the yearly rates vary, the simple average is above the compound rate, and the gap grows with the swings. That is the same drag that makes a volatile bond fund's average return look better than what its investors actually earned. Say which average you are quoting: the compound rate describes the path, the simple average only describes the list of numbers.

    Where candidates lose it

    Adding the rates to get 17% is the trap the test is built around, and the wrong option is always on the list. Under time pressure the candidate recognises 17 as a number from the question and picks it.

    The second trap is dividing 17 by 3 for the annual rate. Even after getting the total right at 15.92%, dividing by 3 gives 5.31%, which ignores compounding the other way. Take the cube root, or test 1.05 cubed.

    What the interviewer asks next

    • What fall in year 4 would bring volumes back exactly to the start?
    • Volumes rise 20% and then fall 20%. Where do they end, and why?
    • Why do fund fact sheets report compound annual returns rather than simple averages?
  2. 031Without a calculator, estimate 1.07 to the power 5. Then check how close the rule of 72 gets to the doubling time at 7%.Mental maths and numeracyCoreFixed income asset management

    Try it first

    Your first instinct for 1.07 to the power 5?

    Show the worked solution

    1.07 to the fifth is about 1.40, 1.4026 exactly, and the rule of 72 gives a doubling time of 10.29 years against a true 10.24. Expand it: 1 plus 5 x 0.07 plus 10 x 0.0049 is already 1.399. The rule of 72 is out by about 15 days over a decade, well inside any rounding a desk cares about.

    How do you get 1.07 to the fifth in your head?

    A savings account paying 7% a year gives you Rs 7 on every Rs 100 in the first year. In the second year you earn Rs 7 on the original Rs 100 again, plus 49 paise on the first year's Rs 7. Compounding is simple interest plus interest on interest, and for small rates over a few years the first correction does almost all the work. That is why the expansion below settles so quickly.

    The relationship
    1.075=1+5(0.07)+10(0.07)2+10(0.07)3+…=1+0.35+0.049+0.0034+…≈1.40261.07^5 = 1 + 5(0.07) + 10(0.07)^2 + 10(0.07)^3 + \ldots = 1 + 0.35 + 0.049 + 0.0034 + \ldots \approx 1.4026
    5, 10, 10the number of ways to pick one, two or three of the five years, from the binomial expansion
    0.07^2 = 0.0049interest earned on one year's interest
    What it says in wordsAdd simple interest, then the pairs of years, then the triples; each layer is about a tenth the size of the last.

    A second route checks the first. Square 1.07 to get 1.1449, square again to get about 1.3108, then multiply by 1.07: 1.3108 plus 7% of it, which is 0.0918, gives 1.4026. Two methods landing on the same number is the thing to say out loud; it tells the interviewer you do not trust a single chain of mental arithmetic.

    Rs 1 at 7%: 1.40 after five years, doubled just after year ten1.01.41.82.203571012Yearssimple interest, 1.84year 5: 1.4026doubles at 10.24yrule of 72: 10.29y1.07 to the 5th, term by term11.0000+ 5 x 0.071.3500+ 10 x 0.00491.3990+ 10 x 0.000341.4024+ 5 x 0.0000241.4026Three terms get 1.399
    One rupee at 7% compounds to 1.4026 after five years and to 2 at 10.24 years, a whisker before the rule of 72's 10.29 years, while simple interest reaches only 1.70 by year ten.

    How good is the rule of 72 at 7%?

    The exact doubling time is the log of 2 over the log of 1.07, 10.245 years. The rule of 72 gives 72 over 7, 10.286 years. The error is about 15 days in ten years, because 72 is tuned to rates near 8% and 7% is close enough. At 2% the rule of 70 does better; at 20% the rule of 72 starts to drift, since the true figure is 3.8 years against its 3.6. Knowing where a shortcut breaks is worth as much as the shortcut.

    The desk use is fast sanity checking. If a 7% bond's coupons are reinvested at 7%, Rs 100 grows to about Rs 140 in five years and Rs 200 in a little over ten. Simple interest would reach only Rs 170 by year ten, so the gap between the two lines in the picture is what reinvestment is worth.

    Where candidates lose it

    The common wrong answer is 1.35: five times 7% with no compounding. Candidates who know better sometimes jump to 1.5 because it feels bigger, without a method behind it.

    The other loss is reciting the rule of 72 without checking it. The interviewer wants the exact figure or a reason the rule is close here; say that 72 over 7 is 10.29 and the truth is about 10.24.

    What the interviewer asks next

    • Estimate 1.07 to the power 10 from your answer.
    • At what rate does money double in exactly 8 years, roughly?
    • Why is the rule of 69.3 exact for continuous compounding?
  3. 052You invest Rs 1 lakh in a bond fund when its unit price is Rs 10 and another Rs 1 lakh when it is Rs 20. What is your average cost per unit, and why is it not Rs 15?Mental maths and numeracyCoreFixed income asset management

    Try it first

    Pick the average cost per unit before you calculate.

    Show the worked solution

    Your average cost is Rs 13.33 a unit, not Rs 15. Rs 1 lakh at Rs 10 buys 10,000 units and Rs 1 lakh at Rs 20 buys 5,000, so Rs 2 lakh buys 15,000 units. Equal money buys more units when the price is low, so the low price carries more weight. The average cost is the harmonic mean of the two prices, which always sits below the simple average when the prices differ.

    Why does equal money not mean equal weight?

    Suppose you fill Rs 1,000 of petrol at Rs 100 a litre one week and Rs 1,000 at Rs 125 the next. You got 10 litres and then 8, so 18 litres for Rs 2,000, about Rs 111 a litre, not the Rs 112.50 midpoint. When you spend the same money each time, you buy more units at the lower price, so the lower price counts for more in your average cost. The bond fund works the same way, only with a wider gap between the prices, so the effect is larger.

    Equal money buys more units at the low price, so the average cost leans low10,000 unitsRs 10Rs 1 lakh at Rs 105,000 unitsRs 20Rs 1 lakh at Rs 20Rs 15: the simple average of pricesRs 13.33: weighted by unitsbalance pointAveraging the two prices(10 + 20) / 2 = Rs 15treats each price as buying the same unitsMoney spent over units bought2,00,000 / 15,000 = Rs 13.33the price you actually paid per unit
    Rs 1 lakh at Rs 10 buys 10,000 units and Rs 1 lakh at Rs 20 buys 5,000, so the balance point of the 15,000 units sits at Rs 13.33, well left of the Rs 15 midpoint of the two prices.

    What is the general rule, and why is it called a harmonic mean?

    Divide total money by total units: 2,00,000 over 15,000 gives Rs 13.33. Written as a formula, the units are money over price, so the average cost is the number of instalments divided by the sum of one over each price. That is the harmonic mean, and for any set of unequal prices it lies below the simple average. The simple average here overstates your cost by 12.5%.

    The relationship
    Pˉ=2110+120=20.15=13.33\bar{P} = \frac{2}{\tfrac{1}{10} + \tfrac{1}{20}} = \frac{2}{0.15} = 13.33
    2the number of equal instalments
    1/10, 1/20units bought per rupee at each price
    P-barthe average cost per unit
    What it says in wordsWith equal money each time, the average cost is the harmonic mean of the prices.

    This is the arithmetic behind the claim that investing a fixed sum on a schedule lowers your average cost. It is true, but only against the simple average of the prices you happened to pay. It does not make the investment itself cheaper or safer: it says nothing about whether Rs 10 or Rs 20 was the fair price. Say that limit, because the interviewer is often checking whether you oversell the result.

    Where candidates lose it

    The fast wrong answer is Rs 15, which averages the prices as if you had bought the same number of units twice. The question says the same money twice, and the whole puzzle turns on that difference.

    The second loss is getting Rs 13.33 and then claiming the method guarantees a good outcome. It only guarantees an average cost below the average price paid; it cannot tell you whether the fund was cheap.

    What the interviewer asks next

    • If you had bought 10,000 units at each price instead, what would your average cost be?
    • A yield moves between 6% and 8% and you buy equal money at each. Which average of the yields describes your purchase?
    • Three instalments at Rs 10, Rs 20 and Rs 40: what is the average cost?