Debt Capital Markets puzzles, solved step by step

Puzzles
100
Traced to a firm
16
Topics
13
Hard
30
Topic
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
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–3 of 3 · filtered from 100Clear filters
  1. 037You have a 3-litre and a 4-litre bottle and unlimited water. How do you measure exactly 2 litres and exactly 5 litres, and which whole-litre amounts up to 7 can you make?Logic and brainteasersWarm upNomuraNew York · 2026

    Try it first

    How many of the amounts 1 to 7 litres can you measure, counting water held across both bottles?

    Show the worked solution

    For 2 litres: fill the 3, pour it into the 4, fill the 3 again and top up the 4; exactly 2 litres stay in the 3-litre bottle. For 5: fill the 4, pour into the 3 to leave 1, empty the 3, move the 1 into it, then fill the 4, for 1 plus 4. Every whole amount from 1 to 7 is possible, because 3 and 4 differ by 1.

    What moves are you actually allowed?

    Three: fill a bottle to the top, empty it, or pour from one into the other until the first is empty or the second is full. The only amounts you can know for certain are full bottles and what is left after a pour stops at a full bottle, so every measurement is built from 3s and 4s. Think of it as making change with only Rs 3 and Rs 4 coins, where you are also allowed to hand coins back.

    Every fill, pour and empty moves water in steps of 3 and 42 litres0/30/4Start3/30/4Fill the 30/33/4Pour 3 into 43/33/4Fill the 3 again2/34/4Pour into 4 until full5 litres0/34/4Fill the 43/31/4Pour 4 into 30/31/4Empty the 31/30/4Pour the 1 into 31/34/4Fill the 4Read each pair as litres in the 3-litre bottle / 4-litre bottle. Lime marks the target amount.
    Four moves leave 2 litres in the 3-litre bottle after the 4-litre bottle is topped up, and five moves put 1 litre in the 3-litre bottle beside a full 4-litre bottle, which together hold exactly 5 litres.

    Why can you reach every amount from 1 to 7?

    Because 4 minus 3 is 1, and once you can make 1 you can make anything by adding bottles. The amounts you can measure are exactly the combinations of 3 and 4 that fit in the bottles, and since the two sizes share no common factor, every whole litre up to their total of 7 is reachable. One litre: fill the 4 and pour into the 3. Three and four: fill one bottle. Five: 1 plus a full 4. Six: 3 in each. Seven: both full.

    The relationship
    gcd⁡(3,4)=1  ⇒  3a+4b can equal any whole number\gcd(3, 4) = 1 \;\Rightarrow\; 3a + 4b \text{ can equal any whole number}
    gcdthe greatest common divisor, the largest number dividing both sizes
    a, bhow many times you add or remove each bottle's volume, positive or negative
    What it says in wordsWhen the bottle sizes share no factor, their combinations reach every whole number.

    The same rule tells you when a puzzle has no answer. With a 4-litre and a 6-litre bottle, every amount you can make is even, so 5 litres is impossible, and you can say so without trying a single pour. Interviewers like that sentence more than the pouring itself, because it shows you found the structure rather than a lucky sequence.

    Where candidates lose it

    Candidates start pouring at random and lose track of the state, which in a phone interview is fatal because the interviewer cannot see your paper. Say each state as a pair, litres in the 3 then litres in the 4, after every move.

    The second miss is solving 2 litres and freezing on 5, which cannot fit in either bottle. The question is asking for water held across both bottles, and saying that out loud is half the answer.

    What the interviewer asks next

    • With a 5-litre and a 7-litre bottle, what is the fewest number of moves to measure 1 litre?
    • Can you measure 5 litres with a 4-litre and a 6-litre bottle? Prove it either way.
    • How does this relate to what bond sizes you can build from fixed lot sizes?

    Asked at Nomura, Equity Capital Markets, New York, 2026 (Wall Street Oasis): How much water can you fill using 1 3liter and 1 4liter bottle using each other?

  2. 046A company has 50 of debt and 50 of equity at market value, trades at 10x earnings and pays 6% on its debt. What is its WACC before and after a 25% tax rate on interest?Cost of capital and valuation riddlesWarm upCitiNew York · 2026

    Try it first

    What cost of equity does a P/E of 10 suggest, as a quick proxy?

    Show the worked solution

    WACC is 8.0% before tax and 7.25% after. Read the cost of equity as the earnings yield, 1 over a P/E of 10, which is 10%. Half the capital costs 10% and half costs 6%, so the blend is 8.0%. The tax shield cuts debt to 6% x 0.75 = 4.5%, and the blend falls to 7.25%. The P/E shortcut assumes no growth; with growth the true cost of equity is higher.

    Where does the cost of equity come from when you are not given a beta?

    If a shop earns Rs 10,000 a year and sells for Rs 1 lakh, a buyer earns 10% on the price. Nobody would pay more unless they expected growth, so 10% is roughly what buyers of such a shop demand. A P/E of 10 means earnings are a tenth of the price, an earnings yield of 10%, and with no growth that is a quick proxy for the cost of equity. Say out loud that it is a proxy: it is the fastest honest route when the interviewer gives you only a multiple.

    The relationship
    WACC=EVke+DVkd(1−t)=0.5×10%+0.5×6%×0.75=7.25%WACC = \tfrac{E}{V}k_e + \tfrac{D}{V}k_d(1-t) = 0.5 \times 10\% + 0.5 \times 6\% \times 0.75 = 7.25\%
    E/V, D/Vequity and debt as shares of total capital at market value, 0.5 each
    k_ecost of equity, here the earnings yield 1 / 10 = 10%
    k_dcost of debt, 6%
    ttax rate on interest, 25%
    What it says in wordsWeight each funder's required return by its share of the capital, and cut the debt cost by the tax it saves.
    Weight each funder's cost by its share of the capitalEquity 50cost 10% = 1 / P/EDebt 50cost 6%, 4.5% after taxMarket valuesBefore tax: 0.5 x 10% + 0.5 x 6%equity 5.0debt 3.0= 8.00%After tax: 0.5 x 10% + 0.5 x 4.5%equity 5.0debt 2.25= 7.25%0%5%10%The tax shield on interest takes 0.75 points off the blend
    Equity of 50 costing 10% and debt of 50 costing 6% blend to 8.0% before tax, and to 7.25% once interest is deducted at 25%, because the debt's after-tax cost falls to 4.5%.

    Why does tax only touch the debt half?

    Interest is paid before tax and dividends after it. Every rupee of interest reduces taxable profit, so the government in effect pays a quarter of the interest bill at a 25% tax rate, and debt that costs 6% on paper costs 4.5% to the company. Equity gets no such relief. That is why the after-tax WACC is 0.75 points lower: half the capital times the 1.5 point shield. The difference compounds into valuation, since a lower discount rate raises the value of every future cash flow.

    State the limit of the shortcut before the interviewer does. The earnings yield equals the cost of equity only for a company that does not grow and pays out all its earnings. If earnings grow at 3% a year, investors paying 10x are expecting roughly 10% plus 3%, about 13%, and the WACC rises with it. For a real company you would build the cost of equity from a risk-free rate, a beta and a market premium instead.

    Where candidates lose it

    The common error is using 10 as a percentage or treating the P/E itself as a cost. Invert it: a multiple becomes a yield only when you flip it.

    The second loss is giving one WACC and not saying whether it is before or after tax. The question asked for both on purpose; give 8.0% and 7.25% and say where the 0.75 points went.

    What the interviewer asks next

    • The company re-levers to 70% debt at a 7% cost. What happens to WACC, and what should happen to the cost of equity?
    • If earnings grow at 4% a year, what cost of equity does a P/E of 10 imply?
    • Why do you use market values rather than book values for the weights?

    Asked at Citi, Capital Markets, New York, 2026 (Wall Street Oasis): $50 debt, $50 equity, P/E 10x, Cost of Debt 6%, what is WACC

  3. 083If a given date is a Monday this year, what day of the week will the same date be one year from now? When does the answer change?Mental maths and numeracyWarm upOaktree Capital ManagementLos Angeles · 2022

    Try it first

    Answer inside five seconds.

    Show the worked solution

    Tuesday, or Wednesday if 29 February falls in between. A normal year is 365 days, which is 52 weeks and 1 day, so every date moves on one weekday. When the year you step across contains 29 February, it is 366 days, 52 weeks and 2 days, and the date moves on two. The remainder after dividing by 7 does all the work.

    Why does the weekday move by exactly one?

    A clock that runs for 25 hours ends up one hour past where it started, because 24 of those hours bring it round in a full circle. Weeks are the same circle with seven stops. Only the remainder after dividing the day count by 7 moves the weekday, and 365 divided by 7 leaves a remainder of 1. The 52 full weeks bring you back to Monday; the one extra day carries you to Tuesday.

    52 full weeks change nothing; only the leftover days move the weekday365 days = 52 weeks + 1 day+1364 days: every weekday comes round exactly 52 timesTuesday366 days = 52 weeks + 2 days+1+1364 days: every weekday comes round exactly 52 timesWednesdayCount forward from MondayMonTueWedThuFriSatSunnormal year: +129 Feb in between: +2
    A normal year of 365 days is 52 full weeks plus one day, so a Monday date falls on a Tuesday the next year; a year containing 29 February has 366 days and moves the same date to a Wednesday.

    When exactly does the answer become Wednesday?

    When 29 February sits between the two dates. That depends on the date as well as the year. 15 March 2027 is a Monday, and 15 March 2028 is a Wednesday, because 29 February 2028 falls in between: 366 days. Step on again and 15 March 2029 is a Thursday, only 365 days later, because the next 29 February is still years away. Ask whether 29 February lies inside the year you are stepping across, not whether either year is a leap year. Leap years are those divisible by 4, except century years, which count only when divisible by 400.

    The relationship
    365=52×7+1366=52×7+2365 = 52 \times 7 + 1 \qquad 366 = 52 \times 7 + 2
    52 x 7364 days, which leaves the weekday unchanged
    +1, +2the leftover days that move the weekday on
    What it says in wordsDivide the day count by seven and the remainder is how far the weekday moves.

    Why would a credit interviewer bother with this?

    Because bonds and loans live on calendars. A coupon due on the 15th of March lands on a different weekday every year, and when it lands on a weekend or holiday the documents say whether it moves to the next business day and whether interest accrues for the extra days. Day count conventions such as actual/365 turn the leap day into a slightly larger coupon in some years. Quick calendar arithmetic is how you catch a payment schedule that has been rolled wrongly.

    Where candidates lose it

    The instinctive answer is Monday, because it is the same date. It takes one sentence about 52 weeks being 364 days to fix, but candidates who say Monday first rarely recover the room.

    The second loss is the leap-year rule stated too broadly. What matters is whether 29 February falls between the two dates, so a date in January of a leap year and a date in March of the year before can both move by two.

    What the interviewer asks next

    • If today is a Monday, what day is it 100 days from now?
    • A 5-year bond pays annual coupons on 15 March. How many of those five coupon dates can fall on a weekend?
    • Why was 1900 not a leap year but 2000 was?

    Asked at Oaktree Capital Management, Generalist, Los Angeles, 2022 (Wall Street Oasis): If [a date] is a Monday what week of the day would [same date] be a year from now?