Debt Capital Markets puzzles, solved step by step

Puzzles: 100
Traced to a firm: 16
Topics: 13
Hard: 30

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076A conglomerate owns three businesses worth Rs 1,200 crore, Rs 800 crore and Rs 500 crore of enterprise value. The market applies a 15% holding company discount to the sum of the parts. Net debt is Rs 600 crore and there are 50 crore shares. What is the value per share, and what would it be with no discount?Cost of capital and valuation riddlesCoreDeutsche BankMumbai · 2024
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Before you calculate: the discount is 15% of enterprise value. How much does the value per share fall?
Show the worked solution
Rs 30.5 per share with the discount and Rs 38.0 without it. The parts add to Rs 2,500 crore. A 15% discount takes off Rs 375 crore, leaving Rs 2,125 crore of enterprise value. Less Rs 600 crore of net debt gives equity of Rs 1,525 crore over 50 crore shares. With no discount equity is Rs 1,900 crore, or Rs 38.0. A 15% discount costs shareholders 19.7%.
Why does the discount come off before the debt?
Think of a house worth Rs 1 crore with a Rs 60 lakh home loan on it. If buyers suddenly pay 15% less for houses on that street, the house is worth Rs 85 lakh, the bank is still owed Rs 60 lakh, and the owner's share drops from Rs 40 lakh to Rs 25 lakh. A conglomerate works the same way. The holding company discount is a haircut on what the whole group is worth, and the lenders' claim is a fixed number that sits ahead of the shareholders, so the haircut is taken from enterprise value and passes through to equity untouched.
So the order is: value each business, add them to get the sum of the partsValuing each business on its own, usually against its own peers, and adding the values together., apply the holding company discountThe gap between what a group trades at and what its businesses would be worth separately, often blamed on head office costs, capital allocation or tax leakage. to the total, subtract net debt, and divide by shares. Rs 2,500 crore less Rs 375 crore is Rs 2,125 crore; less Rs 600 crore is Rs 1,525 crore; over 50 crore shares is Rs 30.5.
The three businesses add to Rs 2,500 crore; the 15% discount removes Rs 375 crore and net debt removes Rs 600 crore, leaving equity of Rs 1,525 crore, or Rs 30.5 a share against Rs 38.0 with no discount.
The relationship
$\text{Value per share} = \frac{(1-d)\sum EV_i - ND}{N} = \frac{0.85 \times 2{,}500 - 600}{50} = 30.5$
d the holding company discount, 15%
EV_i the enterprise value of each business
ND net debt, Rs 600 crore
N shares outstanding, 50 crore
What it says in wordsDiscount the sum of the businesses, take off what the lenders are owed, and share what is left.
Why does a 15% discount cost the shareholders more than 15%?
Equity is the thin slice left after debt, so any fall in enterprise value is a bigger fall as a share of equity. The Rs 375 crore haircut is 15% of Rs 2,500 crore but 19.7% of Rs 1,900 crore. Push net debt to Rs 1,500 crore and the same haircut takes 37.5% of the equity. This is why a credit analyst reading a holding company cares about the discount even though no lender ever sees it on a statement: it is a measure of how much equity cushion the market will actually pay for.
What should you say about where the discount is applied?
Some desks apply the discount to equity value instead of enterprise value. On equity of Rs 1,900 crore, 15% gives Rs 1,615 crore and Rs 32.3 a share. The two conventions differ by Rs 1.8 a share here, so name your convention in the first sentence rather than let the interviewer find it. The enterprise value version is the more common reading when the discount is described as a discount to the sum of the parts, and it is the one that treats the lenders' claim as fixed.
Where candidates lose it
The frequent slip is subtracting the net debt first and then taking 15% off what is left. That gives Rs 32.3 a share and quietly assumes the lenders share the discount, which they do not when the discount is on the sum of the parts. Interviewers accept either convention only if you name it.
The second slip is telling the interviewer that shareholders lose 15%. Say 19.7%, and give the reason in one line: the debt does not shrink, so the whole haircut falls on equity.
What the interviewer asks next
With net debt of Rs 1,500 crore, what discount would halve the equity value?
The group sells unit C for Rs 500 crore of cash and repays debt. If the discount stays at 15% on what remains, what happens to value per share?
Why would a lender to the holding company watch the discount at all?
Asked at Deutsche Bank, Equity Capital Markets, Mumbai, 2024 (Wall Street Oasis): working capital, leases and SOTP with conglomerate discount question
077You have two bowls, 60 white balls and 40 black balls. Split all 100 balls between the bowls any way you like. One bowl is then chosen at random and one ball drawn from it. How do you maximise the chance of drawing white, and what is that chance?Probability and expected valueCoreDeutsche BankMumbai · 2024
Try it first
Pick the best chance you think a clever split can reach.
Show the worked solution
Put one white ball alone in bowl A and the other 99 balls, 59 white and 40 black, in bowl B. Bowl A gives white every time and bowl B gives white 59 times in 99. Each bowl is chosen half the time, so the chance is 0.5 x 1 + 0.5 x 59/99, about 79.8%, against 60% for an even split.
Why does a bowl with one ball count as much as a bowl with 99?
Picture two teams tossing a coin to bat first: the coin does not care that one side has eleven players and the other has one. The bowls are chosen the same way. The coin picks a bowl, not a ball, so a bowl holding a single white ball gets the same half of the draws as a bowl holding ninety nine. That makes one lone white ball the cheapest way to buy certainty for half of all outcomes, and it costs bowl B only one white ball out of sixty.
Bowl A with one white ball gives white every time; bowl B with 59 white and 40 black gives white 59.6% of the time, so the average is 79.8%, well above the 60% an even split gives.
How do you show that nothing beats it?
Bowl A cannot do better than 100%, and one white ball is the least that gets it there. Every further white ball moved into bowl A is wasted there and missed in bowl B, and every black ball moved into bowl A drags it below 100%. So bowl A is one white ball and bowl B takes what is left. Checking all 2,499 possible splits by computer agrees: the best is 79.80%, and only the one-ball split reaches it.
The relationship
$P(\text{white}) = \tfrac{1}{2}\cdot 1 + \tfrac{1}{2}\cdot\frac{W-1}{W+B-1} = \tfrac{1}{2} + \tfrac{1}{2}\cdot\frac{59}{99} \approx 0.798$
W white balls, 60
B black balls, 40
1/2 the chance each bowl is chosen
(W-1)/(W+B-1) the white share in bowl B once one white ball is set aside
What it says in wordsHalf the time you get the certain bowl, half the time you get everything else.
The formula also shows how the answer moves. With 50 white and 50 black, the version a candidate reported from an interview, it gives 74.7%. More white in the pool lifts bowl B and the total; bowl A is already at its ceiling. Here the split adds about 20 percentage points because one ball is turned into half of the probability.
Where candidates lose it
Most people answer 60% because the pool is 60% white and they assume a split cannot change the pool. It cannot change the pool, but it changes the weights, because the bowl is chosen before the ball.
The second loss is stopping at the number. Give the one-line reason no other split does better, then generalise: one half plus one half of (W minus 1) over (W plus B minus 1).
What the interviewer asks next
What if the bowl is chosen with probability proportional to how many balls it holds?
With three bowls and the same 100 balls, what is the best split and the best chance?
You are paid Rs 100 for a white ball and nothing for black. What is the most you would pay to play once?
Asked at Deutsche Bank, Equity Capital Markets, Mumbai, 2024 (Wall Street Oasis): After the distribution, one bowl will be selected at random, and then one ball will be randomly drawn from that bowl
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
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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.
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
$365 = 52 \times 7 + 1 \qquad 366 = 52 \times 7 + 2$
52 x 7 364 days, which leaves the weekday unchanged
+1, +2 the 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?
093A Rs 1,000 crore fund charges a 2% management fee and a 20% performance fee on returns above fees, and earns a 10% gross return. Investors push the management fee down to 1.5%. What performance fee rate keeps the manager's revenue the same?Compounding, PIK and feesCoreTwo SigmaNew York · 2026
Try it first
Roughly what performance fee replaces the lost 0.5%?
Show the worked solution
A performance fee of about 24.7%. Today the manager earns Rs 20 crore of management fee plus 20% of the Rs 80 crore return after fees, Rs 16 crore, for Rs 36 crore. At 1.5% the fixed fee is Rs 15 crore and the return after fees rises to Rs 85 crore, so the performance fee must bring Rs 21 crore: 21 / 85 = 24.7%. The swap is exact only at a 10% gross return.
Why does the performance fee base change?
Think of a salesperson paid a fixed salary plus a share of profit after salary. Cut the salary and the profit after salary goes up, so the share needed to make up the difference is smaller than a straight swap would suggest. The performance fee is charged on the return left after the management fee, so lowering the management fee enlarges the base the performance fee is charged on. That is why the answer is 24.7%, not 25%.
At a 10% gross return both fee terms pay the manager Rs 36 crore, but at 5% the old terms pay 26.0 against 23.6 and at 15% the new terms pay 48.4 against 46.0, so the trade shifts revenue from weak years to strong ones.
The relationship
$15 + x\,(100 - 15) = 20 + 0.20\,(100 - 20) \;\Rightarrow\; x = \frac{36 - 15}{85} = 24.7\%$
100 the 10% gross return on Rs 1,000 crore, in Rs crore
15, 20 the new and old management fees, Rs crore
x the new performance fee rate
What it says in wordsSet the new fixed fee plus the new share of the bigger base equal to today's total.
Is the manager really indifferent?
Only at a 10% gross return. At 5% gross, the old terms pay Rs 26.0 crore and the new pay Rs 23.6 crore; at 15%, the old pay Rs 46.0 crore and the new Rs 48.4 crore. Swapping fixed fee for performance fee moves the manager's revenue out of weak years and into strong ones, so the trade depends on the return you assume. Investors who push for it pay less when things go badly and more when they go well, which is often exactly what they want. A real fund would also have a hurdle rate, a high-water mark and fees charged on average rather than opening assets; each changes the arithmetic, not the logic.
Where candidates lose it
The common answer is 25% or 22.5%, from treating the two fees as if they sat on the same base. The management fee is on assets and the performance fee is on return after fees, so moving one changes the base of the other.
The second loss is saying the new terms are equivalent. They are equal at one assumed return only; say so, and give the 5% and 15% cases to show which way the risk moved.
What the interviewer asks next
What performance fee keeps revenue the same if the assumed gross return is 15%?
The fund adds a 5% hurdle, with performance fee only on returns above it. How does that change the answer at 10%?
From the investor's side, which fee terms would you prefer in a year you expect to be weak, and why?
Asked at Two Sigma, Equity Capital Markets, New York, 2026 (Wall Street Oasis): the 2/20 rule, and if one part of this equation changed, how would the other variable make up for it
096Two Rs 1,000 crore loans are sold in the broadly syndicated market. A term loan A amortises 20% a year over 5 years. A term loan B amortises 1% a year and repays the rest as a bullet at year 7. What is the weighted average life of each?Leverage, coverage and cash flowCoreScotiabankNew York · 2026
Try it first
Before calculating: roughly what is term loan B's weighted average life?
Show the worked solution
Term loan A has a weighted average life of 3.0 years and term loan B about 6.8 years. Term loan A repays 200 in each of years 1 to 5, so its average is (1 + 2 + 3 + 4 + 5) x 200 / 1,000 = 3.0. Term loan B repays 10 in each of years 1 to 6 and 940 in year 7: (210 + 6,580) / 1,000 = 6.79. Amortisation, not maturity, sets how long the lender's money is out.
What are the two loans, and why do they amortise so differently?
That is the question a candidate reported being asked, and the arithmetic is the best way to answer it. A term loan A is built for banks: it amortises steadily, so the lender's exposure shrinks every year. A term loan B is built for institutional investors such as loan funds: it amortises only nominally and returns almost everything at maturity, which suits investors who want to stay invested. The difference in structure shows up as a difference in weighted average life, and that number drives pricing, investor appetite and how long the credit risk lasts.
Term loan A returns 200 of principal every year and has a weighted average life of 3.0 years, while term loan B returns only 10 a year before a 940 bullet in year 7, giving a weighted average life of 6.79 years.
How is weighted average life calculated?
Think of lending a friend Rs 1,000 who repays Rs 200 a year: on average your money is out about three years, even though the last rupee comes back in year five. Weighted average life weights each repayment date by the share of principal repaid on it, and ignores interest. For term loan B: six payments of 10 contribute 10 x (1 + 2 + 3 + 4 + 5 + 6) = 210 year-rupees, and the bullet contributes 940 x 7 = 6,580. Total 6,790 over 1,000 is 6.79 years.
The relationship
$WAL = \frac{\sum_t t \cdot P_t}{\sum_t P_t}: \quad A = \frac{200(1+2+3+4+5)}{1{,}000} = 3.0 \qquad B = \frac{10(1+\dots+6) + 940 \times 7}{1{,}000} = 6.79$
P_t principal repaid in year t
t the year of repayment
What it says in wordsWeighted average life is the average repayment date, weighted by how much principal comes back on each date.
Why does the desk care?
A loan's pricing is spread over its weighted average life, not its stated maturity, so an upfront fee or discount is spread over 3 years on term loan A and nearly 7 on term loan B. In practice term loans B are often repaid early from refinancings or cash sweeps: if this one were refinanced in full at year 4, its weighted average life would fall to 3.94 years. Weighted average life is also not duration; it ignores interest and discounting, so it always sits above the loan's duration.
Where candidates lose it
The common error is quoting maturity: 5 years and 7 years. The question is about how long the money is out on average, and the answer for term loan A is two years shorter than its maturity.
The second is guessing term loan B's life at half its term because it amortises. It amortises so little that the bullet dominates. Say the 94% bullet share first and the answer follows.
What the interviewer asks next
If term loan A amortised 5%, 10%, 15%, 20% and 50% over the five years, what is its weighted average life?
Why do institutional loan investors prefer the term loan B structure?
A 2 point upfront fee is spread over the weighted average life. What does it add to the yield on each loan?
Asked at Scotiabank, Debt Capital Markets, New York, 2026 (Wall Street Oasis): Tell me about the two different types of loans in the broadly syndicated loan market.
1004% of issuers in a sector default within a year. An early warning model flags 75% of the issuers that will default and 10% of those that will not. An issuer is flagged. What is the probability that it defaults?Probability and expected valueCoreBelvedere TradingChicago · 2022
Try it first
Pick the closest before you calculate.
Show the worked solution
About 23.8%. Picture 1,000 issuers: 40 will default and 960 will not. The model flags 75% of the 40, which is 30, and 10% of the 960, which is 96. Of the 126 flagged issuers, 30 default, so the chance is 30 / 126 = 23.8%. The flag multiplies the risk about six times, but most flags are still false alarms because defaults are rare.
Why is the answer not 75%?
A smoke alarm that goes off for every real fire and also for one in ten batches of toast will ring mostly for toast, because toast is far more common than fire. The early warning model is the same. The chance of default given a flag depends on how common defaults are to begin with, and when the base rate is low, false flags from the large healthy group swamp the true flags from the small defaulting group. 75% answers a different question: how often a defaulter gets flagged.
Of 1,000 issuers, the 40 that will default produce 30 flags and the 960 that will not produce 96 false flags, so only 30 of the 126 flagged issuers default, a probability of 23.8%.
The relationship
$P(D \mid F) = \frac{P(F \mid D)P(D)}{P(F \mid D)P(D) + P(F \mid \bar D)P(\bar D)} = \frac{0.75 \times 0.04}{0.75 \times 0.04 + 0.10 \times 0.96} = \frac{30}{126} = 23.8\%$
P(D) the base rate of default, 4%
P(F | D) the chance a defaulter is flagged, 75%
P(F | not D) the chance a healthy issuer is flagged, 10%
What it says in wordsOf all the flags, the share that come from real defaulters is the chance a flag means default.
How should a credit desk use a 23.8% flag?
As a reason to look harder, not a verdict. A flag lifts the default probability from 4% to 23.8%, roughly six times, which is worth a review but not a sale on its own. A second, independent signal compounds the evidence: if a separate test with the same accuracy also flagged the issuer, the probability would rise to about 70%. The honest limitation is that two warning signals about the same company are rarely independent, so the real lift from a second flag is usually smaller.
Where candidates lose it
The common error is answering 75%, swapping the probability of a flag given default for the probability of default given a flag. It is the most frequent mistake in Bayes questions, and interviewers set it up deliberately.
The second loss is doing the formula silently and producing 23.8% with no picture. Say the 1,000 issuers out loud, 30 true flags and 96 false ones, and the interviewer can follow every step.
What the interviewer asks next
What false alarm rate would the model need for a flag to mean a 50% chance of default?
The sector's default rate doubles to 8%. What does a flag mean now?
Two independent models both flag the issuer. What is the probability of default?
Asked at Belvedere Trading, Capital Markets, Chicago, 2022 (Wall Street Oasis): The technical portion of the interview consisted of probability questions including one questions relating to Bayes' theorem

d	the holding company discount, 15%
EV_i	the enterprise value of each business
ND	net debt, Rs 600 crore
N	shares outstanding, 50 crore

W	white balls, 60
B	black balls, 40
1/2	the chance each bowl is chosen
(W-1)/(W+B-1)	the white share in bowl B once one white ball is set aside

52 x 7	364 days, which leaves the weekday unchanged
+1, +2	the leftover days that move the weekday on

100	the 10% gross return on Rs 1,000 crore, in Rs crore
15, 20	the new and old management fees, Rs crore
x	the new performance fee rate

Try it first

Why does the discount come off before the debt?

Why does a 15% discount cost the shareholders more than 15%?

What should you say about where the discount is applied?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a bowl with one ball count as much as a bowl with 99?

How do you show that nothing beats it?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the weekday move by exactly one?

When exactly does the answer become Wednesday?

Why would a credit interviewer bother with this?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the performance fee base change?

Is the manager really indifferent?

Where candidates lose it

What the interviewer asks next

Try it first

What are the two loans, and why do they amortise so differently?

How is weighted average life calculated?

Why does the desk care?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer not 75%?

How should a credit desk use a 23.8% flag?

Where candidates lose it

What the interviewer asks next