Debt Capital Markets puzzles, solved step by step

About 250 trading days, roughly a year. The fund may buy 2% of Rs 1,000 crore, which is Rs 20 crore a day, and Rs 5,000 crore divided by Rs 20 crore is 250. The position is five full days of the whole market's trading, which makes it illiquid for this fund: it would take about as long to sell as to buy.

Why is the answer not five days?

Imagine wanting to buy 500 kilos of tomatoes from a village market that sells 100 kilos a day. In theory it is five days of the market's supply; in practice, if you take everything, prices jump and the regular buyers go home empty handed. So you decide to take no more than a couple of kilos a day. The limit on your share of daily volume, not the market's total volume, sets how fast you can trade, and here that limit is Rs 20 crore a day. Rs 5,000 crore at Rs 20 crore a day is 250 days.

What does this tell you about liquidity?

Liquidity belongs to a position, not to a market: the same bonds are liquid for a Rs 50 crore holder and illiquid for a Rs 5,000 crore holder. Days to trade is the simplest honest measure. At 250 days, the fund carries a year of market risk while it builds, and faces the same year if it ever needs to sell, which matters most exactly when others are selling too. Daily volume also falls in a stressed market, so the exit could take longer than the entry. A desk would size the position down, accept a higher participation rate and more price impact, or look for a block trade or a new issue where it can take size in one go.

About 4.71% of the portfolio defaulting each year. High yield earns 330 basis points more than investment grade. Each default costs 70% of face, since 30% is recovered, so each 1% of defaults costs 70 basis points. 330 divided by 70 is 4.71%. Below that default rate high yield still earns more; above it, less.

Why divide by the loss, not the default rate?

A shop that sells on credit loses less than the full bill when a customer fails to pay, if it can take back some of the goods. Bonds work the same way. The annual cost of defaults is the default rate times the loss given defaultThe share of face value lost when a borrower defaults, which is one minus the recovery rate., so the extra spread covers defaults up to the extra spread divided by the loss given default. With 30% recovered, each default loses 70%, and 330 basis points of extra spread covers 4.71% a year.

What does the number leave out?

A break-even default rate is a cushion, not a forecast: whether 4.7% is comfortable depends on how often default rates have run above it and for how long. It also assumes investment grade loses nothing; if investment grade lost 10 basis points a year, the break-even would rise to 4.86%. Recovery is the soft number: it tends to fall in exactly the years defaults rise, so a 30% assumption can flatter the cushion. And the calculation ignores price volatility, which matters to anyone who must mark the portfolio before the bonds mature.

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%.

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.

Paying cash leaves the equity about Rs 10 crore better off. PIK debt compounds at 12% to Rs 881 crore. The Rs 55 crore a year that PIK keeps in the business, reinvested at 15%, grows to only Rs 371 crore, so net debt is Rs 510 crore against Rs 500 crore with cash pay. The answer flips if the business earns more than about 16.4% on the cash, or if paying cash would starve it.

What exactly is being compared?

Deferring a credit card bill and investing the money instead only pays if your investment grows faster than the card's balance. PIKPayment in kind: interest is added to the loan balance instead of being paid in cash, so the debt compounds. interest is that deferral, made formal. Under PIK the whole loan compounds at the PIK rate, while only the cash you did not pay compounds at your own return, so the two piles grow on different bases. Line up where each option leaves the company at year 5: debt owed, less any cash built up by reinvesting what was not paid.

When does the answer flip?

Solve for the return that makes the kept cash worth exactly the extra Rs 381 crore of PIK debt: about 16.4%. PIK is cheap only if the cash it saves earns more than the PIK rate, and here more still, because the PIK rate carries a one point premium over cash pay. At a 12% reinvestment return the kept cash reaches only Rs 349 crore. Two further reasons can flip the answer in practice: a borrower whose cash is scarce may not be able to pay 11% without cutting investment it needs, and a borrower expecting to refinance or be sold early carries the PIK compounding for less time. Tax treatment of PIK interest can also differ; confirm it rather than assuming.

Interest cover falls from 3.0x to about 2.61x. All the debt is floating, so the rate goes from 10% to 11.5% and interest from Rs 100 crore to Rs 115 crore. EBITDA is unchanged at Rs 300 crore, so cover is 300 / 115 = 2.61x. A 15% rise in the interest bill cuts cover by about 13%, with no change in the business at all.

Why does a small rate move matter so much?

A household on a floating home loan feels a rate rise in the very next EMI, even though nothing about its salary has changed. Companies with floating debt feel it the same way. Floating rate debt passes a benchmark rise straight into the interest bill, so coverage falls even when the business is doing exactly as well as before. Here 150 basis points on Rs 1,000 crore is Rs 15 crore a year, a 15% rise in interest.

What would a lender ask next?

Two questions: how much headroom is left, and how much is hedged. Cover would reach 2.0x only if interest rose to Rs 150 crore, a rate of 15%, so the benchmark would need to climb another 350 basis points from here. Hedging changes the answer more than anything else: if 60% of the debt were swapped to fixed, the same rise would add only Rs 6 crore of interest and cover would fall only to 2.83x. A banker who sees all-floating debt at 3.0x cover asks about hedging before anything else.

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.

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.

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.

Reject the first two bids, then accept the first bid that beats both of them; you get the best bid 43.3% of the time. Taking the first or last bid wins only 20%. The two rejected bids set a benchmark, and the rule succeeds whenever the best bid comes later and the best of the bids before it sits among the first two. Checking all 120 orderings gives 52 wins, which is 43.3%.

Why reject bids you know nothing wrong with?

House hunting in a city you do not know, you would look at a couple of flats before signing anything, simply to learn what good looks like. Sign too early and you never had a benchmark; look too long and the best one may already be gone. The rejected bids are the price of information: they set the bar that later bids must clear, and the only question is how many to spend. With five bids, spending two is the best trade.

How do you get 43.3% without listing all 120 orders?

Ask where the best bid sits. If it is in the first two, you have already rejected it and lose. If it sits at position j, from 3 to 5, you take it only if no earlier bid after the first two already beat the bar, which happens when the best of the first j minus 1 bids lies in the first two. That chance is 2 out of (j minus 1), so the rule wins with probability one fifth of (2/2 + 2/3 + 2/4), which is 43.3%. Say the structure; the arithmetic takes ten seconds.

What is the limitation for a real loan sale?

Two things. The rule maximises the chance of the very best bid, not the expected price; a seller who cares about the average price would behave differently. And real loan sales rarely force on-the-spot decisions: a desk runs a process that collects bids together, precisely to avoid this problem. With many bids, the rule becomes the well-known look at about 37% and then leap, and the success rate falls towards about 37% too.

10 testers. Number the bottles 1 to 1,000 and write each number in binary with ten digits. Tester n drinks from every bottle whose n-th binary digit is 1. After 24 hours, the testers who fall ill spell out the poisoned bottle's number in binary. Ten testers give 1,024 possible patterns, enough for 1,000 bottles; nine give only 512.

Why is each tester worth a binary digit?

Think of the game twenty questions: every yes-or-no answer halves the possibilities, so twenty answers can pick out one thing in about a million. A tester is one yes-or-no answer: ill or fine. With one round of results, n testers can produce 2 to the power n different patterns, and you need at least as many patterns as bottles. 2^9 = 512 falls short of 1,000; 2^10 = 1,024 covers it. So ten is both enough and the minimum.

How do you read the answer back?

Give tester n the place value 2 to the power (n minus 1): tester 1 is worth 1, tester 2 worth 2, up to tester 10 worth 512. Add the place values of the testers who fall ill and you have the bottle number. If bottle 613 is poisoned, its binary form is 1001100101, so testers 10, 7, 6, 3, 1 fall ill, and 512 + 64 + 32 + 4 + 1 = 613. Numbering 1 to 1,000 fits in ten digits because 1,000 is below 1,024, and the pattern where nobody falls ill is left spare.

The desk version of the lesson: design the test so that every outcome carries information. A tester who drinks from one bottle learns about one bottle; a tester who drinks from half of them splits the problem in two.

You can tighten 25 basis points, to 125, and still be 2 times covered. Each 5 basis point step loses 10% of the original demand, which is 0.4x of cover. Going from 4.0x to 2.0x gives up 2.0x, or five steps, so 5 x 5 = 25 basis points. On a Rs 500 crore deal the book falls from Rs 2,000 crore to Rs 1,000 crore, and the issuer saves Rs 1.25 crore a year.

What is actually being traded off?

A shopkeeper with a queue of forty customers for twenty items can raise the price, and some customers will walk away; the question is how high to go before the queue gets too short to be comfortable. Book-building trades cover for price: each step tighter saves the issuer money and costs orders, and cover is the buffer against a weak aftermarket. Here every 5 basis points costs 0.4x, because the loss is 10% of the original book each time.

Would you really go all the way to 125?

Not automatically. The linear rule is a model of the book; real books drop in lumps, because price-sensitive orders carry limits and the orders that leave first are often the highest quality long-term investors. Landing exactly on the 2.0x floor leaves no margin if a few orders drop after the final terms. A syndicate desk might stop one step short, at 130 with 2.4x cover, giving up Rs 0.25 crore a year of saving on Rs 500 crore for a safer book and a better chance of a positive break. The arithmetic sets the limit; the judgement picks the point inside it.

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.

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.