Debt Capital Markets puzzles, solved step by step

The senior class proposes 98 for itself, 0 for class 2, 1 for class 3, 0 for class 4 and 1 for class 5, and the plan passes. Work backwards. With two classes left, class 4 takes all 100, because its own vote is half. With three, class 3 buys class 5 with 1 unit. With four, class 2 buys class 4 with 1. With five, class 1 needs two more votes and buys classes 3 and 5, who would otherwise get nothing, with 1 unit each.

Why start from the end of the game?

When you must catch a train that leaves at a fixed time, you plan from the departure backwards to when you leave home. A bargaining game played in turns is solved the same way: the last stage has an obvious answer, and each earlier proposal only has to beat what each voter would get one stage later. The end here is two classes, 4 and 5. Class 4 proposes 100 for itself and votes yes, and one yes out of two is half, so the plan passes.

How does each proposer buy the cheapest votes?

With three classes left, class 3 needs two of three votes. Class 5 gets nothing if class 3 is removed, so one unit buys it: 99, 0, 1. At every stage the proposer buys the classes that would get nothing at the next stage, because one unit beats nothing. With four left, class 2 needs two of four votes and class 4 gets nothing in the three-class outcome, so the split is 99, 0, 1, 0. With five, class 1 needs three votes, its own plus classes 3 and 5, who get nothing in the four-class outcome.

What does this teach about a real restructuring?

Bargaining power comes from what each party gets if the deal fails, not from its size or its rank on paper. The senior class keeps 98 because it can see what every other class would get if its plan failed, and it pays only for the votes it needs. Real restructurings do not run like this: priority rules, court oversight and class voting thresholds all change the outcome, and junior classes often hold out for a share because a fight costs everyone. Treat the puzzle as a lesson in reasoning from the fallback, not as a model of a court process.

The bond is rich by about 0.35 per Rs 100. Price each cash flow at its own zero rate: the year-one coupon of 10 at 7% is worth 9.35, and the final 110 at 8% for two years is worth 94.31. Fair value is 103.65, so a price of 104.00 is 0.35 too high. In yield terms the bond yields about 7.76% against a fair 7.95%, roughly 19 basis points too low.

Why price each cash flow at its own rate?

Money due in one year and money due in two years are different goods, the way a train ticket for next week and one for next month carry different prices. A zero rate is the price of money for one specific date, so each of a bond's cash flows is discounted at the zero rate for its own date, and the bond is worth the sum. The year-one coupon of 10 is discounted at 7%; the year-two payment of 110 is discounted at 8% for two years.

What does rich mean, and how would a desk use it?

Rich means the price is above what the curve says the cash flows are worth, so the bond yields less than it should: about 7.76% against a fair 7.95%. An investor would rather own the same cash flows through the curve, and a relative value desk might sell the bond and buy a matching pair of zero-coupon bonds, locking in about 0.35 per Rs 100 if the prices converge. On Rs 100 crore of face that is about Rs 35 lakh.

Say what could make the gap real rather than a mistake. A gap of 0.35 can reflect liquidity, a different tax treatment of high-coupon bonds, or the cost of actually building the matching zero-coupon pair, not only a mispricing. The check is whether similar bonds show the same gap. The limit: this assumes the zero rates are for the same issuer's credit; a government zero curve would value a corporate bond too high.

Put 5/9 of the money, Rs 64.8 crore, in the 3-year zero and 4/9, Rs 51.9 crore, in the 12-year zero. The liability is worth 200 / 1.08^7 = Rs 116.7 crore today, and a zero's duration is its maturity, so the weights solve 3w + 12(1 - w) = 7. The risk left is a curve that does not move in parallel, such as short rates falling while long rates rise.

Why match duration rather than just the money?

Imagine saving for a payment due in seven years when the only deposits on offer run three years or twelve. Buy only the three-year deposit and you must reinvest in year three at whatever rate then holds; buy only the twelve-year one and you must sell it in year seven at whatever price then holds. Mixing them lets the two risks cancel. Duration matching means the assets and the liability gain or lose the same amount when rates move a little in parallel: if rates rise, the assets fall in price but the 3-year proceeds are reinvested at more, and the liability's value falls too.

What risk does duration matching leave behind?

Test it. If every rate rises to 9%, the assets are worth Rs 109.50 crore and the liability Rs 109.41 crore; if every rate falls to 7%, Rs 124.66 crore against Rs 124.55 crore. Against parallel moves the match holds, and the barbell even edges ahead because it has more convexity than a single payment, but a twist in the curve hits it directly. If the 3-year rate falls to 7% and the 12-year rate rises to 9% while the 7-year rate stays at 8%, the assets are worth Rs 113.10 crore against a liability still at Rs 116.70 crore: a shortfall of Rs 3.60 crore.

The same risk shows up as reinvestment: in year 3 the insurer receives Rs 81.7 crore that must earn enough for four more years, at whatever the 4-year rate is then. The only complete fix is a 7-year zero that matches the cash flow itself, which the question rules out.

Receiving fixed is a long fixed rate bond funded by a short floating rate note, so its DV01 is about Rs 3.95 lakh per basis point. The bond leg moves Rs 100 crore x 4.2 x 0.0001, or Rs 4.2 lakh, per basis point. A floater resetting quarterly has a duration of about 0.25, which takes off Rs 0.25 lakh. The principals cancel, and the position gains when rates fall.

Why can a swap be written as two bonds?

Suppose you lend a friend Rs 1 lakh at a fixed 7% and, the same morning, borrow Rs 1 lakh from your bank at its floating rate. The two lakhs cross in the air and cancel; what remains is that you collect 7% and pay floating. A receive-fixed swap is exactly that: the cash flows of owning a fixed rate bond and having issued a floating rate note on the same notional, with the two principals cancelling, which is why a swap needs no money upfront.

Why does the floating leg carry almost no rate risk?

A floating rate noteA bond whose coupon is reset to a market benchmark at regular dates, so its price stays close to par. resets its coupon to the market at each reset date, so on a reset date it is worth about par whatever rates have done. Its price can only drift between resets. With quarterly resets the floater behaves like a bond maturing in three months, with a duration of about 0.25, so it cancels only about 6% of the fixed leg's rate risk. The swap is, for rate purposes, nearly the whole fixed bond.

Which way does the position make money, and who uses it?

Receiving fixed gains when rates fall: a 10 basis point drop in the 5-year swap rate makes the position about Rs 39.5 lakh. That is why a desk calls receiving fixed a long duration position. It is also the everyday DCM use: an issuer that sells a 5-year fixed bond and wants floating rate debt receives fixed on a swap against its own coupon, and ends up paying floating. Treat the 4.2 and 0.25 as illustrations; the real figures depend on the coupon, the curve and the reset dates.

The total saving is 0.7% a year, the difference between the fixed gap of 1.2% and the floating gap of 0.5%. A borrows fixed at 7.0%, B borrows floating at benchmark plus 0.8%, and they swap. Split evenly, B pays A 7.05% fixed and A pays B the benchmark: A ends at benchmark minus 0.05% and B at 7.85% fixed, each 0.35% better than going direct.

Why is there a gain when one firm is cheaper at everything?

Picture two flatmates: one cooks and cleans faster than the other, but is much faster at cooking and only slightly faster at cleaning. The household still gets more done if the fast cook cooks and the other one cleans. Borrowing works the same way. What matters is comparative advantage: firm B is 1.2% worse in the fixed market but only 0.5% worse in floating, so B should borrow floating and A fixed, and the gap between those two penalties, 0.7%, is the gain.

How do you check the split adds up?

Follow each firm's cash. A pays its lenders 7.0% fixed, receives 7.05% fixed from B and pays B the benchmark, so its net cost is the benchmark minus 0.05%, against benchmark plus 0.3% going direct. B pays its lenders benchmark plus 0.8%, receives the benchmark from A and pays A 7.05%, so its net cost is 7.85% fixed, against 8.2% direct. 0.35% plus 0.35% is the full 0.7%, which is the check that nothing was lost or invented. On Rs 1,000 crore that is Rs 7 crore a year between them. In practice a bank sits in the middle and keeps a slice, so each firm gets somewhat less.

What is the honest limitation?

The gain is not free money. Part of the gap exists because floating lenders can reprice or refuse to roll a weaker borrower, while fixed lenders are locked in for years; B's floating spread of 0.8% may not hold if its credit slips. Each firm also takes on counterparty risk to the other or to the bank. Say that one sentence and the answer sounds like someone who has watched a swap book.

The protection buyer pays about 12% of notional upfront. The contract's running coupon is 100 basis points but fair protection costs 400, so the buyer underpays by 300 basis points a year. Multiply by the risky duration of 4.0, the present value of one basis point a year until default or maturity, and the shortfall is 1,200 basis points, or 12 points: Rs 12 crore on Rs 100 crore of protection.

Why does a standard coupon create an upfront payment?

Think of a gym that charges every member the same Rs 1,000 a month but costs Rs 4,000 a month to run for some members. It can only take them if they pay the Rs 3,000 a month difference as a lump sum on joining. Standardised credit default swaps work that way. The running coupon is fixed by the contract rather than set to the market, so the gap between the market spread and the coupon, for every year the protection is expected to run, is settled as one payment on the first day.

Why is the risky duration less than five?

Premiums on a credit default swap stop the moment the name defaults, so the expected number of payments is lower than five, and each one is discounted. A risky duration of 4.0 sits below the 4.33 of a plain five-year annuity at 5% because some of those payments will never be made. The riskier the name, the lower the risky duration, which is why a desk re-computes it rather than assuming it stays fixed as the spread moves. The formula above is the market's first approximation; the full calculation also nets accrued premium and uses a model of default timing.

What happens when the spread is below the coupon?

The sign flips. If the market spread were 50 basis points against a 100 coupon, the buyer would overpay by 50 a year and the seller would pay the buyer about 2 points upfront. Whoever is getting the better running deal pays the difference on day one.

Without the guarantee the bonds recover 50%; with it, about 87.5%. The operating lenders are paid first from the Rs 700 crore, so only Rs 100 crore reaches the holding company for Rs 200 crore of bonds. A guarantee gives the bonds a direct claim on the operating company, equal with its loans if both are unsecured: Rs 700 crore across Rs 800 crore of claims, Rs 175 crore to the bonds.

Why do holding company bondholders get only what is left?

Lend money to a friend whose only asset is a stake in the family shop. If the shop fails, its suppliers and its bank are paid out of the shop's till first, and your friend receives only what is left over, if anything. Holding company bonds sit in exactly that position. The holding company owns shares, not assets, so its lenders are paid only from the equity value of the operating company, after every creditor of the operating company has been paid in full. This is called structural subordinationBeing junior not because of a clause in the documents but because your borrower sits higher up a corporate chain than the assets., and no ranking clause in the bond documents can undo it.

What does the guarantee change, and for whom?

An upstream guarantee from the operating company gives the bondholders a second claim, this time directly against the operating assets. If that claim ranks equally with the operating loans, the two groups share Rs 700 crore across Rs 800 crore of claims: 87.5% each, Rs 175 crore to the bonds and Rs 525 crore to the loans. The bondholders' gain of Rs 75 crore is exactly the operating lenders' loss of Rs 75 crore; a guarantee does not create value, it moves the queue. That is why operating lenders often limit or prohibit such guarantees in their own documents.

What assumption would flip the answer back to 50%?

If the operating loans are secured on all of the operating assets, they are paid in full from the collateral before any unsecured guarantee claim, and the bonds are back to the Rs 100 crore residual. Label and security both matter: a guarantee only helps against creditors it can rank alongside. In real documents guarantees can also be capped or challenged, so a credit analyst reads the guarantee and the intercreditor terms before trusting a number like 87.5%.

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.

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.