Debt Capital Markets puzzles, solved step by step

About 8.9 basis points, or roughly Rs 4 crore, and yes, 10 basis points looks generous. A 0.40 point rise on a duration of 4.5 is 0.40 / 4.5 = 8.9 basis points of tightening. On Rs 1,000 crore, 0.40 points is Rs 4.0 crore of value that went to investors. Almost all of the premium came back on the first trade, so the book would probably have cleared tighter.

How do you turn a price jump into basis points?

A shop that sells out of a new phone in an hour, and then sees it resold online for more, has learned that it priced too low. A bond's first trades tell the issuer the same thing. Modified durationThe approximate percentage price change for a one point change in yield. is the exchange rate between price and yield: here a 1 basis point move in yield moves the price by about 4.5 hundredths of a point. So a price rise of 0.40 points means the market is valuing the bond about 8.9 basis points tighter than where it was priced.

What does the Rs 4 crore actually cost the issuer?

It is paid through the coupon, not written as a cheque. Pricing about 8.9 basis points too wide costs Rs 0.89 crore a year on Rs 1,000 crore, for five years: Rs 4.4 crore in total, or about Rs 3.6 crore in today's money at an assumed 7.5% yield, the same order as the Rs 4.0 crore jump in the bond's price. The first trade after pricing tells you how much of the premium was needed, because it shows where investors will buy the bond once the allocation is done.

So was 10 basis points too much?

On these numbers, most of it. But say two things before you convict the syndicate desk. First, strip out the market: if benchmark yields fell 3 basis points during the same session, only 5.9 basis points of the move belongs to the deal. Second, some positive performance is deliberate. Investors who were allocated want to see the bond trade well, and a flat or weak break makes the issuer's next deal harder to fill. A generous premium is a cost this time and an investment in the next book; the judgement is whether nearly the whole premium needed to be spent.

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.

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.

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.

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.

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.

Light rope A at both ends and rope B at one end at the same moment. Rope A burns out after exactly 30 minutes. At that instant, light the other end of rope B, which has 30 minutes of burn left. With two flames it finishes in 15 minutes, so rope B goes out at exactly 45 minutes.

Why does lighting both ends always give 30 minutes?

Two people painting a fence from opposite ends meet when the whole fence is painted, whatever the pace on each stretch; if the job is one hour for one painter, together they finish in half an hour. A rope's burn time behaves the same way: two flames consume the rope's 60 minutes of burn between them, so they must meet after 30 minutes, even though the meeting point is not the middle of the rope. Unevenness changes where they meet, never when.

Why does rope B have exactly 30 minutes left?

Rope B has burned from one end for 30 minutes, so 30 of its 60 minutes of burn are gone and 30 remain, spread unevenly along whatever length is left. Lighting its other end applies the same halving trick to what remains: 30 minutes of burn with two flames takes 15. 30 plus 15 is 45. The trick works on any remaining burn time, not just a whole rope.

Say what the puzzle is testing, briefly: separating what you know (total burn time) from what you do not (where along the rope that time sits). A desk reading a bond's cash flows faces the same split: the total is fixed, the timing can be lumpy.

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%.

Equity falls 50%, from Rs 40 crore to Rs 20 crore. Enterprise value drops Rs 20 crore to Rs 80 crore, but the Rs 60 crore of debt does not shrink, so the whole Rs 20 crore comes off equity. Equity was only 40% of the business, so a 20% fall in the business is 20% x 2.5 = 50% of the equity.

Why does a 20% fall become a 50% fall?

Buy a Rs 1 crore flat with Rs 40 lakh of your own money and a Rs 60 lakh loan. If flat prices fall 20%, the flat is worth Rs 80 lakh, the bank still wants Rs 60 lakh, and your stake is Rs 20 lakh: half gone. Debt is a fixed claim, so every rupee lost in enterprise value comes off the equity, and the smaller the equity slice, the bigger the percentage fall. The multiplier is enterprise value over equity, here 100 over 40, or 2.5.

How does a lender read the same numbers?

From the lender's chair the question is how much cushion is left. The loan to value ratio climbs from 60% to 75%, so the equity buffer that protects the debt has halved even though the debt itself is untouched. Another 25% fall in enterprise value, to Rs 60 crore, would take equity to zero and put the lenders' own money at risk. That is why credit desks track enterprise value and covenants tied to it, not just whether interest is being paid.

No. The price falls from about 74.41 to 55.84 per Rs 100, a fall of 25.0%, not 50%. A zero's price is 100 divided by one plus the yield, compounded for ten years. Doubling the yield moves one plus the yield only from 1.03 to 1.06, so the price is divided by (1.06/1.03) to the tenth, about 1.33. Halving the price would need a yield near 10.4%.

Why is doubling the yield not doubling the discount?

A shop that raises its markup from 3% to 6% has doubled the markup, but the price tag only goes from 103 to 106. The thing that multiplies through is one plus the rate, not the rate. A bond's price depends on (1 + y) compounded over its life, so doubling y from 3% to 6% changes the base only from 1.03 to 1.06, about 2.9% a year. Ten years of that is a factor of 1.333, which takes about a quarter off the price.

How does duration check the answer?

The modified duration of a 10-year zero at 3% is 10 / 1.03, about 9.71. Multiplied by a 3 point rise that predicts a fall of about 29.1%, against the true 25.0%. The straight-line estimate overshoots on a large move because the price curve bends: each extra point of yield takes off less than the one before. That bend is convexity, and a 3 point move is big enough to show it. For small moves duration is close; for large ones, reprice the bond directly.

The real yield is about 2.39%, against 2.50% from simple subtraction. The Fisher relation divides one plus the nominal yield by one plus inflation: 1.07 / 1.045 - 1 = 2.39%. The shortcut overstates the real yield by about 11 basis points, because inflation also erodes the interest itself, and the gap widens as rates rise.

Why is subtraction only an approximation?

Lend Rs 100 for a year at 7% and get Rs 107 back. If prices have risen 4.5%, a basket that cost Rs 100 now costs Rs 104.50. Your Rs 107 buys 107 / 104.5 baskets, 1.0239 of them: a real gain of 2.39%, not 2.5%. Inflation shrinks the purchasing power of everything you get back, the interest as well as the principal, so the real yield is a ratio, not a difference.

When does the shortcut stop being good enough?

The error is roughly the real yield times inflation, divided by one plus inflation. At low rates it is a few basis points and nobody minds; at high inflation it is tens of basis points and it changes the conclusion. With nominal yields of 15% and inflation of 12%, subtraction says 3.0% but the true real yield is 2.68%. For a desk comparing an inflation-linked bond with a conventional one, a gap of that size is the whole trade. The rates here are illustrations; the relation, not the numbers, is the point.

About 5,000 loans, before any filters. Rs 1,000 crore is 1,00,000 lakh, and 1,00,000 divided by 20 is 5,000. The book cannot all go in because a pool takes only eligible loans: current on payments, held long enough to show a track record, and within concentration limits. If the notes must be overcollateralised by 10%, the pool needs about 5,500 loans.

How do you avoid a units slip?

If you are filling a Rs 1,000 box with Rs 20 notes, you need 50 notes; the only way to get it wrong is to mix up rupees and hundreds. Lakh and crore are the Indian version of that trap. Put both numbers in lakh before dividing: Rs 1,000 crore is 1,00,000 lakh, and 1,00,000 over 20 is 5,000. Say the conversion out loud; interviewers listen for it.

Why can the financier not just use the whole book?

Investors and rating agencies buy a pool with rules attached. Eligibility criteriaThe conditions every loan must meet to enter a securitised pool, such as being current on payments and having a minimum repayment history. typically exclude loans that are overdue, loans too new to have shown a repayment record, and loans that would push the pool over limits on any one borrower, region or vehicle type. So the pool size starts with simple division and ends with eligibility: the question is not how many loans exist but how many qualify. India's securitisation rules also set a minimum holding period and a minimum retention by the originator; confirm the current figures in the RBI's directions rather than relying on memory.

What would change the count?

Two things. First, Rs 20 lakh may be the average loan at disbursal; loans that have been repaying for a year or two have smaller outstanding balances. If the average outstanding is Rs 14 lakh, the pool needs about 7,143 loans. Second, structures often require the pool to exceed the notes sold, so Rs 1,000 crore of notes might need Rs 1,100 crore of loans. The financier also has a reason not to strip its best loans out: what remains on its own balance sheet gets worse.