Debt Capital Markets puzzles, solved step by step

About 38 and about 0.026. A 1% daily gain doubles money roughly every 70 days, so a year holds just over five doublings: 2 to the power 5.2 is about 37.8. A 1% daily loss halves it about every 69 days, leaving about 0.026. The two outcomes end roughly 1,481 times apart: a tiny daily edge, held consistently, is enormous by year end.

Why is 365 times 1% the wrong way to think about it?

Picture a savings jar where every evening you add 1% of whatever is already inside. On day 1 that is one paisa per rupee; by day 200 the jar holds about seven times its start, so 1% of it is seven paise of the original rupee. Compounding means each day's 1% is earned on everything built so far, so the growth accelerates rather than adding up in a straight line. Simple addition, 365 times 1%, gives 4.65 and misses almost all of it.

How do you get to 38 without a calculator?

Use the rule of 70A shortcut: a quantity growing at r per cent a period doubles in about 70 divided by r periods.. At 1% a day, money doubles in about 70 days. A year of 365 days holds about 5.2 doublings, and 2 to the power 5 is 32 while 2 to the power 5.2 is about 37, so the answer is in the high thirties. The exact figure is 37.78. The same shortcut run backwards gives the loss: 1% down a day halves value about every 69 days, a little over five halvings in a year, which is one part in about 40.

Why does a markets interviewer care about this?

Because desks live on small edges repeated many times: a few basis points of carry every day, a slightly better fill on every trade, a small cost leak on every rebalance. An edge or a leak that looks trivial on one day decides the year, and it cuts both ways. Also notice that up 1% then down 1% does not get you back: 1.01 times 0.99 is 0.9999. Volatility on its own drags a compounding path downwards, which is worth one sentence after the numbers. The limit: real returns are not a steady 1% a day, so treat this as arithmetic about compounding, not a model of any market.

On my assumptions, about Rs 1,400 crore. Five lakh riders a day at an average Rs 30 fare is about Rs 548 crore of fares a year. A 40% operating margin leaves about Rs 219 crore of cash; a 1.4x cover allows Rs 156 crore a year of debt service. Twenty years of that at 9% is worth about Rs 1,428 crore. Every lakh of daily riders adds or removes about Rs 286 crore.

What structure do you say before any number?

Say the chain first, so the interviewer can follow every assumption: riders, fares, cash, allowed debt service, debt. Debt capacity is a cash flow estimate divided by a cover ratio and turned into a present value, so the whole answer is only as good as the riders and the margin you assume. A tea stall owner asking for a loan gets the same treatment: cups a day, price a cup, what is left after milk and rent, and how much of that the bank will let go to the EMI.

Where do the assumptions come from, and which one matters most?

Each is an illustration you should defend in a sentence. Five lakh daily riders is a busy urban line, not a flagship. Rs 30 is an average across short and long trips. Forty per cent is the margin after staff, power and maintenance, before depreciation. Ridership drives everything, because it multiplies straight through: at 3 lakh riders the capacity falls to about Rs 857 crore, at 7 lakh it rises to about Rs 1,999 crore. Say the range out loud; it shows you know which input moves the answer.

What would a lender say about this number?

Three things. First, ridership on a new line ramps up over several years, so the early payments are the riskiest and a lender may want a grace period or a lower cover test in the first years. Second, a fare box alone rarely funds the build: the debt it supports is usually a slice of the cost, with the rest from grants, equity or land and advertising income. Third, fares are often set by a public authority, so the lender is exposed to a fare decision it does not control. Closing on that shows you see the loan as a credit, not a spreadsheet.

40% debt, where WACC bottoms at 11.10%. After tax, debt costs 5.25%, 5.625%, 6.75% and 9% at the four levels. Weighting each with equity gives 12.00%, 11.365%, 11.10% and 12.20%. Cheap, tax-shielded debt pulls WACC down at first; by 60% the rise in both debt and equity costs outweighs the extra cheap debt.

Why does WACC fall at first?

Think of funding a house with a mix of a bank loan and money from relatives who expect a share of any gain. The bank is cheaper, and in this example the interest also cuts your tax bill. Replacing expensive equity with cheaper, tax-deductible debt lowers the blended cost, as long as neither cost rises too fast. Going from 0% to 20% debt, the after-tax debt cost is only 5.625% against equity at 12.8%, so WACC drops from 12.00% to 11.365%.

Why does it turn up again?

Every extra rupee of debt makes the company riskier for everyone. Lenders ask for more, and shareholders, now standing behind a bigger claim, ask for more too. Past some point the rising cost of both sources outweighs the benefit of shifting weight onto the cheaper one. From 40% to 60%, the debt weight rises by 20 points, but debt's after-tax cost jumps from 6.75% to 9% and equity's from 14% to 17%, and the blend climbs 1.10 points. That is the trade-off theoryThe idea that a company balances the tax saving from debt against the rising cost of financial distress as leverage grows. in four data points.

What does a lender add after the number?

Two caveats show judgement. The minimum is only as good as the cost schedule, and in real life that schedule is an estimate, especially the equity cost at high leverage. Only four points were given, so the true minimum could sit anywhere between 20% and 60%; say that you would want more points before recommending a structure. And a debt desk also cares about what the WACC table cannot show: whether 40% leverage keeps the rating in a band that keeps the market open in a bad year.

About 1.11%. There are three ways to be in default by the end of year 2. Default in year 1: 0.5%. Stay A, then default: 92% times 0.5%, which is 0.46%. Fall to BBB, then default: 7.5% times 2%, which is 0.15%. They add to 1.11%, more than twice the one year rate, because downgrades move issuers into riskier states.

Why can you not just double 0.5%?

Think of a student who might fail an exam this year or slip into a weaker class and fail next year from there. Multi-year default risk comes from migration as well as direct default: an issuer that is downgraded in year 1 faces a higher default rate in year 2. Doubling 0.5% assumes every survivor is still A, and the independent-survival answer, 1.00%, makes the same assumption. Both miss the 7.5% who drift to BBB.

How do you set it up so nothing is missed?

List where the issuer can be after year 1, then ask for each state what happens in year 2. Default is an absorbing state, so a year 1 default stays counted; the other states each carry their own year 2 default rate. The A branch contributes 0.92 times 0.005 and the BBB branch 0.075 times 0.02. A small matrix multiplication does the same thing: square the one year matrix and read off the A to default cell.

What does the migration path tell a credit investor?

The downgrade path is only 7.5% likely but supplies 14% of the two year default risk. For a high rated issuer, most of the medium-term risk is the chance of becoming a weaker credit, which is why investors watch outlooks and migration as closely as default rates. Downgrades also cost money before any default, through wider spreads. Say the limits: real matrices are estimated from history, the same matrix is unlikely to hold every year, and defaults cluster in downturns, so treat this as the arithmetic, not a forecast.

Senior debt gets Rs 250 crore, the subordinated notes nothing, and trade creditors Rs 50 crore. All three are unsecured, so first share Rs 300 crore pro rata across Rs 600 crore of claims: 150, 100 and 50. The notes then turn their Rs 100 crore over to the senior debt, which is still owed 150, leaving senior at 250. Trade creditors signed nothing, so they keep their 50.

What is contractual subordination, in plain terms?

Two siblings borrow from their parents and from a neighbour. The younger sibling promises the elder that any repayment the younger receives will be handed over until the elder is repaid. The neighbour never heard about that promise and is unaffected. Contractual subordinationAn agreement by one class of creditors to be paid only after a named senior class, enforced by handing over anything received until the senior class is paid in full. is a promise between two classes, so it moves value between them and leaves every other creditor exactly where the law puts them.

Why share pro rata first, before applying the subordination?

Because in the eyes of the insolvency law all three claims are unsecured and of equal rank. The legal waterfall treats them pari passu; the subordination agreement then works on what the noteholders receive, not on the waterfall itself. So compute the pro rata split, 50% each on Rs 600 crore of claims, then apply the turnover: senior is still owed Rs 150 crore, and the notes' Rs 100 crore goes to it in full. Had the notes' share exceeded what senior was owed, the surplus would stay with the notes.

What would a wrong reading cost each class?

Reading the notes as subordinated to everyone, a straight ladder, would pay senior 300 in full, trade nothing and notes nothing, a strict waterfall that takes Rs 50 crore from trade creditors who never agreed to it. The difference between subordinated to a named class and subordinated to all creditors is worth real money, so a credit analyst reads the subordination clause before building any recovery table. In real cases, check also whether senior's post-filing interest counts in the turnover, which the documents decide.

Year 3, at about 2.66x. EBITDA grows to 110, 121 and 133.1, and 40% of each repays debt: 44, 48.4 and 53.2. Debt falls to 456, 407.6 and 354.4, so leverage runs 4.15x, 3.37x and 2.66x. Deleveraging comes from both ends: repayment shrinks debt while growth enlarges EBITDA, and either lever alone would leave leverage near 3.8x in year 3.

Why do both repayment and growth matter?

A family with a home loan feels less stretched both when it prepays the loan and when its salary rises. Leverage is a ratio, so it falls when the debt on top shrinks and when the EBITDA below grows, and here both happen every year. The growth also feeds the repayment: a bigger EBITDA means more cash, so each year's repayment is larger than the last. That compounding is why the answer arrives sooner than a straight-line guess.

How do you show the interviewer which lever did the work?

Run each lever on its own. With flat EBITDA and 40 a year of repayment, leverage is 3.80x in year 3 and only reaches 3.0x in year 5; with growth but no repayment it is 3.76x in year 3. Neither alone gets there, and together they reach 2.66x, because each lever makes the other stronger. That split is the follow-up in most leveraged finance interviews, and it is how a lender judges how much of a deleveraging story depends on the growth forecast.

What would a lender worry about in this path?

The growth assumption carries much of the result. If EBITDA stalls, the same repayments leave leverage near 3.8x in year 3, so a covenant set to step down to 3.0x by year 3 would be breached. The model also assumes all free cash flow repays debt, with nothing for dividends, acquisitions or a working capital squeeze. Close with the sensitivity: a lender would size the covenant step-downs on a cash flow case below the base case, and say so.

6 rolls for one six, and 42 rolls for two sixes in a row. A six comes up one time in six, so the wait averages 6. For two in a row, define E0 as the expected rolls from the start and E1 from one six showing. From E1 the next roll ends it with 1/6 or sends you back with 5/6. Solving E0 = 1 + (5/6)E0 + (1/6)E1 and E1 = 1 + (5/6)E0 gives E0 = 42.

Why is the first answer 6?

If a bus comes with probability one in six each minute, on average you wait six minutes. For a repeated trial with success probability p, the expected number of tries until the first success is 1 over p. With p equal to 1/6, that is 6. Say it quickly; the interviewer is only using it to set up the second part.

Why is two in a row not 12?

Twelve assumes the progress you make is kept. In a row means a single miss after a first six erases it, so you keep paying the six-roll wait again and again, and the answer is driven by those resets. Think of climbing two steps on a slippery staircase where any slip on the second step sends you to the ground: most attempts end on step one. Two states capture this: E0 with no six showing, E1 with one six showing.

Is there a quick check you can say out loud?

Yes: for k in a row with success probability p, the expected wait is 1/p plus 1/p squared, and so on up to 1/p to the k. For two sixes that is 6 plus 36, which is 42; for three sixes in a row it is 6 plus 36 plus 216, which is 258. The pattern shows why streaks get expensive fast. The desk version of the same idea: requiring several conditions to hold consecutively, a covenant tested on two quarters in a row, or a run of clean prints, is far rarer than requiring them separately.

The fair swap rate is about 6.478%, roughly 2.2 basis points below the 6.5% average. Discount factors from the path are 0.9434, 0.8858 and 0.8279. The swap rate is the average of the fixings weighted by those factors. Because the lowest fixing comes first and is discounted least, it counts for more, so the weighted average sits under the simple one.

What does it mean for a swap to be worth zero?

Picture two friends agreeing to swap rent for three years: one pays a flat amount, the other pays whatever the market rent turns out to be. The deal is fair only if, in today's money, the flat payments are worth the same as the expected market ones. A swap is worth zero at the start when the present value of the fixed leg equals the present value of the expected floating leg. Present value is the key phrase: a rupee in year one is worth more today than a rupee in year three.

Build the discount factorsThe value today of one rupee paid at a future date. A factor of 0.943 means a rupee in a year is worth 94.3 paise now. from the same path. Year one: 1 over 1.06, or 0.9434. Year two: that divided by 1.065, 0.8858. Year three: divided again by 1.07, 0.8279. Their sum, 2.6571, is the value today of receiving one rupee a year for three years.

Why does the average overshoot?

Look at the two outer years. At 6.5% fixed, the fixed payer overpays by 0.5 in year one and is overpaid by 0.5 in year three. Equal in plain rupees, but the year one amount is discounted by 0.943 and the year three amount by 0.828. The early overpayment is worth more today than the late refund, so at 6.5% the fixed payer loses about 0.058 per Rs 100, and the fair rate must come down until the discounted gaps cancel. On an upward sloping path the swap rate always sits a little under the simple average; on a downward one it sits a little over.

Check it the short way. The ratio of one less the last discount factor to the sum of factors gives the same 6.478%, because a floating leg plus the return of notional at the end is worth exactly par. The gap to 6.5% is small here, about two basis points, but on a steep ten-year curve the difference between the plain and weighted averages becomes large enough to misprice a trade.

About Rs 1 lakh crore of new two-wheeler loans a year, of which perhaps Rs 31,000 crore could reach a securitisation desk. Assume 30 crore households, 55% owning 1.2 vehicles each: 19.8 crore in use. Replacing them every 10 years plus 0.3 crore first-time buyers gives 2.28 crore sales. At 60% financed and Rs 75,000 a loan, that is about Rs 1,02,600 crore; the 30% pool-eligible share is an assumption.

Where do you start a sizing question without a known number?

Start from people and the things they own, then ask how often those things are bought. A family that runs one scooter for ten years buys a scooter every tenth year, so a street of a hundred such families buys about ten a year. Annual sales of a durable good are roughly the number in use divided by how many years each one lasts, plus whatever first-time buyers add. That structure lets you build the answer from facts you can defend instead of a figure half remembered from a newspaper.

Say each assumption out loud as an assumption. Households: 30 crore, and say you would confirm the current census estimate. Ownership: 55% of households, with 1.2 vehicles each where they own one, which gives 19.8 crore two-wheelers on the road. Life: 10 years, so replacement demand is 1.98 crore a year. New owners: if ownership climbs one point a year, 0.3 crore households buy their first. Total: 2.28 crore a year.

How do you turn units into a rupee flow the desk can buy?

Two more links. Share financed: assume 60% of buyers borrow. Ticket: an average vehicle of Rs 1 lakh with a 75% loan-to-valueThe loan as a share of the price of the asset it buys. A 75% loan-to-value on a Rs 1 lakh scooter is a Rs 75,000 loan. is a Rs 75,000 loan. So 1.37 crore loans at Rs 75,000 each is about Rs 1,02,600 crore a year. Then cut to what a securitisation desk can actually buy: only loans made by lenders who sell pools, which at an assumed 30% is about Rs 30,780 crore. Banks that keep loans on their books never reach the desk.

How do you show the interviewer the number is sane?

Give a range and name the link that drives it. Moving the financed share between 50% and 70% and the ticket between Rs 65,000 and Rs 85,000 spreads the answer from Rs 74,100 crore to Rs 1,35,660 crore. A range with a named driver is more credible than a single precise figure. Then say the check you would run outside the room: compare 2.28 crore units with the industry body's published domestic sales, and the rupee total with the originators' disclosed disbursements. The chain is only as good as its weakest assumption, which here is the replacement cycle.

Yes: settling leaves the seniors Rs 361 crore against Rs 342 crore by fighting, Rs 19 crore better. A fight that costs 10% of Rs 380 crore shrinks value to Rs 342 crore before anyone is paid. Giving the juniors 5% of Rs 380 crore costs Rs 19 crore. The seniors should pay anything up to 10% of the equity to avoid the fight, before counting six months of delay.

Why pay anything to a class that is out of the money?

Two heirs argue over a house worth Rs 38 lakh. One has the clear legal claim; the other can drag the case through court for a year and run up lawyers' fees of Rs 3.8 lakh. The strong heir who hands over Rs 1.9 lakh to end it keeps more than one who wins in court. A settlement payment to a junior class is rational whenever it costs less than the value the fight would destroy. Being right about priority does not make fighting free: the costs come out of the same pot.

Here the seniors are owed Rs 400 crore against Rs 380 crore of value, so under absolute priorityThe rule that each class of creditor is paid in full before the class below it gets anything. the juniors get nothing and the seniors take the whole company. The juniors' only lever is delay and cost. Fighting burns 10% of value, Rs 38 crore, leaving Rs 342 crore.

How big a gift would still make sense?

The seniors break even at a gift of 10%, the same as the share of value the fight would destroy. At 5% they are comfortably inside that line, and the six months of delay tips it further, because a recovery received now is worth more than the same recovery received later and a distressed business often loses customers and staff while it waits. Senior recovery rises from 85.5% to 90.25% of their claim.

Say the limits too. The answer assumes the seniors would win the fight outright; if there is any chance the juniors win something, settling looks better still. It also assumes the 10% cost estimate is honest. And a gift to one class can invite others to threaten the same thing, which is why seniors often pair the equity with warrants or tie it to the juniors voting for the plan.