Debt Capital Markets puzzles, solved step by step

A naive scan takes about n K (K minus 1) comparisons; a min-heap takes about n K log2 K. For 64 investors with 1,000 orders each, that is about 4.03 million against 0.384 million, roughly 10 times fewer. The heap holds only each list's current best order, so finding the next order costs a few steps down one branch rather than a look at every list.

What is the naive way, and where does it waste effort?

Imagine 64 queues at a bank, each already in order of arrival, and you must call people one at a time in overall order. The naive clerk walks along all 64 queue fronts every time to find the earliest. Every time one order leaves the book, the naive merge re-compares all K front orders, even though only one of them changed. That is K minus 1 comparisons for each of n K orders: 64,000 orders times 63 is 4,032,000 comparisons.

The other naive route is to merge lists one at a time: merge list 1 and 2, then merge in list 3, and so on. Each merge re-reads everything merged so far, which costs about n times K squared over 2, here about 2.08 million. Better than scanning, but it still grows with the square of K.

Why does a heap fix it?

A min-heapA tree in which every parent is smaller than its children, so the smallest item is always at the top and can be removed and replaced in a number of steps equal to the tree height. keeps the K front orders only partly sorted: the best is always at the top, and the rest are arranged so that fixing the tree after a change touches one path from top to bottom. Taking the best order and inserting that investor's next one costs about log2 K comparisons instead of K, which is 6 instead of 63 at K of 64. Total work becomes n K log2 K, about 384,000 comparisons.

Say where the heap does not matter. With four or five lists, scanning is about as fast and simpler to code, and the orders arrive as fast as a person can read them anyway. The heap earns its place when K is large or the lists do not fit in memory, which is the version in the reported question: arrays read from disk, where only the front of each list is held at once. A careful heap counts about two comparisons per level on the way down, so treat log2 K as the order of the cost, not an exact count.

50% with a fair coin, and about 64.6% with a 55% win rate. With a fair coin, your capital is a fair bet, so the chance of reaching 6 from 3 is 3 over 6. With an edge, the chance is 1 over 1 plus (q over p) cubed, where q over p is 0.45 over 0.55. That gives 1 over 1.548, or 64.6%. Spread the same edge over walls ten times further away and it rises to about 99.8%.

Why is the fair-coin answer simply 3 over 6?

Picture a game where you and a friend toss a coin for Rs 1 until one of you is broke; you start with Rs 3, the friend with Rs 3. Every toss is fair, so on average nobody gains, and your expected wealth at the end must still be Rs 3. If the game ends at 0 or 6 and your expected ending wealth is 3, you must reach 6 exactly half the time. In general the fair-coin chance of reaching N from i is i over N.

How does a 55% edge change it?

This is the classic gambler's ruinA random walk that stops at two walls, used to find the chance of hitting one wall before the other. set-up. A 55% edge on each trade turns into a 64.6% chance of doubling before going broke, a modest lift because the walls are only three steps away. The fair game takes 9 trades on average to finish, so the edge only gets a handful of chances to work.

That is the lesson a desk wants. The same edge, bet in smaller pieces relative to capital, almost removes the risk of ruin: starting at 30 with a target of 60 and 1-unit trades, the chance of success is 99.8%. Betting a large share of capital on each trade throws the edge away, because variance gets to end the game before the edge shows. The limit is the model itself: real trades do not win or lose exactly one unit, and edges are estimated, not known.

Pre-funding costs about Rs 13 crore of negative carry, before tax. The issuer pays 8.2% on the new bond while earning 6.9% on the parked cash, a 1.3% gap on Rs 1,000 crore for twelve months. In return it removes refinancing risk: it no longer needs the market to be open, spreads to hold and its rating to survive the final year. On a five-year deal, waiting would have to cost more than about 26 basis points a year for pre-funding to lose.

What exactly does pre-funding cost?

A family renting a flat signs a new lease a year before the old one ends because rents are rising and good flats are scarce. For twelve months they pay a little extra holding the new flat, but they are not searching in a panic the week the old lease ends. Pre-funding costs the gap between what the new money costs and what the parked cash earns, for as long as both exist: the negative carryThe cost of holding a position whose income is lower than its funding cost, here cash earning less than the bond that raised it.. Here 8.2% less 6.9% is 1.3%, on Rs 1,000 crore for a year, Rs 13 crore, about Rs 1.08 crore a month.

What does that Rs 13 crore buy?

Certainty. Without pre-funding, the issuer must refinance in a narrow window before maturity, and if markets are shut, spreads have widened or its rating has slipped in that window, it either pays far more or cannot repay. Pre-funding is insurance with a known premium. A useful breakeven: spread over a five-year replacement bond, Rs 13 crore is about 26 basis points a year before discounting and about 33 after. If waiting risked a rise larger than that, or a missed maturity, the premium was cheap.

Say the refinements. Interest paid is usually tax deductible and interest earned is taxable, so at an assumed 25% rate on both the after-tax carry is about Rs 9.75 crore. The issuer also carries a bigger balance sheet for a year, gross debt rises before the old bond goes, which can briefly worsen reported leverage. And the parked cash must be invested safely and kept liquid; reaching for yield on it defeats the purpose of paying for certainty.

About 35.7%. Set the price equal to the probability-weighted payoff: p times 100 plus 1 minus p times 30 equals 55. The 30 is paid either way, so the price only has to cover 25 of the 70-point gap between recovery and par, and 25 over 70 is 35.7%. The market thinks restructuring is nearly twice as likely as repayment.

How do you turn a price into a probability?

Suppose a raffle ticket pays Rs 100 if you win and Rs 30 as a consolation prize if you lose, and it sells for Rs 55. Everyone gets the Rs 30, so you are really paying Rs 25 for a shot at an extra Rs 70. A price between two outcomes is a probability-weighted average of them, so you can solve for the weight. The consolation prize matters: ignore it and you would read the Rs 55 as a 55% chance.

What does ignoring discounting hide?

Both payoffs arrive in the future, often a year or more away, and a distressed investor wants a high return for waiting. If the payoffs arrive in a year and the buyer demands 12%, the price must equal the expected payoff divided by 1.12, which lifts the implied repayment probability to about 45.1%. So the 35.7% is a floor: part of the discount to par is a charge for time and risk, not only for default. That is why desks quote these as risk-neutral probabilities, not forecasts.

The other soft spot is the recovery of 30. It is an estimate, and the implied probability is sensitive to it: at a recovery of 40, p is 15 over 60, 25%. Before trading on the number, a credit analyst would test the recovery from the balance sheet and the claims ranking ahead of this bond.

The barbell does better both ways, by about 0.065 per Rs 100 when rates rise and 0.075 when they fall. Up 100: the bullet loses 5.43, the barbell 5.36. Down 100: the bullet gains 5.80, the barbell 5.87. Duration is matched, so the difference is convexity, which the barbell has more of because its cash flows sit further apart.

If the durations match, why do the results differ?

Balance a see-saw with one child sitting in the middle, or with two children at the far ends. Both are balanced, but push the ends and the version with weight at the tips swings further. Duration measures where the weight of the cash flows sits on average; convexity measures how spread out it is, and the barbell's cash at years 2 and 10 is spread much wider than the bullet's single payment at year 6. For a zero, convexity grows roughly with maturity squared, so the 10-year leg adds far more than the 2-year leg takes away.

How big is the advantage, and what does it cost in a real market?

Small on a 100 basis point move: under a tenth of a rupee per Rs 100, about 6.5 paise on a rise and 7.5 on a fall. The advantage grows with roughly the square of the move, so at 300 basis points it is 0.52 to 0.77 per Rs 100, which is why convexity matters most in volatile markets. Price the positions by discounting each cash flow at the new flat yield, which is what the figures above do exactly rather than by the duration and convexity approximation.

Now the honest limit. A flat curve where both positions yield 7% and convexity is free cannot last: buyers would all pick the barbell. In real markets the barbell usually yields a little less than the bullet, and that lower yield is the price of its convexity; whether it pays depends on how much rates actually move. The barbell also loses if the curve twists, for example if 2-year and 10-year yields rise while 6-year yields do not, a non-parallel move that duration and convexity for a single yield do not capture.

WACC is 8.0% before tax and 7.25% after. Read the cost of equity as the earnings yield, 1 over a P/E of 10, which is 10%. Half the capital costs 10% and half costs 6%, so the blend is 8.0%. The tax shield cuts debt to 6% x 0.75 = 4.5%, and the blend falls to 7.25%. The P/E shortcut assumes no growth; with growth the true cost of equity is higher.

Where does the cost of equity come from when you are not given a beta?

If a shop earns Rs 10,000 a year and sells for Rs 1 lakh, a buyer earns 10% on the price. Nobody would pay more unless they expected growth, so 10% is roughly what buyers of such a shop demand. A P/E of 10 means earnings are a tenth of the price, an earnings yield of 10%, and with no growth that is a quick proxy for the cost of equity. Say out loud that it is a proxy: it is the fastest honest route when the interviewer gives you only a multiple.

Why does tax only touch the debt half?

Interest is paid before tax and dividends after it. Every rupee of interest reduces taxable profit, so the government in effect pays a quarter of the interest bill at a 25% tax rate, and debt that costs 6% on paper costs 4.5% to the company. Equity gets no such relief. That is why the after-tax WACC is 0.75 points lower: half the capital times the 1.5 point shield. The difference compounds into valuation, since a lower discount rate raises the value of every future cash flow.

State the limit of the shortcut before the interviewer does. The earnings yield equals the cost of equity only for a company that does not grow and pays out all its earnings. If earnings grow at 3% a year, investors paying 10x are expecting roughly 10% plus 3%, about 13%, and the WACC rises with it. For a real company you would build the cost of equity from a risk-free rate, a beta and a market premium instead.

Interest cover is 5.0x and fixed charge cover is 1.8x; a lender trusts the fixed charge cover more. Interest cover is EBITDA of 150 over interest of 30. Fixed charge cover takes off capex of 40 and tax of 20, leaving 90, and sets it against interest plus the principal that must be repaid this year, 50. It counts the cash that really has to go out, so it shows how thin the cushion is.

Why can a company look five times covered and still be tight?

A salaried person earning Rs 1.5 lakh a month with a Rs 30,000 loan interest bill looks comfortable. But rent and school fees of Rs 60,000 come out first, and the loan also needs Rs 20,000 of principal each month. What is left, Rs 90,000, against Rs 50,000 owed, is the true cushion. Interest cover ignores the cash a business must spend before lenders are paid and ignores the principal it owes, so it flatters the picture. Fixed charge coverCash flow after capex and tax, divided by all the debt payments due in the period, interest and scheduled principal together. fixes both.

Which ratio would you put in a covenant?

The fixed charge ratio, for a lender who cares about being repaid on schedule. At 1.8x the company has Rs 40 crore of spare cash a year after paying lenders, and a 44% fall in that cash would leave it unable to meet its schedule; at 5.0x interest cover the same fall would still look safe. Interest cover is still useful: it is quick, comparable across companies, and a lender pricing a bullet bond with no amortisation cares less about principal. But for an amortising term loan, fixed charge cover is the test that fails first.

Say the limits. Definitions vary by document: some deduct only maintenance capex, some add lease payments to the fixed charges, some take tax paid rather than tax charged. Always read the covenant's own definition before comparing a borrower's ratio with a peer's; two companies at 1.8x on different definitions are not equally safe.

The last passenger gets their own seat with probability exactly 1/2. Every random choice picks seat 1, seat 100, or another seat that just passes the problem on. Seat 1 and seat 100 are always equally likely, and whichever goes first decides it. For passenger k, the same argument over the seats that still matter gives (n minus k plus 1) over (n minus k plus 2): 99 in 100 for passenger 2, falling to 1/2 for the last.

Which seats actually matter to the last passenger?

Think of a game of musical chairs where every displaced person grabs a random empty chair. It looks like a mess, but most grabs only move the mess along to someone else. For the last passenger, only two seats matter: seat 1, the drunk's own, and seat 100, their own. If a displaced passenger takes seat 1, the chain stops and everyone after sits correctly. If someone takes seat 100, the last passenger loses. Any other seat hands the same situation to a later passenger.

How do you get the answer for any passenger?

Apply the same logic to passenger k. For them, the deciding seats are seat 1 and the seats of passengers k to n, which are still unclaimed by their owners when the chain reaches them. Passenger k loses only if their own seat is picked before seat 1, and among the n minus k plus 2 seats that matter, only one of them is theirs. The chance of the chain ending well for them is therefore n minus k plus 1 over n minus k plus 2.

Check the ends of the formula, which is what an interviewer will do. Passenger 2 loses only if the drunk sits in seat 2, a 1 in 100 chance, so 99 in 100 is right. Passenger 99 has a 2 in 3 chance, and passenger 100 has 1 in 2. The answer does not depend on the size of the plane for the last passenger: with 10 seats or 1,000 it is still one half, which is the fact worth saying out loud.

A total return of +1%. You paid 100. During the year you received a coupon of 6, and the bond is now worth 95, so you hold 101 of value against 100 paid. Total return is income plus price change, 6 minus 5, divided by the price you paid. Quoting minus 5% counts only the price and forgets the coupon, which for a bond is most of the return.

What counts as return on a bond?

If you buy a flat for Rs 1 crore, collect Rs 6 lakh of rent over the year and the flat's market value slips to Rs 95 lakh, you are not down 5%. You have Rs 95 lakh of flat and Rs 6 lakh of cash: Rs 1.01 crore. Total return is everything the investment paid you plus the change in what it is worth, divided by what you paid. For a bond, the coupon is the rent.

Why would the price have fallen, and what does it mean for next year?

A fixed coupon bond falls in price when market yields rise. If the bond had four years left after the coupon, a price of 95 means new buyers earn about 7.49% a year to maturity, so the loss this year is partly paid back as extra yield in later years if the bond is held. That is the other reason not to fixate on the minus 5: a holder to maturity still gets 100 at the end, and the price fall is a mark-to-market loss, not money gone for good.

Two limits worth saying. This assumes the coupon is simply held as cash; reinvesting it would add a little more. And it assumes the issuer is still sound: if the price fell because default risk rose, the loss may not come back. A desk reports both numbers, the price change and the total return, because they answer different questions.

Worth more: about Rs 2,134 crore, Rs 134 crore above the unlevered value. Permanent debt of Rs 800 crore at a 25% tax rate creates a tax shield worth 0.25 x 800, Rs 200 crore, lifting value to Rs 2,200 crore. Distress would destroy 30% of that, Rs 660 crore, but with a 10% chance the expected cost is Rs 66 crore. The shield wins unless the distress probability passes about 30%.

Where does debt add value, and where does it take it away?

A family that takes a home loan gets a tax deduction on the interest every year, a steady saving. But if the loan is too big and a job is lost, they may have to sell the house in a hurry at a poor price. Debt adds value through the tax saved on interest and subtracts value through the expected cost of financial distress; the net of the two decides whether borrowing helps. That balance is the {term('trade-off theory', 'The idea that a company chooses its debt level by weighing the tax benefit of interest against the expected costs of financial distress.')} of capital structure.

Why is the tax shield worth t times D?

For permanent debt, the company pays interest of r times D every year forever and saves t times that in tax. Discounting a perpetual saving of t r D at the debt's own rate r gives t D. So Rs 800 crore of permanent debt at a 25% tax rate is worth Rs 200 crore in tax savings, whatever the interest rate, as long as the debt is truly permanent and the company always has profits to deduct against. Both conditions are generous, which is why real tax shields are usually worth less than t D.

Now find where the answer flips. The shield is 200; the distress cost is the probability times 660. The firm is worse off once the distress probability passes 200 over 660, about 30.3%, so debt at this level is value-adding only while distress stays well below a one-in-three chance. Adding more debt raises both lines, the shield in a straight line and the distress probability faster, which is why an optimal level exists. The limits: the 30% cost and the 10% probability are estimates, and indirect distress costs, lost customers and staff, are the hardest part to measure.