Debt Capital Markets puzzles, solved step by step

The implied one-year rate a year from now is about 8.20%. Rs 100 at 7.6% for two years grows to 115.78. Rs 100 at 7% for one year grows to 107, so the second year must turn 107 into 115.78: a rate of 8.20%. It is above both spot rates because the two-year rate is an average, and the second year has to be high enough to pull the 7% first year up to 7.6%.

What does a forward rate actually mean?

You can reach the next town by the highway in one go or by a side road with a stop halfway. If both leave at the same time and arrive at the same time, the second leg of the side road must make up whatever speed the first leg lost. A forward rate is the rate for the second leg that makes lending for two years in one go pay exactly the same as lending for one year and rolling over. If the two routes paid differently, everyone would take the better one until they did not.

Why must the forward sit above both spot rates?

Because the two-year rate is roughly the average of the two one-year legs. If the first leg earns 7% and the average is 7.6%, the second leg must earn about 7.6% plus the 0.6 the first leg fell short, roughly 8.2%. The quick approximation is two times 7.6 less 7, which is 8.2%; compounding adds the last few hundredths to give 8.20%. On an upward sloping curve, forwards always sit above the spot rates, and the steeper the curve, the further above.

Say what the forward is not. It is not a forecast of where one-year rates will be; it is the break-even rate that makes the two routes equal today. If investors demand extra yield for locking money up longer, a term premium, the forward will sit above what the market truly expects. A desk uses it to price a forward starting loan or to judge whether rolling short or locking long is cheaper.

The price is 125 at 6.4% and 83.33 at 9.6%: a gain of 25 against a loss of 16.67. A perpetuity is worth its coupon divided by the yield, so 8 over 0.064 and 8 over 0.096. The price-yield curve is bowed, not straight, which is convexity: for the same move either way, the price rises more than it falls, and the gap grows with the size of the move.

How do you price a bond that never matures?

Think of a shop that pays you Rs 8 of rent every year forever. If you want an 8% return, you will pay Rs 100 for it; if 6.4% is enough, you will pay Rs 125, because 6.4% of 125 is 8. A perpetuity's price is simply its annual payment divided by the yield investors demand. No maturity, no final repayment, no discounting table: one division does all the work.

Why is the gain bigger than the loss?

Because the price curve bends toward you. When yields fall, each extra basis point lifts the price more than the last, and when yields rise, each basis point cuts it less than the last, so equal moves produce unequal price changes in the holder's favour. That bend is convexityThe curvature of the price-yield relationship; positive convexity means prices gain more when yields fall than they lose when yields rise by the same amount.. The average of 125 and 83.33 is 104.17, above the starting 100, which is the whole idea in one number.

A perpetuity has a lot of it, because its cash flows stretch to infinity. At 8% its duration is 1 over the yield, about 12.5 years, so a straight-line estimate would say 1.6 points moves the price about 20 either way. The truth, +25 and -16.67, shows how badly a duration-only estimate does on large moves for long bonds. The limit is that convexity helps only a holder of an option-free bond; a callable perpetual would lose most of that upside.

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.

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 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.

Your average cost is Rs 13.33 a unit, not Rs 15. Rs 1 lakh at Rs 10 buys 10,000 units and Rs 1 lakh at Rs 20 buys 5,000, so Rs 2 lakh buys 15,000 units. Equal money buys more units when the price is low, so the low price carries more weight. The average cost is the harmonic mean of the two prices, which always sits below the simple average when the prices differ.

Why does equal money not mean equal weight?

Suppose you fill Rs 1,000 of petrol at Rs 100 a litre one week and Rs 1,000 at Rs 125 the next. You got 10 litres and then 8, so 18 litres for Rs 2,000, about Rs 111 a litre, not the Rs 112.50 midpoint. When you spend the same money each time, you buy more units at the lower price, so the lower price counts for more in your average cost. The bond fund works the same way, only with a wider gap between the prices, so the effect is larger.

What is the general rule, and why is it called a harmonic mean?

Divide total money by total units: 2,00,000 over 15,000 gives Rs 13.33. Written as a formula, the units are money over price, so the average cost is the number of instalments divided by the sum of one over each price. That is the harmonic mean, and for any set of unequal prices it lies below the simple average. The simple average here overstates your cost by 12.5%.

This is the arithmetic behind the claim that investing a fixed sum on a schedule lowers your average cost. It is true, but only against the simple average of the prices you happened to pay. It does not make the investment itself cheaper or safer: it says nothing about whether Rs 10 or Rs 20 was the fair price. Say that limit, because the interviewer is often checking whether you oversell the result.

Weigh three certificates against three, then one against one inside the suspect group. If the first weighing tips, the forgery is on the heavy side; if it balances, it is among the three set aside. Take that group of three and weigh one against another: the heavier one is the forgery, and if they balance, the third is. Each weighing has three outcomes, so two weighings separate nine cases.

Why split into thirds rather than halves?

Think of a quiz where each answer can be yes, no or maybe, instead of just yes or no. Every question now splits the possibilities three ways, so you get to the answer in fewer questions. A balance scale is a three-answer question: left heavy, right heavy or level, and a good weighing uses all three answers. Splitting in halves throws the level answer away, which is why it needs more weighings.

How do you prove two weighings is the minimum, and the limit?

Count the outcomes. One weighing has 3 outcomes and two weighings have 3 x 3 = 9, so two weighings can pick out at most 9 certificates, and nine is exactly what you have. One weighing cannot do it, because 3 outcomes cannot separate 9 suspects. The same count tells you the scale for any number: three weighings handle up to 27, four handle up to 81.

Now say why a DCM interviewer asks it. The puzzle rewards the habit of asking how much information each step gives before choosing the step. Due diligence on a bond issue works the same way: the best question to ask a management team is the one whose possible answers split the risks most evenly, not the one whose answer you already expect. The analogy is loose, so keep it to one sentence.

The 7.85% semi-annual coupon is marginally more expensive: an effective 8.004% a year against 8.000%. Paying 3.925 every six months means the first half coupon reaches investors early and can itself earn 3.925% for six months. Compounded, 1.03925 squared minus 1 is 8.004%. The two are almost equal because the exact semi-annual equivalent of 8% annual is 7.846%, just below 7.85%.

Why can you not compare 8.00 with 7.85 directly?

A landlord offered Rs 12,000 once a year or Rs 6,000 every six months should prefer the six-monthly cheques: the first Rs 6,000 can sit in a deposit for half a year. A rate only means something together with how often it is paid, because money paid earlier starts earning sooner. 8.00% paid annually and 7.85% paid semi-annually are quoted on different bases, so they must be put on one basis before anyone can say which is cheaper.

How do you convert both to an effective annual rate?

Take Rs 100. The annual bond pays 8.00 at the end of the year, so its effective rate is 8.00%. The semi-annual bond pays 3.925 at six months and 3.925 at twelve. Reinvest the first 3.925 at the same 3.925% for six months and it grows to 4.079, so the investor holds 8.004 per Rs 100 at the year end, an effective 8.004%. Reinvesting at the coupon's own rate is the convention that defines an effective rate.

How big is the difference in money, and when does it matter?

The gap is about 0.4 of a basis point, roughly Rs 2 lakh a year on a Rs 500 crore issue: real, but tiny next to the 15 basis points the headline rates suggest. The exact semi-annual rate matching 8% annual is 2 x (square root of 1.08 minus 1), 7.846%. The comparison matters in India, where government securities usually pay semi-annually and many corporate bonds annually, so a spread between them is only fair once both sit on one basis. The limit: the conversion assumes the early coupon is reinvested at the same rate, a convention rather than a promise.

About Rs 30,000 to 37,500 crore of debt, with Rs 33,750 crore at the midpoint. 10 GW is 10,000 MW; at Rs 4 to 5 crore per MW the build costs Rs 40,000 to 50,000 crore, and 75% of that is debt. The answer moves one for one with the cost per MW: every Rs 0.5 crore per MW shifts the debt by Rs 3,750 crore, more than the effect of moving the gearing five points.

How do you structure the estimate before any arithmetic?

Estimating a home loan starts with the price per square foot, times the floor area, times the share the bank will fund. A project debt estimate is the same chain: capacity times cost per unit of capacity gives the project cost, and the debt share of that cost gives the borrowing. Say the chain out loud first, with its units: megawatts, crore per megawatt, then a percentage. The most common slip in this question is a units error, not an arithmetic one.

Which assumption moves the answer most?

Flex each input across a sensible range while holding the other at its midpoint. Moving the cost from Rs 4 to Rs 5 crore per MW swings the debt by Rs 7,500 crore; moving the gearing from 70% to 80% swings it by Rs 4,500 crore. So the cost assumption is the one to defend, or to ask about. A lender would also want to know how much of the cost is modules against land, grid connection and construction, because those pieces move for different reasons.

Close with the range, not a single number. An estimate built on a range of inputs should come out as a range, with the midpoint named and the driver identified. At the midpoint, Rs 33,750 crore at an assumed 9% would carry roughly Rs 3,038 crore of interest a year. The costs here are the question's assumptions, not current market figures; in a real pitch you would confirm the cost per MW and typical gearing against recent project financings before using them.