Debt Capital Markets puzzles, solved step by step

The Macaulay duration is 13.5 years and the modified duration is 12.5. For a perpetuity, Macaulay duration is (1 + y) / y, 1.08 / 0.08 = 13.5, and modified duration divides by 1 + y, leaving 1 / y = 12.5. The bond never matures, but duration weights each coupon by its value today, and at 8% a coupon sixty years out is worth under 1% of its face. Half the bond's value arrives within the first nine years.

Why does a bond that never ends have a finite duration?

Picture a very long seesaw. A heavy child sits near the pivot and a row of ever lighter children stretches towards the far end. The plank balances not far from the heavy child, however long it is. Duration is the balance point of a bond's cash flows, each weighted by its present value, and distant coupons are discounted so hard that they barely pull on it. At 8%, the coupon in year 30 is worth about 10% of its face today and the coupon in year 60 about 1%.

How do you get 13.5 without summing forever?

Use the shortcut and then check it against the price. For a level perpetuity, Macaulay duration is (1 + y) / y, so at 8% it is 1.08 / 0.08 = 13.5 years, and modified duration is 1 / y = 12.5. The price formula confirms it: a perpetual is worth coupon over yield, 8 / 0.08 = 100, and at 8.01% it is worth 8 / 0.0801, a fall of 0.125. That is 12.5 x 0.01, exactly what a modified duration of 12.5 predicts for one basis point.

Now say the implication, which is what the interviewer is after. Duration falls as the yield rises: the same perpetual at 4% has a Macaulay duration of 26 years, and at 12% only 9.3. Higher yields shorten every bond, because they shrink the weight of distant cash flows. The limit: many perpetual bonds carry an issuer call after five or ten years, and a callable perpetual trades on a much shorter effective duration than this formula gives.

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.

Yes, pitch: the expected value is plus Rs 70 lakh, and the pitch breaks even at a 12.5% win probability. A 30% chance of Rs 4 crore is worth Rs 1.2 crore in expectation, against a certain Rs 50 lakh cost. Break-even is where the probability times Rs 4 crore equals Rs 50 lakh: 0.5 divided by 4, or 12.5%. The 30% estimate clears that by a wide margin, so even a rough probability supports pitching.

How do you compare an uncertain fee with a certain cost?

A shopkeeper deciding whether to print Rs 500 of flyers asks how much extra business they might bring, times how likely that is. Expected value multiplies each outcome by its probability and adds them up, so a 30% chance of Rs 4 crore is worth Rs 1.2 crore before costs. The cost of pitching is paid whether you win or lose, so it comes off in full: Rs 1.2 crore minus Rs 0.5 crore is plus Rs 0.7 crore.

What win probability makes the pitch worth it, and how robust is the answer?

Break-even is the probability at which the expected fee just covers the cost: Rs 50 lakh over Rs 4 crore, 12.5%. The 30% estimate is more than twice that, so the decision survives a lot of error in the estimate. That is the useful part of the calculation: you rarely know a win probability precisely, but you often know whether it is comfortably above or below the break-even.

Name what the simple sum leaves out. Losing is the most likely outcome, 70% of the time, so a desk that pitches only once can easily end Rs 50 lakh down; expected value pays off across many pitches. The team's time also has an opportunity cost if it could pitch a better mandate instead, and a win may bring follow-on business that the Rs 4 crore does not capture. Each of those shifts the break-even; none reverses this answer.

The two-year zero rate is about 7.035%. The one-year par bond pays 106 in a year for 100, so the one-year zero rate is 6%. The two-year par bond pays 7 and then 107. Strip the first coupon at 6%: 7 / 1.06 = 6.604. The remaining 93.396 must be the value of 107 in two years, so (1 + z) squared = 107 / 93.396, and z = 7.035%.

What does bootstrapping actually do?

A fruit seller sells one mango for Rs 20 and a bag of one mango and one papaya for Rs 70. You can price the papaya without ever seeing it sold alone: Rs 50. Bootstrapping prices each future date the same way: use the shorter bond to value the early cash flows, and whatever is left of the longer bond's price belongs to its final cash flow. A one-year par bond at 6% pays 106 in a year for 100, so the one-year zero rate is simply 6%.

Why is the zero rate above the par yield?

A par yield is one rate that blends the rates on every cash flow, so when the curve slopes up, the last cash flow must carry a rate above the blend. The two-year bond's first coupon is discounted at 6%, which makes it worth a little more than a 7% discount would say. To keep the whole bond at 100, the final 107 has to be discounted a little harder: 7.035% rather than 7%. The gap is small because only one coupon of 7 sits at the lower rate.

Say what the zero curve is for, because it is why the question is asked. With zero rates you can price any bond by discounting each cash flow at its own rate, which is how a desk checks whether a bond is fairly priced. The same two rates give the one-year rate one year forward: 1.07035 squared over 1.06, minus 1, about 8.08%. The limit: real bootstrapping uses many bonds with different coupons and settlement conventions, and a two-point curve says nothing about the shape in between.

The 2-year fixed bond carries far more rate risk: a duration of about 1.9 years against about 0.25 for the floater. The floater's coupon resets to the market every quarter, so a rate move can only hurt it until the next reset, three months away. The fixed bond is locked at 7% for two years, and its Macaulay duration is 1.93 years. A 1 point rise in rates costs the floater about 0.25 points and the fixed bond about 1.78.

Why is a 10-year floater so short on rate risk?

Think of two landlords. One has let a flat for two years at a fixed rent; the other has a ten-year lease whose rent is reset to the market every three months. If market rents jump, the first is stuck for two years and the second catches up within a quarter. A floater's coupon resets to the market rate on every reset date, so its value can only drift from par for the few months until the next reset, whatever its final maturity. Just after a reset, that is about a quarter of a year.

What does each lose if rates rise one point?

Reprice both at a rate 1 point higher: the fixed bond falls from 100 to 98.22, a loss of 1.78, while the floater loses only 0.25, because it earns the old coupon for one quarter and then resets. The fixed bond carries about 8 times the rate risk of a note five times its length. This is why a bank funding itself with three-month deposits is comfortable holding floaters: the asset and the liability reprice together.

Keep the other risk in view. The floater's short rate duration says nothing about credit: its spread duration runs to maturity, about 7 years here, so a widening in the issuer's spread hits it far harder than it hits the 2-year bond. Name both numbers if the interviewer pushes, because the claim that floaters are low risk is only half true. The limit: this assumes a flat 7% curve and an issuer whose credit does not change.

The issuer paid a new issue premium of about 15 basis points. Its own curve runs from 8.10% at 5 years to 8.40% at 7 years, so a fair 6-year yield, interpolated halfway, is 8.25%. The new bond priced at 8.40%, 15 basis points wide of that. On a Rs 1,000 crore issue that is Rs 1.5 crore a year of extra coupon, about Rs 6.9 crore in today's money over six years.

What is fair value for a bond that does not exist yet?

If flats in one building sell for Rs 80 lakh on the fifth floor and Rs 84 lakh on the seventh, a sixth-floor flat is fairly priced near Rs 82 lakh; asking Rs 84 lakh for it is a premium. A new bond's fair value is read off the issuer's own curve at the new bond's maturity, because that curve already prices this issuer's credit. With points at 5 and 7 years, the 6-year point is the straight-line midpoint: 8.10% plus half of 30 basis points, 8.25%.

Why do issuers pay a premium at all, and what does it cost?

Investors want a small concession for buying a new bond rather than the existing ones: they take size, commit before the bond trades, and often sell other holdings to make room. The syndicate desk's job is to keep that premium as small as the order book allows. Here 15 basis points on Rs 1,000 crore is Rs 1.5 crore of extra coupon a year, about Rs 6.9 crore today over six years at 8.40%, which is why syndicate desks argue over five basis points.

State the assumptions behind the answer. Linear interpolation assumes the curve is straight between 5 and 7 years; a curve that bows would move fair value by a few basis points. The outstanding bonds may also be small or rarely traded, so their yields can carry a liquidity premium of their own. A desk would check the answer against a second yardstick, such as the spread over the government curve, before calling the premium high or low.

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.

At Rs 60 crore the lender gets 60 and shareholders nothing; at Rs 140 crore the lender gets 80 and shareholders 60. Shareholders prefer riskier assets. The lender's payoff is capped at the Rs 80 crore it is owed, while shareholders keep everything above 80 and cannot fall below zero. Equity works like a call option on the assets with a strike of 80, so more volatility moves value from the lender to the shareholders.

Who gets what in each outcome?

A student borrows Rs 80,000 from a relative to start a small business and owes it back in a year. If the business ends up worth Rs 60,000, the relative gets Rs 60,000 and the student nothing; if it is worth Rs 1,40,000, the relative gets Rs 80,000 and the student keeps Rs 60,000. The lender's payoff is the smaller of the asset value and what it is owed; the shareholders' payoff is whatever is left above the debt, never below zero. Here that is 60 or 80 for the lender and 0 or 60 for the shareholders.

Why do shareholders like risk and lenders dislike it?

Keep the average at Rs 100 crore but widen the outcomes to Rs 20 or Rs 180 crore, each equally likely. The lender now gets 20 or 80, an average of 50 instead of 70, while shareholders get 0 or 100, an average of 50 instead of 30. Nothing about the business improved on average; Rs 20 crore of value simply moved from the lender to the shareholders. That is the shape of a call optionThe right, not the obligation, to buy something at a fixed price, so the holder keeps the upside above that price and loses nothing more below it.: capped loss, open-ended gain, so volatility helps the holder.

Say why a lender cares, because it is the point of the question. Loan documents restrict what a borrower can do with its assets, with limits on new debt, asset sales, dividends and changes of business, to stop shareholders swapping safe assets for risky ones after the loan is made. The pull is strongest near distress, as here, because shareholders have little left to lose. The limit: the 50/50 odds are an assumption, and the option view ignores that managers may also care about keeping the company alive.

Independent: 19% that at least one defaults and 1% that both do. Perfectly correlated: 10% for at least one and 10% for both. Independent loans survive together 0.9 x 0.9 = 81% of the time, so one or more defaults 19% of the time. Perfectly correlated loans default together or not at all. The expected number of defaults is 0.2 in both cases; what changes is how defaults bunch, and the chance of losing both loans is ten times higher.

Why is the answer not simply 10% plus 10%?

A 10% chance of rain on Saturday and 10% on Sunday is not a 20% chance of a wet weekend: some outcomes rain on both days and would be counted twice. The chance that at least one of two events happens is the sum of their chances minus the chance that both happen. For independent loans both default 0.1 x 0.1 = 1% of the time, so at least one defaults 10% + 10% - 1% = 19%. The quicker route is through survival: 1 minus 0.9 squared.

What does perfect correlation change?

With perfect correlation the two loans default in exactly the same states of the world, so at least one defaults 10% of the time, and whenever one does, both do. The average is unchanged: 0.2 loans default on average in both worlds. But the independent pair loses both loans only 1% of the time, while the correlated pair loses both 10% of the time, ten times as often.

Why does a portfolio lender care?

A lender holds capital against bad years, not average years, and correlation decides how bad the bad years are. Loans to one sector, one city or one promoter group tend to default together, which is why banks cap their exposure by sector and by group. Diversification works only when defaults are not strongly linked: ten independent loans with a 10% default chance all repay 35% of the time, but ten loans to one industry can behave like one big loan. The limit: real correlations sit between zero and one and rise in downturns, exactly when they hurt most, so the two cases here are the bounds, not the answer.

About 47.5%, a little under a coin flip. Exactly three anchors can happen in 4 ways, each with probability 0.6 x 0.6 x 0.6 x 0.4, so 4 x 0.0864 = 34.6%. All four come in with probability 0.6 to the fourth, 13.0%. Adding them gives 47.5%. Each anchor is more likely than not to order, yet needing most of a small group pulls the chance of launching below half.

How do you count the ways to get three of four?

Four friends each say they will probably come to dinner, 60% each, and you need three to keep the booking. The chance of any one pattern, say the first three come and the fourth does not, is 0.6 x 0.6 x 0.6 x 0.4 = 8.64%, and there are four such patterns, one for each friend who stays away. So exactly three come 4 x 8.64% = 34.6% of the time. All four come 0.6 to the fourth, 13.0% of the time. The booking survives 47.5% of the time.

Why is the answer below half when each anchor is likely?

Requiring most of a small group multiplies probabilities together, and products of numbers below one shrink fast. The most likely outcomes are two or three anchors, 34.6% each, so the launch hinges on which side of that line you land. If you needed only two anchors the chance would jump to 82.1%; if you needed all four it would fall to 13.0%. The threshold matters more than any single investor.

Say what a syndicate desk does with this. Either lower the threshold, by lining up a fifth anchor or sizing the deal so that fewer anchors are enough, or raise each anchor's probability before launch through early soundings. A fifth anchor at 60% lifts the chance of at least three from 47.5% to 68.3%; lifting each of the four to 70% gives 65.2%. The limit: anchors are rarely independent; they read the same market, so on a bad day they tend to drop out together, and the real chance is lower than this.