Debt Capital Markets puzzles, solved step by step

Carry of 0.90%, roll-down of about 1.05%, so roughly 1.95% over funding. Carry is the bond's 7.40% yield less the 6.50% cost of funding it. Roll-down comes from the bond ageing into a 4 year bond priced at 7.10%: its yield falls 30 basis points, which at a duration of about 3.5 is a 1.05% price gain. On an unchanged steep curve, standing still earns both.

What is carry, and what is roll-down?

Carry is the simple part: you earn the bond's yield and pay your funding rate, like renting out a flat for more than its loan EMI. Roll-down is the price gain that comes purely from time passing on an upward sloping curve: the bond gets shorter, and shorter maturities are priced at lower yields. Nothing about the market has to change for either to happen; that is why desks call them the return from standing still.

How do you turn 30 basis points into a return?

Use duration. A yield change converts to a price change of roughly duration times the change, so a 30 basis point fall at a duration of 3.5 is a gain of about 1.05%. Use the duration the bond will have after the year, when it is a 4 year bond, because that is the bond being repriced; the question gave 3.5 for exactly that reason. Then total return is the yield earned, 7.40%, plus the roll gain, about 8.45% before funding.

How much can the market move before the trade loses money?

Divide the cushion by the duration. A 1.95% return cushion at duration 3.5 absorbs a rise of about 56 basis points in the 4 year yield by year end before the position breaks even. That breakeven is the number a portfolio manager actually trades on. Say the limits too: the curve rarely stands still, the roll-down depends on the curve's shape between the two points, and the duration figure is an approximation that ignores convexity.

Ten doors end open: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Door n is toggled once on each pass whose number divides n, so it is toggled as many times as n has divisors. Divisors come in pairs, 2 with 6 for 12, so most numbers have an even count and end closed. Only a perfect square has a divisor paired with itself, giving an odd count, so only the squares stay open.

What decides whether one door ends open?

Think of a light switch pressed by everyone who walks past: whether the light ends on depends only on whether an odd or even number of people pressed it. Door n is toggled on pass k exactly when k divides n, so the number of toggles equals the number of divisors of n, and the door ends open only if that number is odd. This changes a hundred passes into one question about numbers.

Why do only perfect squares have an odd number of divisors?

Divisors come in pairs that multiply to the number: for 12 they are 1 and 12, 2 and 6, 3 and 4, six in all. Every divisor has a partner, so the count is even, unless a divisor is its own partner, which happens only when the number is a perfect square. For 16, the pairs are 1 and 16, 2 and 8, and 4 on its own, five in all. Squares up to 100 run from 1 times 1 to 10 times 10, so ten doors.

How do you show your working in the room?

Start by tracking one small door aloud, say door 6: passes 1, 2, 3 and 6, four toggles, closed. Moving from a single example to the rule about divisor pairs is what the interviewer is really scoring, more than the final count. Then generalise: with N doors, the open ones are the squares up to N, so the count is the whole part of the square root of N. For 1,000 doors, 31 stay open. The limit is simply that the puzzle is pure logic; its value on a desk is the habit of reducing a big process to one property.

The ask is about 2.5 basis points above the issuer's fair level, a small concession. Interpolating between 8.10% at 3 years and 8.40% at 5 years puts 4.5 years at 8.325%. Paying 8.35% costs the issuer 2.5 basis points on Rs 300 crore, about Rs 7.5 lakh a year and Rs 33.75 lakh over the life, which is often less than a public deal would need to clear.

What do you compare a reverse enquiry against?

A buyer walks into a tailor and asks for a jacket in a size the shop never stocks. The tailor prices it from the sizes either side, not from a different shop's rate card. A reverse enquiry, where an investor asks the issuer for a tailored bond, is judged against the issuer's own curve at that exact tenor. The issuer's 3-year and 5-year yields bracket the request, so the fair yield at 4.5 years comes from reading between them.

Is 2.5 basis points a lot?

Put it in rupees. One basis point on Rs 300 crore is Rs 3 lakh a year, so 2.5 basis points is Rs 7.5 lakh a year, about Rs 33.75 lakh over 4.5 years before discounting. Against that, a reverse enquiry saves the issuer the cost of a public deal: no roadshow, no marketing, and no new issue concessionThe extra yield a new bond usually offers over the issuer existing bonds so that investors take it up.. If a public 4.5-year deal would need a concession wider than 2.5 basis points to clear, the enquiry is the cheaper route.

Say the assumption and its limit. Linear interpolation treats the curve as a straight line between 3 and 5 years. Real curves usually bend, flattening as tenor lengthens, so the true 4.5-year point may sit a fraction above 8.325%, which would shrink the gap. An odd tenor also leaves the issuer with a bond that matches nothing else outstanding and trades less easily, which is one reason investors pay a little for the tailoring and issuers are willing to give a little back.

1.07 to the fifth is about 1.40, 1.4026 exactly, and the rule of 72 gives a doubling time of 10.29 years against a true 10.24. Expand it: 1 plus 5 x 0.07 plus 10 x 0.0049 is already 1.399. The rule of 72 is out by about 15 days over a decade, well inside any rounding a desk cares about.

How do you get 1.07 to the fifth in your head?

A savings account paying 7% a year gives you Rs 7 on every Rs 100 in the first year. In the second year you earn Rs 7 on the original Rs 100 again, plus 49 paise on the first year's Rs 7. Compounding is simple interest plus interest on interest, and for small rates over a few years the first correction does almost all the work. That is why the expansion below settles so quickly.

A second route checks the first. Square 1.07 to get 1.1449, square again to get about 1.3108, then multiply by 1.07: 1.3108 plus 7% of it, which is 0.0918, gives 1.4026. Two methods landing on the same number is the thing to say out loud; it tells the interviewer you do not trust a single chain of mental arithmetic.

How good is the rule of 72 at 7%?

The exact doubling time is the log of 2 over the log of 1.07, 10.245 years. The rule of 72 gives 72 over 7, 10.286 years. The error is about 15 days in ten years, because 72 is tuned to rates near 8% and 7% is close enough. At 2% the rule of 70 does better; at 20% the rule of 72 starts to drift, since the true figure is 3.8 years against its 3.6. Knowing where a shortcut breaks is worth as much as the shortcut.

The desk use is fast sanity checking. If a 7% bond's coupons are reinvested at 7%, Rs 100 grows to about Rs 140 in five years and Rs 200 in a little over ten. Simple interest would reach only Rs 170 by year ten, so the gap between the two lines in the picture is what reinvestment is worth.

Maximum debt is about Rs 872.7 crore, and the interest cover test binds. Cover of 2.5x on EBITDA of 240 allows interest of 96; at 11% that is Rs 872.7 crore of debt. Leverage of 4.0x would allow Rs 960 crore, but the rate is too high for the business to carry it. The two tests meet at a 10% rate; above that, coverage is the tighter limit.

Why run both tests instead of just the leverage one?

A household can be told it may borrow up to four years of income, but it also has to afford the monthly instalment. When interest rates are high, the instalment test fails first, long before the four-years rule does. Leverage measures how much debt there is; interest cover measures whether the business can pay for it, and the rate decides which one runs out first. A lender sizing debt computes both and lends the lower.

At what rate do the two tests swap?

Set them equal: 96 divided by the rate equals 960, so the rate is 10%. Above 10%, coverage binds; below it, leverage binds. That is the general rule to leave the interviewer with: the crossover rate is one over the leverage multiple times the cover multiple, 1 over 4.0 times 2.5, which is 10%. When rates rise, the same business can borrow less even though its EBITDA has not changed, and lenders who size on leverage alone find their borrowers failing cover tests a year later.

Say the limits. EBITDA interest cover ignores capex and tax, so a lender who worries about cash will also run a fixed charge test, which can bind lower still. And the effective leverage here is 3.64x, not the 4.0x the term sheet headline suggests; quoting 4.0x to the borrower would promise debt the second test will not allow.

Roughly Rs 270 crore of bank working capital credit outstanding, within a range of about Rs 150 to Rs 430 crore. A city of 1 crore has about 25 lakh households; at one shop per 100 households that is 25,000 shops. At Rs 50 lakh of sales each, they sell Rs 12,500 crore a year. Thirty days of stock less ten of supplier credit leaves 20 days to fund, Rs 685 crore, and banks fund an assumed 40%.

What exactly are you sizing?

A grocer pays the wholesaler for rice and oil before customers buy them. The money stuck on the shelves in between is the working capital, and a bank limit fills the part that the owner's own cash and the wholesaler's credit do not cover. The answer is a balance outstanding, the credit tied up at any moment through the year, not the total of every rupee drawn and repaid. Saying that first stops the interviewer wondering whether you are sizing sales or credit.

Now build it from people. A city of 1 crore at 4 people a household is 25 lakh households. Assume one small grocery shop per 100 households: 25,000 shops. Assume each sells Rs 50 lakh a year, so all of them sell Rs 12,500 crore. Check that against households: Rs 12,500 crore over 25 lakh households is about Rs 4,167 a month each at the local shop, a plausible share of a family's grocery bill. If it looked absurd, one of the first three links would be wrong.

How do stock days turn into a rupee need?

Each shop then carries about Rs 1.1 lakh of bank limit. A range is more honest than a point here, because the bank share is the weakest assumption: much small-shop credit comes from wholesalers, family and informal lenders. Moving net days between 15 and 25 and the bank share between 30% and 50% spreads the answer from about Rs 154 crore to Rs 428 crore. Customer credit, the running tab a grocer gives regular families, would add to the need; five days of it at the same bank share adds about Rs 68 crore.

The levered beta is 1.4, and the cost of equity rises from 11.8% to 15.4%. Multiply the unlevered beta of 0.8 by 1 plus the after-tax debt-to-equity ratio, 0.75 x 1.0, which is 1.75. At a 7% risk-free rate and a 6% premium, equity then costs 7% plus 1.4 x 6%. Debt is cheaper than equity, but adding it makes the remaining equity riskier, so shareholders demand more.

Why does borrowing make the shares riskier?

Two friends each own half of a Rs 10 lakh tea stall. If sales drop and the stall's value falls 10%, each loses Rs 50,000. Now suppose the stall was bought with Rs 5 lakh of loan and only Rs 5 lakh of their own money. The same 10% fall costs Rs 1 lakh, and the lender still gets repaid in full, so the owners lose 20% of their stake. Debt does not change how risky the business is; it concentrates that risk onto a smaller slice of equity. Beta measures that sensitivity, so it rises with leverage.

If debt is cheaper, why does the cost of equity go up?

Because shareholders are paid last. The cost of equity rises from 11.8% to 15.4%, and the 3.6 points added are the price of standing behind the lenders. The overall cost of capital may still fall, because half the funding is now debt that costs less and whose interest is tax deductible. That is the trade-off the desk is testing: cheaper money in, pricier equity as a result, and the tax shield decides the net.

Say the assumptions. This formula, usually credited to Robert Hamada, treats the debt as riskless, a beta of zero; for a heavily indebted company that overstates how much risk the equity carries alone. It also assumes the debt ratio stays at 1.0. Without the tax term the levered beta would be 1.6, so 1.4 sits between the unlevered 0.8 and that upper bound.

Loan B, with an expected loss of Rs 0.45 crore against Rs 0.10 crore for Loan A. Expected loss is exposure times default probability times loss given default. Loan A: 50 x 1% x 20% is Rs 10 lakh. Loan B: 20 x 3% x 75% is Rs 45 lakh, 4.5 times as much on a loan less than half the size. Security and borrower quality matter more than the headline amount.

What goes into expected loss?

Imagine lending your bicycle to two friends. One borrows the expensive one but always returns things and leaves his phone as a deposit; the other borrows a cheap one, loses things often, and leaves nothing. What you expect to lose depends on how much you lend, how likely it is to go wrong, and how much you get back if it does, multiplied together. Lenders write those three as EAD, PD and LGD.

What does the answer mean for pricing?

Express each loss as a rate on the loan. Loan A costs 0.2% a year in expected losses; Loan B costs 2.25%, so B needs roughly two points more of spread just to break even on credit losses. Two factors did that. The default probability tripled, and the lack of security nearly quadrupled the loss when things go wrong. Security is why a secured loan to a weaker borrower can be safer than an unsecured loan to a stronger one.

Say the limit. Expected loss is an average; it covers the cost of doing business, not the bad year. A credit portfolioA book of many loans, where losses depend on how many default together, not only on each loan alone. with a few large loans can lose far more than its expected loss when one of them fails. Loan A's Rs 10 lakh expected loss hides a Rs 10 crore hit if it defaults, which is why lenders hold capital for unexpected loss as well.

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.