Debt Capital Markets puzzles, solved step by step

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.

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.

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.

The yield has moved 18 basis points, a 2.5% relative change, and the desk quotes the 18 basis points. 7.38 minus 7.20 is 0.18 percentage points, and one basis point is one hundredth of a point. The relative change is 0.18 over 7.20, or 2.5%. Saying the yield rose 2.5% invites a listener to hear 9.70%, so rate moves are quoted as absolute changes in basis points.

Why are there two different answers to one move?

Think of a batting side's run rate going from 6 an over to 9. You can say it rose by 3 runs an over or by 50%, and both are true. Yields work the same way, except the thing being measured is already a percentage. An absolute change in a yield is measured in percentage points; a relative change is a percentage of a percentage, and spoken aloud the two sound alike. Here the absolute change is 0.18 points and the relative change is 0.18 over 7.20, or 2.5%.

Why does a bond desk insist on basis points?

A basis point is one hundredth of a percentage point, a unit small enough to price with and impossible to confuse with a relative change. Say 18 basis points and nobody on the call can hear 9.70%. The unit also makes the money easy to see: on a Rs 500 crore issue, 18 basis points of extra coupon is 0.0018 x 500 crore, Rs 90 lakh a year. Spreads, new issue premiums and fee cuts are all quoted in basis points for the same reason.

The relative figure is not useless. It compares moves in proportion: a 3.00% yield rising to 3.18% is also 18 basis points, but it is a 6% relative jump against 2.5% here. In the room, give the basis point number first, give the relative number only if asked, and label each one.

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 note falls to about 98.7. The benchmark part of the coupon resets every quarter, so benchmark moves barely touch the price. The 100 basis point margin is fixed, and the market now wants 150. That 50 basis point shortfall runs for 12 quarters; discounted, it is worth about 1.32 per Rs 100, so the price is roughly 100 minus 0.50 x 2.64, about 98.7.

Which part of a floater's coupon resets, and which does not?

Picture a shop lease with rent set at inflation plus a fixed Rs 5,000. When inflation rises, the rent rises with it, so the landlord is protected. If rents on the street jump to Rs 8,000 over inflation, the landlord is still stuck at Rs 5,000 until the lease ends. A floating rate note resets its benchmark every quarter, but its credit margin is fixed at issue for the life of the note. Interest rate risk is almost gone; credit spread risk is not.

How much is a 50 basis point shortfall worth today?

The note pays benchmark plus 1.00%; a new note from the same issuer would pay benchmark plus 1.50%. The holder is short 0.50% a year, 0.125 per Rs 100 each quarter, for 12 quarters, and the price falls by today's value of that stream. Assume the benchmark sits flat at 6.50%, so the discount rate is 8.00% a year, 2% a quarter. Twelve payments of 0.125 at 2% a quarter are worth 1.32, so the note trades near 98.68.

Say the shortcut first, then the refinement. A floater's spread duration is a little under its remaining life, so a 50 basis point widening on a 3-year note costs about 1.3 to 1.5 points. Using 3 years gives 98.5; discounting each shortfall gives 98.7, because the later shortfalls are worth less today. Either is a good answer if you say which one you did. The limit: this assumes a flat 6.50% benchmark, and a different curve moves the second decimal, not the story.

The make-whole price is about 105.25, and calling to refinance at 7.3% loses the issuer about 0.81 per Rs 100. Discount the remaining 9, 9 and 109 at 6.5% plus 50 basis points, 7.0%, to get 105.25. Valued at the 7.3% refinancing rate, the same payments are worth 104.44, so paying 105.25 to retire them destroys value. Calling only saves money if the issuer can borrow below 7.0%.

What does a make-whole price actually compute?

Suppose you repay a relative's loan early and they say: fine, but pay me today everything I would have earned, discounted at a low rate. That is a make-whole callAn issuer option to repay a bond early at the present value of its remaining payments, discounted at the government yield plus a fixed spread set in the documents.. The call price is the present value of every remaining coupon and the principal, discounted at the government yield plus a small fixed spread, here 6.5% plus 0.5%, which is 7.0%. The low discount rate is what pushes the price above par: the lender is compensated for giving up a 9% coupon.

Why does refinancing at 7.3% not pay?

Compare two ways of carrying the same obligation. Keeping the old bond means payments worth 104.44 at the issuer's 7.3% borrowing rate; calling means paying 105.25 in cash today, raised with new 7.3% debt. The issuer would give up 0.81 per Rs 100, about Rs 81 lakh on every Rs 100 crore, to swap into debt that only looks cheaper. The coupon saving from 9% to 7.3% is real, but the make-whole has already charged for it, valued at 7.0%, a lower rate than 7.3% and therefore a higher price.

That gives a clean rule to say out loud. A make-whole call only saves money if the issuer can refinance below the make-whole discount rate, the government yield plus 50 basis points. Few corporates borrow that tightly, which is why make-wholes are usually exercised for other reasons: to escape a restrictive covenant, to complete a merger, or to tidy the capital structure. The limit: this ignores the fees on the new bond, which make calling worse still.

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.