Debt Capital Markets puzzles, solved step by step

The EBITDA cure needs Rs 20 crore; the prepayment needs Rs 80 crore. To reach 4.0x with debt of 800, EBITDA must be 200, so the cure adds 20. To reach 4.0x with EBITDA of 180, debt must be 720, so the prepayment is 80. The cure is four times cheaper because each rupee of EBITDA supports four rupees of debt, which is why lenders cap how often and how much a cure can count.

Why does the same covenant give two very different cheque sizes?

A bank lets you borrow up to four times your monthly salary. If you are over the limit, you can either repay some of the loan or show a higher salary. Showing Rs 1,000 more salary makes room for Rs 4,000 of loan, so it takes a quarter as much money to fix. A leverage ratio has debt on top and EBITDA underneath, and at a 4.0x test a rupee added below does the work of four rupees removed above. An equity cureA right in a loan agreement letting the owner inject cash to fix a covenant breach, with the cash treated as if it were extra EBITDA for the test. exploits exactly that.

Why do lenders limit equity cures?

Because the cure is cosmetic for the covenant. Rs 20 crore of one-off equity counted as EBITDA does not make the business earn more; it makes the ratio pass for the test periods it is counted in. The lenders still hold Rs 800 crore against a business earning Rs 180 crore. So documents typically cap how many cures are allowed over the life, forbid them in consecutive quarters, cap the amount at what is needed to pass, and stop the cure cash counting as EBITDA for anything else, such as the room to pay dividends.

One wrinkle is worth saying. Some documents also let the cure cash sit on the balance sheet and be netted against debt. Then you solve 800 minus x over 180 plus x equals 4.0, and x falls to Rs 16 crore. Lenders resist that double count for the same reason: one rupee should not fix both halves of the ratio.

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.

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.

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 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.

At the end of year 5 the borrower owes Rs 121.67 crore, the lender has received Rs 43.33 crore of cash interest, and the IRR at par is exactly 12%. The 4% PIK is added to the balance each year, so it grows to 100 x 1.04 to the fifth. Cash interest is 8% of each opening balance, rising from Rs 8.00 crore to Rs 9.36 crore. Both pieces earn on the full balance, so the lender earns 12% a year.

What does PIK actually do to the balance?

Think of a friend who borrows from you, pays part of the interest in cash each year, and says: add the rest to what I owe. Next year you charge interest on the bigger amount. PIKPayment in kind: interest settled by adding it to the amount owed instead of paying it in cash. interest is not paid; it is added to the loan, so the balance grows every year and every later charge is worked out on the bigger number. Here the balance grows 4% a year, from Rs 100 crore to 100 x 1.04 to the fifth, Rs 121.67 crore.

Why is the IRR exactly 12% when only 8% arrives in cash?

Each year the lender earns 12% on the whole balance: 8% arrives as cash and 4% is added to the balance, which then earns 12% itself. A lender earning 12% a year on every rupee outstanding, and repaid in full, has an IRR of 12%. The cash yield on the original Rs 100 crore starts at 8% and reaches 9.36% in year five, because the cash coupon is charged on the accreted balance.

What is the catch the interviewer wants you to name?

A cash-plus-PIK loan earns the same 12% on paper as a 12% cash loan, but more of the return waits until year five. The lender collects Rs 43.33 crore in cash along the way against Rs 60 crore from a 12% cash-pay loan, and Rs 121.67 crore rides on the final repayment instead of Rs 100 crore. If the borrower defaults in year four, the PIK accrued so far is just a larger claim in the recovery, not cash in hand. That is why PIK paper is priced wider: the 12% holds only if the borrower pays at the end.

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.

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.

Start around 10.5, say 9.5 bid and 11.5 offered, then move to about 12.2 bid, 13.8 offered once the 6 is shown. Each fair die is worth 3.5 on average, so three are worth 10.5. After the reveal the value is 6 plus 3.5 plus 3.5, which is 13.0. The market should also narrow, because one die's uncertainty has gone: the standard deviation falls from 2.96 to 2.42, so the width shrinks by about a fifth, not a third.

Where does the first quote come from?

A shop that buys and sells used phones offers to buy below what it thinks a phone is worth and to sell above it, and earns the gap. A market maker quotes around the expected value, with a bid below and an offer above, and the width reflects how uncertain that value is. Each fair die averages 3.5, so three dice average 10.5. A quote of 9.5 bid, 11.5 offered straddles that value; the sum can still land anywhere from 3 to 18, so the market cannot be tight.

What changes when the first die is shown?

Two things, and candidates usually say only one. The expected value jumps to 6 plus 3.5 plus 3.5, which is 13.0, and the uncertainty shrinks because only two dice are still hidden. Variance adds across independent dice, 35/12 for each, so it falls from 8.75 to 5.83, a third lower; the standard deviation falls from 2.96 to 2.42, about 18% lower. Scale the width by that: 2 points becomes about 1.6, so quote roughly 12.2 bid, 13.8 offered.

What does the interviewer want to hear beyond the numbers?

Move the market the moment information arrives, because a stale quote is a free option for everyone else at the table. If you stay at 9.5 to 11.5 after a 6 shows, every player buys your offer at 11.5 against a fair value of 13.0. Then say what you would do if you suspected the other side knew more than you, for example had seen a second die: widen, or lean your quote towards the risk. The limit: the width here is a choice; a real desk sets it by competition and by how much risk it can hold, not by a formula.