Debt Capital Markets puzzles, solved step by step

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.

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.

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.

Rs 25 lakh a year, Rs 6.25 lakh a quarter and about Rs 6,850 a day. A basis point is 0.0001, so Rs 2,500 crore times 0.0001 is Rs 0.25 crore, Rs 25 lakh a year. Divide by four for a quarter. Divide by 365 for a day, which gives Rs 6,849 on an actual/365 basis, or Rs 6,944 on a 360-day basis.

Why would a desk ask something this simple?

A shopkeeper who knows that one rupee off a product's price costs her Rs 3,000 a month in margin negotiates very differently from one who has to work it out after the customer leaves. Every pricing conversation in debt markets is in basis points, and the listener has to turn them into rupees on the deal in front of them instantly. An issuer who hears that 5 basis points is on the table wants to know that means Rs 1.25 crore a year, not to wait for a spreadsheet.

What changes when the basis point is on price instead of coupon?

The numbers above are one basis point of coupon or spread, paid each year. A basis point of yield on a bond's price is a different quantity. A one basis point move in yield changes a bond's value by roughly its duration times the notional times 0.0001, so on a five-year bond with duration near 4 it is about Rs 1 crore, not Rs 25 lakh. The two agree once you see why: five years of Rs 25 lakh, discounted at 8%, is worth about Rs 99.8 lakh today. Duration is roughly the present value of a stream of basis points.

Day counts are the last detail worth a sentence. Many rupee bonds count actual days over 365, while money market instruments and many floating rate loans elsewhere use 360; confirm the convention in the term sheet rather than assume it. The difference on one day is about Rs 95 here, trivial on a day and material over a large book, which is why the convention is always named in the documents.

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.

For 2 litres: fill the 3, pour it into the 4, fill the 3 again and top up the 4; exactly 2 litres stay in the 3-litre bottle. For 5: fill the 4, pour into the 3 to leave 1, empty the 3, move the 1 into it, then fill the 4, for 1 plus 4. Every whole amount from 1 to 7 is possible, because 3 and 4 differ by 1.

What moves are you actually allowed?

Three: fill a bottle to the top, empty it, or pour from one into the other until the first is empty or the second is full. The only amounts you can know for certain are full bottles and what is left after a pour stops at a full bottle, so every measurement is built from 3s and 4s. Think of it as making change with only Rs 3 and Rs 4 coins, where you are also allowed to hand coins back.

Why can you reach every amount from 1 to 7?

Because 4 minus 3 is 1, and once you can make 1 you can make anything by adding bottles. The amounts you can measure are exactly the combinations of 3 and 4 that fit in the bottles, and since the two sizes share no common factor, every whole litre up to their total of 7 is reachable. One litre: fill the 4 and pour into the 3. Three and four: fill one bottle. Five: 1 plus a full 4. Six: 3 in each. Seven: both full.

The same rule tells you when a puzzle has no answer. With a 4-litre and a 6-litre bottle, every amount you can make is even, so 5 litres is impossible, and you can say so without trying a single pour. Interviewers like that sentence more than the pouring itself, because it shows you found the structure rather than a lucky sequence.

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.