Debt Capital Markets puzzles, solved step by step

The Macaulay duration is about 2.74 years. Discount each cash flow at 10%: 9.09 at year 1, 8.26 at year 2 and 82.64 at year 3, adding to the price of 100. Weight each year by its share of the price: 1 x 9.09 plus 2 x 8.26 plus 3 x 82.64 is 273.55, over 100. It is below 3 because the coupons are paid before maturity and pull the average in.

What is duration actually averaging?

Suppose a friend owes you money and pays a little next month, a little the month after, and most of it in the third month. Asked when, on average, you got paid, you would not say the third month: some of the money came earlier. Macaulay duration is the average time until you are paid, with each payment weighted by its present value as a share of the bond's price. Present values rather than face amounts, because a rupee that arrives later is worth less today and should carry less weight.

Why is it always below maturity for a coupon bond?

Any payment before the final date pulls the balance point towards today, so only a zero-coupon bond has a duration equal to its maturity. Raise the coupon and the pull grows: the same 3-year bond with a 15% coupon, still discounted at 10%, has a duration of 2.65 years. Divide by one plus the yield and you get the modified durationMacaulay duration divided by one plus the yield. It gives the approximate percentage price change for a one point change in yield., 2.49: a 1 point rise in yield should take about 2.49% off the price. The exact price at 11% is 97.56, a fall of 2.44%, close to the estimate.

About 8.9 basis points, or roughly Rs 4 crore, and yes, 10 basis points looks generous. A 0.40 point rise on a duration of 4.5 is 0.40 / 4.5 = 8.9 basis points of tightening. On Rs 1,000 crore, 0.40 points is Rs 4.0 crore of value that went to investors. Almost all of the premium came back on the first trade, so the book would probably have cleared tighter.

How do you turn a price jump into basis points?

A shop that sells out of a new phone in an hour, and then sees it resold online for more, has learned that it priced too low. A bond's first trades tell the issuer the same thing. Modified durationThe approximate percentage price change for a one point change in yield. is the exchange rate between price and yield: here a 1 basis point move in yield moves the price by about 4.5 hundredths of a point. So a price rise of 0.40 points means the market is valuing the bond about 8.9 basis points tighter than where it was priced.

What does the Rs 4 crore actually cost the issuer?

It is paid through the coupon, not written as a cheque. Pricing about 8.9 basis points too wide costs Rs 0.89 crore a year on Rs 1,000 crore, for five years: Rs 4.4 crore in total, or about Rs 3.6 crore in today's money at an assumed 7.5% yield, the same order as the Rs 4.0 crore jump in the bond's price. The first trade after pricing tells you how much of the premium was needed, because it shows where investors will buy the bond once the allocation is done.

So was 10 basis points too much?

On these numbers, most of it. But say two things before you convict the syndicate desk. First, strip out the market: if benchmark yields fell 3 basis points during the same session, only 5.9 basis points of the move belongs to the deal. Second, some positive performance is deliberate. Investors who were allocated want to see the bond trade well, and a flat or weak break makes the issuer's next deal harder to fill. A generous premium is a cost this time and an investment in the next book; the judgement is whether nearly the whole premium needed to be spent.

Light rope A at both ends and rope B at one end at the same moment. Rope A burns out after exactly 30 minutes. At that instant, light the other end of rope B, which has 30 minutes of burn left. With two flames it finishes in 15 minutes, so rope B goes out at exactly 45 minutes.

Why does lighting both ends always give 30 minutes?

Two people painting a fence from opposite ends meet when the whole fence is painted, whatever the pace on each stretch; if the job is one hour for one painter, together they finish in half an hour. A rope's burn time behaves the same way: two flames consume the rope's 60 minutes of burn between them, so they must meet after 30 minutes, even though the meeting point is not the middle of the rope. Unevenness changes where they meet, never when.

Why does rope B have exactly 30 minutes left?

Rope B has burned from one end for 30 minutes, so 30 of its 60 minutes of burn are gone and 30 remain, spread unevenly along whatever length is left. Lighting its other end applies the same halving trick to what remains: 30 minutes of burn with two flames takes 15. 30 plus 15 is 45. The trick works on any remaining burn time, not just a whole rope.

Say what the puzzle is testing, briefly: separating what you know (total burn time) from what you do not (where along the rope that time sits). A desk reading a bond's cash flows faces the same split: the total is fixed, the timing can be lumpy.

No. The price falls from about 74.41 to 55.84 per Rs 100, a fall of 25.0%, not 50%. A zero's price is 100 divided by one plus the yield, compounded for ten years. Doubling the yield moves one plus the yield only from 1.03 to 1.06, so the price is divided by (1.06/1.03) to the tenth, about 1.33. Halving the price would need a yield near 10.4%.

Why is doubling the yield not doubling the discount?

A shop that raises its markup from 3% to 6% has doubled the markup, but the price tag only goes from 103 to 106. The thing that multiplies through is one plus the rate, not the rate. A bond's price depends on (1 + y) compounded over its life, so doubling y from 3% to 6% changes the base only from 1.03 to 1.06, about 2.9% a year. Ten years of that is a factor of 1.333, which takes about a quarter off the price.

How does duration check the answer?

The modified duration of a 10-year zero at 3% is 10 / 1.03, about 9.71. Multiplied by a 3 point rise that predicts a fall of about 29.1%, against the true 25.0%. The straight-line estimate overshoots on a large move because the price curve bends: each extra point of yield takes off less than the one before. That bend is convexity, and a 3 point move is big enough to show it. For small moves duration is close; for large ones, reprice the bond directly.

About 250 trading days, roughly a year. The fund may buy 2% of Rs 1,000 crore, which is Rs 20 crore a day, and Rs 5,000 crore divided by Rs 20 crore is 250. The position is five full days of the whole market's trading, which makes it illiquid for this fund: it would take about as long to sell as to buy.

Why is the answer not five days?

Imagine wanting to buy 500 kilos of tomatoes from a village market that sells 100 kilos a day. In theory it is five days of the market's supply; in practice, if you take everything, prices jump and the regular buyers go home empty handed. So you decide to take no more than a couple of kilos a day. The limit on your share of daily volume, not the market's total volume, sets how fast you can trade, and here that limit is Rs 20 crore a day. Rs 5,000 crore at Rs 20 crore a day is 250 days.

What does this tell you about liquidity?

Liquidity belongs to a position, not to a market: the same bonds are liquid for a Rs 50 crore holder and illiquid for a Rs 5,000 crore holder. Days to trade is the simplest honest measure. At 250 days, the fund carries a year of market risk while it builds, and faces the same year if it ever needs to sell, which matters most exactly when others are selling too. Daily volume also falls in a stressed market, so the exit could take longer than the entry. A desk would size the position down, accept a higher participation rate and more price impact, or look for a block trade or a new issue where it can take size in one go.

About 4.71% of the portfolio defaulting each year. High yield earns 330 basis points more than investment grade. Each default costs 70% of face, since 30% is recovered, so each 1% of defaults costs 70 basis points. 330 divided by 70 is 4.71%. Below that default rate high yield still earns more; above it, less.

Why divide by the loss, not the default rate?

A shop that sells on credit loses less than the full bill when a customer fails to pay, if it can take back some of the goods. Bonds work the same way. The annual cost of defaults is the default rate times the loss given defaultThe share of face value lost when a borrower defaults, which is one minus the recovery rate., so the extra spread covers defaults up to the extra spread divided by the loss given default. With 30% recovered, each default loses 70%, and 330 basis points of extra spread covers 4.71% a year.

What does the number leave out?

A break-even default rate is a cushion, not a forecast: whether 4.7% is comfortable depends on how often default rates have run above it and for how long. It also assumes investment grade loses nothing; if investment grade lost 10 basis points a year, the break-even would rise to 4.86%. Recovery is the soft number: it tends to fall in exactly the years defaults rise, so a 30% assumption can flatter the cushion. And the calculation ignores price volatility, which matters to anyone who must mark the portfolio before the bonds mature.

A performance fee of about 24.7%. Today the manager earns Rs 20 crore of management fee plus 20% of the Rs 80 crore return after fees, Rs 16 crore, for Rs 36 crore. At 1.5% the fixed fee is Rs 15 crore and the return after fees rises to Rs 85 crore, so the performance fee must bring Rs 21 crore: 21 / 85 = 24.7%. The swap is exact only at a 10% gross return.

Why does the performance fee base change?

Think of a salesperson paid a fixed salary plus a share of profit after salary. Cut the salary and the profit after salary goes up, so the share needed to make up the difference is smaller than a straight swap would suggest. The performance fee is charged on the return left after the management fee, so lowering the management fee enlarges the base the performance fee is charged on. That is why the answer is 24.7%, not 25%.

Is the manager really indifferent?

Only at a 10% gross return. At 5% gross, the old terms pay Rs 26.0 crore and the new pay Rs 23.6 crore; at 15%, the old pay Rs 46.0 crore and the new Rs 48.4 crore. Swapping fixed fee for performance fee moves the manager's revenue out of weak years and into strong ones, so the trade depends on the return you assume. Investors who push for it pay less when things go badly and more when they go well, which is often exactly what they want. A real fund would also have a hurdle rate, a high-water mark and fees charged on average rather than opening assets; each changes the arithmetic, not the logic.

Term loan A has a weighted average life of 3.0 years and term loan B about 6.8 years. Term loan A repays 200 in each of years 1 to 5, so its average is (1 + 2 + 3 + 4 + 5) x 200 / 1,000 = 3.0. Term loan B repays 10 in each of years 1 to 6 and 940 in year 7: (210 + 6,580) / 1,000 = 6.79. Amortisation, not maturity, sets how long the lender's money is out.

What are the two loans, and why do they amortise so differently?

That is the question a candidate reported being asked, and the arithmetic is the best way to answer it. A term loan A is built for banks: it amortises steadily, so the lender's exposure shrinks every year. A term loan B is built for institutional investors such as loan funds: it amortises only nominally and returns almost everything at maturity, which suits investors who want to stay invested. The difference in structure shows up as a difference in weighted average life, and that number drives pricing, investor appetite and how long the credit risk lasts.

How is weighted average life calculated?

Think of lending a friend Rs 1,000 who repays Rs 200 a year: on average your money is out about three years, even though the last rupee comes back in year five. Weighted average life weights each repayment date by the share of principal repaid on it, and ignores interest. For term loan B: six payments of 10 contribute 10 x (1 + 2 + 3 + 4 + 5 + 6) = 210 year-rupees, and the bullet contributes 940 x 7 = 6,580. Total 6,790 over 1,000 is 6.79 years.

Why does the desk care?

A loan's pricing is spread over its weighted average life, not its stated maturity, so an upfront fee or discount is spread over 3 years on term loan A and nearly 7 on term loan B. In practice term loans B are often repaid early from refinancings or cash sweeps: if this one were refinanced in full at year 4, its weighted average life would fall to 3.94 years. Weighted average life is also not duration; it ignores interest and discounting, so it always sits above the loan's duration.

You can tighten 25 basis points, to 125, and still be 2 times covered. Each 5 basis point step loses 10% of the original demand, which is 0.4x of cover. Going from 4.0x to 2.0x gives up 2.0x, or five steps, so 5 x 5 = 25 basis points. On a Rs 500 crore deal the book falls from Rs 2,000 crore to Rs 1,000 crore, and the issuer saves Rs 1.25 crore a year.

What is actually being traded off?

A shopkeeper with a queue of forty customers for twenty items can raise the price, and some customers will walk away; the question is how high to go before the queue gets too short to be comfortable. Book-building trades cover for price: each step tighter saves the issuer money and costs orders, and cover is the buffer against a weak aftermarket. Here every 5 basis points costs 0.4x, because the loss is 10% of the original book each time.

Would you really go all the way to 125?

Not automatically. The linear rule is a model of the book; real books drop in lumps, because price-sensitive orders carry limits and the orders that leave first are often the highest quality long-term investors. Landing exactly on the 2.0x floor leaves no margin if a few orders drop after the final terms. A syndicate desk might stop one step short, at 130 with 2.4x cover, giving up Rs 0.25 crore a year of saving on Rs 500 crore for a safer book and a better chance of a positive break. The arithmetic sets the limit; the judgement picks the point inside it.

About 23.8%. Picture 1,000 issuers: 40 will default and 960 will not. The model flags 75% of the 40, which is 30, and 10% of the 960, which is 96. Of the 126 flagged issuers, 30 default, so the chance is 30 / 126 = 23.8%. The flag multiplies the risk about six times, but most flags are still false alarms because defaults are rare.

Why is the answer not 75%?

A smoke alarm that goes off for every real fire and also for one in ten batches of toast will ring mostly for toast, because toast is far more common than fire. The early warning model is the same. The chance of default given a flag depends on how common defaults are to begin with, and when the base rate is low, false flags from the large healthy group swamp the true flags from the small defaulting group. 75% answers a different question: how often a defaulter gets flagged.

How should a credit desk use a 23.8% flag?

As a reason to look harder, not a verdict. A flag lifts the default probability from 4% to 23.8%, roughly six times, which is worth a review but not a sale on its own. A second, independent signal compounds the evidence: if a separate test with the same accuracy also flagged the issuer, the probability would rise to about 70%. The honest limitation is that two warning signals about the same company are rarely independent, so the real lift from a second flag is usually smaller.