Debt Capital Markets puzzles, solved step by step

Puzzles: 100
Traced to a firm: 16
Topics: 13
Hard: 30

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002Estimate 1.01 to the power 365 and 0.99 to the power 365 in your head. What do the two answers say about small daily edges?Mental maths and numeracyHardFixed income asset management
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Gut call first: roughly where does 1.01 to the power 365 land?
Show the worked solution
About 38 and about 0.026. A 1% daily gain doubles money roughly every 70 days, so a year holds just over five doublings: 2 to the power 5.2 is about 37.8. A 1% daily loss halves it about every 69 days, leaving about 0.026. The two outcomes end roughly 1,481 times apart: a tiny daily edge, held consistently, is enormous by year end.
Why is 365 times 1% the wrong way to think about it?
Picture a savings jar where every evening you add 1% of whatever is already inside. On day 1 that is one paisa per rupee; by day 200 the jar holds about seven times its start, so 1% of it is seven paise of the original rupee. Compounding means each day's 1% is earned on everything built so far, so the growth accelerates rather than adding up in a straight line. Simple addition, 365 times 1%, gives 4.65 and misses almost all of it.
How do you get to 38 without a calculator?
Use the rule of 70A shortcut: a quantity growing at r per cent a period doubles in about 70 divided by r periods.. At 1% a day, money doubles in about 70 days. A year of 365 days holds about 5.2 doublings, and 2 to the power 5 is 32 while 2 to the power 5.2 is about 37, so the answer is in the high thirties. The exact figure is 37.78. The same shortcut run backwards gives the loss: 1% down a day halves value about every 69 days, a little over five halvings in a year, which is one part in about 40.
On a log scale both paths are straight lines: growing 1% a day doubles about every 70 days and ends the year at 37.8, shrinking 1% a day halves about every 69 days and ends at 0.026, leaving the two roughly 1,481 times apart.
The relationship
$1.01^{365} = e^{365\ln 1.01} \approx e^{3.63} \approx 37.8 \qquad 0.99^{365} = e^{365\ln 0.99} \approx e^{-3.67} \approx 0.026$
ln 1.01 about 0.00995, a shade under 0.01
ln 0.99 about minus 0.01005, a shade beyond minus 0.01
e the base of natural growth, about 2.718
What it says in wordsA small rate compounded many times is close to e raised to the rate times the number of periods.
Why does a markets interviewer care about this?
Because desks live on small edges repeated many times: a few basis points of carry every day, a slightly better fill on every trade, a small cost leak on every rebalance. An edge or a leak that looks trivial on one day decides the year, and it cuts both ways. Also notice that up 1% then down 1% does not get you back: 1.01 times 0.99 is 0.9999. Volatility on its own drags a compounding path downwards, which is worth one sentence after the numbers. The limit: real returns are not a steady 1% a day, so treat this as arithmetic about compounding, not a model of any market.
Where candidates lose it
The common loss is saying 4.65 for the first number, adding the percentages instead of compounding them. A close second is freezing, because nobody computes a 365th power by hand and the candidate does not reach for a doubling rule.
The other miss is treating the down case as the mirror image. It is not symmetric: the up path gains about 37 times its start, the down path loses about 97% and can never go below zero. Say both numbers and then the asymmetry.
What the interviewer asks next
What is 1.01 to the power 365 times 0.99 to the power 365, and why is it below 1?
How many days of 1% gains does it take to recover from 30 days of 1% losses?
Where does this compounding asymmetry show up in a bond portfolio's returns?
009Quick maths: what is the accrued coupon on Rs 350 crore of a 7.25% bond for 73 days, on an actual/365 basis?Mental maths and numeracyWarm upPrivate credit
Try it first
Before multiplying: what fraction of a year is 73 days on an actual/365 basis?
Show the worked solution
Rs 5.075 crore. The annual coupon is 7.25% of Rs 350 crore, Rs 25.375 crore. On actual/365, 73 days is exactly one fifth of a year, because 73 times 5 is 365. One fifth of Rs 25.375 crore is Rs 5.075 crore, which a buyer settling on day 73 pays the seller on top of the clean price.
What do you look for before you multiply?
A friendly fraction. Mental maths on a desk is mostly spotting which number makes the rest easy. 73 is exactly a fifth of 365, so the whole calculation collapses to one fifth of the annual coupon. It is like splitting a Rs 25,375 bill between five friends: nobody reaches for a calculator once they see it is a clean fifth. Say the fraction out loud first, so the interviewer hears the shortcut rather than watching you grind through 7.25 times 350 times 73.
Seventy three days is exactly one fifth of a 365 day year, so the accrued coupon is one fifth of the Rs 25.375 crore annual coupon on Rs 350 crore at 7.25%, which is Rs 5.075 crore.
How do you get the annual coupon quickly?
Split the rate. 7% of 350 is 24.5 and a quarter per cent of 350 is 0.875, so the annual coupon is Rs 25.375 crore. Breaking an awkward rate into a round rate plus a small piece keeps each multiplication in your head. Then divide by five: 25 divided by 5 is 5, and 0.375 divided by 5 is 0.075, so Rs 5.075 crore. Another route is 350 divided by 5 first, which is 70, and 7.25% of 70 is 5.075. Either way, two steps.
The relationship
$AI = F \times c \times \frac{d}{365} = 350 \times 7.25\% \times \frac{73}{365} = 25.375 \times 0.2 = 5.075$
AI accrued interest, Rs crore
F face value held, Rs 350 crore
c annual coupon rate, 7.25%
d days since the last coupon, 73
What it says in wordsAccrued interest is the annual coupon times the share of the year that has passed since the last payment.
Why does accrued interest matter on a trade?
Because the seller has earned 73 days of coupon but the buyer will receive the whole next coupon. The buyer pays the accrued interest to the seller at settlement, so the cash that changes hands is the clean price plus accrued, the dirty price. On Rs 350 crore, Rs 5.08 crore is real money. Also say the limit: the day count convention varies by market and by instrument, and actual/365, actual/actual and 30/360 give slightly different answers, so check the bond's terms before you settle.
Where candidates lose it
The loss here is speed, not knowledge. Candidates multiply 350 by 7.25% by 73 and divide by 365 the long way, drop a decimal somewhere, and give 50.75 or 0.5075 instead of 5.075.
The second slip is quietly switching to a 360 day year, which gives about 5.145. The question said actual/365; repeat the convention back as you answer so the interviewer knows you heard it.
What the interviewer asks next
What is the accrued interest after 146 days on the same holding?
The bond is quoted at a clean price of 101.20. What does the buyer pay in total per Rs 100 of face?
Why do markets quote bonds on a clean price rather than a dirty price?
016Numerical reasoning: an issuer's bond volumes grow 12%, then 15%, then fall 10% over three years. What is the total change over the period, and what is the average annual growth rate?Mental maths and numeracyCoreCorporate banking
Try it first
What is the total change over the three years?
Show the worked solution
A total rise of about 15.9%, or about 5.05% a year. Percentage changes multiply: 1.12 times 1.15 is 1.288, and a 10% fall from there leaves 1.1592. The steady yearly rate that gets to the same place is the cube root of 1.1592, about 1.0505. Adding the rates gives 17% and an average of 5.67%, both too high, because the fall hits a bigger base than the gains did.
Why can you not add the percentages?
A shop raises a price by 10% and later cuts it by 10%, and the tag ends below where it started, because the cut is taken on the higher price. Each percentage change is measured on the level it starts from, so a chain of changes multiplies rather than adds. Here the 10% fall comes after two good years, so it removes 12.88 points of the index, more than the 12 the first year added.
Indexed to 100, volumes rise to 112, then by 16.8 to 128.8, then fall by 12.88 to 115.92, a total change of 15.92% rather than the 17% that adding the three rates suggests.
How do you do it fast under test conditions?
Break each multiplication into easy pieces. 1.12 times 1.15 is 1.12 plus 15% of 1.12, which is 1.12 plus 0.168, so 1.288; taking 10% off is 1.288 minus 0.1288, which is 1.1592. For the yearly rate, test a round guess: 1.05 cubed is 1.1025 times 1.05, which is 1.1576, just under 1.1592, so the answer is a shade above 5%. On a timed test, that is enough to pick the right option in seconds.
The relationship
$1 + G = 1.12 \times 1.15 \times 0.90 = 1.1592 \qquad g = (1+G)^{1/3} - 1 \approx 5.05\%$
G the total change over three years
g the compound annual growth rate
1/3 one third, because there are three years
What it says in wordsMultiply the yearly growth factors for the total, then take the cube root for the steady yearly rate.
Why does the average of 5.67% overstate growth?
The simple average of 12, 15 and minus 10 is 5.67%, but three years at 5.67% would give 18.0%, not 15.9%. Whenever the yearly rates vary, the simple average is above the compound rate, and the gap grows with the swings. That is the same drag that makes a volatile bond fund's average return look better than what its investors actually earned. Say which average you are quoting: the compound rate describes the path, the simple average only describes the list of numbers.
Where candidates lose it
Adding the rates to get 17% is the trap the test is built around, and the wrong option is always on the list. Under time pressure the candidate recognises 17 as a number from the question and picks it.
The second trap is dividing 17 by 3 for the annual rate. Even after getting the total right at 15.92%, dividing by 3 gives 5.31%, which ignores compounding the other way. Take the cube root, or test 1.05 cubed.
What the interviewer asks next
What fall in year 4 would bring volumes back exactly to the start?
Volumes rise 20% and then fall 20%. Where do they end, and why?
Why do fund fact sheets report compound annual returns rather than simple averages?
031Without a calculator, estimate 1.07 to the power 5. Then check how close the rule of 72 gets to the doubling time at 7%.Mental maths and numeracyCoreFixed income asset management
Try it first
Your first instinct for 1.07 to the power 5?
Show the worked solution
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.
The relationship
$1.07^5 = 1 + 5(0.07) + 10(0.07)^2 + 10(0.07)^3 + \ldots = 1 + 0.35 + 0.049 + 0.0034 + \ldots \approx 1.4026$
5, 10, 10 the number of ways to pick one, two or three of the five years, from the binomial expansion
0.07^2 = 0.0049 interest earned on one year's interest
What it says in wordsAdd simple interest, then the pairs of years, then the triples; each layer is about a tenth the size of the last.
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.
One rupee at 7% compounds to 1.4026 after five years and to 2 at 10.24 years, a whisker before the rule of 72's 10.29 years, while simple interest reaches only 1.70 by year ten.
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.
Where candidates lose it
The common wrong answer is 1.35: five times 7% with no compounding. Candidates who know better sometimes jump to 1.5 because it feels bigger, without a method behind it.
The other loss is reciting the rule of 72 without checking it. The interviewer wants the exact figure or a reason the rule is close here; say that 72 over 7 is 10.29 and the truth is about 10.24.
What the interviewer asks next
Estimate 1.07 to the power 10 from your answer.
At what rate does money double in exactly 8 years, roughly?
Why is the rule of 69.3 exact for continuous compounding?
034What is 1 basis point worth on Rs 2,500 crore for a year, for a quarter and for one day?Mental maths and numeracyWarm upSyndicate desks
Try it first
One basis point on Rs 2,500 crore for a year is:
Show the worked solution
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.
The relationship
$2{,}500 \text{ crore} \times 0.0001 = 0.25 \text{ crore} = 25 \text{ lakh a year}$
0.0001 one basis point, a hundredth of one per cent
0.25 crore a quarter of Rs 1 crore, which is Rs 25 lakh
What it says in wordsMultiply the deal size by one ten-thousandth, then convert crore to lakh by multiplying by 100.
One basis point on Rs 2,500 crore is Rs 25 lakh a year, Rs 6.25 lakh a quarter and Rs 6,849 a day on an actual/365 basis, and five years of it discounted at 8% is worth about Rs 99.8 lakh today.
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.
Where candidates lose it
The common slip is a zero: Rs 2.5 lakh or Rs 2.5 crore, because crore, lakh and 0.0001 are three unit changes in one line. Say the steps aloud: 0.0001 of 2,500 crore is 0.25 crore, which is 25 lakh.
The second loss is treating a basis point of yield and a basis point of coupon as the same thing. On a five-year deal they differ by a factor of about four, and a syndicate desk will ask which one you mean.
What the interviewer asks next
An issuer saves 7 basis points on a Rs 2,500 crore, 5-year deal. What is that worth today at 8%?
What is the price value of a basis point on Rs 2,500 crore of a bond with modified duration 6.5?
Why do money market instruments often use a 360-day year?
051A bond's yield moves from 7.20% to 7.38%. By how many basis points has it moved, and by what percentage? Which number does a bond desk quote?Mental maths and numeracyWarm upFixed income asset management
Try it first
How would a DCM banker describe this move on a call?
Show the worked solution
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%.
The yield moved from 7.20% to 7.38%, which is 0.18 percentage points or 18 basis points, and also a 2.5% relative change; said carelessly, a 2.5% rise can be heard as a move to 9.70%, a 250 basis point error.
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 relationship
$\Delta y = 7.38\% - 7.20\% = 0.18\ \text{pp} = 18\ \text{bp} \qquad \frac{0.18}{7.20} = 2.5\%$
pp percentage points, the absolute gap between two yields
bp basis points, one hundredth of a percentage point
0.18 / 7.20 the relative change, the move as a share of the starting yield
What it says in wordsThe absolute move is the gap between the two yields; the relative move is that gap divided by where the yield started.
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.
Where candidates lose it
The trap is answering 2.5% and stopping, or saying the yield rose 2.5% without saying relative to what. On a desk that sentence gets corrected at once, because the natural hearing is that the yield went up 2.5 percentage points, to 9.70%.
The second loss is a slip in the conversion: 0.18 points is 18 basis points, not 1.8 and not 180. Say one basis point is one hundredth of a point as you convert, and the number checks itself.
What the interviewer asks next
A spread tightens from 250 to 210 basis points. What is the relative change, and when would a credit investor quote that instead?
On Rs 750 crore of issuance, what is 12 basis points of coupon worth each year?
A yield doubles from 0.50% to 1.00%. How many basis points is that, and why does the relative change mislead here?
052You invest Rs 1 lakh in a bond fund when its unit price is Rs 10 and another Rs 1 lakh when it is Rs 20. What is your average cost per unit, and why is it not Rs 15?Mental maths and numeracyCoreFixed income asset management
Try it first
Pick the average cost per unit before you calculate.
Show the worked solution
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.
Rs 1 lakh at Rs 10 buys 10,000 units and Rs 1 lakh at Rs 20 buys 5,000, so the balance point of the 15,000 units sits at Rs 13.33, well left of the Rs 15 midpoint of the two prices.
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%.
The relationship
$\bar{P} = \frac{2}{\tfrac{1}{10} + \tfrac{1}{20}} = \frac{2}{0.15} = 13.33$
2 the number of equal instalments
1/10, 1/20 units bought per rupee at each price
P-bar the average cost per unit
What it says in wordsWith equal money each time, the average cost is the harmonic mean of the prices.
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.
Where candidates lose it
The fast wrong answer is Rs 15, which averages the prices as if you had bought the same number of units twice. The question says the same money twice, and the whole puzzle turns on that difference.
The second loss is getting Rs 13.33 and then claiming the method guarantees a good outcome. It only guarantees an average cost below the average price paid; it cannot tell you whether the fund was cheap.
What the interviewer asks next
If you had bought 10,000 units at each price instead, what would your average cost be?
A yield moves between 6% and 8% and you buy equal money at each. Which average of the yields describes your purchase?
Three instalments at Rs 10, Rs 20 and Rs 40: what is the average cost?

ln 1.01	about 0.00995, a shade under 0.01
ln 0.99	about minus 0.01005, a shade beyond minus 0.01
e	the base of natural growth, about 2.718

AI	accrued interest, Rs crore
F	face value held, Rs 350 crore
c	annual coupon rate, 7.25%
d	days since the last coupon, 73

G	the total change over three years
g	the compound annual growth rate
1/3	one third, because there are three years

5, 10, 10	the number of ways to pick one, two or three of the five years, from the binomial expansion
0.07^2 = 0.0049	interest earned on one year's interest

Try it first

Why is 365 times 1% the wrong way to think about it?

How do you get to 38 without a calculator?

Why does a markets interviewer care about this?

Where candidates lose it

What the interviewer asks next

Try it first

What do you look for before you multiply?

How do you get the annual coupon quickly?

Why does accrued interest matter on a trade?

Where candidates lose it

What the interviewer asks next

Try it first

Why can you not add the percentages?

How do you do it fast under test conditions?

Why does the average of 5.67% overstate growth?

Where candidates lose it

What the interviewer asks next

Try it first

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

How good is the rule of 72 at 7%?

Where candidates lose it

What the interviewer asks next

Try it first

Why would a desk ask something this simple?

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

Where candidates lose it

What the interviewer asks next

Try it first

Why are there two different answers to one move?

Why does a bond desk insist on basis points?

Where candidates lose it

What the interviewer asks next

Try it first

Why does equal money not mean equal weight?

What is the general rule, and why is it called a harmonic mean?

Where candidates lose it

What the interviewer asks next