Debt Capital Markets puzzles, solved step by step

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

Showing 51–60 of 100

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?
053Nine bond certificates look identical, but one is a forgery printed on slightly heavier paper. With a balance scale and only two weighings, how do you find the forgery?Logic and brainteasersCoreSyndicate desks
Try it first
What should the first weighing be?
Show the worked solution
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.
The first weighing of three against three sends each of its three outcomes to a group of three suspects, and the second weighing of one against one inside that group sends each outcome to a single certificate, so two weighings cover all nine cases.
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.
The relationship
$\text{certificates you can search} = 3^{w} \qquad 3^{2} = 9$
w the number of weighings allowed
3 outcomes per weighing: left heavy, right heavy, level
What it says in wordsEach weighing multiplies the cases you can tell apart by three.
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.
Where candidates lose it
Most candidates start with four against four because halving feels natural. If the scale tips, four suspects remain, and one weighing cannot finish the job, so they need three. The interviewer is waiting to see whether you notice the level outcome is information too.
The second loss is solving it but not proving two is the minimum. Have the counting argument ready: 3 outcomes per weighing, 3 x 3 = 9, and one weighing gives only 3.
What the interviewer asks next
You now have 12 certificates and do not know whether the forgery is heavier or lighter. How many weighings do you need?
What is the largest number of certificates you could search with three weighings?
If two of the nine are forged, can you still find them in two weighings?
054A 3-year floating rate note pays the benchmark plus 100 basis points, resetting quarterly. The issuer's market spread widens to 150 basis points. Roughly what price does the note fall to, and why does a floater still lose money?Credit spreads and default probabilityHardSyndicate desksFixed income asset management
Try it first
Before any maths: what happens to the note's price?
Show the worked solution
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.
The benchmark part of the coupon resets, but the fixed 100 basis point margin now falls 50 basis points short of what the market wants; twelve quarterly shortfalls of 0.125, discounted, are worth 1.32 per Rs 100, so the note trades near 98.68.
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.
The relationship
$P \approx 100 - \Delta s \times D_s = 100 - 0.50 \times 2.64 \approx 98.68$
Delta s the widening in the issuer's spread, 0.50 percentage points
D_s spread duration, the price change per point of spread, about 2.64 years here
100 the price at a reset when the margin is fair
What it says in wordsA floater loses the spread change times its spread duration, and a floater's spread duration is close to its remaining life.
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.
Where candidates lose it
The trap is saying a floater always trades near par because it resets. That is true for rate moves and false for credit moves: the margin is locked at issue, so a wider market spread leaves the note underpaying every quarter until maturity.
The second loss is using the maturity for both risks. Rate duration runs only to the next reset, about 0.25 years; spread duration runs to maturity, about 2.6 years here. Keep the two apart out loud.
What the interviewer asks next
The same 150 basis point widening hits a 7-year floater from the same issuer. Roughly how far does it fall?
What is this note's interest rate duration, and why is it so much smaller than its spread duration?
If the issuer's spread tightens back to 100 basis points after a year, where does the note trade?
055A 9% annual coupon bond has 3 years left and a make-whole call at the government yield plus 50 basis points. The 3-year government yield is 6.5%. What is the make-whole price, and should the issuer call if it can refinance at 7.3%?Issuance and refinancing arithmeticHardSyndicate desksCorporate banking
Try it first
At a 7.3% refinancing rate, does calling save the issuer money?
Show the worked solution
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.
The remaining 9, 9 and 109 discounted at 7.0% are worth 8.41, 7.86 and 88.98, a make-whole price of 105.25; the same payments valued at the issuer's 7.3% refinancing rate are worth 104.44, so calling costs 0.81 per Rs 100 more than keeping the bond.
The relationship
$P_{MW} = \frac{9}{1.07} + \frac{9}{1.07^2} + \frac{109}{1.07^3} = 8.41 + 7.86 + 88.98 = 105.25$
1.07 one plus the make-whole discount rate, government 6.5% plus 0.5%
9 the annual coupon per Rs 100
109 the last coupon plus the principal
What it says in wordsThe make-whole price is the bond's remaining cash flows valued at a rate barely above the government yield.
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.
Where candidates lose it
Candidates see a 9% coupon and a 7.3% refinancing rate and say call, because 1.7% a year sounds like free money. They forget the issuer must first pay a premium price that already contains the value of that saving.
The second loss is discounting at the wrong rate: using 6.5% or 9% instead of the government yield plus the 50 basis point spread named in the documents. Say the rate before you discount.
What the interviewer asks next
At what refinancing rate is the issuer indifferent between calling and keeping the bond?
How would a fixed-price call at 102 change the answer?
Why do investors prefer a make-whole to a fixed-price call, and what does it do to the bond's price when rates fall?
056An issuer can pay 8.00% once a year or 7.85% in two semi-annual instalments of 3.925% each. Which is more expensive for the issuer, measured as an effective annual rate?Bond pricing and yieldCoreIndian debt capital markets
Try it first
Which costs the issuer more?
Show the worked solution
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.
Per Rs 100, the annual bond delivers 8.000 at the year end, while the semi-annual bond's first 3.925 grows to 4.079 by the year end and adds to the second 3.925 for 8.004, so the semi-annual structure costs the issuer 0.4 basis points more.
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.
The relationship
$\text{EAR} = \left(1 + \frac{0.0785}{2}\right)^2 - 1 = 1.03925^2 - 1 = 8.004\%$
0.0785 / 2 the rate paid each half year
squared two half years compounded into one year
EAR effective annual rate, the one-payment-a-year equivalent
What it says in wordsCompound the half-yearly rate over two periods to compare it with a rate paid once a year.
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.
Where candidates lose it
The fast wrong answer is that 8.00% costs more because it is the bigger number. That ignores when the money moves: a coupon paid six months early is worth more to the investor and costs more to the issuer.
The opposite slip is treating compounding as a large effect. At these rates it is worth about 15 basis points in total, which is exactly why 7.85% semi-annual lands almost on top of 8.00% annual. Give the size of the gap, not just its direction.
What the interviewer asks next
What quarterly rate is equivalent to 8.00% paid annually?
A government security yields 7.10% semi-annual and a corporate bond 7.90% annual. What is the spread on a like-for-like basis?
Why does the compounding effect grow as rates rise?
057A Rs 100 crore private credit loan pays 8% cash interest plus 4% PIK. The PIK compounds annually and the cash interest is paid on the accreted balance. What is owed at the end of year 5, how much cash interest has the lender received, and what is its IRR if it lent at par?Compounding, PIK and feesHardAres ManagementLos Angeles · 2026
Try it first
What IRR does the lender earn over the five years?
Show the worked solution
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.
The Rs 100 crore balance accretes at 4% a year to Rs 121.67 crore by year five, and the 8% cash coupon charged on that growing balance rises from Rs 8.00 crore to Rs 9.36 crore, Rs 43.33 crore in total, for an IRR of 12% at par.
Year Opening balance Cash interest, 8% PIK added, 4% Closing balance
1 100.00 8.00 4.00 104.00
2 104.00 8.32 4.16 108.16
3 108.16 8.65 4.33 112.49
4 112.49 9.00 4.50 116.99
5 116.99 9.36 4.68 121.67
Total 43.33 21.67 121.67
Rs crore. The lender receives Rs 43.33 crore of cash interest over five years and Rs 121.67 crore at maturity, of which Rs 21.67 crore is accrued PIK.
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.
The relationship
$100 = \sum_{t=1}^{5} \frac{8 \times 1.04^{t-1}}{1.12^{t}} + \frac{121.67}{1.12^{5}} \;\Rightarrow\; \text{IRR} = 12\%$
8 x 1.04^(t-1) cash interest in year t, 8% of the opening balance
121.67 the accreted balance repaid at the end of year 5
1.12 one plus the IRR that makes both sides equal
What it says in wordsDiscounting the cash coupons and the accreted repayment at 12% gives back exactly the Rs 100 crore lent.
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.
Where candidates lose it
The common mistake is to add the pieces as simple interest: 4% x 5 years is Rs 20 crore of PIK and 8% x Rs 100 crore x 5 is Rs 40 crore of cash. Both understate, because the PIK compounds and the cash coupon is charged on the growing balance: the right figures are Rs 21.67 crore and Rs 43.33 crore.
The second is saying the IRR is below 12% because part of the interest arrives late. Late is not lost: the PIK earns the full 12% while it waits. Say that, then name the real cost, which is credit risk concentrated at maturity.
What the interviewer asks next
The lender bought the loan at 97 instead of par. Roughly what is the IRR now?
What if the PIK is simple rather than compounding and cash interest is charged only on the original Rs 100 crore?
The borrower can choose each year between 12% in cash and 13% in PIK. When would it choose PIK, and what does that tell the lender?
Asked at Ares Management, Credit, Los Angeles, 2026 (Wall Street Oasis): First 1v1 they said was mainly behavioral had PIK question
058A perpetual bond yields 8%. What are its Macaulay duration and its modified duration, and how can a bond that never matures have a duration of only about 13.5 years?Duration and convexityHardFixed income asset management
Try it first
What is the Macaulay duration of a perpetual yielding 8%?
Show the worked solution
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%.
The present value of each year's coupon falls from 7.41 in year one to 0.80 in year thirty, so the first nine years hold half the value and the value-weighted balance point, the Macaulay duration, sits at 13.5 years.
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.
The relationship
$D_{Mac} = \frac{1+y}{y} = \frac{1.08}{0.08} = 13.5 \qquad D_{mod} = \frac{D_{Mac}}{1+y} = \frac{1}{y} = 12.5$
y the bond's yield, 8%
D_Mac Macaulay duration, the value-weighted average time to the cash flows, in years
D_mod modified duration, the percentage price change for a one point change in yield
What it says in wordsA perpetual's duration depends only on its yield: one plus the yield, over the yield.
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.
Where candidates lose it
The instinctive answer is infinite, because the bond never matures. It confuses maturity, the date of the last cash flow, with duration, the value-weighted average date of all of them.
The second slip is mixing up the two durations. 13.5 is the Macaulay figure, a time in years; 12.5 is the modified figure, the price change per point of yield. Give both and say which is which.
What the interviewer asks next
What is the modified duration of the same perpetual if its yield falls to 5%?
A perpetual is callable at par in year 5 and trades above par. Which duration would you use to hedge it?
Why does a zero-coupon bond's duration equal its maturity while a perpetual's does not?
059Estimate the debt needed to build 10 GW of new solar capacity, assuming a project cost of Rs 4 to 5 crore per MW and 75% debt funding. How sensitive is the answer to the cost assumption?Estimation and market sizingCoreCorporate bankingIndian debt capital markets
Try it first
Roughly how much debt is that?
Show the worked solution
About Rs 30,000 to 37,500 crore of debt, with Rs 33,750 crore at the midpoint. 10 GW is 10,000 MW; at Rs 4 to 5 crore per MW the build costs Rs 40,000 to 50,000 crore, and 75% of that is debt. The answer moves one for one with the cost per MW: every Rs 0.5 crore per MW shifts the debt by Rs 3,750 crore, more than the effect of moving the gearing five points.
How do you structure the estimate before any arithmetic?
Estimating a home loan starts with the price per square foot, times the floor area, times the share the bank will fund. A project debt estimate is the same chain: capacity times cost per unit of capacity gives the project cost, and the debt share of that cost gives the borrowing. Say the chain out loud first, with its units: megawatts, crore per megawatt, then a percentage. The most common slip in this question is a units error, not an arithmetic one.
The relationship
$\text{Debt} = 10{,}000\ \text{MW} \times \text{Rs } 4 \text{ to } 5\ \text{crore per MW} \times 75\% = \text{Rs } 30{,}000 \text{ to } 37{,}500\ \text{crore}$
10,000 MW 10 GW of capacity, since 1 GW is 1,000 MW
Rs 4 to 5 crore per MW the all-in project cost range given in the question
75% the share of project cost funded with debt
What it says in wordsCapacity times cost per MW gives the project cost; the debt share of that is the borrowing.
10,000 MW at Rs 4 to 5 crore per MW with 75% debt needs Rs 30,000 to 37,500 crore of borrowing, and the cost assumption swings the answer by Rs 7,500 crore against Rs 4,500 crore for a ten point swing in gearing.
Which assumption moves the answer most?
Flex each input across a sensible range while holding the other at its midpoint. Moving the cost from Rs 4 to Rs 5 crore per MW swings the debt by Rs 7,500 crore; moving the gearing from 70% to 80% swings it by Rs 4,500 crore. So the cost assumption is the one to defend, or to ask about. A lender would also want to know how much of the cost is modules against land, grid connection and construction, because those pieces move for different reasons.
Close with the range, not a single number. An estimate built on a range of inputs should come out as a range, with the midpoint named and the driver identified. At the midpoint, Rs 33,750 crore at an assumed 9% would carry roughly Rs 3,038 crore of interest a year. The costs here are the question's assumptions, not current market figures; in a real pitch you would confirm the cost per MW and typical gearing against recent project financings before using them.
Where candidates lose it
The usual loss is units. Candidates multiply 10 by Rs 4 to 5 crore and answer Rs 40 to 50 crore, or forget the 75% and give the whole project cost as the debt.
The second is a single number with false precision. The cost input is a range, so the output is a range: say Rs 30,000 to 37,500 crore, name the midpoint, and say which assumption drives the spread.
What the interviewer asks next
Roughly what annual interest bill does the midpoint debt carry at 9%, and what cash flow would cover it 1.3 times?
How would you split this debt between bank loans and bonds, and why?
Construction costs fall 20% while gearing rises to 80%. What is the new debt estimate?
060Pitching for a bond mandate costs Rs 50 lakh of team time. You estimate a 30% chance of winning a mandate worth Rs 4 crore in fees. Should you pitch, and what win probability makes the pitch break even?Probability and expected valueWarm upSyndicate desks
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What is the expected value of pitching, net of the cost?
Show the worked solution
Yes, pitch: the expected value is plus Rs 70 lakh, and the pitch breaks even at a 12.5% win probability. A 30% chance of Rs 4 crore is worth Rs 1.2 crore in expectation, against a certain Rs 50 lakh cost. Break-even is where the probability times Rs 4 crore equals Rs 50 lakh: 0.5 divided by 4, or 12.5%. The 30% estimate clears that by a wide margin, so even a rough probability supports pitching.
How do you compare an uncertain fee with a certain cost?
A shopkeeper deciding whether to print Rs 500 of flyers asks how much extra business they might bring, times how likely that is. Expected value multiplies each outcome by its probability and adds them up, so a 30% chance of Rs 4 crore is worth Rs 1.2 crore before costs. The cost of pitching is paid whether you win or lose, so it comes off in full: Rs 1.2 crore minus Rs 0.5 crore is plus Rs 0.7 crore.
The relationship
$EV = 0.30 \times 4.0 - 0.5 = +0.7\ \text{crore} \qquad p^{*} = \frac{0.5}{4.0} = 12.5\%$
0.30 your estimated chance of winning
4.0 the fee if you win, Rs crore
0.5 the cost of pitching, paid either way, Rs crore
p* the break-even win probability
What it says in wordsExpected fee less the certain cost; break-even is the cost divided by the prize.
The expected fee of Rs 1.2 crore less the Rs 0.5 crore cost leaves plus Rs 0.7 crore, and the expected fee line crosses the cost at a 12.5% win probability, well below the 30% estimate.
What win probability makes the pitch worth it, and how robust is the answer?
Break-even is the probability at which the expected fee just covers the cost: Rs 50 lakh over Rs 4 crore, 12.5%. The 30% estimate is more than twice that, so the decision survives a lot of error in the estimate. That is the useful part of the calculation: you rarely know a win probability precisely, but you often know whether it is comfortably above or below the break-even.
Name what the simple sum leaves out. Losing is the most likely outcome, 70% of the time, so a desk that pitches only once can easily end Rs 50 lakh down; expected value pays off across many pitches. The team's time also has an opportunity cost if it could pitch a better mandate instead, and a win may bring follow-on business that the Rs 4 crore does not capture. Each of those shifts the break-even; none reverses this answer.
Where candidates lose it
The trap is anchoring on the most likely outcome. You lose 70% of the time, so candidates say the pitch loses money. Expected value is not the most likely outcome; it is the probability-weighted average, and here that is plus Rs 70 lakh.
The second loss is forgetting to subtract the cost and answering Rs 1.2 crore, or dividing the wrong way for break-even. Say break-even as cost over prize, then check it: 12.5% of Rs 4 crore is Rs 50 lakh.
What the interviewer asks next
If you win you must share the mandate with a second bank, halving your fee. Does the answer change?
You can pitch three mandates like this but can only staff two. How do you choose?
How would you estimate the 30% win probability in the first place?

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

2	the number of equal instalments
1/10, 1/20	units bought per rupee at each price
P-bar	the average cost per unit

w	the number of weighings allowed
3	outcomes per weighing: left heavy, right heavy, level

Delta s	the widening in the issuer's spread, 0.50 percentage points
D_s	spread duration, the price change per point of spread, about 2.64 years here
100	the price at a reset when the margin is fair

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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

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Why does equal money not mean equal weight?

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

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What the interviewer asks next

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Why split into thirds rather than halves?

How do you prove two weighings is the minimum, and the limit?

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What the interviewer asks next

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

How much is a 50 basis point shortfall worth today?

Where candidates lose it

What the interviewer asks next

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What does a make-whole price actually compute?

Why does refinancing at 7.3% not pay?

Where candidates lose it

What the interviewer asks next

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Why can you not compare 8.00 with 7.85 directly?

How do you convert both to an effective annual rate?

How big is the difference in money, and when does it matter?

Where candidates lose it

What the interviewer asks next

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What does PIK actually do to the balance?

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

What is the catch the interviewer wants you to name?

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Why does a bond that never ends have a finite duration?

How do you get 13.5 without summing forever?

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What the interviewer asks next

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How do you structure the estimate before any arithmetic?

Which assumption moves the answer most?

Where candidates lose it

What the interviewer asks next

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How do you compare an uncertain fee with a certain cost?

What win probability makes the pitch worth it, and how robust is the answer?

Where candidates lose it

What the interviewer asks next