Debt Capital Markets puzzles, solved step by step

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

Showing 71–80 of 100

071You hold Rs 100 crore face of a 5-year corporate bond with a DV01 of 0.043 per Rs 100 of face. You want to hedge its interest rate risk by shorting a 10-year government bond whose DV01 is 0.068 per Rs 100. How much face value of the government bond do you short?Duration and convexityCoreFixed income asset managementSyndicate desks
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How much of the government bond do you short?
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Short about Rs 63.2 crore face of the government bond. The corporate position moves Rs 100 crore x 0.043%, Rs 4.3 lakh, for each basis point. Each Rs 1 crore of the 10-year government bond moves Rs 6,800 per basis point. Matching the two takes 4.3 lakh over 6,800, about 63.2 lots of Rs 1 crore. The hedge ratio is the ratio of the DV01s, 0.043 over 0.068, not one for one on face value.
What exactly are you trying to balance?
Two children on a seesaw balance when weight times distance from the pivot matches on both sides, not when they weigh the same. A rate hedge balances when the rupee change per basis point matches on both sides, so you hedge DV01 against DV01, not face value against face value. The corporate bond's DV01The change in a bond position value for a one basis point change in yield, in rupees. is 0.043 per Rs 100, so Rs 100 crore of it gains or loses Rs 4.3 lakh for every basis point. That is the number the short has to offset.
Rs 100 crore of the corporate bond at a DV01 of 0.043 balances Rs 63.2 crore of the government bond at a DV01 of 0.068, both carrying Rs 4.3 lakh per basis point, while shorting a matching Rs 100 crore would over-hedge by Rs 2.5 lakh per basis point.
The relationship
$F_{hedge} = F \times \frac{DV01_{corp}}{DV01_{gov}} = 100 \times \frac{0.043}{0.068} = 63.2\ \text{crore}$
F the face value you hold, Rs 100 crore
DV01_corp the corporate bond's price change per basis point, per Rs 100
DV01_gov the government bond's price change per basis point, per Rs 100
What it says in wordsScale the hedge so that its rupee move per basis point equals the position's.
Why do you need less of the government bond than you hold?
The 10-year government bond has a longer duration, so each rupee of it moves more when rates move: 0.068 per Rs 100 against 0.043. A smaller position in the more sensitive bond carries the same rupee risk. Check the balance: Rs 63.2 crore x 0.068% is Rs 4.3 lakh a basis point, the same as the corporate position. If rates fall 20 basis points in parallel, the corporate bond gains about Rs 86 lakh and the short loses about Rs 86 lakh.
Then say what the hedge does not cover, because that is the follow-up. A DV01 hedge removes the risk of a parallel move in rates, but it leaves you exposed to the curve twisting between 5 and 10 years and to the corporate bond's credit spread moving. If 5-year yields rise while 10-year yields stay put, the hedge does nothing for you. Desks often hedge with a bond or swap closer in maturity for exactly this reason, and rebalance as DV01s drift with time and rates.
Where candidates lose it
The instinctive answer is Rs 100 crore, shorting the same face value you hold. With a longer, more sensitive hedging bond that over-hedges by more than half and turns a long rate position into a short one.
The second slip is inverting the ratio and shorting Rs 158 crore. Check the direction with intuition: the hedging bond is more sensitive, so you need less of it, not more.
What the interviewer asks next
Rates fall 20 basis points in a parallel move. What is the profit or loss on each leg?
The 5-year yield rises 10 basis points and the 10-year is unchanged. What happens to the hedged position?
Why might you hedge with a 5-year interest rate swap instead?
072A bond trades 300 basis points over the risk-free curve, and investors expect 40% recovery if it defaults. Using the credit triangle, what annual default probability is the market implying?Credit spreads and default probabilityCoreCredit researchFixed income asset management
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What annual default probability does the spread imply?
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About 5% a year. The credit triangle says the spread pays for expected loss: spread = probability of default x loss given default. With 40% recovery the loss given default is 60%, so 3.0% = PD x 60% and PD = 5.0%. Treat it as an upper-bound reading, because real spreads also pay for liquidity and for bearing risk, so the default rate the market truly expects is usually lower.
Why does a spread imply a default probability at all?
An insurer charging Rs 3,000 a year to cover a Rs 1 lakh motorbike, where a stolen bike is usually recovered at 40% of its value, is pricing in some chance of theft; work backwards and you can read that chance off the premium. A credit spread is the premium a lender demands for expected loss: roughly, the chance of default each year times the share of the money lost when it happens. Everything else equal, a lender earning 300 basis points over the risk-free curve is being paid to lose about 3% a year on average.
The 300 basis point spread equals the default probability times a 60% loss given default, which implies 5.0% a year; the same spread implies 3.0% if nothing is recovered and 10.0% if 70% is recovered.
The relationship
$s \approx PD \times (1 - R) \;\Rightarrow\; PD = \frac{3.0\%}{1 - 0.40} = 5.0\%$
s the credit spread, 3.0% a year
PD the annual probability of default
R the expected recovery rate, 40%
1 - R loss given default, 60%
What it says in wordsThe spread roughly equals the yearly chance of default times the share lost when it happens.
How does the recovery assumption change the answer?
The higher the recovery, the higher the default probability a given spread implies, because each default costs the lender less. At 40% recovery, 300 basis points implies 5%; at 0% recovery it implies only 3%; at 70% recovery it implies 10%. So the recovery assumption is not a detail. Two analysts reading the same spread with different recovery views can disagree by a factor of two on how risky the credit is.
Expected recovery Loss given default Implied annual PD
0% 100% 3.0%
40% 60% 5.0%
70% 30% 10.0%
The same 300 basis point spread implies a 3.0%, 5.0% or 10.0% annual default probability depending on whether investors expect to recover nothing, 40% or 70% in default.
What does the credit triangle leave out?
The triangle treats the whole spread as payment for expected loss, but part of any spread pays for liquidity and for the discomfort of carrying credit risk. Default probabilities read off spreads therefore tend to sit above the default rates actually seen for similar credits. Use the triangle to compare credits and to sanity-check a spread, and say out loud that it gives an upper bound, not a forecast. Held for five years, 5% a year compounds to about 23% cumulative. The limit: it is a one-period shortcut that ignores when in the year default happens.
Where candidates lose it
The quick wrong answer is 3%, reading the spread as the default probability. That assumes lenders lose everything in default; with 40% recovery they lose 60%, so the same spread implies more defaults, not fewer.
The second slip is dividing by the recovery rate instead of the loss rate and getting 7.5%. Say loss given default out loud, 1 minus recovery, before you divide.
What the interviewer asks next
A secured bond from the same issuer has expected recovery of 70%. Roughly what spread should it trade at?
Why do default probabilities implied by spreads usually sit above historical default rates?
At 5% a year, what is the cumulative chance of default over five years?
073A company has EBITDA of Rs 200 crore, gross debt of Rs 900 crore and cash of Rs 300 crore. It uses Rs 200 crore of its cash to repay debt. What happens to its gross leverage and to its net leverage?Leverage, coverage and cash flowWarm upLeveraged financeCorporate banking
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What happens to net leverage?
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Gross leverage falls from 4.5x to 3.5x; net leverage stays at 3.0x. Gross debt drops from Rs 900 crore to Rs 700 crore against EBITDA of Rs 200 crore. Net debt is 900 minus 300 before and 700 minus 100 after, Rs 600 crore both times, because cash and debt fall together. The repayment cuts interest and tidies the balance sheet, but it does not change what the company owes net of what it holds.
Why does one ratio move and the other stay put?
If you owe a friend Rs 9,000 and have Rs 3,000 in your wallet, you are Rs 6,000 in the hole. Hand over Rs 2,000 and you owe Rs 7,000 with Rs 1,000 left: still Rs 6,000 in the hole. Net debt already counts the cash as if it could repay debt, so actually using the cash to repay debt changes nothing in the net figure. Gross debt ignores the cash, so it falls by the full Rs 200 crore.
Repaying Rs 200 crore of debt from cash takes gross leverage from 4.5x to 3.5x, but debt and cash fall together, so net debt stays at Rs 600 crore and net leverage stays at 3.0x.
The relationship
$\text{Gross: } \frac{900}{200} = 4.5\times \to \frac{700}{200} = 3.5\times \qquad \text{Net: } \frac{900-300}{200} = \frac{700-100}{200} = 3.0\times$
900, 700 gross debt before and after, Rs crore
300, 100 cash before and after, Rs crore
200 EBITDA, Rs crore, unchanged by the repayment
What it says in wordsGross leverage sees only the debt; net leverage sees debt less cash, and both fall by the same amount.
Why would a company repay debt with cash at all?
Because the money still changes: interest on Rs 200 crore of debt usually costs more than the same cash earns on deposit. If the debt costs 9% and the cash earns 6%, repaying saves about Rs 6 crore a year of net interest. It also matters where covenants are written on gross debt, as some loan documents are, and where the cash sits in a subsidiary or abroad and could not easily reach lenders in a crisis. That is why credit analysts ask whether cash is truly available before netting it.
Now look at the other side. Spending the cash also removes a cushion: a company with Rs 100 crore of cash has less room to absorb a bad quarter than one with Rs 300 crore, at the same net leverage. Lenders and rating analysts look at liquidity alongside leverage for this reason. The limit: this assumes EBITDA is untouched by the repayment, which holds, because interest sits below EBITDA.
Where candidates lose it
The fast wrong answer is that both ratios improve, because repaying debt sounds like deleveraging. Net leverage already assumed the cash could repay debt, so doing it changes nothing on that measure.
The opposite slip is saying the repayment achieved nothing. It lowers gross leverage, cuts net interest cost and spends liquidity; name all three.
What the interviewer asks next
The company instead raises Rs 200 crore of new debt and holds it as cash. What happens to each ratio?
Which leverage figure would you write into a covenant, and why?
When is it risky to net cash against debt?
074One-year and two-year zero rates are 7% and 8%. A two-year bond with a 10% annual coupon trades at 104.00. Is it rich or cheap to the zero curve, and by how much per Rs 100?Bond pricing and yieldHardSyndicate desksFixed income asset management
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Against the zero curve, the bond is:
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The bond is rich by about 0.35 per Rs 100. Price each cash flow at its own zero rate: the year-one coupon of 10 at 7% is worth 9.35, and the final 110 at 8% for two years is worth 94.31. Fair value is 103.65, so a price of 104.00 is 0.35 too high. In yield terms the bond yields about 7.76% against a fair 7.95%, roughly 19 basis points too low.
Why price each cash flow at its own rate?
Money due in one year and money due in two years are different goods, the way a train ticket for next week and one for next month carry different prices. A zero rate is the price of money for one specific date, so each of a bond's cash flows is discounted at the zero rate for its own date, and the bond is worth the sum. The year-one coupon of 10 is discounted at 7%; the year-two payment of 110 is discounted at 8% for two years.
The coupon of 10 at 7% is worth 9.35 and the final 110 at 8% for two years is worth 94.31, a fair value of 103.65, so the market price of 104.00 is 0.35 rich and the bond yields about 19 basis points less than the curve says it should.
The relationship
$P^{*} = \frac{10}{1.07} + \frac{110}{1.08^{2}} = 9.346 + 94.307 = 103.65 \qquad 104.00 - 103.65 = 0.35$
P* the fair price implied by the zero curve
1.07 one plus the one-year zero rate
1.08^2 two years of growth at the two-year zero rate
What it says in wordsFair value is each cash flow discounted at its own date's zero rate; rich or cheap is the market price against that.
What does rich mean, and how would a desk use it?
Rich means the price is above what the curve says the cash flows are worth, so the bond yields less than it should: about 7.76% against a fair 7.95%. An investor would rather own the same cash flows through the curve, and a relative value desk might sell the bond and buy a matching pair of zero-coupon bonds, locking in about 0.35 per Rs 100 if the prices converge. On Rs 100 crore of face that is about Rs 35 lakh.
Say what could make the gap real rather than a mistake. A gap of 0.35 can reflect liquidity, a different tax treatment of high-coupon bonds, or the cost of actually building the matching zero-coupon pair, not only a mispricing. The check is whether similar bonds show the same gap. The limit: this assumes the zero rates are for the same issuer's credit; a government zero curve would value a corporate bond too high.
Where candidates lose it
The instinctive answer is that a 10% coupon against 7% and 8% rates must be cheap. The high coupon is already in the price; the question is whether 104.00 is the right price for those cash flows, and it is not.
The second slip is discounting both cash flows at 8%, which gives 103.57 and overstates how rich the bond is. Match each cash flow to its own date's rate.
What the interviewer asks next
At what yield would the bond need to trade to be fair to the curve?
How would you lock in the gap if you believed it was a pure mispricing?
If the two-year zero rate rises to 8.5%, is the bond still rich at 104.00?

Expected recovery	Loss given default	Implied annual PD
0%	100%	3.0%
40%	60%	5.0%
70%	30%	10.0%

075Revenue is Rs 800 crore, up 10% on last year's Rs 727.3 crore. The EBITDA margin is 18%, depreciation is Rs 30 crore, the tax rate is 25%, capex is Rs 45 crore, and the net working capital build is 12% of the change in revenue. Walk from revenue to unlevered free cash flow.Leverage, coverage and cash flowCoreRBC Capital MarketsLondon · 2026

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Roughly what is unlevered free cash flow?

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Unlevered free cash flow is about Rs 61.8 crore. EBITDA is 18% of Rs 800 crore, Rs 144 crore. Less Rs 30 crore of depreciation gives EBIT of Rs 114 crore, and tax at 25% leaves NOPAT of Rs 85.5 crore. Add back depreciation, which is not cash, to reach Rs 115.5 crore, then take off capex of Rs 45 crore and the working capital build of 12% of the Rs 72.7 crore revenue increase, Rs 8.7 crore.

What order do you walk the lines in?

Think of a shopkeeper's year: sales come in, stock and staff are paid, the taxman takes a share, some money goes on a new fridge, and more stock sits on the shelves because the shop is busier. What is left is what the owner could take out. Unlevered free cash flow follows the same order: operating profit, less tax on that profit, plus non-cash charges, less investment in fixed assets and in working capital. It is unlevered because interest is left out: the cash belongs to lenders and shareholders together.

EBITDA of Rs 144 crore loses Rs 30 crore of depreciation and Rs 28.5 crore of tax to reach NOPAT of Rs 85.5 crore, gets the depreciation back, then loses Rs 45 crore of capex and Rs 8.7 crore of working capital, leaving Rs 61.8 crore of unlevered free cash flow.

Line	Rs crore
Revenue	800.0
EBITDA at 18%	144.0
Less depreciation	(30.0)
EBIT	114.0
Less tax at 25% of EBIT	(28.5)
NOPAT	85.5
Add back depreciation	30.0
Less capex	(45.0)
Less working capital build, 12% x 72.7	(8.7)
Unlevered free cash flow	61.8

Unlevered free cash flow is Rs 61.8 crore, 43% of EBITDA, after tax on operating profit, capex of 1.5 times depreciation and a working capital build on the Rs 72.7 crore rise in revenue.

Why tax EBIT rather than profit after interest?

Unlevered cash flow is the cash the business makes before any financing choice, so tax is charged as if there were no debt: 25% of EBIT, Rs 28.5 crore. Real tax is lower when there is interest to deduct, and that saving is counted elsewhere, in the discount rate or in a separate tax shield line. Taxing profit after interest and then discounting at a rate that already includes the shield counts the same benefit twice.

Where does the cash go between EBITDA and free cash flow?

Of Rs 144 crore of EBITDA, only Rs 61.8 crore, about 43%, becomes unlevered free cash flow: tax takes 28.5, capex 45 and working capital 8.7. Capex is 1.5 times depreciation, so the company is investing to grow, and working capital rises with sales because more revenue means more receivables and stock. A lender sizing debt on EBITDA should look at this conversion, because interest and repayments are paid from the Rs 61.8 crore, not the Rs 144 crore. The limit: one year of cash conversion can mislead when capex is lumpy.

Where candidates lose it

The common slip is stopping at EBITDA less capex and calling it free cash flow, forgetting tax and working capital; that gives Rs 99 crore, well above the true Rs 61.8 crore.

The second is charging working capital on the whole revenue, 12% of Rs 800 crore, instead of on the change. Working capital is a balance; only the increase uses cash this year.

What the interviewer asks next

Revenue falls 10% next year instead. What happens to the working capital line?
How would you get from unlevered to levered free cash flow?
Capex falls to the level of depreciation. What is free cash flow now, and is that sustainable?

Asked at RBC Capital Markets, Leveraged Finance, London, 2026 (Wall Street Oasis): Walk me through Revenue to unleveraged FCF

076A conglomerate owns three businesses worth Rs 1,200 crore, Rs 800 crore and Rs 500 crore of enterprise value. The market applies a 15% holding company discount to the sum of the parts. Net debt is Rs 600 crore and there are 50 crore shares. What is the value per share, and what would it be with no discount?Cost of capital and valuation riddlesCoreDeutsche BankMumbai · 2024
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Before you calculate: the discount is 15% of enterprise value. How much does the value per share fall?
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Rs 30.5 per share with the discount and Rs 38.0 without it. The parts add to Rs 2,500 crore. A 15% discount takes off Rs 375 crore, leaving Rs 2,125 crore of enterprise value. Less Rs 600 crore of net debt gives equity of Rs 1,525 crore over 50 crore shares. With no discount equity is Rs 1,900 crore, or Rs 38.0. A 15% discount costs shareholders 19.7%.
Why does the discount come off before the debt?
Think of a house worth Rs 1 crore with a Rs 60 lakh home loan on it. If buyers suddenly pay 15% less for houses on that street, the house is worth Rs 85 lakh, the bank is still owed Rs 60 lakh, and the owner's share drops from Rs 40 lakh to Rs 25 lakh. A conglomerate works the same way. The holding company discount is a haircut on what the whole group is worth, and the lenders' claim is a fixed number that sits ahead of the shareholders, so the haircut is taken from enterprise value and passes through to equity untouched.
So the order is: value each business, add them to get the sum of the partsValuing each business on its own, usually against its own peers, and adding the values together., apply the holding company discountThe gap between what a group trades at and what its businesses would be worth separately, often blamed on head office costs, capital allocation or tax leakage. to the total, subtract net debt, and divide by shares. Rs 2,500 crore less Rs 375 crore is Rs 2,125 crore; less Rs 600 crore is Rs 1,525 crore; over 50 crore shares is Rs 30.5.
The three businesses add to Rs 2,500 crore; the 15% discount removes Rs 375 crore and net debt removes Rs 600 crore, leaving equity of Rs 1,525 crore, or Rs 30.5 a share against Rs 38.0 with no discount.
The relationship
$\text{Value per share} = \frac{(1-d)\sum EV_i - ND}{N} = \frac{0.85 \times 2{,}500 - 600}{50} = 30.5$
d the holding company discount, 15%
EV_i the enterprise value of each business
ND net debt, Rs 600 crore
N shares outstanding, 50 crore
What it says in wordsDiscount the sum of the businesses, take off what the lenders are owed, and share what is left.
Why does a 15% discount cost the shareholders more than 15%?
Equity is the thin slice left after debt, so any fall in enterprise value is a bigger fall as a share of equity. The Rs 375 crore haircut is 15% of Rs 2,500 crore but 19.7% of Rs 1,900 crore. Push net debt to Rs 1,500 crore and the same haircut takes 37.5% of the equity. This is why a credit analyst reading a holding company cares about the discount even though no lender ever sees it on a statement: it is a measure of how much equity cushion the market will actually pay for.
What should you say about where the discount is applied?
Some desks apply the discount to equity value instead of enterprise value. On equity of Rs 1,900 crore, 15% gives Rs 1,615 crore and Rs 32.3 a share. The two conventions differ by Rs 1.8 a share here, so name your convention in the first sentence rather than let the interviewer find it. The enterprise value version is the more common reading when the discount is described as a discount to the sum of the parts, and it is the one that treats the lenders' claim as fixed.
Where candidates lose it
The frequent slip is subtracting the net debt first and then taking 15% off what is left. That gives Rs 32.3 a share and quietly assumes the lenders share the discount, which they do not when the discount is on the sum of the parts. Interviewers accept either convention only if you name it.
The second slip is telling the interviewer that shareholders lose 15%. Say 19.7%, and give the reason in one line: the debt does not shrink, so the whole haircut falls on equity.
What the interviewer asks next
With net debt of Rs 1,500 crore, what discount would halve the equity value?
The group sells unit C for Rs 500 crore of cash and repays debt. If the discount stays at 15% on what remains, what happens to value per share?
Why would a lender to the holding company watch the discount at all?
Asked at Deutsche Bank, Equity Capital Markets, Mumbai, 2024 (Wall Street Oasis): working capital, leases and SOTP with conglomerate discount question
077You have two bowls, 60 white balls and 40 black balls. Split all 100 balls between the bowls any way you like. One bowl is then chosen at random and one ball drawn from it. How do you maximise the chance of drawing white, and what is that chance?Probability and expected valueCoreDeutsche BankMumbai · 2024
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Pick the best chance you think a clever split can reach.
Show the worked solution
Put one white ball alone in bowl A and the other 99 balls, 59 white and 40 black, in bowl B. Bowl A gives white every time and bowl B gives white 59 times in 99. Each bowl is chosen half the time, so the chance is 0.5 x 1 + 0.5 x 59/99, about 79.8%, against 60% for an even split.
Why does a bowl with one ball count as much as a bowl with 99?
Picture two teams tossing a coin to bat first: the coin does not care that one side has eleven players and the other has one. The bowls are chosen the same way. The coin picks a bowl, not a ball, so a bowl holding a single white ball gets the same half of the draws as a bowl holding ninety nine. That makes one lone white ball the cheapest way to buy certainty for half of all outcomes, and it costs bowl B only one white ball out of sixty.
Bowl A with one white ball gives white every time; bowl B with 59 white and 40 black gives white 59.6% of the time, so the average is 79.8%, well above the 60% an even split gives.
How do you show that nothing beats it?
Bowl A cannot do better than 100%, and one white ball is the least that gets it there. Every further white ball moved into bowl A is wasted there and missed in bowl B, and every black ball moved into bowl A drags it below 100%. So bowl A is one white ball and bowl B takes what is left. Checking all 2,499 possible splits by computer agrees: the best is 79.80%, and only the one-ball split reaches it.
The relationship
$P(\text{white}) = \tfrac{1}{2}\cdot 1 + \tfrac{1}{2}\cdot\frac{W-1}{W+B-1} = \tfrac{1}{2} + \tfrac{1}{2}\cdot\frac{59}{99} \approx 0.798$
W white balls, 60
B black balls, 40
1/2 the chance each bowl is chosen
(W-1)/(W+B-1) the white share in bowl B once one white ball is set aside
What it says in wordsHalf the time you get the certain bowl, half the time you get everything else.
The formula also shows how the answer moves. With 50 white and 50 black, the version a candidate reported from an interview, it gives 74.7%. More white in the pool lifts bowl B and the total; bowl A is already at its ceiling. Here the split adds about 20 percentage points because one ball is turned into half of the probability.
Where candidates lose it
Most people answer 60% because the pool is 60% white and they assume a split cannot change the pool. It cannot change the pool, but it changes the weights, because the bowl is chosen before the ball.
The second loss is stopping at the number. Give the one-line reason no other split does better, then generalise: one half plus one half of (W minus 1) over (W plus B minus 1).
What the interviewer asks next
What if the bowl is chosen with probability proportional to how many balls it holds?
With three bowls and the same 100 balls, what is the best split and the best chance?
You are paid Rs 100 for a white ball and nothing for black. What is the most you would pay to play once?
Asked at Deutsche Bank, Equity Capital Markets, Mumbai, 2024 (Wall Street Oasis): After the distribution, one bowl will be selected at random, and then one ball will be randomly drawn from that bowl
078A 3-year bond pays a 10% annual coupon and yields 10%, so it prices at par. Work out its Macaulay duration from a table of present values, and explain why the answer is less than 3 years.Duration and convexityCoreFixed income asset management
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Before building the table: roughly where does the duration land?
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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.
Year Cash flow Discount factor at 10% Present value Share of price Year x share
1 10 0.9091 9.09 9.09% 0.0909
2 10 0.8264 8.26 8.26% 0.1653
3 110 0.7513 82.64 82.64% 2.4793
Total 130 100.00 100.00% 2.7355
The present values add to the price of 100, and the year-weighted shares add to a duration of 2.7355 years.
Placed as weights on a time line, the present values 9.09, 8.26 and 82.64 balance at 2.74 years, a little inside the 3 year maturity, because the two early coupons pull the balance point towards today.
The relationship
$D_{Mac} = \sum_t t\,\frac{PV_t}{P} = \frac{1(9.09) + 2(8.26) + 3(82.64)}{100} = 2.74$
t the year a cash flow arrives
PV_t the present value of that cash flow at the 10% yield
P the bond price, the sum of the present values
What it says in wordsWeight each payment date by how much of today's price arrives on that date.
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.
Where candidates lose it
The common error is weighting the years by the raw cash flows, 10, 10 and 110, instead of their present values. On this bond it gives 2.77 years: close enough to look right and wrong enough to fail the first follow-up, because it ignores that later money is worth less.
The other loss is saying duration equals maturity for any bond. It does only for a zero. Say that once, with the reason, and you have answered the why before it is asked.
What the interviewer asks next
What is the modified duration, and what price change does it predict for a 50 basis point rise in yield?
Without calculating, is the duration of a 3-year zero-coupon bond higher or lower than this one?
What happens to this bond's duration if its yield rises to 15%, and why?
079An insurer owes Rs 200 crore in 7 years. It may hold only 3-year and 12-year zero-coupon bonds, and all rates are 8%. How much does it put in each so that the portfolio's duration matches the liability, and what risk is left?Duration and convexityHardFixed income asset managementRisk management
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What share of the money goes into the 3-year bond?
Show the worked solution
Put 5/9 of the money, Rs 64.8 crore, in the 3-year zero and 4/9, Rs 51.9 crore, in the 12-year zero. The liability is worth 200 / 1.08^7 = Rs 116.7 crore today, and a zero's duration is its maturity, so the weights solve 3w + 12(1 - w) = 7. The risk left is a curve that does not move in parallel, such as short rates falling while long rates rise.
Why match duration rather than just the money?
Imagine saving for a payment due in seven years when the only deposits on offer run three years or twelve. Buy only the three-year deposit and you must reinvest in year three at whatever rate then holds; buy only the twelve-year one and you must sell it in year seven at whatever price then holds. Mixing them lets the two risks cancel. Duration matching means the assets and the liability gain or lose the same amount when rates move a little in parallel: if rates rise, the assets fall in price but the 3-year proceeds are reinvested at more, and the liability's value falls too.
The 3-year zero, worth Rs 64.8 crore, sits 4 years short of year 7 and the 12-year zero, worth Rs 51.9 crore, sits 5 years long, so a 5 to 4 split balances exactly where the Rs 200 crore liability falls due.
The relationship
$3w + 12(1-w) = 7 \;\Rightarrow\; w = \frac{12-7}{12-3} = \frac{5}{9}$
w the share of present value in the 3-year zero
3, 12 the durations of the two zeros, equal to their maturities
7 the duration of the liability, a single payment in year 7
What it says in wordsChoose the mix whose weighted average maturity lands exactly on the liability date.
Holding Share Invested today, Rs crore Repays, Rs crore Repays in year
3-year zero 5/9 64.83 81.67 3
12-year zero 4/9 51.87 130.61 12
Liability 116.70 200.00 7
Rs 116.70 crore invested today in a 5 to 4 split grows to Rs 81.67 crore in year 3 and Rs 130.61 crore in year 12.
What risk does duration matching leave behind?
Test it. If every rate rises to 9%, the assets are worth Rs 109.50 crore and the liability Rs 109.41 crore; if every rate falls to 7%, Rs 124.66 crore against Rs 124.55 crore. Against parallel moves the match holds, and the barbell even edges ahead because it has more convexity than a single payment, but a twist in the curve hits it directly. If the 3-year rate falls to 7% and the 12-year rate rises to 9% while the 7-year rate stays at 8%, the assets are worth Rs 113.10 crore against a liability still at Rs 116.70 crore: a shortfall of Rs 3.60 crore.
Parallel moves of one point either way leave a surplus of only about Rs 10 lakh, but a twist that lowers the 3-year rate and raises the 12-year rate leaves the insurer Rs 3.60 crore short.
The same risk shows up as reinvestment: in year 3 the insurer receives Rs 81.7 crore that must earn enough for four more years, at whatever the 4-year rate is then. The only complete fix is a 7-year zero that matches the cash flow itself, which the question rules out.
Where candidates lose it
The frequent slip is putting the heavier weight on the 12-year bond because it is further out, or splitting by face value rather than present value. The weights are present values, and the nearer bond takes more because it is nearer to year 7.
The second loss is declaring the insurer fully hedged. Say parallel moves only, then name the twist and the reinvestment of the 3-year proceeds, and you have answered the question behind the question.
What the interviewer asks next
Rates stay at 8% for a year. Is the portfolio still matched, and why?
Why does the barbell gain slightly when rates move in parallel?
Which single bond would remove the curve risk entirely, and why might an insurer not be able to buy it?
080You receive fixed on a Rs 100 crore 5-year interest rate swap. Explain the position as a long 5-year fixed rate bond and a short floating rate note, and estimate its DV01 if the fixed bond leg has a modified duration of about 4.2.Yield curve and forward ratesHardSyndicate desksFixed income asset management
Try it first
Roughly what is the DV01 of receiving fixed on Rs 100 crore?
Show the worked solution
Receiving fixed is a long fixed rate bond funded by a short floating rate note, so its DV01 is about Rs 3.95 lakh per basis point. The bond leg moves Rs 100 crore x 4.2 x 0.0001, or Rs 4.2 lakh, per basis point. A floater resetting quarterly has a duration of about 0.25, which takes off Rs 0.25 lakh. The principals cancel, and the position gains when rates fall.
Why can a swap be written as two bonds?
Suppose you lend a friend Rs 1 lakh at a fixed 7% and, the same morning, borrow Rs 1 lakh from your bank at its floating rate. The two lakhs cross in the air and cancel; what remains is that you collect 7% and pay floating. A receive-fixed swap is exactly that: the cash flows of owning a fixed rate bond and having issued a floating rate note on the same notional, with the two principals cancelling, which is why a swap needs no money upfront.
The swap's cash flows equal a long fixed bond minus a short floater once the Rs 100 crore principals cancel, and its rate sensitivity is the bond's Rs 4.20 lakh per basis point less the floater's Rs 0.25 lakh, about Rs 3.95 lakh.
Why does the floating leg carry almost no rate risk?
A floating rate noteA bond whose coupon is reset to a market benchmark at regular dates, so its price stays close to par. resets its coupon to the market at each reset date, so on a reset date it is worth about par whatever rates have done. Its price can only drift between resets. With quarterly resets the floater behaves like a bond maturing in three months, with a duration of about 0.25, so it cancels only about 6% of the fixed leg's rate risk. The swap is, for rate purposes, nearly the whole fixed bond.
The relationship
$DV01 \approx N \times (D_{fix} - D_{flt}) \times 0.0001 = 10{,}000 \times (4.20 - 0.25) \times 0.0001 = 3.95 \text{ lakh}$
N notional in lakh: Rs 100 crore is 10,000 lakh
D_fix modified duration of the fixed leg, 4.2
D_flt duration of the floating leg, about 0.25 for quarterly resets
0.0001 one basis point as a decimal
What it says in wordsThe swap's sensitivity is the fixed bond's sensitivity minus the small sensitivity of the floater.
Which way does the position make money, and who uses it?
Receiving fixed gains when rates fall: a 10 basis point drop in the 5-year swap rate makes the position about Rs 39.5 lakh. That is why a desk calls receiving fixed a long duration position. It is also the everyday DCM use: an issuer that sells a 5-year fixed bond and wants floating rate debt receives fixed on a swap against its own coupon, and ends up paying floating. Treat the 4.2 and 0.25 as illustrations; the real figures depend on the coupon, the curve and the reset dates.
Where candidates lose it
The usual slip is a power of ten: Rs 42 lakh or Rs 42,000 instead of Rs 4.2 lakh. Convert before you multiply: Rs 100 crore is 10,000 lakh, and one basis point is 0.0001.
The second is saying a swap has no risk because it is worth zero when it is struck. Value and sensitivity are different things; a swap worth nothing today carries almost as much rate risk as a Rs 100 crore bond.
What the interviewer asks next
An issuer sells a 5-year fixed bond and swaps it to floating. Which side of the swap is it on?
Just before a reset date, what is the floating leg's duration?
How many government bond futures would you need to hedge this DV01, if one contract has a DV01 of Rs 50,000?

F	the face value you hold, Rs 100 crore
DV01_corp	the corporate bond's price change per basis point, per Rs 100
DV01_gov	the government bond's price change per basis point, per Rs 100

P*	the fair price implied by the zero curve
1.07	one plus the one-year zero rate
1.08^2	two years of growth at the two-year zero rate

d	the holding company discount, 15%
EV_i	the enterprise value of each business
ND	net debt, Rs 600 crore
N	shares outstanding, 50 crore

W	white balls, 60
B	black balls, 40
1/2	the chance each bowl is chosen
(W-1)/(W+B-1)	the white share in bowl B once one white ball is set aside

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What exactly are you trying to balance?

Why do you need less of the government bond than you hold?

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Why does a spread imply a default probability at all?

How does the recovery assumption change the answer?

What does the credit triangle leave out?

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Why does one ratio move and the other stay put?

Why would a company repay debt with cash at all?

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Why price each cash flow at its own rate?

What does rich mean, and how would a desk use it?

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What order do you walk the lines in?

Why tax EBIT rather than profit after interest?

Where does the cash go between EBITDA and free cash flow?

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Why does the discount come off before the debt?

Why does a 15% discount cost the shareholders more than 15%?

What should you say about where the discount is applied?

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Why does a bowl with one ball count as much as a bowl with 99?

How do you show that nothing beats it?

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What is duration actually averaging?

Why is it always below maturity for a coupon bond?

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Why match duration rather than just the money?

What risk does duration matching leave behind?

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What the interviewer asks next

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Why can a swap be written as two bonds?

Why does the floating leg carry almost no rate risk?

Which way does the position make money, and who uses it?

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What the interviewer asks next