Debt Capital Markets puzzles, solved step by step

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

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070A company trades at 8x EV/EBITDA on EBITDA of Rs 100 crore. Net debt is Rs 300 crore at 10% interest, depreciation is Rs 20 crore and the tax rate is 25%. What P/E is that?Cost of capital and valuation riddlesCoreCredit researchLeveraged finance
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Which is closest to the P/E?
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About 13.3x. Equity value is enterprise value less net debt: 8 x 100 = 800, less 300, is Rs 500 crore. Net income is EBITDA of 100, less depreciation of 20 and interest of 30, which leaves 50 before tax and 37.5 after 25% tax. Rs 500 crore over Rs 37.5 crore is 13.3x. The same business looks cheap at 8x EBITDA and dearer on earnings, because depreciation, interest and tax all sit between the two profit lines.
Why can the two multiples not be compared directly?
The price of a flat and the price of the owner's stake in it differ when there is a home loan: an Rs 80 lakh flat with a Rs 30 lakh loan leaves Rs 50 lakh of equity. EV/EBITDA prices the whole business against profit available to all funders; P/E prices only the equity against profit left after lenders and the tax authority are paid. To move from one to the other, bridge the value by taking off net debt, and bridge the profit by taking off depreciation, interest and tax.
Enterprise value of Rs 800 crore less Rs 300 crore of net debt leaves Rs 500 crore of equity, and EBITDA of Rs 100 crore less depreciation, interest and tax leaves Rs 37.5 crore of net income, so 8.0x EBITDA becomes 13.3x earnings.
The relationship
$P/E = \frac{8 \times 100 - 300}{(100 - 20 - 0.10 \times 300)(1 - 0.25)} = \frac{500}{37.5} = 13.3\times$
8 x 100 - 300 equity value: enterprise value less net debt
100 - 20 - 30 profit before tax: EBITDA less depreciation and interest
(1 - 0.25) what is left after 25% tax
What it says in wordsBridge the numerator from enterprise to equity value and the denominator from EBITDA to net income, then divide.
Where does the gap between 8x and 13.3x come from?
Bridge it in two moves. Depreciation and tax alone turn 8x EBITDA into 13.3x earnings: with no debt, EBIT of 80 taxed at 25% is 60, and 800 over 60 is 13.3x. So the gap here comes from the lines between EBITDA and net income, not from the debt. A business with heavy depreciation or a high tax rate always looks dearer on P/E than on EV/EBITDA.
Does the debt change the P/E at all?
With these numbers, no, and that is worth noticing. Leverage raises the P/E when the after-tax cost of debt is above the business's unlevered earnings yield, and lowers it when the cost is below. The unlevered earnings yield is 60 over 800, 7.5%; the after-tax cost of debt is 10% x 0.75, also 7.5%. Swapping equity for debt removes value and earnings in the same proportion, so the P/E stays at 13.3x. At 8% interest it would be 11.9x; at 12%, 15.2x.
Interest rate on Rs 300 crore Net income P/E on Rs 500 crore
8% 42.0 11.9x
10%, as given 37.5 13.3x
12% 33.0 15.2x
Rs crore. With the after-tax cost of debt equal to the 7.5% unlevered earnings yield, leverage leaves the P/E at 13.3x; cheaper debt lowers it and dearer debt raises it.
State the lesson for a credit desk. Never compare a P/E with an EV/EBITDA, or P/Es across companies with very different leverage, without bridging them first. The limit: this treats net debt as the only claim between enterprise and equity value; in real companies minority interests, leases and pension deficits also sit in that bridge.
Where candidates lose it
The fast wrong answer is 8x, treating the two multiples as interchangeable. They price different claims against different profit lines, so they match only by accident.
The second loss is forgetting the interest, which gives net income of 60 and a P/E of 8.3x, or dividing the whole enterprise value by net income, which gives 21.3x. Bridge the numerator and the denominator separately, and say each bridge out loud.
What the interviewer asks next
At what interest rate on the debt would the P/E be exactly 10x?
The company refinances at 12%. What happens to the P/E, and why?
Why do lenders size leverage off EBITDA rather than earnings?
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?
Show the worked solution
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?
Show the worked solution
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?
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
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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?
081A Rs 1,000 crore 5-year bond is priced at 100 and trades at 100.40 on the break. Its modified duration is 4.5. How many basis points did the issuer leave on the table, what is that in rupees, and was a 10 basis point new issue premium too generous?Issuance and refinancing arithmeticCoreSyndicate desks
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How many basis points of yield is a 0.40 point price rise on this bond?
Show the worked solution
About 8.9 basis points, or roughly Rs 4 crore, and yes, 10 basis points looks generous. A 0.40 point rise on a duration of 4.5 is 0.40 / 4.5 = 8.9 basis points of tightening. On Rs 1,000 crore, 0.40 points is Rs 4.0 crore of value that went to investors. Almost all of the premium came back on the first trade, so the book would probably have cleared tighter.
How do you turn a price jump into basis points?
A shop that sells out of a new phone in an hour, and then sees it resold online for more, has learned that it priced too low. A bond's first trades tell the issuer the same thing. Modified durationThe approximate percentage price change for a one point change in yield. is the exchange rate between price and yield: here a 1 basis point move in yield moves the price by about 4.5 hundredths of a point. So a price rise of 0.40 points means the market is valuing the bond about 8.9 basis points tighter than where it was priced.
A 0.40 point rise on a duration of 4.5 is 8.9 basis points of tightening and Rs 4.0 crore of value, which is nearly all of the 10 basis point premium the issuer paid, leaving only 1.1 basis points that the market still required.
The relationship
$\Delta y \approx \frac{\Delta P / P}{D_{mod}} = \frac{0.40\%}{4.5} = 0.089\% \approx 8.9\text{ bp}$
Delta P / P the percentage price change on the break, 0.40%
D_mod modified duration, 4.5
Delta y the implied change in yield
What it says in wordsDivide the percentage price move by duration to get the yield move that caused it.
What does the Rs 4 crore actually cost the issuer?
It is paid through the coupon, not written as a cheque. Pricing about 8.9 basis points too wide costs Rs 0.89 crore a year on Rs 1,000 crore, for five years: Rs 4.4 crore in total, or about Rs 3.6 crore in today's money at an assumed 7.5% yield, the same order as the Rs 4.0 crore jump in the bond's price. The first trade after pricing tells you how much of the premium was needed, because it shows where investors will buy the bond once the allocation is done.
So was 10 basis points too much?
On these numbers, most of it. But say two things before you convict the syndicate desk. First, strip out the market: if benchmark yields fell 3 basis points during the same session, only 5.9 basis points of the move belongs to the deal. Second, some positive performance is deliberate. Investors who were allocated want to see the bond trade well, and a flat or weak break makes the issuer's next deal harder to fill. A generous premium is a cost this time and an investment in the next book; the judgement is whether nearly the whole premium needed to be spent.
Where candidates lose it
The fast wrong answer treats 0.40 points as 40 basis points. Points are price, basis points are yield, and duration converts one into the other. Say the division out loud so the interviewer hears the method.
The second loss is a one-sided verdict. Calling the premium wasted without adjusting for the market move, or without mentioning why desks want some aftermarket performance, sounds like arithmetic without judgement.
What the interviewer asks next
The bond had instead traded at 99.80 on the break. What does that tell you, and who is unhappy?
How would you separate the deal's own performance from a market rally on the same day?
Why might an issuer accept paying a premium it does not strictly need?
082Firm A can borrow fixed at 7.0% or floating at benchmark plus 0.3%. Firm B can borrow fixed at 8.2% or floating at benchmark plus 0.8%. A wants floating rate debt and B wants fixed. How much can they save in total with an interest rate swap, and how could the saving be split?Issuance and refinancing arithmeticHardSyndicate desksCorporate banking
Try it first
Firm A is cheaper in both markets. Is there anything to gain from a swap?
Show the worked solution
The total saving is 0.7% a year, the difference between the fixed gap of 1.2% and the floating gap of 0.5%. A borrows fixed at 7.0%, B borrows floating at benchmark plus 0.8%, and they swap. Split evenly, B pays A 7.05% fixed and A pays B the benchmark: A ends at benchmark minus 0.05% and B at 7.85% fixed, each 0.35% better than going direct.
Why is there a gain when one firm is cheaper at everything?
Picture two flatmates: one cooks and cleans faster than the other, but is much faster at cooking and only slightly faster at cleaning. The household still gets more done if the fast cook cooks and the other one cleans. Borrowing works the same way. What matters is comparative advantage: firm B is 1.2% worse in the fixed market but only 0.5% worse in floating, so B should borrow floating and A fixed, and the gap between those two penalties, 0.7%, is the gain.
B pays 1.2% more than A in fixed but only 0.5% more in floating, so A borrows fixed, B borrows floating and they swap at 7.05%, leaving A at benchmark minus 0.05% and B at 7.85% fixed, each 0.35% better than borrowing directly.
The relationship
$\text{Gain} = (8.2 - 7.0) - (0.8 - 0.3) = 1.2 - 0.5 = 0.7\%$
8.2 - 7.0 B's extra cost in the fixed market
0.8 - 0.3 B's extra cost in the floating market
What it says in wordsThe saving is the difference between the two credit gaps, not either gap on its own.
How do you check the split adds up?
Follow each firm's cash. A pays its lenders 7.0% fixed, receives 7.05% fixed from B and pays B the benchmark, so its net cost is the benchmark minus 0.05%, against benchmark plus 0.3% going direct. B pays its lenders benchmark plus 0.8%, receives the benchmark from A and pays A 7.05%, so its net cost is 7.85% fixed, against 8.2% direct. 0.35% plus 0.35% is the full 0.7%, which is the check that nothing was lost or invented. On Rs 1,000 crore that is Rs 7 crore a year between them. In practice a bank sits in the middle and keeps a slice, so each firm gets somewhat less.
What is the honest limitation?
The gain is not free money. Part of the gap exists because floating lenders can reprice or refuse to roll a weaker borrower, while fixed lenders are locked in for years; B's floating spread of 0.8% may not hold if its credit slips. Each firm also takes on counterparty risk to the other or to the bank. Say that one sentence and the answer sounds like someone who has watched a swap book.
Where candidates lose it
The trap is saying A should simply borrow floating because it is cheaper in both markets. That reasons from absolute advantage and misses the 0.7% entirely.
The second loss is a split that does not reconcile. Candidates pick a swap rate out of the air and never check that the two firms' savings add back to 0.7%. Walk each firm's net cost out loud; the arithmetic is the proof.
What the interviewer asks next
A bank intermediates and keeps 0.1%. What swap rates give each firm an equal share of the rest?
Why might B's floating spread be closer to A's than its fixed spread is?
Which firm is exposed if the other defaults halfway through the swap, and to what?

8 x 100 - 300	equity value: enterprise value less net debt
100 - 20 - 30	profit before tax: EBITDA less depreciation and interest
(1 - 0.25)	what is left after 25% tax

Interest rate on Rs 300 crore	Net income	P/E on Rs 500 crore
8%	42.0	11.9x
10%, as given	37.5	13.3x
12%	33.0	15.2x

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

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%

Try it first

Why can the two multiples not be compared directly?

Where does the gap between 8x and 13.3x come from?

Does the debt change the P/E at all?

Where candidates lose it

What the interviewer asks next

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What exactly are you trying to balance?

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

Where candidates lose it

What the interviewer asks next

Try it first

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?

Where candidates lose it

What the interviewer asks next

Try it first

Why does one ratio move and the other stay put?

Why would a company repay debt with cash at all?

Where candidates lose it

What the interviewer asks next

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Why price each cash flow at its own rate?

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

Where candidates lose it

What the interviewer asks next

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What is duration actually averaging?

Why is it always below maturity for a coupon bond?

Where candidates lose it

What the interviewer asks next

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Why match duration rather than just the money?

What risk does duration matching leave behind?

Where candidates lose it

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?

Where candidates lose it

What the interviewer asks next

Try it first

How do you turn a price jump into basis points?

What does the Rs 4 crore actually cost the issuer?

So was 10 basis points too much?

Where candidates lose it

What the interviewer asks next

Try it first

Why is there a gain when one firm is cheaper at everything?

How do you check the split adds up?

What is the honest limitation?

Where candidates lose it

What the interviewer asks next