Debt Capital Markets puzzles, solved step by step

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

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045A barbell of equal market values in 2-year and 10-year zeros and a bullet 6-year zero both yield 7% and have the same duration of 6. Rates move 100 basis points up, then separately 100 down. Which position does better in each case, and by how much?Duration and convexityHardFixed income asset managementSyndicate desks
Try it first
Rates jump 100 basis points in parallel. Which does better?
Show the worked solution
The barbell does better both ways, by about 0.065 per Rs 100 when rates rise and 0.075 when they fall. Up 100: the bullet loses 5.43, the barbell 5.36. Down 100: the bullet gains 5.80, the barbell 5.87. Duration is matched, so the difference is convexity, which the barbell has more of because its cash flows sit further apart.
If the durations match, why do the results differ?
Balance a see-saw with one child sitting in the middle, or with two children at the far ends. Both are balanced, but push the ends and the version with weight at the tips swings further. Duration measures where the weight of the cash flows sits on average; convexity measures how spread out it is, and the barbell's cash at years 2 and 10 is spread much wider than the bullet's single payment at year 6. For a zero, convexity grows roughly with maturity squared, so the 10-year leg adds far more than the 2-year leg takes away.
The relationship
$C_{\text{bullet}} = \frac{6 \times 7}{1.07^2} \approx 36.7 \qquad C_{\text{barbell}} = \tfrac{1}{2} \cdot \frac{2 \times 3 + 10 \times 11}{1.07^2} \approx 50.7$
T(T+1)/(1+y)^2 convexity of a zero-coupon bond maturing in T years at yield y
1/2 each leg is half the barbell's market value
What it says in wordsThe barbell averages a small and a very large convexity, which beats the single middle one.
For a 100 basis point rise the barbell loses 5.36 per Rs 100 against 5.43 for the bullet, and for a 100 basis point fall it gains 5.87 against 5.80; the barbell's advantage grows with the size of the move, to about 0.52 at plus 300.
How big is the advantage, and what does it cost in a real market?
Small on a 100 basis point move: under a tenth of a rupee per Rs 100, about 6.5 paise on a rise and 7.5 on a fall. The advantage grows with roughly the square of the move, so at 300 basis points it is 0.52 to 0.77 per Rs 100, which is why convexity matters most in volatile markets. Price the positions by discounting each cash flow at the new flat yield, which is what the figures above do exactly rather than by the duration and convexity approximation.
Now the honest limit. A flat curve where both positions yield 7% and convexity is free cannot last: buyers would all pick the barbell. In real markets the barbell usually yields a little less than the bullet, and that lower yield is the price of its convexity; whether it pays depends on how much rates actually move. The barbell also loses if the curve twists, for example if 2-year and 10-year yields rise while 6-year yields do not, a non-parallel move that duration and convexity for a single yield do not capture.
Where candidates lose it
The common wrong answer is that matched duration means matched results. It does to first order, for small moves, and the interviewer picked 100 basis points to push past that.
The second trap is saying the barbell wins and stopping. Add the two caveats: in a real curve you pay for convexity through a lower yield, and a non-parallel move can hurt the barbell. Those two sentences separate a memorised answer from an understood one.
What the interviewer asks next
What yield give-up on the barbell would make the two break even for a 100 basis point move either way?
The curve steepens: 2-year yields fall 50 and 10-year yields rise 50. Which position wins now?
Why do liability-matching investors often prefer bullets?
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?

063Which carries more interest rate risk: a 10-year floating rate note that resets every quarter, bought just after a reset, or a 2-year bond with a fixed 7% annual coupon yielding 7%? Estimate each one's duration.Duration and convexityWarm upFixed income asset management

Try it first

Which has the longer interest rate duration?

Show the worked solution

The 2-year fixed bond carries far more rate risk: a duration of about 1.9 years against about 0.25 for the floater. The floater's coupon resets to the market every quarter, so a rate move can only hurt it until the next reset, three months away. The fixed bond is locked at 7% for two years, and its Macaulay duration is 1.93 years. A 1 point rise in rates costs the floater about 0.25 points and the fixed bond about 1.78.

Why is a 10-year floater so short on rate risk?

Think of two landlords. One has let a flat for two years at a fixed rent; the other has a ten-year lease whose rent is reset to the market every three months. If market rents jump, the first is stuck for two years and the second catches up within a quarter. A floater's coupon resets to the market rate on every reset date, so its value can only drift from par for the few months until the next reset, whatever its final maturity. Just after a reset, that is about a quarter of a year.

The 10-year floater's coupon is locked for only three months, a rate duration of about 0.25 years, while the 2-year fixed bond's coupon is locked for both years, a duration of 1.93 years, so the shorter bond carries about 8 times the rate risk.

The relationship

D_{fix} = \frac{1 \times \tfrac{7}{1.07} + 2 \times \tfrac{107}{1.07^{2}}}{100} = 1.93 \qquad D_{FRN} \approx 0.25

D_fix	Macaulay duration of the fixed bond, its value-weighted average time to the cash flows
7, 107	the fixed bond's two annual cash flows per Rs 100
D_FRN	the floater's rate duration, about the time to its next reset

What it says in wordsA fixed bond's duration runs to its cash flows; a floater's runs only to its next reset.

What does each lose if rates rise one point?

Reprice both at a rate 1 point higher: the fixed bond falls from 100 to 98.22, a loss of 1.78, while the floater loses only 0.25, because it earns the old coupon for one quarter and then resets. The fixed bond carries about 8 times the rate risk of a note five times its length. This is why a bank funding itself with three-month deposits is comfortable holding floaters: the asset and the liability reprice together.

Keep the other risk in view. The floater's short rate duration says nothing about credit: its spread duration runs to maturity, about 7 years here, so a widening in the issuer's spread hits it far harder than it hits the 2-year bond. Name both numbers if the interviewer pushes, because the claim that floaters are low risk is only half true. The limit: this assumes a flat 7% curve and an issuer whose credit does not change.

Where candidates lose it

The trap is equating maturity with rate risk. A 10-year floater sounds riskier than a 2-year bond, but its coupon catches up with the market every quarter, so a rate move can hurt it for a few months at most.

The second loss is stopping there and calling the floater safe. It has almost no rate duration but a long spread duration; say both, or the follow-up on credit will catch you.

What the interviewer asks next

Halfway between two resets, what is the floater's rate duration?
A bank funds itself with 3-month deposits. Which of the two bonds matches its funding better, and why?
Roughly what is the spread duration of the 10-year floater, and what does a 50 basis point widening cost it?

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
Try it first
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?
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
Try it first
Before building the table: roughly where does the duration land?
Show the worked solution
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
Try it first
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?

T(T+1)/(1+y)^2	convexity of a zero-coupon bond maturing in T years at yield y
1/2	each leg is half the barbell's market value

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

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

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

Try it first

If the durations match, why do the results differ?

How big is the advantage, and what does it cost in a real market?

Where candidates lose it

What the interviewer asks next

Try it first

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

How do you get 13.5 without summing forever?

Where candidates lose it

What the interviewer asks next

Try it first

Why is a 10-year floater so short on rate risk?

What does each lose if rates rise one point?

Where candidates lose it

What the interviewer asks next

Try it first

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

What is duration actually averaging?

Why is it always below maturity for a coupon bond?

Where candidates lose it

What the interviewer asks next

Try it first

Why match duration rather than just the money?

What risk does duration matching leave behind?

Where candidates lose it

What the interviewer asks next