Debt Capital Markets puzzles, solved step by step

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

Showing 1–4 of 4 · filtered from 100Clear filters

015A Rs 500 crore bond portfolio has 60% in bonds with a modified duration of 4 and 40% in bonds with a modified duration of 9. The central bank surprises with a 25 basis point hike and the whole curve moves up in parallel. Roughly what is the mark-to-market loss?Duration and convexityCorePIMCOSan Diego · 2026
Try it first
What is the portfolio's modified duration?
Show the worked solution
About Rs 7.5 crore, 1.5% of the portfolio. Portfolio modified duration is the value-weighted average: 0.6 times 4 plus 0.4 times 9 is 6.0. A 25 basis point parallel rise costs about duration times the move, 6.0 times 0.25%, or 1.5%, and 1.5% of Rs 500 crore is Rs 7.5 crore. The duration 9 block is only 40% of the money but carries 60% of the loss.
What does modified duration convert?
Think of duration as a lever length. A seesaw with a long arm moves more at the end for the same push. Modified duration tells you the approximate percentage fall in a bond's price for a one percentage point rise in its yield, so a duration of 4 means about 4% per 1% move, or 1% for 25 basis points. Once you have that, a rate shock converts straight into rupees: duration times the move times the value held.
How do you combine two blocks of bonds?
Weight each duration by the money in it. Portfolio duration is the value-weighted average of the holdings' durations, because each bond's loss is its own value times its own duration times the move. Rs 300 crore at duration 4 loses Rs 3.0 crore; Rs 200 crore at duration 9 loses Rs 4.5 crore. Together that is Rs 7.5 crore, exactly what Rs 500 crore at duration 6.0 gives. The figure makes the point visually: area is the loss.
With width for rupees held and height for duration, the Rs 300 crore block at duration 4 loses Rs 3.0 crore and the Rs 200 crore block at duration 9 loses Rs 4.5 crore for a 25 basis point rise, together Rs 7.5 crore, the same area as Rs 500 crore at a portfolio duration of 6.0.
The relationship
$\Delta V \approx -D_{p} \times \Delta y \times V = -(0.6 \times 4 + 0.4 \times 9) \times 0.0025 \times 500 = -6.0 \times 0.0025 \times 500 = -7.5$
D_p portfolio modified duration, the value-weighted average
\Delta y the parallel rise in yields, 0.25%
V portfolio value, Rs 500 crore
What it says in wordsThe rupee change is roughly minus duration times the yield move times the money held.
What would make the true loss differ from Rs 7.5 crore?
Three things, and naming them is what separates a desk answer from a formula. Convexity makes the true loss slightly smaller than the duration estimate for a rise in yields, though for a 25 basis point move the difference is tiny. Curves rarely move in parallel after a surprise hike: short yields usually jump more, which would hurt the duration 4 block more than this sum assumes. And spreads on corporate bonds can move on top of the base rate. Say that Rs 7.5 crore is the first-order answer to a parallel shift, then name which of these you would check first.
Where candidates lose it
The common error is averaging 4 and 9 to get 6.5, giving Rs 8.12 crore. Durations combine by money weight, and the interviewer set 60 and 40 precisely to see whether you use them.
The second loss is getting the percentage right and the rupees wrong: 6 times 0.25 is 1.5%, and 1.5% of Rs 500 crore is Rs 7.5 crore, not Rs 75 crore. Say the percentage first, then convert.
What the interviewer asks next
How much of the duration 9 bonds would you sell into cash to cut the loss for the same shock to Rs 5 crore?
If the short end rises 40 basis points and the long end only 10, which block loses more?
How would you hedge this portfolio's duration with a bond future or an interest rate swap?
Asked at PIMCO, Debt Capital Markets, San Diego, 2026 (Wall Street Oasis): Which is cheaper us bonds or us equities How does duration affect interest rtes
026A callable bond is priced at 101.20. If yields fall 50 basis points it rises to 102.40; if they rise 50 basis points it falls to 99.10. What is its effective duration, and what does the asymmetry tell you about its convexity?Duration and convexityCoreAmundiLondon · 2018
Try it first
Before you divide anything: which way does this bond's convexity point?
Show the worked solution
Effective duration is about 3.26, and convexity is negative. Take the price gap between the two shocked prices, 102.40 less 99.10, which is 3.30, and divide by twice the price times the 0.005 yield move: 3.30 over 1.012. A fall in yields adds only 1.20 while a rise costs 2.10, so the call is capping the upside: effective convexity works out near -356.
Why do you shock the yield both ways instead of using a formula?
Think of a landlord who can ask the tenant to leave whenever a better tenant turns up. The tenant's lease is worth less when the market is hot, because that is exactly when it gets cut short. A callable bond works the same way: the issuer takes the bond back when rates fall. Because the cash flows change with the yield, no fixed schedule of payments exists to plug into the usual duration formula, so you move the yield up and down and watch what the price actually does. That is effective durationDuration measured by repricing the bond after a small yield shift each way, so that options that change the cash flows are captured.: the percentage price change per unit of yield, measured from the model rather than from a cash flow list.
The relationship
$D_{\text{eff}} = \frac{P_{-} - P_{+}}{2\,P_0\,\Delta y} = \frac{102.40 - 99.10}{2 \times 101.20 \times 0.005} \approx 3.26$
P_- price after yields fall 50 basis points, 102.40
P_+ price after yields rise 50 basis points, 99.10
P_0 today's price, 101.20
\Delta y the yield shock as a decimal, 0.005
What it says in wordsAverage the two price moves, express them as a share of today's price, and scale to a one-point yield change.
The callable bond rises only 1.20 when yields fall 50 basis points but loses 2.10 when they rise 50, because its price flattens as it approaches the call level, while the same bond without the call keeps climbing.
What does the lopsided move say about convexity?
For an ordinary bond the price curve bends upward, so a fall in yields adds more than an equal rise takes away. Here it is the reverse. A smaller gain than loss for the same shock means the price curve bends downward, which is negative convexity. The effective convexity formula makes it a number: 102.40 plus 99.10 less twice 101.20 is minus 0.90, divided by 101.20 times 0.005 squared, about -356. The sign is the point; the size depends on the shock you chose.
Say what it means for the holder. You are short an option to the issuer: you keep the losses when rates rise but hand back most of the gains when rates fall. The extra yield a callable bond pays over a straight bond is the premium for that option. In the illustrative model behind the picture, the same bond without the call has an effective duration of about 4.5, higher than 3.26, because the call shortens the bond's expected life when yields drop.
Where candidates lose it
The common slip is quoting a modified duration from the bond's final maturity, as if the call did not exist. That overstates the price gain from falling yields, which is exactly the move where the call bites.
The second loss is computing 3.26 and stopping. The interviewer gave you two different price moves on purpose: say the gain is smaller than the loss, name negative convexity, and say who owns the option.
What the interviewer asks next
Why would the effective duration shrink further if yields fell another 100 basis points?
Mortgage-backed securities show the same pattern. What plays the role of the issuer's call?
Would you use a 10 basis point or a 100 basis point shock to measure effective duration, and why?
Asked at Amundi, Rates, London, 2018 (Wall Street Oasis): What would your allocation be in today's market? What is effective duration?
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?

D_p	portfolio modified duration, the value-weighted average
\Delta y	the parallel rise in yields, 0.25%
V	portfolio value, Rs 500 crore

P_-	price after yields fall 50 basis points, 102.40
P_+	price after yields rise 50 basis points, 99.10
P_0	today's price, 101.20
\Delta y	the yield shock as a decimal, 0.005

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

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

Try it first

What does modified duration convert?

How do you combine two blocks of bonds?

What would make the true loss differ from Rs 7.5 crore?

Where candidates lose it

What the interviewer asks next

Try it first

Why do you shock the yield both ways instead of using a formula?

What does the lopsided move say about convexity?

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