Debt Capital Markets puzzles, solved step by step

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  1. 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.

    Area = rupees x duration: the long block is 40% of the money and 60% of the lossRs 300 cr x 4x 0.25% = Rs 3.0 cr60% of the moneyRs 200 cr x 9x 0.25% = Rs 4.5 cr40% of the moneyportfolio duration 6.0 across all Rs 500 crore469durationLoss for +25 bpsRs 7.5 crore1.5% of the book
    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
    ΔV≈−Dp×Δy×V=−(0.6×4+0.4×9)×0.0025×500=−6.0×0.0025×500=−7.5\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_pportfolio modified duration, the value-weighted average
    \Delta ythe parallel rise in yields, 0.25%
    Vportfolio 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

  2. 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
    Deff=P−−P+2 P0 Δy=102.40−99.102×101.20×0.005≈3.26D_{\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_0today's price, 101.20
    \Delta ythe 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.
    Yields fall and the callable bond barely rises; yields rise and it falls in full96100104108-150-100-500+50+100+150Change in yield, basis pointsthe call price caps the upsideSame bond, no call102.40: +1.20101.2099.10: -2.10Effective duration3.26, convexity -356
    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?