Portfolio Management puzzles, solved step by step

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  1. 047A Rs 100 crore bond portfolio holds Rs 40 crore of a bond with duration 2, Rs 35 crore with duration 5 and Rs 25 crore with duration 12. What is the portfolio's duration, and how much must move from the 2-year bond into the 12-year bond to lift it to 7?Bond mathsHardPIMCOLos Angeles · 2026

    Try it first

    How much must move from the duration-2 bond to the duration-12 bond to lift portfolio duration from 5.55 to 7?

    Show the worked solution

    Duration is 5.55, and Rs 14.5 crore must move from the 2-year bond to the 12-year bond. Portfolio duration is the value-weighted average: (40 x 2 + 35 x 5 + 25 x 12) over 100, which is 555 over 100, or 5.55. Each Rs 1 crore switched gains 10 years of duration on a hundredth of the portfolio, adding 0.1. Closing a gap of 1.45 needs Rs 14.5 crore.

    Why is portfolio duration a simple weighted average?

    Think of the average age of people in a room: each person counts in proportion to how many of them there are. Duration measures how much a bond's price moves for a one-point change in yield, and for a small parallel move the portfolio's rupee loss is just the sum of each bond's rupee loss, so its duration is the value-weighted average of the bonds' durations. The Rs 25 crore in the 12-year bond is only a quarter of the money but supplies more than half the duration: 3.00 of the 5.55.

    Duration is a value-weighted average, so one switch moves it by a set amount0.80Rs 40.0 cr x 21.75Rs 35.0 cr x 53.00Rs 25.0 cr x 125.55Before0.51Rs 25.5 cr x 21.75Rs 35.0 cr x 54.74Rs 39.5 cr x 127.00After the switch2-yr bond5-yr bond12-yr bondEach rupee movedfrom 2-yr to 12-yradds 10 years xits weight1.45 / 0.10= Rs 14.5 crof Rs 100 cr
    Before the switch, the three bonds contribute 0.80, 1.75 and 3.00 years for a portfolio duration of 5.55. Moving Rs 14.5 crore from the 2-year to the 12-year bond changes the contributions to 0.51, 1.75 and 4.74, which total exactly 7.00.

    What changes for the portfolio when duration goes from 5.55 to 7?

    Solve for the switch with one line: the shift x changes duration by x times (12 minus 2) over 100, and that must equal 1.45. After the switch, a one-point parallel rise in yields costs about 7% of the portfolio, Rs 7 crore, instead of about 5.55%, Rs 5.55 crore. The portfolio gains more if yields fall and loses more if they rise. Its cash-flow profile also becomes a barbell, heavier at the long end, which gives it more convexity than a single bond with the same duration but more exposure to the long end of the curve if the curve steepens.

    The relationship
    Dp=∑iwiDi=40(2)+35(5)+25(12)100=5.55x=(7−5.55)×10012−2=14.5D_p = \sum_i w_i D_i = \frac{40(2) + 35(5) + 25(12)}{100} = 5.55 \qquad x = \frac{(7 - 5.55) \times 100}{12 - 2} = 14.5
    w_ieach bond's share of portfolio value
    D_ieach bond's duration, in years
    xthe Rs crore switched from the 2-year to the 12-year bond
    What it says in wordsDuration averages by value, so a switch moves it by the amount moved times the duration gap, over the portfolio's size.

    Say the limits: a weighted average of durations describes small, parallel shifts in yields. If short and long yields move by different amounts, a portfolio at duration 7 built from a barbell behaves differently from one built from 7-year bonds, and the switch changes the portfolio's yield and credit mix as well. For large moves, convexity adds a second-order correction.

    Where candidates lose it

    The most common slip is taking a simple average of the three durations, 19 over 3, about 6.3, ignoring the amounts held. The other is solving for the switch but forgetting to divide by the portfolio size, which gives Rs 1.45 crore or some other scale error.

    Say weighted by value first, give 5.55, then set up the switch as one equation. Close with what the higher duration means in rupees for a one-point move in yields; that sentence is the part the question actually asks about.

    What the interviewer asks next

    • How would you reach duration 7 without selling any of the 2-year bond?
    • Why might the barbell at duration 7 behave differently from a bullet 7-year bond if the curve steepens?
    • What does a one-point parallel fall in yields do to the portfolio after the switch?

    Asked at PIMCO, Generalist, Los Angeles, 2026 (Wall Street Oasis): Given a portfolio of these 3 bonds (I forgot exactly what they were) explain how the portfolio changes if duration increases.

  2. 075A callable bond is priced at 100.0. If yields rise 50 basis points its price falls to 98.9; if yields fall 50 basis points it rises only to 100.6. What are its effective duration and effective convexity?Bond mathsHardAmundiLondon · 2018

    Try it first

    What sign does the convexity take?

    Show the worked solution

    Effective duration is about 1.7 and effective convexity about -200. Duration is the price difference across the two shocks over twice the price times the shock: 1.7 over 1.0, which is 1.7. Convexity is the sum of the shocked prices less twice the base, over the price times the shock squared: minus 0.5 over 0.0025, which is -200. The call caps the price, so it gains less than it loses.

    Why effective duration rather than the usual formula?

    Think of renting out a flat on a lease the tenant can cancel whenever cheaper flats appear. When rents fall, the tenant leaves and you do not keep the high rent; when rents rise, you are stuck. A callable bond's cash flows change with yields, because the issuer calls it when rates fall, so you measure its duration from how its price actually moves, not from a fixed schedule of coupons. That is effective duration: shock the yield both ways, reprice, and read the slope.

    The relationship
    Deff=P−−P+2P0Δy=100.6−98.92×100×0.005=1.7Ceff=P−+P+−2P0P0Δy2=−0.50.0025=−200D_{eff} = \frac{P_- - P_+}{2P_0\Delta y} = \frac{100.6 - 98.9}{2 \times 100 \times 0.005} = 1.7 \qquad C_{eff} = \frac{P_- + P_+ - 2P_0}{P_0 \Delta y^2} = \frac{-0.5}{0.0025} = -200
    P_-price when yields fall 50 bp, 100.6
    P_+price when yields rise 50 bp, 98.9
    P_0the starting price, 100.0
    Delta ythe shock, 0.005
    What it says in wordsDuration is the average slope across the two shocks; convexity is how much the two moved prices bend away from a straight line.
    Price against yield change: the call caps the upside, so the curve bends the wrong way9698100102104call price 101straight bondcallable bond100.6100.098.9102.0-500+50+100Change in yield, basis points
    The callable bond rises only to 100.6 when yields fall 50 basis points, because the call caps it below 101, but falls to 98.9 when yields rise, so its effective duration is 1.7 and its convexity is negative, about -200.

    What does negative convexity cost the holder?

    It means the bond loses more when yields rise than it gains when they fall. The holder has sold the issuer an option to refinance, and the price of that option is the upside given up when rates fall. A straight bond with a duration of about 4 would gain about 2.0 points for a 50 basis point fall; this one gains 0.6. The holder is paid for this through a higher yield than an equivalent straight bond, and the question for a portfolio manager is whether that extra yield covers the option given away.

    Two more things are worth saying. The duration of 1.7 is not fixed: as yields fall towards the level where a call becomes likely, duration shrinks further, and as they rise, it lengthens towards the bond's straight duration. That shifting is why callable bonds, and mortgage securities with the same feature, need effective measures rather than the textbook formulas. The curve in the figure is a stylised fit around the three prices given; a real one would come from an option pricing model.

    Where candidates lose it

    The trap is computing a positive convexity by habit, or dividing by the shock rather than the shock squared and getting minus 1. Write the formula, put the sign of 100.6 plus 98.9 minus 200 on the page, and the negative number is obvious.

    The second loss is giving the numbers without the story. Say that the call caps the upside, which is why duration is short and convexity negative.

    What the interviewer asks next

    • Estimate the price change for a 100 basis point fall using this duration and convexity. Why might it be wrong?
    • Why does a callable bond's effective duration lengthen when yields rise?
    • What would the price-yield curve of a putable bond look like?

    Asked at Amundi, Rates, London, 2018 (Wall Street Oasis): What would your allocation be in today's market? What is effective duration?