Portfolio Management puzzles, solved step by step

Puzzles
100
Traced to a firm
31
Topics
13
Hard
30
Topic
All topicsStatistics and forecasting9Portfolio risk maths10Logic brainteasers7Behavioural and decision traps7Probability and expected value8Bond maths10Valuation riddles8Performance measurement8Private and real asset maths8Funds, ETFs and implementation7Compounding and fee drag7Market sizing and estimation6Currency and global returns5
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–3 of 3 · filtered from 100Clear filters
  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?

  3. 100You can hold a 7-year bullet yielding 7.0%, or a barbell of 2-year bonds at 6.5% and 12-year bonds at 7.2% with the same duration. Which does better if yields shift 100 basis points in parallel, either way? What does the barbell give up?Bond mathsHardFixed incomeInstitutional asset management

    Try it first

    On an immediate 100 basis point parallel move, up or down, which wins?

    Show the worked solution

    The barbell wins both ways on an immediate parallel move, by about 0.12 points if yields fall and 0.10 if they rise, but it yields about 0.15 points less a year. With duration matched, convexity is the only difference: 71 for the barbell against 49 for the bullet. Over a year the barbell needs a move of roughly 113 to 123 basis points to earn back its lower yield, and it loses if the curve steepens.

    Why does spreading money to the two ends add convexity?

    Balancing a plank on a see-saw with one heavy child in the middle is steady; balancing it with a child at each end makes it swing more for the same push. Price sensitivity grows faster than linearly with maturity, so a mix of short and long bonds with the same average duration as a middle bond has more curvature than the middle bond: here about 71 against 49. More convexity means gaining a little more when yields fall and losing a little less when they rise.

    The barbell's convexity edge is real, small, and paid for in yield-300 bp+300 bpprice change, %Bullet (solid) and barbell (dashed):almost the same line-300-100+100+3000.51.0yield given up:0.15 a year+0.12+0.10Barbell minus bullet,points of priceparallel shift, basis points
    The bullet and barbell price changes almost coincide across parallel shifts, but the barbell's advantage grows with the square of the move, from about 0.10 to 0.12 points at 100 basis points to over 0.7 at 300; it only clears the 0.15 points of yield it gives up each year beyond moves of roughly 113 to 123 basis points.

    What does the barbell give up, and when does it lose?

    First, yield. Weighting the 2-year at about 50% to match duration gives a blended yield near 6.85%, about 0.15 points below the bullet's 7.0%. That is the price of convexity: if yields sit still, the bullet simply earns more, and over a year the barbell's edge only pays for itself on a parallel move of more than about 1.2 points. Second, curve shape. If the curve steepens, with short yields falling and long yields rising, the barbell's long leg loses and the bullet wins; if the 7-year sector rallies against the two ends, the bullet wins too.

    BulletBarbell
    Yield, blended7.00%6.85%
    Modified duration6.546.54
    Convexity48.970.6
    Yields fall 100 bp+6.79%+6.91%
    Yields rise 100 bp-6.30%-6.20%
    Priced as zero-coupon bonds with modified duration matched at 6.54, the barbell gains 6.91% against 6.79% on a 100 basis point fall and loses 6.20% against 6.30% on a rise, while yielding 0.15 points less.

    State the assumptions: zero-coupon bonds, a blended yield taken as a weighted average, which is an approximation, and an instantaneous shift. The conclusion survives all three: the barbell buys protection against large parallel moves with yield, and takes on curve-shape risk.

    Where candidates lose it

    The common slip is saying same duration, same result. Duration matching removes only the first-order difference, and the question is about what is left over.

    The second loss is praising the barbell without the cost. It gives up yield and carries curve risk, and a candidate who names both sounds like someone who has run a bond book.

    What the interviewer asks next

    • How would you build the barbell to match both duration and yield? What would you give up?
    • Which does better if the curve flattens with no change in the 7-year yield?
    • Why do liability-driven investors often prefer bullets matched to their payment dates?