Portfolio Management puzzles, solved step by step

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  1. 006What are the Macaulay duration and the modified duration of a three-year bond paying a 6% annual coupon and priced at par?Bond mathsCorePIMCOLos Angeles · 2024

    Try it first

    Before you calculate: where does the Macaulay duration sit?

    Show the worked solution

    Macaulay duration about 2.83 years and modified duration about 2.67. At par the yield equals the 6% coupon, so the cash flows discount to 5.66, 5.34 and 89.00, which add to 100. Weight each year by its share of the price: 1 x 0.0566 + 2 x 0.0534 + 3 x 0.8900 gives 2.83. Divide by 1.06 for modified duration: a one point rise in yield cuts the price by roughly 2.67%.

    What is duration, if not the time to maturity?

    Picture a seesaw with three children sitting at the one, two and three metre marks. If the child at three metres is much heavier, the pivot that balances the seesaw sits close to three, not at the middle. Macaulay duration is that pivot: the average time you wait for your money, with each payment weighted by its present value. Here the three weights are the discounted coupons of 5.66 and 5.34 and the discounted final payment of 89.00. The last payment is so heavy that the balance point, 2.83 years, sits only two months short of maturity.

    Duration is where the discounted cash flows balance on a timelineyear 0year 1year 2year 35.665.3489.00balance point: 2.83 yearsEach weight is a cash flow's present valueYear 1: 6 / 1.06 = 5.66Year 2: 6 / 1.06 squared = 5.34Year 3: 106 / 1.06 cubed = 89.00Price = 5.66 + 5.34 + 89.00 = 100.00Modified = 2.83 / 1.06 = 2.67
    The bond's discounted cash flows of 5.66, 5.34 and 89.00 sit at years 1, 2 and 3 and balance at 2.83 years, which is the Macaulay duration; dividing by 1.06 gives a modified duration of 2.67.
    YearCash flowPresent value at 6%Share of priceYear x share
    165.660.05660.0566
    265.340.05340.1068
    310689.000.89002.6700
    Total118100.001.00002.8334
    Weighting each payment date by its share of the price gives a Macaulay duration of 2.8334 years.

    Why divide by 1.06 to get modified duration?

    Macaulay duration is a time. Modified duration is a price sensitivity: the percentage change in price for a one point change in yield. With annual compounding the two differ by a factor of one plus the yield, so 2.833 divided by 1.06 is 2.673. Say what it means: if yields rise from 6% to 7%, the bond loses about 2.67% of its price, a little less in reality because the price-yield curve bends. That bend is convexity, and it is the natural next question.

    The relationship
    Dmac=∑tt PVtP=1(5.66)+2(5.34)+3(89.00)100≈2.83Dmod=Dmac1+y≈2.67D_{mac}=\sum_t t\,\frac{PV_t}{P}=\frac{1(5.66)+2(5.34)+3(89.00)}{100}\approx 2.83 \qquad D_{mod}=\frac{D_{mac}}{1+y}\approx 2.67
    PV_tthe present value of the payment at year t
    Pthe bond's price, 100 at par
    ythe yield, 6%
    What it says in wordsMacaulay duration is the value-weighted average payment date; modified duration divides it by one plus the yield to turn it into a price sensitivity.

    Where candidates lose it

    The fast wrong answer is three years, which is true only of a zero coupon bond. The second is averaging the dates without weighting them, which gives two. The interviewer wants the words present value weighted before any number.

    Candidates also mix up the two durations. Say which is a time and which is a sensitivity, and use modified duration for any question about how much the price moves.

    What the interviewer asks next

    • What happens to the duration if the coupon rises to 10% and the bond still trades at par?
    • Estimate the price if yields jump to 7%, then say whether the true price is higher or lower.
    • What is the duration of a three-year zero coupon bond?

    Asked at PIMCO, Product & Strategy, Los Angeles, 2024 (Wall Street Oasis): Lots of random bond math questions -- duration of this bond with x coupon sold at par

  2. 064A corporate bond yields 180 basis points more than a government bond of the same maturity and has a spread duration of 5. How far can its spread widen over the next year before it earns no more than the government bond?Bond mathsCoreAQR Capital ManagementGreenwich · 2021

    Try it first

    How much widening does the 180 basis point spread absorb?

    Show the worked solution

    About 36 basis points. Over a year the bond earns 180 basis points more than the government bond. Every basis point the spread widens knocks about 5 basis points off its price, because its spread duration is 5. The extra yield is used up when widening times 5 equals 180, which is at 36 basis points. Beyond that the corporate bond does worse than the government bond.

    What is the cushion, and what eats it?

    Think of a shop that earns a steady margin on every sale but whose stock loses value when fashions change. The margin comes in slowly; a markdown hits all at once. A credit spread pays carry slowly over the year, while widening hits the price immediately, in proportion to spread duration. The carry is 180 basis points. A spread duration of 5 means a 1 basis point widening costs about 5 basis points of price.

    Excess return over government bonds for a year, against spread widening+2%+1%0%-1%-2%no widening: carry of +1.80%break-even: 180 / 5 = 36 bp60 bp wider: -1.20%each 1 bp of wideningcosts 5 bp of price020366080Spread widening over the year, basis points
    The bond earns 1.80% more than the government bond if spreads do not move, loses 0.05% for every basis point of widening, and so falls behind once spreads widen more than 36 basis points.
    The relationship
    Δs∗=spreadspread duration=1805=36 bp\Delta s^* = \frac{\text{spread}}{\text{spread duration}} = \frac{180}{5} = 36 \text{ bp}
    Delta s*the widening at which the extra return is zero
    spreadthe extra yield over the government bond, 180 basis points
    spread durationthe percentage price change for a 1 point change in spread, 5
    What it says in wordsDivide the extra yield by the spread duration to find how much widening it can absorb.

    What would you add to sound like a credit investor?

    Two refinements, both worth a sentence. First, part of the spread pays for defaults, not risk. If expected default losses were 50 basis points a year, an illustration, only 130 is true cushion and the break-even falls to 26 basis points. The spread is not all profit, so the honest break-even uses the spread after expected losses. Second, the price loss is felt at the end of the year, when the bond is shorter; at a spread duration of about 4.2 then, the break-even is closer to 43. The 36 is the conservative, quick answer.

    The ratio also compares bonds quickly. A short bond with a small spread can have a wider break-even than a long bond with a big one, because the long bond's duration magnifies every move. Credit portfolio managers often rank bonds by spread per unit of spread duration for exactly this reason.

    Where candidates lose it

    The trap is saying 180 basis points, as if the spread could widen by its own size before the bond loses out. Candidates forget that duration multiplies every basis point of widening into a larger price loss.

    The second loss is treating the whole spread as profit. Mention expected default losses; it shows you know why the spread exists in the first place.

    What the interviewer asks next

    • A 2-year bond at 90 bp and a 10-year at 220 bp with spread duration 8: which has the wider break-even?
    • How does roll-down along the credit curve change this answer?
    • Why might a portfolio manager hold the bond even if she expects 50 bp of widening?

    Asked at AQR Capital Management, Investment Research, Greenwich, 2021 (Wall Street Oasis): Discussion on credit spreads on fixed income products and duration.