Debt Capital Markets puzzles, solved step by step

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  1. 045A barbell of equal market values in 2-year and 10-year zeros and a bullet 6-year zero both yield 7% and have the same duration of 6. Rates move 100 basis points up, then separately 100 down. Which position does better in each case, and by how much?Duration and convexityHardFixed income asset managementSyndicate desks

    Try it first

    Rates jump 100 basis points in parallel. Which does better?

    Show the worked solution

    The barbell does better both ways, by about 0.065 per Rs 100 when rates rise and 0.075 when they fall. Up 100: the bullet loses 5.43, the barbell 5.36. Down 100: the bullet gains 5.80, the barbell 5.87. Duration is matched, so the difference is convexity, which the barbell has more of because its cash flows sit further apart.

    If the durations match, why do the results differ?

    Balance a see-saw with one child sitting in the middle, or with two children at the far ends. Both are balanced, but push the ends and the version with weight at the tips swings further. Duration measures where the weight of the cash flows sits on average; convexity measures how spread out it is, and the barbell's cash at years 2 and 10 is spread much wider than the bullet's single payment at year 6. For a zero, convexity grows roughly with maturity squared, so the 10-year leg adds far more than the 2-year leg takes away.

    The relationship
    Cbullet=6×71.072≈36.7Cbarbell=12⋅2×3+10×111.072≈50.7C_{\text{bullet}} = \frac{6 \times 7}{1.07^2} \approx 36.7 \qquad C_{\text{barbell}} = \tfrac{1}{2} \cdot \frac{2 \times 3 + 10 \times 11}{1.07^2} \approx 50.7
    T(T+1)/(1+y)^2convexity of a zero-coupon bond maturing in T years at yield y
    1/2each leg is half the barbell's market value
    What it says in wordsThe barbell averages a small and a very large convexity, which beats the single middle one.
    Same duration, but the barbell comes out ahead whichever way rates jumpValue change per Rs 100, 100 bp moves+5.80bullet+5.87barbellRates -100 bpbarbell ahead by 0.075-5.43bullet-5.36barbellRates +100 bpbarbell ahead by 0.0650.20.40.60.8-300-1000+100+300Parallel move, basis pointsBarbell minus bullet, per Rs 1000.770.52
    For a 100 basis point rise the barbell loses 5.36 per Rs 100 against 5.43 for the bullet, and for a 100 basis point fall it gains 5.87 against 5.80; the barbell's advantage grows with the size of the move, to about 0.52 at plus 300.

    How big is the advantage, and what does it cost in a real market?

    Small on a 100 basis point move: under a tenth of a rupee per Rs 100, about 6.5 paise on a rise and 7.5 on a fall. The advantage grows with roughly the square of the move, so at 300 basis points it is 0.52 to 0.77 per Rs 100, which is why convexity matters most in volatile markets. Price the positions by discounting each cash flow at the new flat yield, which is what the figures above do exactly rather than by the duration and convexity approximation.

    Now the honest limit. A flat curve where both positions yield 7% and convexity is free cannot last: buyers would all pick the barbell. In real markets the barbell usually yields a little less than the bullet, and that lower yield is the price of its convexity; whether it pays depends on how much rates actually move. The barbell also loses if the curve twists, for example if 2-year and 10-year yields rise while 6-year yields do not, a non-parallel move that duration and convexity for a single yield do not capture.

    Where candidates lose it

    The common wrong answer is that matched duration means matched results. It does to first order, for small moves, and the interviewer picked 100 basis points to push past that.

    The second trap is saying the barbell wins and stopping. Add the two caveats: in a real curve you pay for convexity through a lower yield, and a non-parallel move can hurt the barbell. Those two sentences separate a memorised answer from an understood one.

    What the interviewer asks next

    • What yield give-up on the barbell would make the two break even for a 100 basis point move either way?
    • The curve steepens: 2-year yields fall 50 and 10-year yields rise 50. Which position wins now?
    • Why do liability-matching investors often prefer bullets?
  2. 058A perpetual bond yields 8%. What are its Macaulay duration and its modified duration, and how can a bond that never matures have a duration of only about 13.5 years?Duration and convexityHardFixed income asset management

    Try it first

    What is the Macaulay duration of a perpetual yielding 8%?

    Show the worked solution

    The Macaulay duration is 13.5 years and the modified duration is 12.5. For a perpetuity, Macaulay duration is (1 + y) / y, 1.08 / 0.08 = 13.5, and modified duration divides by 1 + y, leaving 1 / y = 12.5. The bond never matures, but duration weights each coupon by its value today, and at 8% a coupon sixty years out is worth under 1% of its face. Half the bond's value arrives within the first nine years.

    Why does a bond that never ends have a finite duration?

    Picture a very long seesaw. A heavy child sits near the pivot and a row of ever lighter children stretches towards the far end. The plank balances not far from the heavy child, however long it is. Duration is the balance point of a bond's cash flows, each weighted by its present value, and distant coupons are discounted so hard that they barely pull on it. At 8%, the coupon in year 30 is worth about 10% of its face today and the coupon in year 60 about 1%.

    Each coupon is worth less today the later it comes, so the balance sits at 13.5 years0102030405060Years from todayBalance point: 13.5 years (Macaulay duration)Year 1: 7.41Year 30: 0.80Year 60: 0.08Years 1 to 9 (green) hold half the valueModified duration = 1 / 0.08 = 12.5
    The present value of each year's coupon falls from 7.41 in year one to 0.80 in year thirty, so the first nine years hold half the value and the value-weighted balance point, the Macaulay duration, sits at 13.5 years.

    How do you get 13.5 without summing forever?

    Use the shortcut and then check it against the price. For a level perpetuity, Macaulay duration is (1 + y) / y, so at 8% it is 1.08 / 0.08 = 13.5 years, and modified duration is 1 / y = 12.5. The price formula confirms it: a perpetual is worth coupon over yield, 8 / 0.08 = 100, and at 8.01% it is worth 8 / 0.0801, a fall of 0.125. That is 12.5 x 0.01, exactly what a modified duration of 12.5 predicts for one basis point.

    The relationship
    DMac=1+yy=1.080.08=13.5Dmod=DMac1+y=1y=12.5D_{Mac} = \frac{1+y}{y} = \frac{1.08}{0.08} = 13.5 \qquad D_{mod} = \frac{D_{Mac}}{1+y} = \frac{1}{y} = 12.5
    ythe bond's yield, 8%
    D_MacMacaulay duration, the value-weighted average time to the cash flows, in years
    D_modmodified duration, the percentage price change for a one point change in yield
    What it says in wordsA perpetual's duration depends only on its yield: one plus the yield, over the yield.

    Now say the implication, which is what the interviewer is after. Duration falls as the yield rises: the same perpetual at 4% has a Macaulay duration of 26 years, and at 12% only 9.3. Higher yields shorten every bond, because they shrink the weight of distant cash flows. The limit: many perpetual bonds carry an issuer call after five or ten years, and a callable perpetual trades on a much shorter effective duration than this formula gives.

    Where candidates lose it

    The instinctive answer is infinite, because the bond never matures. It confuses maturity, the date of the last cash flow, with duration, the value-weighted average date of all of them.

    The second slip is mixing up the two durations. 13.5 is the Macaulay figure, a time in years; 12.5 is the modified figure, the price change per point of yield. Give both and say which is which.

    What the interviewer asks next

    • What is the modified duration of the same perpetual if its yield falls to 5%?
    • A perpetual is callable at par in year 5 and trades above par. Which duration would you use to hedge it?
    • Why does a zero-coupon bond's duration equal its maturity while a perpetual's does not?
  3. 079An insurer owes Rs 200 crore in 7 years. It may hold only 3-year and 12-year zero-coupon bonds, and all rates are 8%. How much does it put in each so that the portfolio's duration matches the liability, and what risk is left?Duration and convexityHardFixed income asset managementRisk management

    Try it first

    What share of the money goes into the 3-year bond?

    Show the worked solution

    Put 5/9 of the money, Rs 64.8 crore, in the 3-year zero and 4/9, Rs 51.9 crore, in the 12-year zero. The liability is worth 200 / 1.08^7 = Rs 116.7 crore today, and a zero's duration is its maturity, so the weights solve 3w + 12(1 - w) = 7. The risk left is a curve that does not move in parallel, such as short rates falling while long rates rise.

    Why match duration rather than just the money?

    Imagine saving for a payment due in seven years when the only deposits on offer run three years or twelve. Buy only the three-year deposit and you must reinvest in year three at whatever rate then holds; buy only the twelve-year one and you must sell it in year seven at whatever price then holds. Mixing them lets the two risks cancel. Duration matching means the assets and the liability gain or lose the same amount when rates move a little in parallel: if rates rise, the assets fall in price but the 3-year proceeds are reinvested at more, and the liability's value falls too.

    Five parts at year 3 and four parts at year 12 balance at year 7today5/93-year zeroPV 64.8, repays 81.74/912-year zeroPV 51.9, repays 130.64 years short5 years longLiability: Rs 200 crore due in year 7, worth Rs 116.7 crore today
    The 3-year zero, worth Rs 64.8 crore, sits 4 years short of year 7 and the 12-year zero, worth Rs 51.9 crore, sits 5 years long, so a 5 to 4 split balances exactly where the Rs 200 crore liability falls due.
    The relationship
    3w+12(1−w)=7  ⇒  w=12−712−3=593w + 12(1-w) = 7 \;\Rightarrow\; w = \frac{12-7}{12-3} = \frac{5}{9}
    wthe share of present value in the 3-year zero
    3, 12the durations of the two zeros, equal to their maturities
    7the duration of the liability, a single payment in year 7
    What it says in wordsChoose the mix whose weighted average maturity lands exactly on the liability date.
    HoldingShareInvested today, Rs croreRepays, Rs croreRepays in year
    3-year zero5/964.8381.673
    12-year zero4/951.87130.6112
    Liability116.70200.007
    Rs 116.70 crore invested today in a 5 to 4 split grows to Rs 81.67 crore in year 3 and Rs 130.61 crore in year 12.

    What risk does duration matching leave behind?

    Test it. If every rate rises to 9%, the assets are worth Rs 109.50 crore and the liability Rs 109.41 crore; if every rate falls to 7%, Rs 124.66 crore against Rs 124.55 crore. Against parallel moves the match holds, and the barbell even edges ahead because it has more convexity than a single payment, but a twist in the curve hits it directly. If the 3-year rate falls to 7% and the 12-year rate rises to 9% while the 7-year rate stays at 8%, the assets are worth Rs 113.10 crore against a liability still at Rs 116.70 crore: a shortfall of Rs 3.60 crore.

    Parallel moves leave the match intact; a twist does notParallel rise to 9%assets 109.50, liability 109.41+0.09, barely above zeroParallel fall to 7%assets 124.66, liability 124.55+0.11, barely above zeroTwist: 3-year 7%, 12-year 9%assets 113.10, liability 116.70-3.60 shortfallzero surplus
    Parallel moves of one point either way leave a surplus of only about Rs 10 lakh, but a twist that lowers the 3-year rate and raises the 12-year rate leaves the insurer Rs 3.60 crore short.

    The same risk shows up as reinvestment: in year 3 the insurer receives Rs 81.7 crore that must earn enough for four more years, at whatever the 4-year rate is then. The only complete fix is a 7-year zero that matches the cash flow itself, which the question rules out.

    Where candidates lose it

    The frequent slip is putting the heavier weight on the 12-year bond because it is further out, or splitting by face value rather than present value. The weights are present values, and the nearer bond takes more because it is nearer to year 7.

    The second loss is declaring the insurer fully hedged. Say parallel moves only, then name the twist and the reinvestment of the 3-year proceeds, and you have answered the question behind the question.

    What the interviewer asks next

    • Rates stay at 8% for a year. Is the portfolio still matched, and why?
    • Why does the barbell gain slightly when rates move in parallel?
    • Which single bond would remove the curve risk entirely, and why might an insurer not be able to buy it?