Mutual Fund Mastery puzzles, solved step by step

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  1. 014A bond has a modified duration of 7 and a convexity of 80. Yields rise by 150 basis points. Estimate the percentage price change with duration alone, then with convexity added.Bond maths and durationHardFixed income desksIndian AMCs

    Try it first

    With convexity added, what is the estimated price change?

    Show the worked solution

    About -10.5% with duration alone and about -9.6% with convexity. Duration gives -7 times 0.015, which is -10.5%. Convexity adds one half times 80 times 0.015 squared, or +0.90%. The price-yield curve bends upward, so a straight-line duration estimate overstates losses when yields rise and understates gains when they fall, and the gap grows with the square of the move.

    Why is duration alone not enough for a big move?

    Think of estimating how far a ball thrown upward will fall back by looking only at its speed at the start. For a moment the straight line is fine; over a longer stretch the curve of its path takes over. Duration is the slope of the price-yield curve at today's yield, so it is exact for tiny moves and drifts further from the true price the larger the move. The curve bends upward for an ordinary bond, so it always sits above the tangent.

    Duration is the tangent; the bond sits on the curve above it808590951001050+100+150+200+300Change in yield, basis pointsPrice90.4 with convexity89.5 duration aloneYields up 150 bpDuration term-10.5%Convexity term+0.9%Estimate-9.6%the gap grows with thesquare of the move
    For a 150 basis point rise, the duration tangent puts the price at 89.5 while the curve with convexity puts it at 90.4, so duration alone overstates the loss by 0.9 points, and the gap widens as the move grows.

    How do you compute both estimates quickly?

    Write the move as a decimal first, 0.015. The duration term is minus duration times the move, -7 times 0.015, which is -10.5%; the convexity term is half the convexity times the move squared, 40 times 0.000225, which is +0.90%. Add them: -9.60%. For a 150 basis point fall the same two terms give +10.5% plus 0.90%, a gain of 11.40%, so convexity helps in both directions.

    The relationship
    ΔPP≈−Dmod Δy+12 C (Δy)2=−7(0.015)+12(80)(0.015)2=−10.5%+0.9%=−9.6%\frac{\Delta P}{P} \approx -D_{mod}\,\Delta y + \tfrac{1}{2}\,C\,(\Delta y)^2 = -7(0.015) + \tfrac{1}{2}(80)(0.015)^2 = -10.5\% + 0.9\% = -9.6\%
    D_{mod}modified duration, 7
    Cconvexity, 80
    \Delta ythe change in yield as a decimal, 0.015
    What it says in wordsThe price change is the straight-line duration estimate plus a correction for the curve that grows with the square of the move.

    The squared term is why size matters. At 50 basis points the convexity term is only +0.10%, easy to ignore; at 300 basis points it is +3.6%, too large to leave out. The limitation is that this is still a two-term approximation of a curve; for very large moves, or for bonds with call features whose convexity can turn negative, a desk reprices the bond in full rather than trusting the formula.

    Where candidates lose it

    The common slip is subtracting the convexity term because yields went up, giving -11.4%. For an ordinary bond the convexity term is positive in both directions; the squared move has no sign.

    The other slip is a units error: using 1.5 instead of 0.015, which makes the convexity term 90 and the answer absurd, or forgetting the one half. Convert basis points to a decimal before anything else and say the half out loud.

    What the interviewer asks next

    • Why is convexity valuable to a bondholder, and what does the market charge for it?
    • Why can a callable bond show negative convexity when yields fall?
    • Which has more convexity at the same duration, a bullet bond or a barbell of short and long bonds?
  2. 054At an 8% yield, which has the highest duration: a 10-year zero-coupon bond, a 10-year 9% coupon bond, or a 15-year 12% coupon bond?Bond maths and durationHardFixed income desksIndian AMCs

    Try it first

    Gut call first: which bond carries the most interest rate risk?

    Show the worked solution

    The 10-year zero, with a duration of 10.0 years. A zero pays everything at maturity, so its duration equals its maturity. The 15-year 12% bond has a duration of about 8.6 years, because its large coupons bring value forward, and the 10-year 9% bond is about 7.1. The zero moves most when rates move, even though it is not the longest bond.

    Why is maturity the wrong ruler?

    Think of two loans to a friend. One friend repays everything in one go after ten years. The other repays a large slice every year for fifteen years. Most of the second loan is back in your hands long before the first friend pays a rupee. Duration measures when, on average, a bond's value comes back to you, weighted by the present value of each payment, and rate risk follows that average, not the final date. A zero returns all of its value at one date, so its duration is exactly its maturity.

    Duration is the balance point of the present values10-year zeroDuration 10.0 years10-year, 9% couponDuration 7.1 years15-year, 12% couponDuration 8.6 yearsyr 0yr 5yr 10yr 15Bars: present value of each payment at 8%. Triangle: where the present values balance, the Macaulay duration.
    At an 8% yield the 10-year zero balances at 10.0 years, the 10-year 9% bond at 7.1 years and the 15-year 12% bond at 8.6 years, because the large coupons of the longest bond pull its balance point inside that of the zero.

    How do you rank them without a calculator?

    Use two rules and one check. A zero's duration is its maturity. A coupon bond's duration is always below its maturity, and the higher the coupon, the further below. So the only real contest is between the zero at 10 and the 15-year bond, and a 12% coupon pulls hard: about 60% of that bond's present value arrives in the first ten years. The 10-year 9% bond cannot beat the 10-year zero, because it has the same final date and pays some value earlier.

    The relationship
    D=∑tt⋅CFt(1+y)t∑tCFt(1+y)tD = \frac{\sum_t t \cdot \frac{CF_t}{(1+y)^t}}{\sum_t \frac{CF_t}{(1+y)^t}}
    tthe year a cash flow arrives
    CF_tthe cash flow in year t, coupon plus face at maturity
    ythe yield, 8% here
    What it says in wordsDuration is the average arrival time of a bond's cash flows, each weighted by what it is worth today.

    Add the sizing, because desks price risk in rupees. Modified durationMacaulay duration divided by one plus the yield. It gives the approximate percentage price change for a one point move in yield. is Macaulay duration over 1.08, so the zero loses about 9.3% of its price for a one point rise in yield and the 15-year bond about 7.9%. The limit: this is a small-move estimate, and it ignores the curve changing shape. Even a 30-year 12% bond only reaches about 11.5 years, because a coupon bond's duration can never exceed that of a perpetuity, 13.5 years at 8%.

    Where candidates lose it

    The instinct is to pick the 15-year bond because it is the longest. That confuses the last payment date with the average one, and on a desk it means hedging the wrong book.

    The quieter miss is treating coupon size as a detail. At 12% the coupons are large enough to pull the duration down to 8.6 years; at a 2% coupon the same 15-year bond would have a duration well above 10.

    What the interviewer asks next

    • What coupon on the 15-year bond would make its duration equal to the zero's?
    • Why does a floating-rate bond have a duration close to its next reset date?
    • Yields rise from 8% to 10%. Which of the three loses the most in rupees per Rs 100 of face?
  3. 086A client needs her money in exactly three years. Why does a bond fund with a duration of about three years protect her whether interest rates rise or fall? Show it with Rs 1 crore and yields jumping from 7% to 8% on the first day.Bond maths and durationHardPIMCOLondon · 2022

    Try it first

    Yields jump to 8% on day one and the fund's value falls. Where does she stand at year three?

    Show the worked solution

    Because a fall in price and a rise in reinvestment income cancel at a horizon equal to the duration. If yields jump to 8%, Rs 1 crore at a three-year duration drops about Rs 2.74 lakh on day one, then compounds at 8% instead of 7%. By year three it is worth Rs 122.53 lakh against Rs 122.50 lakh had nothing moved. If yields fall to 6%, the gain today offsets the lower reinvestment rate in the same way.

    Why do rising rates both hurt and help a bond investor?

    Think of a tenant who has locked in a lease. If market rents rise, the lease itself is worth less to sell on, because a new buyer could get a better deal elsewhere, but any new space the tenant takes from now on costs more too. A bond investor faces the mirror image. When yields rise, bonds already held fall in price, but every coupon and every maturing bond is reinvested at the new, higher yield. One effect is immediate and the other builds over time. Which one wins depends only on how long you hold.

    Make it concrete with a fund that has a duration of exactly three years. Picture its Rs 100 lakh as two holdings of Rs 50 lakh each at 7%: paper that matures in one year, and paper that matures in five. The value-weighted average of one and five is three, so the fund's Macaulay durationThe average time until a bond portfolio pays back its cash, with each payment weighted by its share of present value. is 3.0 years. Left alone at 7%, both halves grow to Rs 61.25 lakh by year three, Rs 122.50 lakh in all.

    Rs 1 crore with a three-year duration: the gap to the no-change path-3-2-1+1+2+30012345Years after the rate moveRs lakh above or below Rs 100 lakh growing at 7%her date, year 3day one: -2.74 if yields go to 8%day one: +2.88 if yields go to 6%yields 8%yields 6%At year 3, yields at 8%1-year paper, rolled at 8%62.40 (+1.15)5-year paper, sold early60.12 (-1.13)Fund at year 3122.53Target, no change122.50Rs lakhAlone, each leg fails:all short, yields 6%: 119.10all long, yields 8%: 120.25
    A jump to 8% knocks Rs 2.74 lakh off the fund on day one, but reinvesting at 8% wins it back by about year 3.0, so at her three-year date she holds Rs 122.53 lakh against a plan of Rs 122.50 lakh; a fall to 6% gives the mirror image.

    What happens to each holding when yields jump to 8%?

    The one-year paper matures at Rs 53.50 lakh and is rolled for two more years at 8%, reaching Rs 62.40 lakh, Rs 1.15 lakh more than the plan. The five-year paper still has two years to run at year three and must be sold at an 8% yield, fetching Rs 60.12 lakh, Rs 1.13 lakh less than the plan. One leg carries reinvestment risk and the other carries price risk, and a duration equal to the horizon sets them against each other in equal size. Held alone, each leg would fail: all short paper with yields falling to 6% gives only Rs 119.10 lakh, and all five-year paper with yields at 8% gives only Rs 120.25 lakh.

    The relationship
    V3=P(y) (1+y)3P(8%)=53.501.08+70.131.085=97.26  ⇒  97.26×1.083=122.53V_3 = P(y)\,(1+y)^3 \qquad P(8\%) = \frac{53.50}{1.08} + \frac{70.13}{1.08^5} = 97.26 \;\Rightarrow\; 97.26 \times 1.08^3 = 122.53
    P(y)the fund's value today at yield y, Rs lakh
    53.50, 70.13the amounts the one-year and five-year paper pay at maturity
    V_3the value at year three if all cash is reinvested at y
    What it says in wordsWhatever the new yield, the fund's value at year three is today's repriced value grown at that yield, and at a horizon equal to the duration the two changes offset.

    Where does this protection stop working?

    In three places, and naming them is what the interviewer is after. Her horizon shrinks by exactly a year every year, but the fund's duration does not keep pace on its own: maturing paper has to be reinvested and coupons and rate moves shift the average, so the fund has to be rebalanced to stay matched. The offset assumes all yields move together; if short yields rise while long yields fall, the two legs no longer cancel. And the match only protects someone whose date is fixed: an open-ended fund whose manager keeps duration at three years forever is matched to nobody's date in particular. A product that runs down towards a fixed maturity close to her date does the matching more naturally. The small surplus either way, about Rs 0.02 lakh, comes from convexity and is a bonus, not the point.

    Where candidates lose it

    The common answer is that a shorter fund is safer, so she should use a liquid fund. That removes price risk but leaves her fully exposed to falling rates: if yields drop to 6% and stay there, three years of rolling short paper leaves her at Rs 119.10 lakh, about Rs 3.4 lakh short of plan.

    The other loss is saying duration is a measure of price sensitivity and stopping there. This question is about the second meaning of duration, a time horizon at which price risk and reinvestment risk balance. Say both meanings, then show the cancellation with one number.

    What the interviewer asks next

    • After one year, what has happened to the fund's duration and to her horizon, and what should change?
    • Short yields rise 1% and long yields fall 1% on the same day. Does the protection hold?
    • Why does the matched fund end slightly above plan whichever way yields move?

    Asked at PIMCO, Sales, London, 2022 (Wall Street Oasis): Typically the product interview was toughest with questions regarding applications of duration