Mutual Fund Mastery puzzles, solved step by step

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All topicsCompounding and time value9Statistics, correlation and diversification8Bond maths and duration10Performance measurement and returns8Costs and fee drag8Valuation riddles11Logic and numeracy brainteasers6Estimation and market sizing7Probability and expected value8NAV, units and fund mechanics7Risk, volatility and drawdown8Behavioural traps6Withdrawals and after-tax arithmetic4
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  1. 026A 5-year government bond yields 7.6% and a 4-year bond yields 7.2%. You buy the 5-year bond at par. If the yield curve does not move at all over the next year, what return do you earn for the year?Bond maths and durationCoreFixed income desksIndian AMCs

    Try it first

    Before you work it: roughly what does a year of holding the 5-year bond return?

    Show the worked solution

    About 9.0%, not 7.6%. You collect the 7.6% coupon. A year later the bond has four years left, and on an unchanged curve four-year bonds yield 7.2%, so its yield has fallen 0.40% with no market move at all. With a duration of about 3.4, that lifts the price about 1.35%, from 100 to 101.35. Coupon plus this roll-down gain is about 8.95%.

    Why does the bond's yield change if the curve does not move?

    Stand halfway down a slope and take one step towards the bottom. The slope has not moved, but you are lower than you were. A bond on an upward sloping yield curve is doing the same thing every day it is held. The curve stays put, but the bond's own maturity shortens by a year, so a year later it is priced off a lower point on the same curve. Nothing happened in the market; the bond simply got older.

    That slide has a name, roll-downThe price gain a bond earns as it ages and its yield moves down an unchanged, upward sloping yield curve., and a fund manager counts it as part of expected return just as surely as the coupon. Today the bond is a 5-year bond at 7.6%. A year from now it is a 4-year bond, and 4-year bonds yield 7.2%. A 7.6% coupon discounted at 7.2% is worth more than par.

    Same curve, one year older: the bond slides down to a lower yield6.0%6.5%7.0%7.5%8.0%1y2y3y4y5yYears to maturityToday: 5y at 7.6%A year on: 4y at 7.2%yield falls 0.40%with no market move7.607.6% coupon+1.35 roll-down8.95%One year's returncurve unchanged
    On an unchanged, upward sloping curve the bond starts at the 5-year point yielding 7.6% and a year later sits at the 4-year point yielding 7.2%. That 0.40% fall in its yield adds about 1.35% of price gain to the 7.6% coupon, for a one-year return near 8.9%.

    How do you size the roll-down gain in your head?

    Use duration. A bond's price moves by roughly its modified duration times the change in its yield, and the duration that matters is the bond's duration at the end of the year, when it is a 4-year bond. A 4-year bond with a 7.6% coupon has a modified duration of about 3.36. Multiply by the 0.40% fall and you get about 1.34%. Pricing the bond exactly, four coupons of 7.6 and 100 at maturity discounted at 7.2%, gives 101.35, a gain of 1.35%. The shortcut is within a hundredth of a per cent.

    The relationship
    r1y≈ybuy+Dend×(y5−y4)=7.6%+3.36×0.40%≈8.95%r_{1y} \approx y_{\text{buy}} + D_{\text{end}} \times (y_{5} - y_{4}) = 7.6\% + 3.36 \times 0.40\% \approx 8.95\%
    y_buythe yield you bought at, 7.6%
    D_endmodified duration of the bond a year later, as a 4-year bond
    y_5 - y_4how far the yield rolls down the curve, 0.40%
    What it says in wordsOne year's return on an unchanged curve is the yield you bought plus duration times the yield you roll down.

    When does the roll-down vanish or turn against you?

    Roll-down is only as good as the slope. On a flat curve there is nothing to slide down and the return is the coupon. On an inverted curve, where shorter bonds yield more, the bond rolls up to a higher yield and loses price as it ages. And the curve rarely stays still: if the 4-year yield ends the year at 7.6% instead of 7.2%, the roll-down is gone and you earn roughly the coupon alone. Say this limit out loud; it shows you treat roll-down as an expected return on an assumption, not a promise.

    Where candidates lose it

    The common answer is 7.6%, because candidates treat yield to maturity as the return for any holding period. Yield to maturity is the return only if you hold to maturity; over one year, the price at the end of the year matters, and on a sloped curve that price has moved.

    The second loss is getting the direction wrong: the yield falls, so some candidates say the return falls. A falling yield means a rising price. Say that link explicitly before you size the gain.

    What the interviewer asks next

    • What is the one-year return if the curve is flat at 7.6%?
    • How much would the 4-year yield have to rise for the year's return to fall to 7.6%?
    • Why might a debt fund manager prefer the 5-year bond to a 4-year bond at 7.2% even with no view on rates?
  2. 045A target maturity debt fund has a portfolio yield to maturity of 7.4% and an expense ratio of 0.2%. What return should an investor who holds to maturity expect, and why will it not be exactly that?Bond maths and durationCoreFixed income desksIndian AMCs

    Try it first

    What is the best single estimate of the hold-to-maturity return, per year?

    Show the worked solution

    About 7.2% a year: the portfolio's yield to maturity less the expense ratio. Holding to the fund's maturity removes most of the price risk, so the bonds earn roughly their 7.4% yield and the fund keeps 0.2% a year. It will not be exactly 7.2% because coupons are reinvested at future yields nobody knows, the fund tracks its index imperfectly, and the yield you lock in is the one on the day you invest. On these numbers the drift is about plus or minus 0.13% a year.

    Why is yield to maturity a fair starting point?

    A fixed deposit tells you its rate because the bank promises to hold it to maturity for you. A bond does the same if you hold it to maturity: price swings along the way wash out, and what remains is the yield to maturityThe single discount rate that makes the present value of a bond coupons and principal equal its price today; the return earned if the bond is held to maturity and coupons are reinvested at that same rate.. A target maturity fund holds bonds that mature near one date and then pays out, so an investor who stays to that date earns close to the portfolio's yield to maturity, less the fund's costs. 7.4% less 0.2% is about 7.2% a year.

    Yield to maturity, less cost, is the estimate; reinvestment moves it6.8%7.0%7.2%7.4%7.6%7.40%-0.207.20%7.337.07reinvest6.4% to 8.4%trackingPortfolio YTMExpense ratioBest estimateWhat moves itScale starts at 6.8%, not zero, so the small effects are visible.
    The portfolio's 7.40% yield to maturity less the 0.20% expense ratio gives a best estimate of 7.20% a year, and reinvesting coupons at 6.4% to 8.4% instead of 7.4% would move the realised return to between 7.07% and 7.33% after costs, with tracking effects of a few basis points on top.

    Why will the realised return not be exactly 7.2%?

    Yield to maturity quietly assumes every coupon is reinvested at the same yield, and the future reinvestment rate is unknown. Take a 5-year bond bought at par with a 7.4% coupon. If its coupons can only be reinvested at 6.4%, the realised return is about 7.27% a year; at 8.4% it is about 7.53%. After the 0.2% fee that is 7.07% to 7.33%. The fund also holds bonds that do not mature on exactly one date, keeps some cash, and replicates its index imperfectly, each worth a few basis points either way.

    The relationship
    rhold≈YTM−TER=7.4%−0.2%=7.2%r_{\text{hold}} \approx \text{YTM} - \text{TER} = 7.4\% - 0.2\% = 7.2\%
    YTMthe portfolio's yield to maturity on the day you invest, 7.4%
    TERthe total expense ratio, 0.2% a year
    r_holdthe return a hold-to-maturity investor can reasonably expect
    What it says in wordsFor an investor who stays to maturity, the best estimate of the return is today's portfolio yield less the yearly cost.

    What else should the investor be told?

    Three things. The estimate holds only to maturity: an investor who exits after two years takes the market price then, which can be well above or below the path to 7.2%. The yield locked in is the one on the day of investment, not the one quoted at launch. And the figure is before tax; how a debt fund's gains are taxed has changed in recent years, so confirm the current treatment before turning 7.2% into an after-tax number. The estimate is the honest best guess, not a promise.

    Where candidates lose it

    The trap is quoting 7.4% as the return, forgetting that the expense ratio is taken out of the NAV every day. The fund's yield is the bonds' yield; the investor's return is that less the cost.

    The second loss is treating 7.2% as a promise because the fund holds to maturity. It is an estimate: reinvestment, tracking and timing all move it, and an exit before maturity exposes the investor to price risk.

    What the interviewer asks next

    • Yields rise 1% the day after you invest. What happens to your NAV and to your return if you hold to maturity?
    • Why do coupon reinvestment effects matter less for a 2-year fund than a 10-year fund?
    • How would you compare this fund with a 5-year bank deposit at 7.1%?