Mutual Fund Mastery puzzles, solved step by step

Modified duration is about 3.99. Bracket first: a coupon bond's Macaulay duration sits below its 5-year maturity but not far, because the principal dominates, so somewhere in the low fours; dividing by 1.08 takes it just under 4. Exactly, the Macaulay duration is 4.312 years, and 4.312 divided by 1.08 is 3.993: a 1 point rise in yield cuts the price by roughly 4%.

How do you bracket it before doing any arithmetic?

Picture a see-saw with one heavy child at the far end and four small children spread along the plank. The balance point sits close to the heavy child but is pulled in a little by the others. A bond's Macaulay duration is the balance point of its discounted cash flows, so it can never exceed maturity and sits close to it when the final payment dominates. A zero coupon 5-year bond sits exactly at 5; this one pays 8 a year along the way, so it lands a little inside.

The final payment of 108 is worth 73.5 today, about 74% of the price of 100. The four coupons carry the rest at years 1 to 4. A balance point roughly three quarters of the way at 5 and a quarter spread between 1 and 4 lands a bit above 4, and dividing by 1 plus the yield lands just under 4.

Is there a shortcut for the exact figure?

For a bond priced at par there is a closed form. At par, Macaulay duration equals (1 + y) over y, times one minus the discount factor at maturity: 13.5 times (1 minus 0.6806), which is 4.312 years. Divide by 1.08 for modified duration, 3.993. Saying you know the par shortcut, then checking it against the bracket, is a strong answer in the room.

Say what the number is for. A modified duration of 3.99 means a 1 percentage point rise in yield costs roughly 3.99% of price, and a 0.25 point rise roughly 1%. The estimate is linear, so it drifts for large moves; convexity handles that.

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.

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.

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.

A duration of 4.05, rising to 7.25 after the switch. Weight each duration by its share of the Rs 100 crore: 0.40 x 1 + 0.35 x 4 + 0.25 x 9 = 4.05. Move the Rs 40 crore of 1-year duration into the 9-year bonds and it becomes 0.35 x 4 + 0.65 x 9 = 7.25. A 1% rise in yields now costs about Rs 7.25 crore instead of Rs 4.05 crore: the rate risk is about 79% higher.

Why is portfolio duration a weighted average?

A household's average commute is not the average of the three commutes, it depends on who travels most. If the person with the one-kilometre walk makes most of the trips, the household's average is short. A portfolio's duration is the money-weighted average of its bonds' durations, because each bond's price change counts in proportion to the rupees held in it. Here the biggest holding is the shortest, so the portfolio sits at 4.05, below the simple average of 4.67.

What changes in the portfolio when its duration rises?

The portfolio becomes more sensitive to interest rates in both directions: a 1% fall in yields gains about 7.25% instead of 4.05%, and a 1% rise loses as much. On Rs 100 crore, that is a swing of about Rs 7.25 crore for each 1% move instead of Rs 4.05 crore. On an upward sloping curve the portfolio's yield usually rises too, because longer bonds pay more, so the switch buys extra income with extra rate risk. Convexity also rises, which slightly cushions large moves, and the portfolio loses the cash-like buffer the short bonds gave it for meeting redemptions.

One caveat worth saying: duration is a linear estimate. For a 1% move it is close; for a 3% move, convexity makes the true loss smaller and the true gain larger than duration alone suggests. The durations here are given, and they drift as bonds age and yields move, so a portfolio's duration has to be re-measured, not set once.

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.

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.

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.

About 8.00% for next year, and about 6.5% if the curve is inverted. Investing for two years at 7.5% must give the same as one year at 7.0% rolled into next year's rate, or someone could profit from the gap. So 1.075 squared equals 1.07 times (1 + f), and f is 8.00%. Flip the curve to 7.5% then 7.0% and the implied rate falls to 6.50%: the market is pricing lower short rates ahead.

Why must the two paths give the same answer?

Two routes to the same railway station, one direct and one with a change, must cost about the same, or everyone would take the cheaper one until the prices matched. Money for two years can be locked in at the two-year rate, or invested for one year and rolled; if one path paid more, investors would crowd into it, so the rate the curve implies for the second year is the one that makes them equal. That rate is the forward rateThe interest rate for a future period that is locked in today by the current yield curve, found by making a long investment equal to a chain of shorter ones.. Rs 100 at 7.5% for two years becomes Rs 115.56. Rs 100 at 7.0% becomes Rs 107 after one year, so the second year must turn Rs 107 into Rs 115.56.

What does an inverted curve tell you?

In your head, the forward is about twice the long rate minus the short rate: 2 x 7.5 - 7.0 = 8.0%. Turn the curve upside down, 7.5% for one year and 7.0% for two, and the same step gives 2 x 7.0 - 7.5 = 6.5%, exactly 6.50%. An inverted curve says the market expects short rates to fall, which usually happens when it expects the central bank to cut, often because it expects growth to slow. That is why inversions are watched as a recession signal.

State the limit, because it is the follow-up. Forward rates are not pure forecasts. Lenders usually want extra pay for tying money up longer, a term premium, so an upward curve partly reflects that premium rather than expected hikes. The forward rate is the break-even: if next year's actual one-year rate comes in below it, the investor who locked in two years did better.