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.

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 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.