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.

The yield to maturity is below 7%, and the current yield is 6.73%. Current yield is the coupon over the price: 7 / 104 = 6.73%. The yield to maturity is lower still, because you pay 104 and get back only 100, so the Rs 4 premium is a loss spread over the bond's life. For an assumed five years left, the yield to maturity is about 6.05%.

Why must the yield be below the coupon when the price is above 100?

Pay Rs 104 for a gift voucher worth Rs 100 that also pays Rs 7 of cashback a year. The cashback is generous, but you have overpaid for the voucher by Rs 4. A bond priced above par returns less than its coupon, because the buyer pays more than the Rs 100 that comes back at maturity, and that premium is lost along the way. The coupon of 7% is set on the face value of 100 and never changes. The current yieldThe yearly coupon divided by the bond price today. It ignores any gain or loss as the price moves to face value at maturity. divides the same Rs 7 by what you actually pay: 7 / 104 = 6.73%.

How far below 7% is the yield to maturity?

That depends on how long the bond has to run, which the question does not say, so name an assumption. With five years left, the Rs 4 premium is lost at roughly Rs 0.80 a year. A quick estimate takes the coupon less that yearly loss, Rs 6.20, over the average of the purchase and redemption prices, 102: about 6.08%. The exact yield to maturityThe single discount rate that makes all remaining coupons and the final repayment worth exactly the price paid today. is 6.05%. The longer the bond, the thinner the yearly slice of the premium and the closer the yield to maturity sits to the current yield.

For a debt fund this ordering is everyday arithmetic: when rates fall, older high-coupon bonds trade above par, and the fund's quoted portfolio yield sits below the coupons it receives. The limit of yield to maturity: it assumes the bond is held to maturity, never defaults and that coupons are reinvested at the same yield, none of which is guaranteed. A bond callable before maturity at par makes the premium an even bigger risk.

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.

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.

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.