Duration and convexity

When a Debt Long-Short strategy document says "modified duration of 4.2 years", that number tells an investor how much the strategy loses if yields rise 1%. Duration is the single most useful risk measure in fixed income and the basis of every hedge ratio in Chapter 13. The exam asks you to compute it for a short bond, and to say how it changes with coupon, maturity and yield.

Macaulay duration: the weighted average wait

Macaulay duration is the weighted average time, in years, until a bond's cash flows are received, with each payment weighted by its present value as a share of the price. A zero coupon bond's duration equals its maturity because there is only one cash flow. A coupon bond's duration is shorter than its maturity because some money comes back early.

Modified duration: the price sensitivity

Dividing Macaulay duration by (1 + y) gives modified duration, which directly measures the percentage change in price for a 1 percentage point change in yield. For semi-annual bonds divide by (1 + y ÷ 2).

For our bond, if yield rises 50 basis points (0.50%) from 8% to 8.5%, the estimated price change is −2.60 × 0.005 = −1.30%, so the price falls from ₹97.42 to about ₹96.16. The exact recalculated price at 8.5% is ₹96.17. The estimate is slightly too pessimistic, and the gap is convexity. For a 1% rise the estimate is −2.60% and the actual fall is −2.49%; the gap widens as the yield move grows.

The PVBP (price value of a basis point, also called DV01) is the rupee change in price for a 0.01% change in yield: Modified duration × Price × 0.0001. For this bond PVBP = 2.60 × 97.42 × 0.0001 = ₹0.0253 per ₹100 face. On a ₹1 crore face holding, a one-basis-point move is worth about ₹2,530. Traders and hedgers think in PVBP because it converts duration into money.

What moves duration

• Maturity. Longer maturity, higher duration (the cash flows arrive later). A 30-year G-Sec has a duration near 12; a 2-year bond near 1.9.
• Coupon. Higher coupon, lower duration (more money comes back early). A zero coupon bond has the highest duration for its maturity, equal to the maturity itself.
• Yield. Higher yield, lower duration (distant cash flows are discounted more heavily and carry less weight).
• Floating rate bonds have duration close to the time to the next reset, because the coupon adjusts to market rates.

Portfolio duration is the market-value-weighted average of the durations of the holdings. A fund with ₹60 crore at duration 2 and ₹40 crore at duration 5 has duration (60 × 2 + 40 × 5) ÷ 100 = 3.2. Mutual fund categories are defined by these numbers: short duration 1 to 3 years, medium 3 to 4, medium to long 4 to 7, long above 7. A Debt Long-Short SIF can push portfolio duration below zero by shorting interest rate futures, something a mutual fund cannot do.

Convexity: the second-order correction

Duration assumes the price-yield relation is a straight line; it is actually a curve that bends towards the price axis. Convexity measures that curvature. Because of it, the actual price rise for a yield fall is larger than duration predicts and the actual price fall for a yield rise is smaller. Convexity is good for a bond holder, and investors pay for it with slightly lower yields on high-convexity bonds (long maturity, low coupon).

For our 3-year bond the convexity works out to about 9.4. For the 50 basis point rise: convexity term = ½ × 9.4 × (0.005)² = 0.000118, or +0.012%, which lifts the estimate from −1.30% to −1.29% and matches the exact price of ₹96.17. For small moves the correction is tiny; for a 2% move on a 10-year bond it is worth several rupees. Negative convexity appears in callable bonds and mortgage-backed securities, where the issuer's right to prepay caps the price rise when yields fall.

Duration in practice: funds and SIFs

A fund manager who expects yields to fall lengthens duration to capture bigger price gains; one who expects a rise shortens it. Within a mutual fund the only tools are to buy shorter bonds or hold cash. A Debt Long-Short strategy has a third: sell interest rate futures, each of which carries the duration of its underlying bond, to cut portfolio duration quickly without selling holdings. The hedge ratio formula in Chapter 13 is nothing more than matching the PVBP of the portfolio to the PVBP of the futures. When you explain a strategy's duration band to a client, translate it: "duration 4 means roughly a 4% loss for every 1% rise in yields, before the convexity cushion".