When do you get your money back
Two bonds mature in ten years. One pays a fat coupon every year; the other pays nothing until the end. Both have a ten-year maturity, but the first returns most of your money much sooner. Duration is the measure that captures the difference: the weighted average time by which the investor receives the original amount invested, using the present value of each cash flow as its weight.
Macaulay duration = [ Sum over t = 1 to n of ( CF_t / (1 + y)^t ) x t ] / CMP
- t is each year from the date of investment; n is the remaining term to maturity.
- CF_t is the cash flow in year t: a coupon, or the final coupon plus redemption in year n.
- y is the yield to maturity at the current market price; CMP is that price.
Because every year's weight is the present value of that year's cash flow, coupons paid early pull the average forward. Duration is therefore usually less than the term to maturity. Only a bond with a single payment at the end, a zero-coupon bond, has a duration equal to its maturity.
Face value 100, coupon 10 paid annually, yield 10%, price 100. Year 1: cash flow 10, present value 10 / 1.10 = 9.09, times 1 = 9.09. Year 2: cash flow 110, present value 110 / 1.21 = 90.91, times 2 = 181.82. Sum = 190.91; divide by the price of 100: Macaulay duration is 1.91 years, less than the two-year maturity because a tenth of the money came back after year one.
From time to sensitivity
Divide Macaulay duration by (1 + y) and it becomes modified duration, a measure of how much the bond's price responds to a change in interest rates. Bonds with high modified duration gain more when rates fall and lose more when rates rise than bonds with low modified duration.
Modified duration = Macaulay duration / (1 + y)
- For the two-year bond above: 1.91 / 1.10 = 1.74.
- As a rough rule the price changes by about modified duration percent for each one percentage point move in yield, in the opposite direction: a rise of one point in yield would cost this bond roughly 1.74% of its price.
Duration folds three things into one number: tenor, coupon and yield. That makes it the most convenient measure of a bond's interest rate risk.
Holding everything else constant: the higher the time to maturity, the higher the duration and the higher the interest rate risk. The lower the coupon rate, the higher the duration and the risk. The lower the yield, the higher the duration and the risk. Higher modified duration always means higher sensitivity of price to interest rates.
Intuition sometimes runs the wrong way: a small coupon feels like a small bond and therefore a safe one. The opposite holds. With a small coupon, most of the money arrives at maturity, the weighted average time stretches out, and the price becomes more sensitive to rates. The zero-coupon bond is the extreme case.
Duration is not a fixed property. It changes as the bond's tenor and yield change, and as a bond approaches maturity its duration falls, so the bond becomes less risky with time.
- Macaulay duration is the weighted average time at which an investor gets the invested money back, with the present values of the bond's cash flows as weights. It is usually less than the term to maturity.
- Modified duration equals Macaulay duration divided by (1 + y) and measures how sensitive the price is to a change in interest rates: higher modified duration, bigger price moves either way.
- Other things equal: higher time to maturity, lower coupon and lower yield each raise duration and interest rate risk.
- Duration is not static; it falls as the bond approaches maturity, making the bond less risky over time.