Macaulay Duration

The Macaulay duration is defined as the average time it takes to receive all the cash flows of a bond, weighted by the present value of each of the cash flows. It measures the number of years required to recover the true cost of a bond, considering the present value of all coupon and principal payments received in the future. Essentially, it is the payment-weighted point in time at which an investor can expect to recoup his or her original investment.

The Macaulay duration is quoted in “years” and it is the only type of duration with this unit. Interest rates are assumed to be continuously compounded.

Given its relative ability to predict price changes based on changes in interest rates, duration allows for the effective comparison of bonds with different maturities and coupon rates. For example, a 5-year zero coupon bond may be more sensitive to interest rate changes than a 7-year bond with a 6% coupon. By comparing the bonds’ durations, you may be able to anticipate the degree of price change in each bond assuming a given change in interest rates.

Utilizing Duration

Duration can help predict the likely change in the price of a bond given a change in interest rates. As a general rule, for every 1% increase or decrease in interest rates, a bond’s price will change approximately 1% in the opposite direction for every year of duration. For example, if a bond has a duration of 5 years, and interest rates increase by 1%, the bond’s price will decline by approximately 5%. Conversely, if a bond has a duration of 5 years and interest rates fall by 1%, the bond’s price will increase by approximately 5%.

Formula

Example

For example, for a two-year bond with a $1000 face value and one coupon payment every six months of $50, the duration (calculated in years) is:

Rules of Duration

When thinking about duration, a few general rules apply. With everything else being equal:

  • The duration of any bond that pays a coupon will be less than its maturity, because some amount of coupon payments will be received before the maturity date.

  • The lower a bond’s coupon, the longer its duration, because proportionately less payment is received before final maturity. The higher a bond’s coupon, the shorter its duration, because proportionately more payment is received before final maturity.

  • Because zero coupon bonds make no coupon payments, a zero coupon bond’s duration will be equal to its maturity.

  • The longer a bond’s maturity, the longer its duration, because it takes more time to receive full payment. The shorter a bond’s maturity, the shorter its duration, because it takes less time to receive full payment.