Macaulay duration

Duration is an estimation of the average term-to-maturity of a bond taking into account discounting of individual payments value. Therefore duration will always be less or equal to the term to bond’s maturity; it will be equal only to the term to maturity of discount (zero coupon) bonds. Duration is usually measured in years, but on Russian and Ukrainian markets days are more common. The duration formula is shown below:


D – duration
C[i] – current coupon payment i
t[0] – current date
t[i] – date of i coupon payment
N[i] – current par payment i (usually the bonds are redeemed in the end, then N[i]=0, i
P – current price (inclusive of AI)
T – amount of bond payments
r – effective yield-to-maturity

Suppose, that the bond has a 3-year maturity, 10% annual coupon, effective yield – 10% p.a., bond traded at par. The bond’s duration will be as follows:

It is important to say that duration of the money flow depends not only on its structure but also on the current interest rate. The higher the rate, the lower is the cost of long-term payments as compared to the short-term and the smaller is the duration; and visa versa, the lower the rate, the longer the duration of payments.

Duration does not only indicate the average term of the payments flow; it is also a good measure of the interest rate sensitivity of the price. The longer the duration, the higher is the interest rate volatility depending on the price fluctuations. The phrase “bond duration equals 3 years” means the bond in question has the same interest rate sensitivity as a 3-year zero coupon yield.

Modified duration is an even better measure of the price dependence on interest rates.