Annuity factor, 4 years, 2% = 3·808
Equal annual amounts repayable per year = CHF60,000,000/3·808 = CHF15,756,303
Macaulay duration
(15,756,303 x 0·980 x 1 year +
15,756,303 x 0·961 x 2 years +
15,756,303 x 0·942 x 3 years +
15,756,303 x 0·924 x 4 years)/60,000,000
= 2·47 years
Modified duration = 2·47/1·02 = 2·42 years
The equation linking modified duration (D), and the relationship between the change in interest rates (∆i) and change in price or value of a bond or loan (∆P) is given as follows:
∆P = [–D x ∆i x P]
(P is the current value of a loan or bond and is a constant)
The size of the modified duration will determine how much the value of a bond or loan will change when there is a change in interest rates. A higher modified duration means that the fluctuations in the value of a bond or loan will be greater, hence the value of 2·42 means that the value of the loan or bond will change by 2·42 times the change in interest rates multiplied by the original value of the bond or loan.
The relationship is only an approximation because duration assumes that the relationship between the change in interest rates and the corresponding change in the value of the bond or loan is linear. In fact, the relationship between interest rates and bond price is in the form of a curve which is convex to the origin (i.e. non-linear). Therefore duration can only provide a reasonable estimation of the change in the value of a bond or loan due to changes in interest rates, when those interest rate changes are small.
