9.24 Modified Duration
If you wish to (reasonably) accurately measure a bond’s price sensitivity to specific interest rate (yield) movements, we use a formula called “modified duration,” rather than Macaulay Duration. Mac D assumes continuous compounding of the coupon payments, which is not realistic because coupons are paid periodically.
The concept is attributed to Sir John Hicks (1939). The formula for Modified Duration may be found below; the manner in which the formula is used to predict price changes follows on the next pages.
Modified Duration = Macaulay Duration ÷ (1 + YTM/P)
Example:
In a prior example we had a duration of 4.05 and a YTM = .10, where P = 2.
| Modified Duration | = 4.05 ÷ (1 + .05) |
| = 3.85 |
Macaulay assumed that interest on bonds is paid continuously, which is, of course untrue. Modified Duration corrects for the fact that bonds pay periodically, usually, in fact, every six months.
Notes:
1. YTM is divided by 2 for semis; other adjustments are necessary for other frequencies.
2. As compounding frequency increases, Modified Duration approaches Macaulay Duration
| Semi | Modified Duration | = 4.05 ÷ (1 + 10/2) = 3.85 |
| Quarterly | Modified Duration | = 4.05 ÷ (1 + 10/4) = 3.95 |
| Monthly | Modified Duration | = 4.05 ÷ (1 + 10/12) = 4.02 |
| Daily | Modified Duration | = 4.05 ÷ (1 + 10/365) ≅ 4.05 |
3. Notice that daily compounding approximates Mac D.
