John hull options futures and other derivatives 9th pdf

In finance, the duration of a financial asset that consists of fixed cash flows, for example a bond, is the weighted average of the times until those fixed cash flows are received. When the price of an asset is considered as a function of yield, duration also measures the price sensitivity to yield, the rate of change john hull options futures and other derivatives 9th pdf price with respect to yield or the percentage change in price for a parallel shift in yields. The dual use of the word “duration”, as both the weighted average time until repayment and as the percentage change in price, often causes confusion.


Strictly speaking, Macaulay duration is the name given to the weighted average time until cash flows are received, and is measured in years. Modified duration is the name given to the price sensitivity and is the percentage change in price for a unit change in yield. Macaulay duration is a time measure with units in years, and really makes sense only for an instrument with fixed cash flows.

For a standard bond the Macaulay duration will be between 0 and the maturity of the bond. It is equal to the maturity if and only if the bond is a zero-coupon bond. The concept of modified duration can be applied to interest-rate sensitive instruments with non-fixed cash flows, and can thus be applied to a wider range of instruments than can Macaulay duration.

Modified duration is used more often than Macaulay duration. Macaulay and modified duration can be a useful aid to intuition. Macaulay duration, named for Frederick Macaulay who introduced the concept, is the weighted average maturity of cash flows. Consider some set of fixed cash flows.

These terms add to 1. 0 and serve as weights for a weighted average.

The Macaulay duration will equal the final maturity if and only if there is only a single payment at maturity. Macaulay duration will equal the bond maturity only for a zero-coupon bond. Macaulay duration has the diagrammatic interpretation shown in figure 1.

The circles represent the present value of the payments, with the coupon payments getting smaller the further in the future they are, and the final large payment including both the coupon payment and the final principal repayment. 78 years in this case.

Similarities in both values and definitions of Macaulay Duration versus Weighted Average Life can lead to confusing the purpose and calculation of the two. For example, a 5 year fixed-rate interest only bond would have a Weighted Average Life of 5, and a Macaulay Duration that should be very close. Macaulay Duration only measures fixed period cash flows, Weighted Average Life factors in all principal cash flows whether they be in fixed or floating.

Thus for Fixed Period Hybrid ARM mortgages, for modeling purposes, the entire fixed period ends on the date of the last fixed payment or the month prior to reset. Macaulay Duration discounts all cash flows at the corresponding cost of capital. Weighted Average Life does not discount.