2 duration and convexity - What Practitioners Need To Know...

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What Practitioners Need To Know . . . About Duration and Convexity Mark KHtzman, Windham Capital Management In 1938, Frederick Macaulay pub- lished his classic book. Some The- oretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States Since 1865 ^ Al- though Macaulay focused prima- rily on the theory of interest rates, as an aside he introduced the concept of duration as a more precise alternative to maturity for measuring the life of a bond. As with many of the important inno- vations in finance, the investment community was slow to appreci- ate Macaulay's discovery of dura- tion. It was not until the 1970s that professionai investors began to substitute duration for maturity in order to measure a fixed in- come portfolio's exposure to in- terest rate risk.^ Today, duration and convexity—the extent to which duration changes as inter- est rates change—are indispens- able tools for fixed income inves- tors. In this column, I review these important concepts and show how they are applied to manage interest rate risk. Macaulay's Duration A bond's maturity measures the time to receipt of the final princi- pal repayment and, therefore, the length of time the bondholder is exposed to the risk that interest rates will increase and devalue the remaining cash flows. Al- though it is typically the case that, the longer a bond's maturity, the more sensitive its price is to changes in interest rates, this re- lationship does not always hold. Maturity is an inadequate mea- sure of the sensitivity of a bond's price to changes in interest rates, because it ignores the effects of coupon payments and prepay- ment of principal. Consider two bonds, both of which mature in 10 years. Sup- pose the first bond is a zero- coupon bond that pays $2000 at maturity, while the second bond pays a coupon of $100 annually and $1000 at maturity. Although both bonds yield the same total cash flow, the bondholder must wait 10 years to receive the cash flow from the zero-coupon bond, while he receives almost half the cash flow from the coupon- bearing bond prior to its matu- rity. Therefore, the average time to receipt of the cash flow of the coupon-bearing bond is signifi- cantly shorter than it is for the zero-coupon bond. The first cash flow from the cou- pon-bearing bond comes after one year, the second after two years, and so on. On average, the bondholder receives the cash flow in five and one-half years (l-l-2-^•3 + . . . + 10)/10). In the case of the zero-coupon bond, the bondholder receives a single cash flow after 10 years. This computation of the average time to receipt of cash flows is an inadequate measure of the effec- tive life of a bond, because it fails to account for the relative magni- tudes of the cash flows. The prin- cipal repayment of the coupon- bearing bond is 10 times the size of each of the coupon payments. It makes sense to weight the time
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This note was uploaded on 08/13/2010 for the course FINS 2624 R taught by Professor Yippie during the Three '10 term at University of New South Wales.

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2 duration and convexity - What Practitioners Need To Know...

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