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15a_duration - Duration and Convexity The rst and second...

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Duration and Convexity The fi rst and second derivatives of bond prices with respect to interest rates are crucial in hedging interest rate risk. These can be numerically calculated on a computer. But they can be guessed by using the concepts of average life or duration and average life squared or convexity. 1 Duration or Average Life How much does price change when the interest rates change? By what percentage does price change when the interest rate changes ? Is there some way to see the answer simply, in one’s head, rather than having to compute a complicated formula? A brilliantly simple answer was provided by John Hicks in his famous book Value and Capital in 1939. Also discovered by Macauley in 1938. They de fi ned the average life or duration of a bond, and the average life squared or convexity of a bond. 1.1 Average Life or Duration de fi ned Duration (Hicks, Macauley) Average life of a bond should be the average age at which the bond’s value is paid. The average life or duration of a bond with payments ( m 1 , m 2 , ..., m T ) at fi xed interes rate r is de fi ned to be AvgLife ( m, r ) = D ( m, r ) m 1 1 + r 1 + m 2 (1 + r ) 2 2 + m 3 (1 + r ) 3 3 + · · · + m T (1 + r ) T T PV ( m, r ) = T X t =1 m t (1 + r ) t t PV ( m, r ) For example, let m = (0 , ..., 0 , m t , 0 , ..., 0) be a zero coupon bond with bullet payment at t. Then D ( m, r ) = m t (1 + r ) t t PV ( m, r ) = t The fi rst thing to notice about average life is that it does not depend on the level of the payments m t , but rather on their relative sizes. Doubling all the m t would double the numerator, but also double PV(r) and therefore the denominator, leaving average life the same. This is the fi rst reason average life an easy number to work with. The second reason is that average life does not change much with the interest rate. A higher interest rate lowers both numerator and denominator, leaving average life nearly the same. 1
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Third, average life is a weighted average of numbers, and most people are good at doing such calculations in their head, by intuition. Let us consider three examples showing that indeed average life is a very easy number to guess in one’s head. A one year zero has average life 1, no matter whether it is a Washington zero (paying $1) or a Lincoln zero (paying $5), and no matter what the interest rate is. Similarly a ten year zero has average life 10, no matter whether it is a Washington or Lincoln or Hamilton zero (paying $10), and no matter what the interest rate is. Consider a ten year 9% coupon bond at 8% interest. You should be able to guess that its average life is around 7. Clearly the average life is above 5, because payments are spread over ten years (which suggests 5), but the biggest payment is at the end, which suggests more than 5. On the other hand the biggest payment is discounted the most, so the average must be much less than 10. In fact, it can be computed exactly as AvgLife (8%) = 7 . 097508 (See average life spread sheet). If the interest rate were 10%, the average life would be a little di ff erent. Is your intuition good enough to guess what it would be? Clearly
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