1 Duration and Convexity by Binam Ghimire

Learning Objectives  Duration of a bond, how to compute it  Modified duration and the relationship between a bond’s modified duration and its volatility  Convexity for a bond, and computation  Under what conditions is it necessary to consider both modified duration and convexity when estimating a bond’s price volatility?  Excel computation 2

Duration  Developed by Frederick Macaulay, 1938  It combines the properties of maturity and coupon 3

Duration  Example  Two 20 – year bonds, one with an 8% coupon and the other with a 15% coupon, do not have identical life economic times. An investor will recover the original purchase price much sooner with the 15% coupon bond.  Therefore a measure is needed that accounts for the entire pattern (both size and timing) of the cashflows over the life of the bond – the effective maturity of the bond. Such a concept is called Duration 4

Duration Where: t = time period in which the coupon or principal payment occurs C t = interest or principal payment that occurs in period t i = yield to maturity on the bond

Duration  Duration is the average number of years an investor waits to get the money back.  Duration is the weighted average, on a present value basis, of the time to full recovery of the principal and interest payment on a bond. 6

Duration  Calculation of Duration depends on 3 factors  The Coupon Payments  Time to Maturity  The YTM 7

Duration  The Coupon of Payments  Coupon is ………….related to duration. This is logical because higher coupons lead to …………….. recovery of the bond’s value resulting in a ………… duration, relative to lower coupons 8

Duration  The Coupon of Payments  Coupon is inversely related to duration. This is logical because higher coupons lead to quicker recovery of the bond’s value resulting in a shorter duration, relative to lower coupons 9

Duration  Time to Maturity  Duration ………………. with time to maturity but a decreasing rate 10

Duration  Time to Maturity  Duration expands with time to maturity but a decreasing rate 11

Duration  Time to Maturity  Note that for all coupon paying bonds, duration is always less than maturity.  For a zero coupon bond, duration is equal to maturity 12

Duration  YTM  YTM is inversely related to duration 13

Characteristics of Duration  Duration of a bond with coupons is always less than its term to maturity because duration gives weight to these interim payments  A zero-coupon bond’s duration equals its maturity  There is an inverse relation between duration and coupon

Characteristics of Duration  There is a positive relation between term to maturity and duration, but duration increases at a decreasing rate with maturity  There is an inverse relation between YTM and duration  Sinking funds and call provisions can have a dramatic effect on a bond’s duration

Modified Duration and Bond Price Volatility An adjusted measure of duration can be used to approximate the price volatility of a bond Where: m = number of payments a year YTM = nominal YTM

Duration and Bond Price Volatility  Bond price movements will vary proportionally with modified duration for small changes in yields  An estimate of the percentage change in bond prices equals the change in yield time modified duration Where:  P = change in price for the bond P = beginning price for the bond D mod = the modified duration of the bond  i = yield change in basis points divided by 100

18 Convexity  The equation above generally provides an approximate change in price for very small changes in required yield. However, as changes become larger, the approximation becomes poorer.  Modified duration merely produces symmetric percentage price change estimates using equation when, in actuality, the price-yield relationship is not linear but curvilinear. (pls see price-yield graph already covered)  Hence, Convexity is the term used to refer to the degree to which duration changes as the YTM changes.

19 Convexity  Convexity is largest for low coupon bonds, long-maturity bonds, and low YTM.

Convexity The convexity is the measure of the curvature and is the second derivative of price with resect to yield (d 2 P/di 2 ) divided by price Convexity is the percentage change in dP/di for a given change in yield

Convexity  Inverse relationship between coupon and convexity  Direct relationship between maturity and convexity  Inverse relationship between yield and convexity