# Bond pricing theorems. Bond convexity The mathematical relationship between bond yields and prices.

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Bond pricing theorems

Bond convexity The mathematical relationship between bond yields and prices

Duration A measure of the average maturity of the stream of payments generated by a financial asset D = [ (1)CF 1 /(1+ ytm) + (2)CF 2 /(1+ ytm) 2 +....... + (t)CF t /(1+ ytm) t ] /(Price) Very often used: Modified duration: D* = D/(1+ytm)

Exemplification: A 6% coupon bond

Observation: Bond prices and yields move inversely.

Exemplification: A 6% coupon bond Observation: Dollar changes in bond prices are not symmetrical for a given basis point increase/decrease in YTM, other things constant 2.6% decrease in price 2.5% increase in price

Exemplification: A 6% coupon bond Observation The longer the maturity, the longer the duration, other things held constant.

Exemplification: A 6% coupon bond

Observation Longer maturity bonds are more sensitive to yield changes than shorter maturity bonds, other things held constant 3.4% decrease in price 3.2% increase in price 0.94% decrease in price 0.95%increase in price

Exemplification: A 6% coupon bond

Observation As maturity approaches, bond prices converge towards their face value at an increasing rate, other things held constant. 0.7% decrease in price 0.77% decrease in price 0.79% decrease in price 0.94% decrease in price

A 6% coupon bond A 5% coupon bond

A 6% coupon bond A 5% coupon bond Observation The lower the coupon rate the longer the duration

A 6% coupon bond A 5% coupon bond Observation Lower coupon bonds are more sensitive to yield changes than higher coupon bonds 3.4% decrease in price 3.2% increase in price 3.43% decrease in price 3.6% increase in price

Bond Pricing Theorems: A Summary I. Bond prices and yields move inversely. II. As maturity approaches, bond prices converge towards their face value at an increasing rate, other things held constant. III. Dollar changes in bond prices are not symmetrical for a given basis point increase/decrease in YTM, other things constant. IV. Lower coupon bonds are more sensitive to yield changes than higher coupon bonds, other things held constant. V. Longer maturity bonds are more sensitive to yield changes than shorter maturity bonds, other things held constant.

Duration Theorems: A Summary I. The duration of a zero coupon bond always equals its time to maturity. II. The lower the coupon rate the longer the duration, other things held constant. III. The longer the maturity, the longer the duration, other things held constant. IV. The lower the yield to maturity, the longer the duration, other things held constant

Using duration to approximate bond price changes The following formula approximates the change in bond prices for small changes in yields: (P 1 - P 0 )/P 0 = - D* (ytm 1 - ytm 0 ) A better approximation is given by the following formula: (P 1 - P 0 )/P 0 = - D*(ytm 1 - ytm 0 ) + (0.5)(Convexity)(ytm 1 - ytm 0 ) 2 Convexity The rate of change of the rate of change of the bond price (the curvature of the relationship between yields and prices).

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