Question

You used second order approximation (with convexity correction) to establish the following: YTM increases by 1% → Bond A’s price decrease 2.5% YTM decreases by 3% → Bond A’s price increases 8.5% Use 2nd order approximation to calculate Bond A’s volatility for a 0.5% decrease in YTM. State your answer as a percentage with six digits after the decimal point. (ANSWER 1.312500%)

Answer 1

Let's call modified duration as MD and convexity as C.

%change in Price = - MD x %age change in yield + 0.5 x C x (%age change in yield)

Hence, -2.5% = - MD x 1% + 0.5 x C x (1%)² ----------------Equation (1)

And, 8.5% = - MD x (-3%) + 0.5 x C x (-3%)² ---------------Equation (2)

Equation (2) + 3 x Equation (1) give:

(8.5% - 2.5% x 3) = 0.5 x C x [(-3%)² + 3 x 1%²]

Or, 1% = 0.01 = 0.5 x C x 0.0012

Hence, C = 16.67

From equation (1): MD = [0.5 x C x (1%)² + 2.5%] / 1% = 2.58

Hence, Bond A’s volatility for a 0.5% decrease in YTM = %age change in price of the bond = - MD x (-0.5%) + 0.5 x C x (-0.5%)² = - 2.58 x (-0.5%) + 0.5 x 16.67 x (-0.5%)² = 0.013125000 = 1.312500%