Question

A 30-year maturity 6% coupon bond making annual coupon payments selling at a yield to maturity of 8% has a duration of 11.79 years and a convexity of 231.2.

a. Suppose the yield to maturity increases to 9%. What will be the actual percentage capital loss on the bond? What percentage capital loss would be predicted by the duration rule and the duration-with-convexity rule?

b. Repeat part (a), but this time assume the yield to maturity decreases to 7%.

c. Compare the performance of the bond in the two scenarios, one involving an increase in rates, the other a decrease. Based on the comparative investment performance, explain the attraction of convexity.

Answer 1

Answer a. When yield to maturity increases the then the price of the bond will decrease by (change in yield * Duration) = (1% * 11.79 years) = a decrease of 11.79% in the price of the bond which means a loss of 11.79%.

According to the duration rule, the formula for the capital loss = (negative duration) * change in yield = -11.79 * 0.01 = -0.1179

According to the duration with convexity rule, the formula for the capital loss = (negative duration) * change in yield + (0.5*convexity*((change in yield)^2), by substituting the values we will get

= -11.79 * 0.01 + (0.5 * 231.2 * ((0.01)^2))

= -0.1179 + 0.01156

= - 0.10634

=-10.63%

This represents a decrease of 10.63% in the price of the bond and this is considered more accurate.

Answer b. The yield to maturity decreases to 7%, this means a decrease of 1%. We know that price of the bond increases by (change in yield * Duration)= 1% * 11.79 = an increase of 11.79%.

According to the duration rule, the formula for the capital loss = (negative duration) * change in yield = -11.79 * -0.01 = 0.1179 = 11.79%

According to the duration with convexity rule, the formula for the capital loss = (negative duration) * change in yield + (0.5*convexity*((change in yield)^2), by substituting the values we will get

= -11.79 * -0.01 + (0.5 * 231.2 * ((-0.01)^2))

= 0.1179 + 0.01156

= 0.12946

=12.95%

This represents an increase of 10.63% in the price of bond. This is considered more accurate than the duration rule.

Answer c. Convexity tells us that the relationship between the price of a bond and the yield rate in the market is not a linear relationship but is slightly curved. This curvature in the relationship is measured through Convexity. Here we can clearly see the difference between the change in price of the bond when we consider a linear relation vs when we consider the convexity in the relationship of the price vs market yield change. Through the linear relation in Answer a we got a change of -11.79% in the price of the bond with an increase of 1% in the market yield through duration only rule whereas through duration with convexity rule we got a change of -10.63% which is considered to be more accurate. The same thing can be said for answer b. where by duration only formula we got the change of 11.79% with a decrease in market yield whereas through convexity we found out that the more accurate change is an increase of 12.95%.