Question

A 12.75-year maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual yield) has a convexity of 150.3 and a modified duration of 11.81 years. A 30-year maturity 6% coupon bond making annual coupon payments also selling at a yield to maturity of 8% has a nearly identical modified duration—11.79 years—but considerably higher convexity of 231.2.

a. Suppose the yield to maturity on both bonds increases to 9%. What will be the actual percentage of capital loss on each bond? What percentage of capital loss would be predicted by 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 two bonds in the two scenarios, one involving an increase in rates, the other a decrease. Based on their comparative investment performance, explain the attraction of convexity.

d. In view of your answer to ( c ), do you think it would be possible for two bonds with equal duration, but different convexity, to be priced initially at the same yield to maturity if the yields on both bonds always increased or decreased by equal amounts, as in this example? Would anyone be willing to buy the bond with lower convexity under these circumstances?

Short answer

a. Actual % loss (zero-coupon) = 11.06% loss and (discount coupon) 10.63% loss

b. Actual % gain (zero-coupon) = 12.56% gain and (discount coupon) 12.95% gain

c. 6% coupon bond outperforms the zero bond

d. No. This situation will change and the price of the lower convexity bond will fall and its yield to maturity will rise.

Step-by-step solution

Calculation of the price of the zero-coupon bond with $1000 face value

The price of the zero coupon bond: (calculated using financial calculator)

for YTM 8% = $374.84

for YTM 9% = $333.28

The price of the 6% coupon bond: (calculated using financial calculator)

for YTM 8% = $774.84

for YTM 9% = $691.79

Calculation of actual and predicted % loss for zero and coupon bond \using Duration with convexity rule ‘a’

For zero Coupon Bond:

Actual % loss = $333.28 - $374.84 / $374.84

= -0.1106 = 11.06% loss

Predicted % loss by Duration with convexity rule= [(- 11.81 x 0.01 ) + (0.5 x 150.3 x (0.01)²]

= - 0.1106

=11.06% loss

For 6% Coupon Bond:

Actual % loss = $691.79 - $774.84 / $774.84

= - 0.1072 = 10.72% loss

Predicted % loss by Duration with convexity rule= [(- 11.79 x 0.01 ) + (0.5 x 231.2 x (0.01)²]

= - 0.1063

=10.63% loss

Calculation of actual and predicted % loss for zero and coupon bond \using Duration with convexity rule at 7% YTM ‘b’

In this case, the price of the zero-coupon bond: (calculated using a financial calculator)

for YTM 7% = $422.04 and

The price of the coupon bond = $ 875.91

For zero Coupon Bond:

Actual % gain = $422.04 - $374.84 / $374.84

= 0.1259 = 12.59% gain

Predicted % gain by Duration with convexity rule= [(- 11.81 x - 0.01 ) + (0.5 x 150.3 x (-0.01)²]

= -0.1256

=12.56% gain

For 6% Coupon Bond:

Actual % gain = $875.91 - $774.84 / $774.84

= 0.1304 = 13.04% gain

Predicted % loss by Duration with convexity rule = [(- 11.79 x - 0.01 ) + (0.5 x 231.2 x (0.01)²]

= 0.1295

=12.95% gain

Comparison of performance of two bonds ‘c’

From the above calculations, it is clear that the 6% coupon bond outperforms the zero bond. This is also confirmed by the higher convexity bond.

Explanation of the situation 'd'

No. An always underperforming lower convexity bond would not be sought after. This situation will change and the price of the lower convexity bond will fall and its yield to maturity will rise. This will ensure that the lower convexity bond sells at a higher initial yield to maturity i.e compensation for the lower convexity.

The higher yield-lower convexity bond will perform better; if rates change a little. But if they change by a greater amount, the lower yield-higher convexity bond will do better.