D = (1.03/.03) [ 1 – (1/1.03)⁶]

= 5.58 half year periods

= 2.79 years

If the bond’s yield is 10%, use Rule 7, setting the semiannual yield to 5%, and semiannual coupon to 3%:

D = – (1.05/.05) - [1.05 + 6(.03-.05)]/{.03[(1.05)⁶ - 1] + .05}

= 21 – 15.448 = 5.552 half-year periods = 2.776 years
 

8. a. (iv)

b. (ii)

c. (i)

d. (i)

e. (iii)

f. (i)

g. (i)

h. (iii)

18. Using a financial calculator, the price of the bond for a yield to maturity of 7% is $1620.45; for YTM of 8%, the price is $1450.31; and for YTM of 9% the price is $1308.21.

Using the Duration Rule, assuming yield to maturity falls to 7%

Predicted price change = – Duration X (change in y)/(1+y)X P₀

= –11.54 X (-.01)/1.08 X 1450.31 = 154.97

Therefore, predicted new price = 154.97 + 1450.31 = $1605.28

The true price at a 7% yield to maturity is $1620.45. Therefore,

% error =(1620.45 - 1605.28)/1620.45 = .0094 = .94 % (too low)

 
Using the Duration Rule, assuming yield to maturity increases to 9%

Predicted price change = – Duration X (change in y)/(1+y) X P₀

= –11.54 X +.01/1.08 X 1450.31 = –154.97

Therefore, predicted new price = –154.97 + 1450.31 = $1295.34

The true price at a 9% yield to maturity is $1308.21. Therefore,

% error =(1308.21-1295.34)/1308.21 = .0098 = .98 % (too low)

Using Duration-with-Convexity Rule, assuming yield to maturity falls to 7%

Predicted price change = [( – Duration X Dy/(1+y) ) + (0.5 X convexity X Dy²)] X P0

= [–11.54  X -.01/1.08  + 0.5 X 192.4 X (-0.01)²] X 1450.31 = 168.92

 
Therefore, predicted price = 168.92 + 1450.31 = $1619.23

The true price at a 7% yield to maturity is $1620.45. Therefore,

% error = (1620.45-1619.23)/1620.45 = .00075 = .075% (too low)

Using Duration-with-Convexity Rule, assuming yield to maturity rises to 9%

Predicted price change = [( – Duration X Dy/(1+y) ) + (0.5 X convexity X Dy²)] X P₀

= [–11.54 X .01/1.08  + 0.5 X 192.4 X (0.01)²] X 1450.31 = –141.02

Therefore, predicted price = –141.02 + 1450.31 = $1309.29

The true price at a 9% yield to maturity is $1308.21. Therefore,

% error =(1309.29 - 1308.21)/1308.21  = .00083 = .083% (too high)

Conclusion: the duration-with-convexity rule provides more accurate approximations to the true change in price. In this example, the percentage error using convexity with duration is less than one-tenth the error using only duration to estimate the price change.

19. a. The price of the zero coupon bond ($1000 face value) selling at a yield to maturity of 8% is $374.84 and that of the coupon bond is $774.84.

At a YTM of 9% the price of the zero coupon bond is $333.28 and that of the coupon bond is $691.79.

Zero coupon bond

Actual % loss =(333.28 - 374.84)/374.84  = –.1109, an 11.09% loss

The percentage loss predicted by the duration-with-convexity rule is:

Predicted % loss = [( –11.81) X .01 + 0.5 x 150.3 X (0.01)²]

= –.1106, an 11.06% loss

Coupon bond

Actual % loss = (691.79 - 774.84)/774.84 = –.1072, a 10.72% loss

The percentage loss predicted by the duration-with-convexity rule is:

Predicted % loss = [( –11.79) X .01 + 0.5 X 231.2 X (0.01)²]

= –.1063, a 10.63% loss

b. Now assume yield to maturity falls to 7%. The price of the zero increases to $422.04, and the price of the coupon bond increases to $875.91.

Zero coupon bond

Actual % gain = (422.04 - 374.84)/374.84  = .1259, a 12.59% gain

The percentage gain predicted by the duration-with-convexity rule is:

Predicted % gain = [( –11.81) X (–.01) + 0.5 X 150.3 X (0.01)² ]

= .1256, an 12.56% gain

Coupon bond

Actual % gain = (875.91 - 774.84)/774.84 = .1304, a 13.04% gain

The percentage gain predicted by the duration-with-convexity rule is:

Predicted % gain = [ (–11.79) X (–.01) + 0.5 X 231.2 X (0.01)²]

= .1295, a 12.95% gain

c. The 6% coupon bond -- which has higher convexity -- outperforms the zero regardless of whether rates rise or fall. This can be seen to be a general property by noting from the duration-with-convexity formula that the duration effect on the two bonds due to any change in rates will be equal (since their durations are virtually equal), but the convexity effect, which is always positive, will always favor the higher convexity bond. Thus, if the yields on the bonds always change by equal amounts, as we have assumed in this example, the higher convexity bond will always outperform a lower convexity bond with equal duration and initial yield to maturity.

d. This situation cannot persist. No one will be willing to buy the lower convexity bond if it always underperforms the other bond. Its price will fall and its yield to maturity will rise. Thus, the lower convexity bond will sell at a higher initial yield to maturity. That higher yield is compensation for lower convexity. If rates change by only a little, the higher yield-lower convexity bond will do better; if rates change by a lot, the lower yield-higher convexity bond will do better.

20. a. The following spreadsheet shows that the convexity of the bond is 64.933. The present value of each cash flow is obtained by discounting at 7%. (since the bond has a 7% coupon and sells at par, its YTM must be 7%.) Convexity equals the sum of the last column, 7434.175, divided by [P X (1 + y)²] = 100 X (1.07)².
 

c. The duration rule would predict a percentage price change of

– D/(1.07) X .01 = – 7.515/1.07 X .01 = – .0702 = – 7.02%

This overstates the actual percentage decline in price by .31%.

d. The duration with convexity rule would predict a percentage price change of

– 7.515/1.07 X .01 + .5 X 64.933 X (.01)² = .0670 = –6.70%

which results in an approximation error of only .01%, far smaller than the error using the duration rule.