Question

A bond portfolio named VEX, comprises four bonds (face value=$1000):

1. 100 semi-annual bond, 5-year maturity, a coupon rate of 4%
2. 200 annual bonds, 30-year maturity, 8% coupon bond.
3. 300 zero coupon bonds, 10-year maturity.
4. 400 zero coupon bonds, 20-year maturity.

a) If required yield (discount rate) is 6% for all bonds per year, what is the market fair value of VEX? What is each bond’s weight in the portfolio?

b) Given the 6% initial yield, what is the VEX’s duration (use Macaulay’s duration)?

c) What is the time to maturity of VEX? Is VEX’s duration shorter or longer than its time to maturity? What is the meaning of VEX’s duration, how to interpret it?

d) According to the price-duration formula with Macaulay’s duration D, if the yield increases from 6% to 7%, the VEX’s market value should fall by how much ($)?

e) Calculate each bond’s convexity using Excel template. Then calculate VEX’s portfolio convexity. Considering VEX’s convexity, when the yield increases from 6% to 7%, the VEX’s market value should fall by how much ($)?

f) What is the difference of price drop between a formula with and without convexity?

Answer 1

Considering VEX’s convexity, if each bond’s convexity is given as follow:

Bond 1 (semi-annual coupon bond): 46.38

Bond 2 (annual coupon bond): 424.80

Bond 3 (zero coupon bond): 197.94

Bond 4 (zero coupon bond): 214

Convexity is the rate that the duration changes along the price-yield curve, and, thus, is the 1st derivative to the equation for the duration and the 2nd derivative to the equation for the price-yield function. Convexity is always positive for vanilla bonds. Furthermore, the price-yield curve flattens out at higher interest rates, so convexity is usually greater on the upside than on the downside, so the absolute change in price for a given change in yield will be slightly greater when yields decline rather than increase. Consequently, bonds with higher convexity will have greater capital gains for a given decrease in yields than the corresponding capital losses that would occur when yields increase by the same amount.

Some additional properties of convexity include the following:

Convexity increases as yield to maturity decreases, and vice versa.
Convexity decreases at higher yields because the price-yield curve flattens at higher yields, so modified duration is more accurate, requiring smaller convexity adjustments. This is also the reason why convexity is more positive on the upside than on the downside.
Among bonds with the same YTM and term length, lower coupon bonds have a higher convexity, with zero-coupon bonds having the highest convexity.
This results because lower coupons or no coupons have the highest interest rate volatility, so modified duration requires a larger convexity adjustment to reflect the higher change in price for a given change in interest rates.