Advances in Risk Management

76 MANAGING INTEREST RATE RISK UNDER NON-PARALLEL CHANGES

Table 4.1 Bonds included in portfolios 1 and 2

Bond A B C D

Coupon (%) 5.5 10 12 9
Maturity (years) 5 15 20 10

Yield (%) 5.5 10 12 9
Duration with respect tos(Ds) 0.5085 0.5499 0.6344 0.5346

Duration with respect toL(DL) 0.8714 0.9342 1.0750 0.9123
Convexity with respect tos(δs) 0.3686 0.3924 0.4519 0.3826

Convexity with respect toL(δL) 1.1002 1.1644 1.3354 1.1439

see that, because of the equality between their generalized durations, the
relative behavior of these portfolios does not depend on the magnitude or
on the type of changes in the yield curve.
Portfolio 1 consists of bonds A, B and C, and portfolio 2 includes only
the bond D. All these bonds have a nominal value equal to $100 and pay
coupons on a semi-year basis. Additionally, we have the features listed in
Table 4.1.
We choose the bond proportions in portfolio 1 to equate the generalized
durations (per a 100-basis-points change in yield) and the market values for
both portfolios, and obtain the following system of equations:

xADAs+xBDBs+xCDCs=DDs

xADAL+xBDBL+xCDCL=DDL (4.28)

xA+xB+xC= 1

wherexj,j=A,B,Cis the proportion invested on each bond andD
j
i,i=s,L,
j=A,B,Crepresents the generalized duration of the j-th bond with respect
to the i-th factor. Solving this system of equations, we obtain that the pro-
portions for the three bonds are 59.930 percent, 11.203 percent, and 28.866
percent, respectively.
We compute the generalized convexities and the yield for both portfolios.
For portfolio 1, we compute the weighted average of the convexity and the
yield of its bonds; the weights are the proportions of the portfolio invested
on each bond. The results are as shown in Table 4.2.
As expected by design, both portfolios have the same generalized dura-
tions with respect to both factors. It can be seen that the generalized duration
with respect to the short-term interest rate and the yield of portfolio 2 are
greater than those of portfolio 1. The difference in yields suggests that the
best strategy consists of buying portfolio 2 and selling portfolio 1. Thus, we