MANUEL MORENO 77Table 4.2Computations of the generalized convexities and the yield for
both portfolios
Portfolio 1 Portfolio 2Duration with respect tos(Ds) 0.5346 0.5346
Duration with respect toL(DL) 0.9123 0.9123
Duration with respect tor(Dr) 2.8476 3.1155
Convexity with respect tos(δs) 0.3848 0.3826
Convexity with respect toL(δL) 1.1451 1.1439
Yield (%) 7.5272 9
can obtain a gain of 147.2 basis points. On the other hand, the generalized
convexities of portfolio 2 are slightly lower than those of portfolio 1. This
fact suggests that portfolio 1 can provide a greater yield than portfolio 2 if
there are certain shifts in the yield curve.
From now on, we will assume a shift in the yield curve instantaneously
after the acquisition of these portfolios and we will analyse the relative
behavior of both portfolios, measured by the difference between their yields.
We will consider three possibilities: a parallel change and two types of twist
in the slope of the yield curve: flattening and steepening.
For each case, at the end of the investment horizon, we obtain a certain
value for both portfolios. This value is the sum of three terms: the market
value of the portfolio, the coupons paid by the bonds, and the reinvestment
gain generated by the coupons. Changes in interest rates have two effects
on this value and, hence, on the portfolio yield: if interest rates increase,
the market value of the bond decreases (price risk) but the coupons gener-
ate a higher amount of money (reinvestment risk). The opposite situation
happens if interest rates fall. Therefore, the final gain will depend on the
changes in interest rates and the combined effect of both types of risk.
We consider an investment horizon of six months. Therefore, we need no
assumptions on the reinvestment rate of the coupons because each bond
is sold just after providing the first (and last) coupon. Hence, the final
(accumulated) value for each bond is its market price plus its coupon.
Table 4.3 shows this value and the yield (in annual terms) for the bonds
included in the portfolio 1 when there is a parallel change in the yield curve.
The first column of this table includes the change size. The coupons, in
percentage terms, paid by the bondsA, B and C are 2.75, 5 and 6, respectively.
We can observe that, when interest rates are increasing, each bond yield
decreases. The bond C provides the highest yield because it has the longest
maturity. On the contrary, the bond A, with the shortest maturity, shows the
narrowest interval of yields.