Acquire some data on yield, convexity, and duration for five yield curve
buckets19 in the U.S. market, together with the same data for the index as a
whole.
(a) Use solveQP to find a portfolio that maximizes the relative yield
versus the index but stays fully invested and matches duration as well
as convexity.
(b) Use S-PLUS to decompose the yield changes of all five buckets into
three uncorrelated principal components and their associated loadings.
Assume that these three loadings explain most of the variance. Is this a
good assumption?
(c) Calculate the tracking error of the portfolio you found under a). Where
does the risk come from?
(d) Use the mean-variance approach discussed in this chapter to eliminate
active risk. What portfolio do you arrive at?
In order to solve this exercise, you might find the following short digression
useful. Using principal component analysis, we can decompose yield
changes into three uncorrelated principal components ( ∆pci ), which usually
explain most of the variance of the underlying series apart from “odd”
places such as Japan. Together with the associated loadings ( bij ), we can
write
3
1 . j ij i i y b pc = ∆= ∆ ¦
We need to calculate the matrix of factor loadings b and the covariance
matrix of principal components ȍ∆pc . We can combine this information
with the duration vector D to arrive at active risk:
() () 2 . T T T σ = − w w Db b p ȍ∆ c b bDw w−
Risk from the respective principal components can be calculated from
( ) ( ) T pc b a
a
d
d
σ
σ
∆ − = ȍ b Dw w
pc . Hint: See Scherer (2004).