21-Bond Portfolio Man Page 671 Wednesday, February 4, 2004 1:12 PM
Bond Portfolio Management 671the liabilities. This duration is intended in the sense of effective duration
which allows for a shift in the term structure. This condition does not
determine univocally the portfolio.
To determine the portfolio, we can proceed in two ways. The first is
through optimization. Optimization calls for maximizing some function
subject to constraints. In the CFM problem there are two constraints:
(1) The initial present value of cash flows must match the initial present
value of liabilities, and (2) the duration of cash flows must match the
duration of liabilities. A typical objective function is the portfolio’s
return at the final date. It can be demonstrated that this problem can be
approximated by an LP problem.
Optimization might not be ideal as the resulting portfolio might be
particularly exposed to the risk of nonparallel shifts of the term struc-
ture. In fact, it can be demonstrated that the result of the yield maximi-
zation under immunization constraints tends to produce a barbell type
of portfolio. A barbell portfolio is one in which the portfolio is concen-
trated at short-term and long-term maturity securities. A portfolio of
this type is particularly exposed to yield curve risk, i.e., to the risk that
the term structure changes its shape, as described in Chapter 20.
One way to control yield curve risk is to impose second-order con-
vexity conditions. In fact, reasoning as above and taking the second
derivative of both sides, it can be demonstrated that, in order to protect
the portfolio from yield curve risk, the convexity of the cash flow stream
and the convexity of the liability stream must be equal. (Recall from
Chapter 4 that mathematically convexity is the derivative of duration.)
This approach can be generalized^16 by assuming that changes of interest
rates can be approximated as a linear function of a number of risk fac-
tors. Under this assumption we can write
k∆rt = ∑βjt, ∆εfj + t
j = 1
where the fj are the factors and εt is an error term that is assumed to be
normally distributed with zero mean and unitary variance. Factors here
are a simple discrete-time instance of the factors we met in the description
of the term structure in continuous time in Chapter 19. There we assumed
that interest rates were an Itô process function of a number of other Itô
processes. Here we assume that changes in interest rates, which are a dis-
crete-time process, are a linear function of other discrete-time processes
called “factors.” Each path is a vector of real numbers, one for each date.(^16) See Stavros Zenios, Practical Financial Optimization, unpublished manuscript.