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The management of interest rate risk of the balance sheet
is a major task in banks.
This task has both an earning and a risk component. Nowadays,
some treasurers base the
optimal hedge decisions on one-period models, although it
is well-known to practitioners
that distributing risk and earning over time would improve
the performance of the
interest rate risk management.
A key obstacle to the diffusion of multi-period models
in the finance industry is the
absence of economic meaningful and applicable dynamic models.
We propose such a model
in this paper.
Our model is based on the recent work of Li and Ng (2000),
Leippold et al., (2003 and
2004). These authors showed how a standard one-period mean-variance
extends to
a multi-period optimization program. This natural extension
possesses a severe
mathematical drawback: The dynamic model is no longer separable
in the dynamic
programming sense which makes a neat analytical solution
impossible. The key
observation of the authors was to embed the dynamic optimization
program into a higher
dimensional one, and then to prove that the solutions of
the two programs are uniquely
related to each other. A first contribution of this paper
is to prove that indeed the two
programs are equivalent. The authors of the base work only
proved one part of the equivalence and tacitly assumed that
the other part is also true. Given the equivalence,
we provide full analytical/semi-analytical solutions for
a joint balance sheet and hedge
portfolio in the case of linear equalities and inequalities
as restrictions. This is to our
knowledge, a new contribution, which extends the results
in Li and Ng, (2000) and
Leippold et al., (2004) in discrete time and Yong and Zhou
(1999), and Li et al. (2002) in
continuous time, where only linear equality restrictions
are considered. Our final
theoretical contribution is related to the multi-period
optimization program if the term
structure dynamics is specified. We show first that given
a time independent term
structure, the multi-period and the one-period model are
equivalent. Then we prove that,
if the best guess about future interest rates is the forward
rate, the optimal balance sheet
management has to mimic the benchmark.
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