The Mathematics of Financial Modelingand Investment Management

21-Bond Portfolio Man Page 672 Wednesday, February 4, 2004 1:12 PM

672 The Mathematics of Financial Modeling and Investment Management

Ignoring the error term, changes in the present value of the stream of cash

flows are therefore given by the following expression:

n

∆V= – ∑[αiKi, 1 + ...α+ iKimt e m m

• r t
  , m ∆rm ]
  i = 1
  n k

= – ∑ αiKi, 1 + ...αiKimt e

• rmtm

+ , m ∑βjt, m ∆fj

i = 1 j = 1

The derivative of the present value with respect to one of the factors

is therefore given by

n

∂V + e m m

------- = – ∑ αiKi, 1 + ...αiKim, tmβjt, m

• r t
  ∂fj i = 1

The factor duration with respect to the j-th factor is defined as the rela-

tive value sensitivity to that factor:

k = ---- ------^1 ∂V -

j V ∂f

j

The second derivative represents convexity relative to a factor:

1 ∂^2 V

Qj = ---- ----------

V ∂f^2

j

First- and second-order immunization conditions become the equality of

factor duration and convexity relative to cash flows and liabilities.

Scenario Optimization

The above strategies are based on perturbing the term structure of inter-

est rates with a linear function of one or more factors. We allow sto-

chastic behavior as rates can vary (albeit in a controlled way through

factors) and impose immunization constraints. We can obtain a more

general formulation of a stochastic problem in terms of scenarios.^17 Let

the variables be stochastic but assume distributions are discrete. Scenar-

(^17) Ron Dembo, “Scenario Immunization,” in Financial Optimization.