Risk Management and Financial Institutions, 4th Edition

APPENDIX G Taylor Series Expansions

Consider a function z = F(x). When a small change Δx is made to x, there is a corresponding small change Δz in z. A first approximation to the relationship between Δz and Δx is

(G.1)

This relationship is exact if z is a linear function of x and approximate in other situations. A more accurate approximation is

(G.2)

This relationship is exact if z is a quadratic function of x and approximate in other situations. By adding more terms in the series, we can increase accuracy. The full expansion is

where ! denotes the factorial function: 2! = 2 × 1 = 2$; $3! = 3 × 2 × 1 = 6$; $4! = 4 × 3 × 2 × 1 = 24; and so on.

EXAMPLE G.1

Consider the function so that

Suppose that x = 2 and Δx = 0.1 so that . When x = 2,

The first order approximation to Δ ...