Risk Management and Financial Institutions, + Web Site, 3rd Edition

APPENDIX G

Taylor Series Expansions

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

EXAMPLE G.1

Consider the function so that

Suppose that x=2 and Δ x = 0.1 so that . When x=2

The first order approximation to Δ z, given by equation (G.1), is

The second order approximation, given by equation (G.2), is

The third order approximation is