Fixed Income and Interest Rate Derivative Analysis

CONVEXITY 117

expectation. It is perhaps more clearly expressed in mathematical notation:

E[f(x)]

>_ f(E[x])

Let's take the simplest example I can think of. A simple convex function is

f(x) - x 2.

Now let's assume that x has a binomial distribution with equally

likely outcomes of 0 and 2. We can now check Jensen's inequality

E[f(x)] -

0.5 • (x - 0) 2 + 0.5 • (x - 2) 2

-2

and the right-hand side of the inequality is given by

f(E[x]) - f(1)

-1

So the inequality held for this case. Another way of understanding the

inequality more generally is via a second-order Taylor ...