The ActEd notes maintain that the absolute risk aversion function:

A(w) = -U''(w)/U'(w)

Can be ascribed to the fact that "*The absolute value of the certainty equivalent of a fair gamble is proportional to -U''(w)/U'(w)".*

Attempting to verify this for the function U(w)=log(w) does not appear to yield that conclusion. Consider the following where w is the initial level of wealth:

A fair additive gamble with payoff x s.t.

E(Gamble) = 1/2x + 1/2(-x) = 0

Now, we have that the certainty equivalent (CE) is given by:

log(CE) = 1/2 log(w+x) + 1/2 log (w-x)

log(CE) = 1/2 log( (w+x)(w-x) )

log(CE) = log( [(w+x)(w-x)] ^ 1/2)

|c_x| = [(w+x)(w-x)] ^ 1/2 - w

However,

-U''(w)/U'(w) = 1/w

Thus c_x does not appear to be proportional to the absolute risk aversion measure A(w).

Hi Ehrich, thanks for your help. Been wrestling with the question again and trying to wrap my head around it as per your instruction, but still a little bit stuck.

Could you please explain how to expand 1+x/w to 1+0.5*x/w-1/8*x^2/w^2?

Our formula sheet would state the Taylor expansion as follows:

f(w+h)=f(w) + h*f'(w) + h^2/2! f''(w) + ...

I'm assuming that here we take x to be h. So I need to find some way to express 1+x/w as function of (w+x).

What I attempted was to use the function f(r)=1/r since this would give me an approximation for f(w+x) = 1/(w+x).

Multiplying the resulting expression by w and then raising it to the power of -1, would give me an approximation for (w+x)/w = 1+x/w as required.