In the paper: "Risk aversion in the small and in the large" by John Pratt from 1964, a formula is derived for the approximation of the risk premium: rp ≈ 0.5*σ^2*r(x)
The risk can be represented with a binary outcome of either xlow or xhigh.
p=probability of a high outcome
Δ=xhigh - xlow
In which σ^2=Δ^2*p(1-p)
If utility would be quantified with ln(x) then r(x)=1/x
rp ≈ 0.5*Δ^2*p(1-p)*1/x
If you choose the certainty equivalent as x, the approximation is actually more precise than when you choose x. But you need the output of the formula: the risk premium, to use it (CE). So this is clearly not possible.
What do the terms: O() and o( ) mean, and can they be represented with a mathematical formula?
In the attachments I have added the first four pages of the John Pratt paper and two pictures.
The first picture shows the quantification of the most common risk preference: risk aversion.
The second picture shows the risk premia for the whole continuum of the expected x values. The risk premium is the highest somewhere in the middle, since the variance: Δ^2*p(1-p), is the highest at p=50%. The red line shows the exact risk premium. Arp11 shows the approximation in which the CE is used as x and to calculate p (p=(CE-xlow)/(xhigh-xlow)). In Arp22 E(x) is used as the x value and to derive p (p=(E(x)-xlow)/(xhigh-xlow)).
Although Arp11 seems to be the best interpretation of the formula, Arp22 is the closest approximation.