# Meaning terms O() “terms order at most” and o( ) “terms of smaller order than”

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

So,

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.

• Although Arp11 seems to be the best interpretation of the formula, Arp22 is the closest approximation.* mistake: Although Arp22 seems to be the best interpretation of the formula with E(x) as x, Arp11 is the closest approximation. – david laport Jan 9 '19 at 17:18
• To answer the question somewhere in the middle: O() usually refers to the Landau Symbols: en.m.wikipedia.org/wiki/Big_O_notation. I'm a bit confused by all the rest though – nope Jan 9 '19 at 17:37