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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.

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  • $\begingroup$ 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. $\endgroup$ – david laport Jan 9 at 17:18
  • $\begingroup$ 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 $\endgroup$ – nope Jan 9 at 17:37
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You can find more here. For Big-O:

in mathematics, it is commonly used to describe how closely a finite series approximates a given function, especially in the case of a truncated Taylor series or asymptotic expansion

And about little-o:

The difference between the earlier definition for the big-O notation and the present definition of little-o, is that while the former has to be true for at least one constant M, the latter must hold for every positive constant ε, however small.[5] In this way, little-o notation makes a stronger statement than the corresponding big-O notation: every function that is little-o of g is also big-O of g, but not every function that is big-O of g is also little-o of g.

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  • $\begingroup$ The precise derivation of the formula with O() and o() are shown on page 125 in the attachments, in formulas 4a until 5. With final formula: rp = 0.5*σ^2*r(x) + o(σ^2) $\endgroup$ – david laport Jan 9 at 17:46

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