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Topic: Linear approximation to certainty equivalent for small-valued random variable
Replies: 3   Last Post: Nov 21, 2012 5:21 PM

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Paul

Posts: 263
Registered: 2/23/10
Re: Linear approximation to certainty equivalent for small-valued
random variable

Posted: Nov 16, 2012 1:43 AM
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On Nov 15, 11:22 pm, Paul <paul.domas...@gmail.com> wrote:
> I'm following Keeney's interpretation of risk aversion at
> http://tinyurl.com/d2jskgb. For a lottery involving the addition of
> a small-valued zero-mean random variable x~ to a (presumably much
> larger) offset x0, the definition of risk premium (equation 4.15) is
> Taylor expanded (4.16 and 4.17) before dropping all terms beyond
> first order (4.18).
>
> I can see why this is justified in 4.17, but I'm not 100% sure in
> 4.16. Usually, higher order terms are dropped when small numbers
> are raised to high powers. In 4.16 this case, would the reason be
> that pi is small? It is after all the risk premium for x~. Since
> x~ is very small, the expectation and the mean are small. x~ is the
> difference between expectation and mean, and so it must be small?


Correction: Pi is the risk premium for (x0 + x~), not for x~.
However, the idea is the same. That is, x~ has a small value range,
so (x0 + x~) does as well. Since both the mean and the certainty
equivalent must occur within that small value range, the difference
beween them (i.e. the risk premium) must also be small.



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