Financial Economics

Martín Solá

October 2010

Martín Solá (FE)

Measures of Risk Aversion

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Introduction

In this …rst stage we will study the individual decisions of optimal portfolio choice under uncertainty and its consequences in the valuation of risky assets. In short, the Financial Theory rests on the no-arbitrage principle. The idea behind this principle is that it is not possible to make pro…ts without risk, without initial investment.

Martín Solá (FE)

Measures of Risk Aversion

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Risk Aversion (Jensen’ Inequality) s

Consider the following situation: given an initial wealth wo , there is a lottery where the possible results are h1 < 0 with probability p, h2 > 0 with probability (1-p ). A lottery is actuarially fair if the expected payo¤ is 0, h1 p + h2 (1-p ) = 0.

De…nition

An agent is risk averse if this agent is not willing to take an actuarially fair lottery.

Martín Solá (FE)

Measures of Risk Aversion

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Risk Aversion (Jensen’ Inequality) s

So, we have the following inequality U ( wo ) U (wo + h1 )p + U (wo + h2 )(1-p ).

Since, by de…nition, an actuarially fair lottery satis…es wo = (wo + h1 )p + (wo + h2 )(1-p ), we can rewrite the inequality as U ((wo + h1 )p + (w0 + h2 )(1-p )) U (w0 + h1 )p + U (w0 + h2 )(1-p )

This relation proves that the risk aversion implies that the agent’ s utility function to be concave, and a concave utility function implies risk aversion.

Martín Solá (FE)

Measures of Risk Aversion

08/10

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Risk Aversion (Jensen’ Inequality) s

If we de…ne the (ex-ante) wealth as a random variable w = wo + h, b where h is the random variable previously de…ned, the Jensen’ s Inequality can be expressed as U (E (w )) b E (U (w )). b

Remember that concavity implies that the marginal utility of wealth is decreasing. For a risk averse agent is worthless to take a fair lottery.

Martín Solá (FE)