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)

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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

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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)

Measures of Risk Aversion

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Risk Premium (Markowitz)

The risk premium , Rp, is de…ned as the amount of money that an agent is willing to pay to avoid a lottery. So, Rp = E (w ) b wRp ,

where wRp is the certainty equivalent wealth de…ned as U (wRp ) = E (U (w )). b

Martín Solá (FE)

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Risk Premium (Arrow Pratt R. A.)

Instead of de…ning the measure of risk for a lottery we can use another alternative measure. A risk averse agent will be indi¤erent between z }| { z }| { U (wRp ) = U (wo -Rp ) = E (U (w )) = E (U (wo + h)). b 1 2

, 9 a fair lottery h such that h1 p + h2 (1-p ) = 0

If we compute a …rst-order Taylor expansion of the …rst part around wo , we get U (wo -Rp ) = U (wo )-U 0 (wo )(Rp ) + O ((wo -Rp )2 ), where O (( )n ) is the rest of the Lagrangian for an order higher than n. Martín Solá (FE) Measures of Risk Aversion 08/10 7 / 41

Risk Premium (Arrow Pratt R. A.)

If we also expand U (wo + h) by Taylor around wo , we get U ( wo + h ) = U ( w o ) + U 0 ( w o ) h + U 00 (wo ) 2 h + O ((wo -Rp )3 ). 2

Computing the expected value of U (wo + h), E (U (wo + h)) = U (wo ) + U 0 (wo )E (h) + | {z } =0

Then, making equal both expressions, Rp = -

U 00 (wo ) E (h )2 . | {z } 2

=Var (h )

1 U 00 (wo )σ2 h . 2 U 0 ( wo )

Martín Solá (FE)

Measures of Risk Aversion

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Risk Premium (Arrow Pratt R. A.)

Using this last expression, we can de…ne the Coe¢ cient of Absolute Risk Aversion (or Arrow Pratt Coe¢ cient) as RA = U 00 (wo ) . U 0 ( wo )

This coe¢ cient measures the degree of aversion for each unit of risk.

Martín Solá (FE)

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Comparative Statics

Di¤erent utility functions imply a di¤erent behavior of the individual in front of an increase in wealth.

De…nition

A utility function U ( ) exhibits decreasing absolute risk aversion if RA is a A decreasing function in wealth, i.e., dR