# Lotteries and risk aversion

## Introduction

### Lotteries

A choice may not have a certain outcome.

For example an action could have have 50

### Von Neumann-Morgenstern utility theorem

We can model any risk preference as:

$$U[L]=\sum_i p_iu(x_i)$$

If the agent is risk neutral we can use:

$$u(x_i)=x_i$$

If the agent is risk averse:

$$u(x_i)=\ln x_i$$

If the agent is risk loving we can use:

$$u(x_i)=x_i^2$$

## Absolute risk aversion

### Absolute risk aversion

Given a utility function we can calculate the risk aversion.

$$A(x)=-\dfrac{u''(x)}{u'(x)}$$

Constant Absolute Risk Aversion (CARA) is:

$$A(x)=c$$

$$u(x)=1-e^{\alpha x}$$

Hyperbolic Absolute Risk Aversion (HARA) is:

$$A(x)=\dfrac{1}{ax+b}$$

Increasing and Decreasing Absolute Risk Aversion (IARA and DARA):

Risk aversion increase or decreases in $$x$$.

## Relative risk aversion

### Relative risk aversion

Absolute risk aversion is:

$$A(x)=-\dfrac{u''(x)}{u'(x)}$$

$$R(x)=xA(x)$$

$$R(x)=-\dfrac{xu''(x)}{u'(x)}$$

## Risk

### Expected utility

If an agent faces an uncertain world they can make decisions under uncertainty. For example, how would an agent value £10 relative to a 50 There are many different attitudes an agent could have – we need a form which can capture these. A standard approach is expected utility.

We start by taking

$$E[u(x)]=$$

Subjective expected utility

### Risk aversion

HARA Hyperbolic Absolute Risk Aversion

CRRA Constant Relative Risk Aversion

### Prospect theory

Cumulative prospect theory.