Understand the principles of rational choice theory

Principles of Rational Choice Theory in Actuarial Modelling

Welcome to Rational Choice Theory for CM2! In financial economics and actuarial science, we model how economic agents make optimal decisions when faced with uncertainty and risk. Instead of looking at simple consumer goods under certainty, CM2 focuses on decision-makers choosing among uncertain financial payoffs and investment opportunities by maximising expected utility.

The Foundations: Axioms of Rationality

To establish a coherent mathematical framework for choice under uncertainty, rational choice theory assumes that an individual's preferences across uncertain prospects (or lotteries) satisfy fundamental axioms (the von Neumann-Morgenstern axioms):

1. Completeness: For any two risky prospects \( A \) and \( B \), the individual can rank them: either \( A \succ B \), \( B \succ A \), or \( A \sim B \) (indifference).
2. Transitivity: If \( A \succ B \) and \( B \succ C \), then \( A \succ C \).
3. Continuity: If \( A \succ B \succ C \), there exists some probability \( p \in (0, 1) \) such that the decision-maker is indifferent between getting \( B \) for certain and a lottery yielding \( A \) with probability \( p \) and \( C \) with probability \( 1 - p \).
4. Independence: If \( A \succ B \), then for any third prospect \( C \) and any probability \( p \in (0, 1] \), a lottery combining \( A \) and \( C \) is strictly preferred to a lottery combining \( B \) and \( C \): \( pA + (1-p)C \succ pB + (1-p)C \).

The Expected Utility Theorem

When an individual's preferences satisfy these axioms, their behaviour can be represented by a cardinal utility function \( U(w) \) defined over wealth \( w \). Under the Expected Utility Theorem, given a risky prospect with payoffs \( w_i \) occurring with probabilities \( p_i \), the decision-maker chooses the option that maximises the expected value of utility:

\( E[U(W)] = \sum_i p_i U(w_i) \) (in the discrete case) or \( E[U(W)] = \int U(w) f(w) dw \) (in the continuous case).

Key Properties of Utility Functions

1. Non-Satiation

A rational investor prefers more wealth to less. Mathematically, marginal utility is strictly positive:
\( U'(w) > 0 \)

2. Attitudes Toward Risk

An individual's risk preference is determined by the curvature of their utility function (the sign of the second derivative, \( U''(w) \)):

Risk Aversion: The investor dislikes risk and prefers a certain outcome over a risky prospect with the same expected payoff. By Jensen's Inequality, \( E[U(W)] < U(E[W]) \). This corresponds to a strictly concave utility function, where marginal utility decreases as wealth rises:
\( U''(w) < 0 \)
Risk Neutrality: The investor is indifferent between a risky gamble and its expected value: \( E[U(W)] = U(E[W]) \). This corresponds to a linear utility function:
\( U''(w) = 0 \)
Risk Seeking: The investor prefers the gamble to its expected value: \( E[U(W)] > U(E[W]) \). This corresponds to a strictly convex utility function:
\( U''(w) > 0 \)

Measures of Risk Aversion: Arrow-Pratt Measures

To quantify risk aversion independently of positive linear transformations of \( U(w) \), we use the Arrow-Pratt measures:

Absolute Risk Aversion (ARA)

The Coefficient of Absolute Risk Aversion is defined as:
\( A(w) = -\frac{U''(w)}{U'(w)} \)
- Decreasing Absolute Risk Aversion (DARA): \( A'(w) < 0 \). As an individual becomes wealthier, they are willing to hold more absolute currency amounts in risky assets. This is widely considered the most realistic economic behaviour.
- Constant Absolute Risk Aversion (CARA): \( A'(w) = 0 \). Wealth changes do not alter the monetary amount invested in risky assets.
- Increasing Absolute Risk Aversion (IARA): \( A'(w) > 0 \). The individual reduces their dollar exposure to risky assets as wealth increases.

Relative Risk Aversion (RRA)

The Coefficient of Relative Risk Aversion measures risk aversion relative to proportional bets and percentage of wealth:
\( R(w) = w A(w) = -\frac{w U''(w)}{U'(w)} \)
- Constant Relative Risk Aversion (CRRA): \( R'(w) = 0 \). As wealth changes, the individual keeps an unchanged proportion of their wealth in risky assets.
- Decreasing/Increasing Relative Risk Aversion (DRRA/IRRA): The proportion of wealth held in risky assets increases/decreases as wealth grows.

Commonly Used Utility Functions

The syllabus requires familiarity with the mathematical and economic properties of standard utility functions:

1. Quadratic Utility: \( U(w) = w - b w^2 \) (for \( w < \frac{1}{2b} \), \( b > 0 \))
- Derivatives: \( U'(w) = 1 - 2bw > 0 \), \( U''(w) = -2b < 0 \)
- Absolute Risk Aversion: \( A(w) = \frac{2b}{1 - 2bw} \), which increases with \( w \) (IARA). This property is economically counter-intuitive because it suggests rich people take fewer absolute financial risks.
- Key CM2 feature: Expected utility depends only on the mean and variance of wealth: \( E[U(W)] = E[W] - b E[W^2] = E[W] - b(Var(W) + (E[W])^2) \), linking rational choice directly to mean-variance portfolio theory.

2. Logarithmic Utility: \( U(w) = \ln(w) \) (for \( w > 0 \))
- Derivatives: \( U'(w) = \frac{1}{w} > 0 \), \( U''(w) = -\frac{1}{w^2} < 0 \)
- \( A(w) = \frac{1}{w} \) (DARA)
- \( R(w) = w \left(\frac{1}{w}\right) = 1 \) (CRRA)

3. Power Utility: \( U(w) = \frac{w^{1 - \gamma}}{1 - \gamma} \) (for \( \gamma > 0, \gamma \neq 1 \))
- Derivatives: \( U'(w) = w^{-\gamma} \), \( U''(w) = -\gamma w^{-\gamma - 1} \)
- \( A(w) = \frac{\gamma}{w} \) (DARA)
- \( R(w) = \gamma \) (CRRA, where \( \gamma \) is the constant coefficient of relative risk aversion)

4. Exponential Utility: \( U(w) = -e^{-a w} \) (for \( a > 0 \))
- Derivatives: \( U'(w) = a e^{-aw} \), \( U''(w) = -a^2 e^{-aw} \)
- \( A(w) = a \) (CARA, constant absolute risk aversion)
- \( R(w) = a w \) (IRRA)

Current Wealth and State-Dependent Utility

An individual's current initial wealth \( w_0 \) shifts the evaluation point along the utility curve. If an investment yields payoff \( X \), total final wealth is \( W = w_0 + X \), and the investor maximises \( E[U(w_0 + X)] \). Under DARA, an increase in \( w_0 \) makes the investor more willing to accept a given gamble.

In standard utility theory, utility depends solely on final wealth. However, under state-dependent utility, the utility function itself varies across external states of nature: \( U(w, s) \). For example, a payout received in a state of poor health or economic depression may yield higher marginal utility than the same payout received in a prosperous state.

Actuarial Applications: Insurance and Investment

1. Evaluating Investment Opportunities

To compare two risky assets \( X \) and \( Y \), an investor calculates \( E[U(w_0 + X)] \) and \( E[U(w_0 + Y)] \) and selects the one with the higher expected utility. The certainty equivalent \( C \) is the sure amount of wealth such that \( U(C) = E[U(W)] \). The difference \( E[W] - C \) represents the investor's risk premium.

2. Analysing Simple Insurance Problems

Consider an individual with initial wealth \( w \) facing a potential loss \( L \) with probability \( p \). If an insurer offers full insurance for a premium \( P \):
- Expected utility without insurance: \( E[U_{no\_ins}] = (1 - p) U(w) + p U(w - L) \)
- Utility with insurance: \( U_{ins} = U(w - P) \)
For a risk-averse individual (\( U'' < 0 \)), Jensen's inequality guarantees that they are willing to pay a maximum premium \( P_{max} > pL \) (a premium strictly greater than the fair premium) to eliminate the risk.

Summary Box

Core Points for CM2:
- Rational agents act to maximise expected utility \( E[U(W)] \).
- Non-satiation: \( U'(w) > 0 \); Risk aversion: \( U''(w) < 0 \).
- Arrow-Pratt measures: Absolute \( A(w) = -\frac{U''(w)}{U'(w)} \) and Relative \( R(w) = -\frac{w U''(w)}{U'(w)} \).
- Quadratic: IARA, depends only on mean and variance.
- Exponential: CARA.
- Power and Logarithmic: DARA and CRRA.
- Insurance: Risk-averse individuals are willing to pay a premium higher than the expected loss to achieve certainty.