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Home/Financial Mathematics/Course 3

Course 3

Theory

6-7 hours

Utility Theory and Risk Preferences

Expected utility theory, risk aversion, and optimal decision-making under uncertainty

Learning Objectives

•Understand von Neumann-Morgenstern expected utility theory and its axiomatic foundation
•Master the concepts of risk aversion, risk neutrality, and risk seeking
•Derive and apply Arrow-Pratt measures of absolute and relative risk aversion
•Calculate certainty equivalents and risk premiums for various utility functions
•Apply utility theory to portfolio choice and insurance decisions

1. Preferences Under Certainty

We begin with the simpler case of preferences over certain outcomes, which provides the foundation for understanding preferences under uncertainty.

Axiom 1.1: Completeness

For any two outcomes $x$ and $y$ , either $x \succeq y$ (x is weakly preferred to y), or $y \succeq x$ , or both.

This says the agent can always compare any two options.

Axiom 1.2: Transitivity

If $x \succeq y$ and $y \succeq z$ , then $x \succeq z$ .

This ensures consistency in rankings and rules out preference cycles.

Theorem 1.1: Utility Representation Theorem (Certainty)

If preferences $\succeq$ satisfy completeness and transitivity (and continuity), then there exists a utility function $u: X \to \mathbb{R}$ such that:

x \succeq y \iff u(x) \geq u(y)

Moreover, $u$ is unique up to positive affine transformation: if $v$ also represents $\succeq$ , then $v = au + b$ for some $a > 0$ and $b$ .

Remark 1.1: Ordinal vs Cardinal Utility

Under certainty, utility is ordinal - only the ranking matters, not the magnitude of utility differences. We can apply any increasing transformation without changing preferences.

Under uncertainty (expected utility), utility becomes cardinal - the magnitudes matter because we take expectations. Only affine transformations preserve expected utility preferences.

2. von Neumann-Morgenstern Expected Utility Theory

Definition 2.1: Lottery

A lottery (or risky prospect) $L$ is a probability distribution over outcomes. For finite outcomes ${x_1, \ldots, x_n}$ with probabilities ${p_1, \ldots, p_n}$ :

L = (x_1, p_1; x_2, p_2; \ldots; x_n, p_n)

where $p_i \geq 0$ and $\sum_{i=1}^n p_i = 1$ .

Axiom 2.1: Independence Axiom

For any lotteries $L$ , $M$ , $N$ and $\alpha \in (0,1)$ :

L \succeq M \iff \alpha L + (1-\alpha)N \succeq \alpha M + (1-\alpha)N

This says preferences between $L$ and $M$ are unaffected by mixing both with a common lottery $N$ .

Axiom 2.2: Continuity (Archimedean)

If $L \succ M \succ N$ , then there exist $\alpha, \beta \in (0,1)$ such that:

\alpha L + (1-\alpha)N \succ M \succ \beta L + (1-\beta)N

This rules out infinitely preferred or infinitely dispreferred outcomes.

Theorem 2.1: von Neumann-Morgenstern Representation

If preferences over lotteries satisfy completeness, transitivity, continuity, and independence, then there exists a utility function $u$ such that lottery $L$ is preferred to lottery $M$ if and only if:

\mathbb{E}_L[u] > \mathbb{E}_M[u]

For lottery $L = (x_1, p_1; \ldots; x_n, p_n)$ :

\mathbb{E}_L[u] = \sum_{i=1}^n p_i u(x_i)

Moreover, $u$ is unique up to positive affine transformation.

Proof of VNM Representation (Sketch):

Step 1: Normalize utility by setting $u(x_{best}) = 1$ and $u(x_{worst}) = 0$ .

Step 2: For any outcome $x$ , by continuity, there exists $p \in [0,1]$ such that:

x \sim p x_{best} + (1-p)x_{worst}

Define $u(x) = p$ .

Step 3: For lottery $L = (x_1, p_1; \ldots; x_n, p_n)$ , let $u(x_i) = q_i$ . By independence:

L \sim \sum_{i=1}^n p_i[q_i x_{best} + (1-q_i)x_{worst}]

= \left(\sum_{i=1}^n p_i q_i\right) x_{best} + \left(1 - \sum_{i=1}^n p_i q_i\right)x_{worst}

Therefore $u(L) = \sum_{i=1}^n p_i q_i = \sum_{i=1}^n p_i u(x_i) = \mathbb{E}_L[u]$ .

∎

Example 2.1: The St. Petersburg Paradox

A fair coin is tossed repeatedly until heads appears. If heads first appears on toss $n$ , you receive $\$2^n$ .

Expected value:

\mathbb{E}[X] = \sum_{n=1}^\infty \left(\frac{1}{2}\right)^n \times 2^n = \sum_{n=1}^\infty 1 = \infty

Despite infinite expected value, people typically pay only $10-$25 to play. Why?

Using utility $u(w) = \ln(w)$ with initial wealth $w_0 = 100$ :

\mathbb{E}[u] = \sum_{n=1}^\infty \left(\frac{1}{2}\right)^n \ln(100 + 2^n)

This sum converges! The certainty equivalent solves $\ln(CE) = \mathbb{E}[u]$ , yielding a finite willingness to pay.

This paradox motivated Bernoulli (1738) to propose utility theory.

3. Risk Attitudes and Arrow-Pratt Measures

Definition 3.1: Risk Aversion

An agent with utility $u$ is risk-averse if for any lottery $L$ :

u(\mathbb{E}[L]) > \mathbb{E}[u(L)]

By Jensen's inequality, this holds if and only if $u$ is strictly concave: $u'' < 0$ .

Definition 3.2: Risk Neutrality and Risk Seeking

Risk-neutral: $u(\mathbb{E}[L]) = \mathbb{E}[u(L)]$ for all $L$ . Equivalently, $u$ is linear: $u'' = 0$ .
Risk-seeking: $u(\mathbb{E}[L]) < \mathbb{E}[u(L)]$ for all $L$ . Equivalently, $u$ is convex: $u'' > 0$ .

Definition 3.3: Certainty Equivalent

The certainty equivalent $CE$ of lottery $L$ is the certain amount satisfying:

u(CE) = \mathbb{E}[u(L)]

For risk-averse agents: $CE < \mathbb{E}[L]$ . The difference is the risk premium:

\pi = \mathbb{E}[L] - CE

Theorem 3.1: Arrow-Pratt Coefficient of Absolute Risk Aversion

The coefficient of absolute risk aversion is:

A(w) = -\frac{u''(w)}{u'(w)}

For a small fair gamble $\tilde{\epsilon}$ with $\mathbb{E}[\tilde{\epsilon}] = 0$ and variance $\sigma^2$ , the risk premium is approximately:

\pi \approx \frac{1}{2}A(w)\sigma^2

Proof of Risk Premium Approximation:

Consider wealth $w$ and gamble $\tilde{\epsilon}$ . Taylor expand around $w$ :

u(w + \tilde{\epsilon}) \approx u(w) + u'(w)\tilde{\epsilon} + \frac{1}{2}u''(w)\tilde{\epsilon}^2

Taking expectations (using $\mathbb{E}[\tilde{\epsilon}] = 0$ , $\mathbb{E}[\tilde{\epsilon}^2] = \sigma^2$ ):

\mathbb{E}[u(w + \tilde{\epsilon})] \approx u(w) + \frac{1}{2}u''(w)\sigma^2

The certainty equivalent satisfies $u(CE) = \mathbb{E}[u(w + \tilde{\epsilon})]$ . Taylor expanding $u(CE) \approx u(w) + u'(w)(CE - w)$ :

u(w) + u'(w)(CE - w) \approx u(w) + \frac{1}{2}u''(w)\sigma^2

Therefore:

CE - w \approx \frac{u''(w)}{2u'(w)}\sigma^2

Since $\pi = \mathbb{E}[w + \tilde{\epsilon}] - CE = w - CE$ :

\pi \approx -\frac{u''(w)}{2u'(w)}\sigma^2 = \frac{1}{2}A(w)\sigma^2

∎

Definition 3.4: Coefficient of Relative Risk Aversion

The coefficient of relative risk aversion is:

R(w) = -\frac{w u''(w)}{u'(w)} = w \cdot A(w)

This measures risk aversion in proportional terms, useful for comparing agents with different wealth levels.

Example 3.1: Common Utility Functions and Their Risk Aversion

1. Exponential (CARA): $u(w) = -e^{-aw}$

u'(w) = ae^{-aw}, \quad u''(w) = -a^2e^{-aw}

A(w) = \frac{a^2e^{-aw}}{ae^{-aw}} = a \quad (\text{constant!})

R(w) = aw \quad (\text{increasing in } w)

CARA = Constant Absolute Risk Aversion

2. Power (CRRA): $u(w) = \frac{w^{1-\gamma}}{1-\gamma}$

u'(w) = w^{-\gamma}, \quad u''(w) = -\gamma w^{-\gamma-1}

A(w) = \frac{\gamma w^{-\gamma-1}}{w^{-\gamma}} = \frac{\gamma}{w}

R(w) = \gamma \quad (\text{constant!})

CRRA = Constant Relative Risk Aversion. Special case: $\gamma = 1$ gives $u(w) = \ln(w)$ .

3. Quadratic: $u(w) = w - \frac{b}{2}w^2$

u'(w) = 1 - bw, \quad u''(w) = -b

A(w) = \frac{b}{1-bw} \quad (\text{increasing in } w)

R(w) = \frac{bw}{1-bw}

Quadratic utility implies mean-variance preferences but has the unrealistic feature of decreasing marginal utility.

Proposition 3.1: Comparing Risk Aversion

Agent 1 with utility $u_1$ is more risk-averse than agent 2 with utility $u_2$ if and only if:

A_1(w) \geq A_2(w) \quad \text{for all } w

Equivalently, $u_1 = g(u_2)$ for some concave increasing function $g$ .

Example 3.2: Calculating Certainty Equivalent

An investor with wealth $w = 100$ and utility $u(w) = \ln(w)$ faces a 50-50 gamble: win $20 or lose $20.

Expected value: $\mathbb{E}[\tilde{w}] = 0.5(120) + 0.5(80) = 100$

Expected utility:

\mathbb{E}[u] = 0.5\ln(120) + 0.5\ln(80) = \ln(\sqrt{120 \times 80}) = \ln(\sqrt{9600}) \approx \ln(97.98)

Certainty equivalent: $CE$ satisfies:

\ln(CE) = \ln(97.98) \implies CE \approx 97.98

Risk premium:

\pi = 100 - 97.98 = 2.02

The investor would pay up to $2.02 to avoid this risk!

4. Applications to Portfolio Choice and Insurance

Proposition 4.1: Optimal Portfolio with Risk-Free and Risky Asset

Consider an agent with wealth $W$ , allocating fraction $\alpha$ to a risky asset with return $\tilde{r}$ and $(1-\alpha)$ to a risk-free asset with return $r_f$ .

Terminal wealth: $\tilde{W} = W[1 + \alpha \tilde{r} + (1-\alpha)r_f]$

The optimal $\alpha^*$ maximizes $\mathbb{E}[u(\tilde{W})]$ . First-order condition:

\mathbb{E}[u'(\tilde{W})(\tilde{r} - r_f)] = 0

Example 4.1: Portfolio Choice with CARA Utility

With $u(W) = -e^{-aW}$ and $\tilde{r} \sim N(\mu, \sigma^2)$ , the optimal allocation is:

\alpha^* = \frac{\mu - r_f}{a\sigma^2}

Key insights:

Higher risk aversion $a$ leads to lower allocation to risky asset
Higher expected excess return $(\mu - r_f)$ leads to higher allocation
Higher variance $\sigma^2$ leads to lower allocation
Allocation is independent of wealth (CARA property)

Proposition 4.2: Insurance Demand

An agent with wealth $W$ faces potential loss $L$ with probability $p$ . Insurance costs $\lambda pL$ (where $\lambda \geq 1$ is the loading factor) to fully cover the loss.

For actuarially fair insurance ( $\lambda = 1$ ), a risk-averse agent fully insures. For $\lambda > 1$ , partial insurance is optimal.

Remark 4.1: Empirical Evidence on Risk Aversion

Estimates of relative risk aversion coefficient $\gamma$ :

Laboratory experiments: $\gamma \in [0.5, 2]$
Household portfolio data: $\gamma \in [1, 5]$
Asset pricing (equity premium puzzle): $\gamma > 30$ needed

The wide range suggests context-dependent risk preferences and possible violations of expected utility theory.

Key Takeaways

1.von Neumann-Morgenstern axioms provide foundations for expected utility maximization
2.Risk aversion corresponds to concave utility (u'' < 0); most people are risk-averse
3.Arrow-Pratt measures quantify risk aversion: ARA for absolute, RRA for relative
4.Risk premium equals the cost of eliminating risk, proportional to variance and risk aversion
5.CRRA utility is empirically realistic; CARA utility simplifies many portfolio problems
6.Utility theory explains insurance demand and portfolio allocation patterns

Frequently Asked Questions

Why can't we just use expected value to make decisions?

The St. Petersburg Paradox shows that expected value alone leads to absurd conclusions. People value outcomes non-linearly - the utility of $2 million is not twice the utility of $1 million. Expected utility theory accounts for this by using a concave utility function.

What does it mean to be risk-averse?

A risk-averse person prefers a certain outcome to a risky lottery with the same expected value. Mathematically, this corresponds to a concave utility function (u'' < 0). Most people exhibit risk aversion due to diminishing marginal utility of wealth.

How is certainty equivalent different from expected value?

Expected value is the probability-weighted average of payoffs. Certainty equivalent is the guaranteed amount that gives the same utility as the risky lottery. For risk-averse agents, CE < EV; the difference is the risk premium they'd pay to avoid risk.

What's the difference between absolute and relative risk aversion?

Absolute risk aversion (ARA) measures how risk aversion changes with wealth level in absolute terms. Relative risk aversion (RRA) measures it in percentage terms. Empirical evidence suggests constant relative risk aversion (CRRA) is realistic for most people.

Why are the von Neumann-Morgenstern axioms important?

These axioms provide conditions under which preferences can be represented by expected utility. If an agent's preferences satisfy these axioms (completeness, transitivity, continuity, independence), then there exists a utility function representing those preferences.

Practice Quiz

Utility Theory and Risk Preferences

Questions

Correct

Accuracy

If utility function is

u(w) = \ln(w)

, what is the utility of wealth

w = 100

Easy

Not attempted

Which utility function exhibits constant relative risk aversion (CRRA)?

Medium

Not attempted

An agent with

u(w) = \sqrt{w}

faces a 50-50 gamble: win

400 or lose

0. Current wealth is $100. What is the expected utility?

Medium

Not attempted

u(w) = -e^{-0.01w}

, what is the Arrow-Pratt coefficient of absolute risk aversion?

Hard

Not attempted

An agent is indifferent between

\

for sure and a 50-50 lottery of

0

\

110$. The agent is:

Easy

Not attempted

For utility

u(w) = w^{0.5}

, what is the coefficient of relative risk aversion?

Hard

Not attempted

The St. Petersburg Paradox involves a gamble with:

Medium

Not attempted

Which axiom states that preferences don't change when mixed with a common outcome?

Medium

Not attempted

If current wealth is

\

100

, utility is

u(w) = \ln(w)

, and a fair coin flip wins

20

or loses

\

20$, what is the certainty equivalent?

Hard

Not attempted

A risk-neutral agent has utility function:

Easy

Not attempted

For

u(w) = -\frac{1}{w}

, the agent is:

Medium

Not attempted

Insurance demand is primarily driven by:

Easy

Not attempted

1. Preferences Under Certainty

2. von Neumann-Morgenstern Expected Utility Theory

3. Risk Attitudes and Arrow-Pratt Measures

1. Exponential (CARA): u(w)=−e−awu(w) = -e^{-aw}u(w)=−e−aw

2. Power (CRRA): u(w)=w1−γ1−γu(w) = \frac{w^{1-\gamma}}{1-\gamma}u(w)=1−γw1−γ​

3. Quadratic: u(w)=w−b2w2u(w) = w - \frac{b}{2}w^2u(w)=w−2b​w2

4. Applications to Portfolio Choice and Insurance

Key Takeaways

Frequently Asked Questions

Why can't we just use expected value to make decisions?

What does it mean to be risk-averse?

How is certainty equivalent different from expected value?

What's the difference between absolute and relative risk aversion?

Why are the von Neumann-Morgenstern axioms important?

Practice Quiz

1. Exponential (CARA): $u(w) = -e^{-aw}$

2. Power (CRRA): $u(w) = \frac{w^{1-\gamma}}{1-\gamma}$

3. Quadratic: $u(w) = w - \frac{b}{2}w^2$