Risk and Insurance

Introduction and Motivation

Motivation

From the Actuaries Institute:

“Actuaries evaluate risk and opportunity - applying mathematical, statistical, economic, and financial analyses to a wide range of business problems.”

A key concept here is risk; actuaries must understand the uncertainties and potential losses associated with various events, such as insurance claims, investment returns, and retirement benefits. Understanding the principles of risk transfer through insurance is fundamental: transferring a potential loss from the insured to the insurer in exchange for a premium.

Overview

The principles of risk and insurance are not just theoretical; they are applied tools that actuaries use to manage uncertainty and provide financial security. This week covers the basics of insurance economics, risk measurement, and the role of variability in risk assessment. More advanced topics will be covered in later courses, such as ACTL2111: Financial Mathematics and ACTL2131: Stochastic Models.

Some Applications

1. Personal Insurance
   • Health insurance - covering medical expenses
   • Life insurance - providing financial support to beneficiaries
2. Property and Casualty Insurance
   • Home insurance - protecting against property damage or theft
   • Auto insurance - covering vehicle damage and liability
3. Corporate Risk Management
   • Liability insurance - protecting businesses against legal claims
   • Business interruption insurance - covering loss of income due to disruptions

Economics of Risk

“Risk comes from not knowing what you’re doing.” - Warren Buffett

Definitions

• Risk - Future uncertainty that can affect the outcome of an event.
• Insurance - A financial arrangement where an insured pays a premium to transfer the risk of a potential loss to an insurer.

Expected Value and Variability

• Expected Value (E[X]) - The average outcome of a random variable X over many trials.
• Variance (Var[X]) - A measure of how much the outcomes of X deviate from the expected value.

Example: Investment Choices

You have a choice of investing 10,000 in two potential investments:

Outcome	Probability	Investment A	Investment B
Good	1/10	50,000	26,000
Middle	22/25	12,500	15,000
Bad	1/50	0	10,000

Solution

• Expected values:

\[E[A] = \frac{1}{10} \cdot 50,000 + \frac{22}{25} \cdot 12,500 + \frac{1}{50} \cdot 0 = 16,000\] \[E[B] = \frac{1}{10} \cdot 26,000 + \frac{22}{25} \cdot 15,000 + \frac{1}{50} \cdot 10,000 = 16,000\]

• Variance and standard deviation:

\[\text{Var}[A] = 131,500,000 \quad \text{and} \quad \sigma_A = 11,467.34\] \[\text{Var}[B] = 11,600,000 \quad \text{and} \quad \sigma_B = 3,405.88\]

Investment B has lower variability and is usually regarded as less risky.

Expected Utility and Risk Aversion

Utility Functions

• Utility (v(w)) - A measure of satisfaction or value derived from wealth w.
• Expected Utility - The average utility over all possible outcomes.

Risk Aversion

• An individual is risk averse if they prefer the expected value of wealth over the random wealth itself:

\[W \prec E[W] \quad \Longleftrightarrow \quad E[v(W)] < v(E[W]).\]

• Concave Utility Function - Indicates risk aversion:

\[\frac{\partial v}{\partial w} > 0 \quad \text{and} \quad \frac{\partial^2 v}{\partial w^2} < 0.\]

Insurance Example

• With Insurance: Final wealth is $w_0 - k\ell$ with certainty, where $k$ is the premium rate.
• Without Insurance: Final wealth is a random variable depending on whether the loss $\ell$ occurs.

The individual chooses insurance if:

\[v(w_0 - k \ell) > pv( w_0 - \ell) + (1-p)v( w_0 ).\]

Risk Pooling for Independent Risks

Setting and Assumptions

• Assume $n$ individuals each face an independent risk $X_i$ with identical distribution.
• Risk Pooling: Individuals agree to pay the actual average loss into a pool, which then covers the losses.

Expectation and Variance

• Expectation of the average loss:

\[E[A] = E\left[ \sum_{i=1}^{n}X_{i} / n \right] = \mu\]

• Variance of the average loss:

\[\text{Var}[A] = \frac{\sigma^2}{n}.\]

Pooling reduces variability and is beneficial for risk-averse individuals.

Correlation and Dependence

Definitions

• Covariance - Measure of linear dependence between two variables.

\[Cov(X,Y) = E[(X-E[X])(Y-E[Y])] = E[XY] - E[X]E[Y].\]

• Correlation - Standardized measure of covariance:

\[\rho(X,Y) = \frac{Cov(X,Y)}{\sigma_X \sigma_Y}.\]

Pooling of Correlated Risks

• Variance of the average loss for two correlated risks X and Y:

\[Var\left( \frac{X+Y}{2} \right) = \frac{\sigma^2}{2} \left( 1 + \rho \right).\]

• Diversification Benefits: As long as $\rho \neq 1$, pooling reduces risk.

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Note: This content is a simplified representation for educational purposes. Real-world applications involve more complexity and require professional actuarial judgment.