Outline - Pooling - CB

Principles of Risk Management Pooling

Objectives

  • Ability to define "pooling".

  • Ability to calculate effects of pooling on risk management decisions.

I. Review/Complete Measurement to This Point

A. Measures of Central Tendency

  • Represents the "best estimate" of future outcomes.

B. Measures of Dispersion (Risk)

  1. Scenario Analysis: If observations of $0 losses decline by 200, while $200 and $1000 losses rise by 100 each, it suggests an increase in overall risk due to higher expected losses.

  2. If the largest potential loss increases to $100,000 from $10,000, the risk profile becomes more severe, indicating higher potential liability.

C. Definition of Correlation

  • Understanding the significance of the relationship between two or more variables.

II. Effect of Pooling Uncorrelated Events (Law of Large Numbers)

A. Example Distribution for Individual Automobile Accidents

  • Outcome (Loss Value) & Probability:

    • $10,000 (0.05 probability)

    • $0 (0.95 probability)

  1. Expected Loss Per Individual:

    • Calculation: (0.05 * $10,000) + (0.95 * $0) = $500.

  2. Standard Deviation Per Individual:

    • Formula: √(σ²) where σ² = (0.05 * ($10,000 - $500)² + 0.95 * ($0 - $500)²).

B. Impact of Pooling Between Two Individuals

  1. Definition of Pooling: Combining losses and sharing risk.

  2. New Combined Probability Distribution: Outcome and corresponding probability must be calculated for pooled values.

    • Example: Outcomes might be $20,000 with combined probabilities adjusted accordingly.

  3. Expected Loss When Pooling Two Individuals: Must average individual expected losses.

  4. Standard Deviation When Pooling Two Individuals: Must calculate pooled standard deviations considering correlation (or lack thereof).

C. Impact of Pooling Among Many Individuals

  1. Expected Loss Per Individual: Remains constant despite pooling.

  2. Standard Deviation Per Individual: Declines with increased pooling, indicating risk reduction.

III. Central Limit Theorem

A. Relative Dispersion with Increased Observations

  • As sample size increases, relative dispersion diminishes (Law of Large Numbers).

B. Distribution Approaching Normality

  • With more observations, the distribution approximates a Normal Distribution.

IV. Normal Probability Distribution

A. Characteristics

  • Symmetrical and bell-shaped. Not all symmetrical distributions are Normal.

B. Definition by Mean and Standard Deviation

  • Mean ± 1 standard deviation covers 68% of distribution.

  • Mean ± 2 standard deviations covers 95% of distribution.

  • Mean ± 3 standard deviations covers 99% of distribution.

  • Example Case: If Amazon's estimated workers’ compensation losses are $750,000 mean and $200,000 standard deviation:

    • Expected range for 95% observations: $750,000 +/- $400,000 = between $350,000 and $1,150,000.

    • If adjusting leads to mean of $975,000 with standard deviation of $10,000, the reduction in variability is beneficial, indicating worthwhile change.

C. Benefits of Normal Distribution

  • Aids in risk assessment and decision-making with reliable estimations.

V. Effects of Correlated Events on Pooling

  • Understanding how correlations affect risk pooling dynamics and overall risk profile.