lecture5

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Unformatted text preview: Click to edit Master subtitle style Page ‹#› Lecture #5: Decision Making Under Risk and Uncertainty (Part 1) Decision Making under Risk and Uncertainty (Part 1 of 4) 1 1 Lecture #5: Decision Making Under Risk and Uncertainty (Part 1) "The evils of uncertainty must count for something". Alfred Marshall (1920), in Principles of Economics . Page ‹#› Lecture #5: Decision Making Under Risk and Uncertainty (Part 1) What we learned from the last 2 lectures Expected values, variances, standard deviations, covariances, correlations Discrete and continuous probability distributions Central Limit Theorem Normal Distribution 2 2 Lecture #5: Decision Making Under Risk and Uncertainty (Part 1) Page ‹#› Lecture #5: Decision Making Under Risk and Uncertainty (Part 1) Risk Pooling Making decisions under certainty Making decisions under uncertainty: the expected value model The St. Petersburg Paradox Expected utility model 3 3 Lecture #5: Decision Making Under Risk and Uncertainty (Part 1) Today’s Game Plan Page ‹#› Lecture #5: Decision Making Under Risk and Uncertainty (Part 1) Risk Pooling Let E ( L i ) = expected loss for insured i and 1 ( ) ( ) n T i i E L E L = = ∑ = total expected loss of the risk pool. Then E ( L p ) = w E L i i i n ( ) = ∑ 1 (1) = average loss per policy, where w i = E ( L i )/ E ( L T ). Similarly, σ ρ σ σ Lp i j n i n j ij i j w w 2 1 1 = = = ∑ ∑ (2) = average risk per policy, where ρ ij = σ i / σ i σ j = correlation between losses on policy i and policy j . 4 4 Lecture #5: Decision Making Under Risk Page ‹#› Lecture #5: Decision Making Under Risk and Uncertainty (Part 1) Risk Pooling • Let losses be identically distributed; i.e., E ( L i ) = μ, σ ι 2 = σ 2 , and w i = 1/ n for all insureds, while { 2 2 if if . ij i j i j i j σ ρσ ρ σ σ = ≠ = Therefore, (1) and (2) are rewritten E ( L p ) = w E L i i i n ( ) = ∑ 1 = (1/ n ) n μ = μ, and (1a) σ ρ σ L ij j n i n p n 2 2 2 1 1 1 = = = ∑ ∑ 2 2 1 . n n n σ ρσ- = + (2a) 5 5 Lecture #5: Decision Making Under Risk Page ‹#› Lecture #5: Decision Making Under Risk and Uncertainty (Part 1) Risk Pooling • Since 2 2 2 1 p L n n n σ σ ρσ- = + , 2 2 lim ; p L n σ ρσ →∞ = i.e., only covariance risk remains. • Now suppose losses are iid . Then 2 lim 0. p L n σ →∞ = Thus the average loss becomes more predictable as the number of risks pooled becomes large. o By pooling many independent risks, insurers can treat uncertain losses as almost known. o Risk pooling effectively defrays risk by exploiting the law of large numbers. 6 6 Lecture #5: Decision Making Under Risk Page ‹#›...
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