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) Todays 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, Lp i j n i n j ij i j w w 2 1 1 = = = (2) = average risk per policy, where ij = i / i 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 ) = , 2 = 2 , and w i = 1/ n for all insureds, while { 2 2 if if . ij i j i j i j = = Therefore, (1) and (2) are rewritten E ( L p ) = w E L i i i n ( ) = 1 = (1/ n ) n = , and (1a) L ij j n i n p n 2 2 2 1 1 1 = = = 2 2 1 . n n n - = + (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 - = + , 2 2 lim ; p L n = i.e., only covariance risk remains. Now suppose losses are iid . Then 2 lim 0. p L n = 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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