Lecture 2 The Universal Principle of Risk Management Pooling and Hedging of Risk

Probability and Insurance Concept of probability began in 1660s Concept of probability grew from interest in gambling. Mahabarata story (ca. 400 AD) of Nala and Rtuparna, suggests some probability theory was understood in India then. Fire of London 1666 and Insurance

Probability and Its Rules Random variable: A quantity determined by the outcome of an experiment Discrete and continuous random variables Independent trials Probability P, 0<P<1 Multiplication rule for independent events: Prob(A and B) = Prob(A)  Prob(B)

Insurance and Multiplication Rule Probability of n independent accidents = P n Probability of x accidents in n policies (Binomial Distributon):

Expected Value, Mean, Average

Geometric Mean For positive numbers only Better than arithmetic mean when used for (gross) returns Geometric  Arithmetic

Variance and Standard Deviation Variance (  2 )is a measure of dispersion Standard deviation  is square root of variance

Covariance A Measure of how much two variables move together

Correlation A scaled measure of how much two variables move together -1  1

Regression, Beta=.5, corr=.93

Distributions Normal distribution (Gaussian) (bell-shaped curve) Fat-tailed distribution common in finance

Normal Distribution

Normal Versus Fat-Tailed

Expected Utility Pascal’s Conjecture St. Petersburg Paradox, Bernoulli: Toss coin until you get a head, k tosses, win 2 (k-1) coins. With log utility, a win after k periods is worth ln(2 k-1 )

Present Discounted Value (PDV) PDV of a dollar in one year = 1/(1+r) PDV of a dollar in n years = 1/(1+r) n PDV of a stream of payments x 1,..,x n

Consol and Annuity Formulas Consol pays constant quantity x forever Growing consol pays x(1+g)^t in t years. Annuity pays x from time 1 to T

Insurance Annuities Life annuities: Pay a stream of income until a person dies. Uncertainty faced by insurer is termination date T

Problems Faced by Insurance Companies Probabilities may change through time Policy holders may alter probabilities (moral hazard) Policy holders may not be representative of population from which probabilities were derived Insurance Company’s portfolio faces risk