# Lecture 2: Probability and Insurance 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

x f ( x) P (1 P) ( n x) n! /( x!(n x)!) Expected Value, Mean, Average E ( x) x i 1 prob ( x xi ) xi

E ( x) x f ( x) xdx n x xi / n i 1 Geometric Mean For positive numbers only Better than arithmetic mean when used for (gross) returns Geometric Arithmetic n

1/ n G( x) ( xi ) i 1 Variance and Standard Deviation Variance (2)is a measure of dispersion Standard deviation is square root of variance i 1 var( x) prob( x xi )( xi E( x)) 2

2 x n s ( xi x) 2 / n i 1 Covariance A Measure of how much two variables move together n

cov( x, y ) ( x x)( y y ) / n i 1 Correlation A scaled measure of how much two variables move together -1 1 cov( x, y ) /( s x s y ) Regression, Beta=.5, corr=.93 Return XYZ Corporation against Market 1990-2001 25

Return on XYZ Corporation 20 15 Each point represents a year. Linear (Each point represents a year.) 10 5 0 -10

-5 0 5 10 Return on the Market 15 20

25 Distributions Normal distribution (Gaussian) (bell-shaped curve) Fat-tailed distribution common in finance Normal Distribution Norm al Distribution w ith Zero Mean 0.45 0.4 0.35 0.3 0.25

f(x) Standard Dev. = 3 Standard Dev. = 1 0.2 0.15 0.1 0.05 0 -15 -10

-5 0 Return (x) 5 10 15 Normal Versus Fat-Tailed Normal Versus Fat Tailed Distributions 0.45

0.4 0.35 0.3 f(x) 0.25 Normal Distribution Cauchy Distribution 0.2 0.15 0.1

0.05 0 -15 -10 -5 0 Return x 5 10

15 Expected Utility Pascals 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(2k-1) E(U ) prob( x xi )U ( xi ) i 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 x1,..,xn T PDV xt /(1 r ) t t 1 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

Consol PDV x / r Growing Consol PDV x /(r g ) Annuity PDV x 1 1 /(1 r )T r 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 Companys portfolio faces risk