Lecture 2 The Universal Principle of Risk ManagementProbability and InsuranceProbability and Its RulesInsurance and Multiplication RuleExpected Value, Mean, AverageGeometric MeanVariance and Standard DeviationCovarianceCorrelationRegression, Beta=.5, corr=.93DistributionsNormal DistributionNormal Versus Fat-TailedExpected UtilityPresent Discounted Value (PDV)Consol and Annuity FormulasInsurance AnnuitiesProblems Faced by Insurance CompaniesLecture 2The Universal Principle of Risk ManagementPooling and Hedging of RiskProbability 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 InsuranceProbability 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 = Pn•Probability of x accidents in n policies (Binomial Distributon):))!(!/(!)1()()(xnxnPPxfxnxExpected Value, Mean, Average1)()(iixxxxprobxEi xdxxfxEx)()(nii nxx1/Geometric Mean•For positive numbers only•Better than arithmetic mean when used for (gross) returns•Geometric  Arithmeticnniixx/11)()G(Variance and Standard Deviation•Variance (2)is a measure of dispersion•Standard deviation  is square root of variance21))E()(prob()var( xxxxxiiinxxsniix/)(212Covariance•A Measure of how much two variables move togethernyyxxyxni/))((),cov(1Correlation•A scaled measure of how much two variables move together•-1 1)/(),cov(yxssyxRegression, Beta=.5, corr=.93Return XYZ Corporation against Market 1990-20010510152025-10 -5 0 5 10 15 20 25Return on the MarketReturn on XYZ CorporationEach point represents a year.Linear (Each point represents a year.)Distributions•Normal distribution (Gaussian) (bell-shaped curve)•Fat-tailed distribution common in financeNormal DistributionNormal Distribution with Zero Mean00.050.10.150.20.250.30.350.40.45-15 -10 -5 0 5 10 15Return (x)f(x)Standard Dev. = 3Standard Dev. = 1Normal Versus Fat-TailedNormal Versus Fat Tailed Distributions00.050.10.150.20.250.30.350.40.45-15 -10 -5 0 5 10 15Return xf(x)Normal DistributionCauchy DistributionExpected 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(2k-1))()prob()E(1iiixUxxU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,..,xntTttrx )1/(PDV1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)/(PDV Consol Growing/ PDV ConsolgrxrxrrTx)1/(11PDVAnnuity Insurance AnnuitiesLife annuities: Pay a stream of income until a person dies.Uncertainty faced by insurer is termination date TProblems 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