Chapter 2 Redemption of Loan Dr A PHILIP

Chapter 2 Redemption of Loan Dr. A. PHILIP AROKIADOSS Assistant Professor Department of Statistics St. Joseph’s College (Autonomous) Tiruchirappalli-620 002.

Probability and Insurance • Concept of probability began in 1660 s • 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 = Pn • 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, . . , xn

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