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Lecture 2 The Universal Principle of Risk Management

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.

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Lecture 2 The Universal Principle of Risk Management

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

  2. 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

  3. 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)

  4. Insurance and Multiplication Rule • Probability of n independent accidents = Pn • Probability of x accidents in n policies (Binomial Distributon):

  5. Expected Value, Mean, Average

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

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

  8. Covariance • A Measure of how much two variables move together

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

  10. Regression, Beta=.5, corr=.93

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

  12. Normal Distribution

  13. Normal Versus Fat-Tailed

  14. 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(2k-1)

  15. 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

  16. 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

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

  18. 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

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