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Lecture 2 The Universal Principle of Risk Management

This lecture explores foundational concepts in risk management, including the principles of pooling and hedging risks. It traces the historical evolution of probability from gambling roots in the 1660s to its application in modern insurance. Key topics include random variables, independent trials, and the binomial distribution. The session also delves into variance, standard deviation, correlation, and the normal distribution, highlighting their relevance in finance. Finally, we address challenges faced by insurance companies, including changing probabilities and moral hazards.

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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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