Lecture 2 The Universal Principle of Risk Management

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Lecture 2The Universal Principle of Risk
Management

• Pooling and Hedging of Risk

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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

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Probability and Its Rules

• Random variable A quantity determined by the
  outcome of an experiment
• Discrete and continuous random variables
• Independent trials
• Probability P, 0ltPlt1
• Multiplication rule for independent events
  Prob(A and B) Prob(A)?Prob(B)

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Insurance and Multiplication Rule

• Probability of n independent accidents Pn
• Probability of x accidents in n policies
  (Binomial Distributon)

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Expected Value, Mean, Average
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Geometric Mean

• For positive numbers only
• Better than arithmetic mean when used for
  (gross) returns
• Geometric ? Arithmetic

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Variance and Standard Deviation

• Variance (?2)is a measure of dispersion
• Standard deviation ? is square root of variance

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Covariance

• A Measure of how much two variables move together

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Correlation

• A scaled measure of how much two variables move
  together
• -1 ???1

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Regression, Beta.5, corr.93
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Distributions

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

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Normal Distribution
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Normal Versus Fat-Tailed
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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)

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Present Discounted Value (PDV)

• PDV of a dollar in one year 1/(1r)
• PDV of a dollar in n years 1/(1r)n
• PDV of a stream of payments x1,..,xn

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Consol and Annuity Formulas

• Consol pays constant quantity x forever
• Growing consol pays x(1g)t in t years.
• Annuity pays x from time 1 to T

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Insurance Annuities

• Life annuities Pay a stream of income until a
  person dies.
• Uncertainty faced by insurer is termination date
  T

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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