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Intensive Actuarial Training for Bulgaria January 2007

Intensive Actuarial Training for Bulgaria January 2007. Lecture 10 – Risk Theory and Utility Theory By Michael Sze, PhD, FSA, CFA. Some Preliminary Probability Functions. Probability density function f(x): the chance that the random variable X is equal to x

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Intensive Actuarial Training for Bulgaria January 2007

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  1. Intensive Actuarial Training for Bulgaria January 2007 Lecture 10 – Risk Theory and Utility Theory By Michael Sze, PhD, FSA, CFA

  2. Some Preliminary Probability Functions • Probability density function f(x): the chance that the random variable X is equal to x • P(X = x) = f(x); x f(x) = 1 • Some examples • Binomial distribution: P(X = i) = nCi (1/2)n • x = E[X] = x x f(x): average value of X • 2 = Variance = Var(X) = E[(X - x )2]: the variation of X.

  3. Some Properties of Probability Functions • E[aX + b] = a E[X] + b • Var(X) = E[X2] - 2 • Var(X+b) = Var(X) • Var(aX) = a2 Var(X) • For normal distribution N( , 2), E[X] = , Var(X) = 2 • For N( , 2), moment generating function MX(t) = E[e tX] = e t + (t2 2)/2

  4. Desire for Wealth • Most persons are risk averse • They appreciate having more wealth than less wealth • However, as their wealth increases, their desire, or appreciation for additional wealth decreases • This appreciation is called utility function u(w) • u(w) has the following important proporties • u’(w) > 0 • u”(w) < 0

  5. Some Common Utility Functions • Logarithmic function: u(w) = k ln w, w > 0 • Exponential: u(w) = - e - w, for all w,  > 0 • Fractional power: u(w) = w, w>0, 0< <1 • Quadratic: u(w) = w - w2, w<1/(2),  > 0 • For each of the above functions, we can prove that • u’(w) > 0 • u”(w) < 0

  6. Use of Utility Theory • When a person take decision about an action X that has uncertain outcomes • His decision is not based on the expected reward or value of X: E[X] • His decision is based on his expected appreciation of X: E[u(X)] • Some examples are shown in the following spread sheet

  7. Comments on Examples • Of the two investment portfolios A and B: • Portfolio A has greater expected return • Portfolio A has higher Variance • Depending upon the degree of risk aversion of the user (I.e. the person’s utility function) • Either of the portfolios may be considered as more desirable

  8. Further Analysis of the Utility Function u(w) • u’(w) > 0: u is an increasing function, slope is upwards • u”(w) < 0: u is curving downwards • At any point on the curve, the tangent line lies above the curve, I.e. • u(x) < u(w) + u’(w) (x – w), for all x, and all w • In particular, u(x) < u() + u’() (x – ) • E[u(x)] < E[u() + u’() (x – )] = u() + u’()E[(x – )] = u() = uE[X]

  9. Application of Jensen’s Inequality • Jensen’s Inequality: If u’(x) > and u”(x) < 0 for all x, then E[u(X)] < u(E[X]) • This has important application in insurance • It says: a risk averse will accept an arrangement that may charge higher premium than the expected loss • As long as the expected value of u(X) is acceptable • We shall prove this in the next few slides

  10. Application to Insurance • An insured person with initial wealth w seeks to purchase insurance to cover a contingent event X • If the premium is G, his expected utility value must be the same with both arrangements • Pay G to get insurance coverage • Not buy insurance and get expected loss

  11. Formulas for Above Concepts • E[u(w – G)] = E[u(w – X)] • LS = u(w – G) • RS = E[u(w – X)] < u(E[w – X]) = u(w - ) • Thus, u(w – G) < u(w - ) • But u is increasing function, so • w – G < w -  • Which implies that G >  • In other words, the insured will accept premium higher than the expected loss

  12. Application to Insurance Company • For an insurance company with initial wealth w and utility function u(w) • It will be happy to receive premium H to cover contingency X, if • E[u(w)] = E[u(w + H – X)] • LS = u(w) • RS < u(E[w + H – X]) = u(w + H - ) • This is w < w + H - , or H >  • An insurance company will accept insurance is the premium is greater than the expected loss

  13. For Exponential Utility Function • u(w) = - e - w, for all w,  > 0 • u(w – G) = E[u(w – X)] • LS = - e - (w – G) = - e - w e G • RS = E[-e - (w – X) ] = -e - w E[eX]= MX() • Thus e G = MX(), the moment generating function of X • So, G = (ln MX())/ • Similarly, H = (ln MX())/

  14. In particular, for N(, 2) • f(x) = 1/(2)½ x e – (x-)2/22 • MX(t) = e t+t22/2 • For u(w) = - e – 5w, and X is N(5,2) then • G = [- 5x5 + 52x2/2]/5 = 0

  15. Fractional Power Utility • For u(w) = w .5, and X a uniform distribution on [0, 10], and initial w = 10, • (10 – G) .5 = E[(10 – X) .5] • = 10 (10 – X)½/ 10 dx = 2/310 0 • G = 5.5556

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