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) = n C i (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[X 2 ] -  2 Var(X+b) = Var(X) Var(aX) = a 2 Var(X) For normal distribution N( ,  2 ), E[X] = , Var(X) =  2 For N( ,  2 ), moment generating function M X (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 -  w 2, w 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  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 ]= M X (  ) Thus e  G = M X (  ), the moment generating function of X So, G = (ln M X (  ))/  Similarly, H = (ln M X (  ))/ 

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