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

• 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 X2 = Variance = Var(X) = E[(X - x )2]: the

variation of X.

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

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

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

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

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

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]

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

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

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

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

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())/

In particular, for N(, 2)

• f(x) = 1/(2)½ x e – (x-)2/22

• MX(t) = e t+t22/2

• For u(w) = - e – 5w, and X is N(5,2) then

• G = [- 5x5 + 52x2/2]/5 = 0

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