By Dickson D.C.M.
ISBN-10: 0521846404
ISBN-13: 9780521846400
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6 Notes and references A comprehensive reference on utility theory is Gerber and Pafumi (1998), which discusses applications in both risk theory and finance. For a more general discussion of the economics of insurance, including applications of utility theory, see Borch (1990). 7 Exercises 1. An insurer, whose current wealth is W , uses the utility function u(x) = x − x2 , 2β where x < β, for decision making purposes. Show that the insurer is risk averse, and that the insurer’s risk aversion coefficient, r (x), is an increasing function of x.
A random variable X has a logarithmic distribution with parameter θ , where 0 < θ < 1, if its probability function is Pr(X = x) = −1 θx log(1 − θ) x for x = 1, 2, 3, . . Show that M X (t) = log(1 − θet ) log(1 − θ) for t < − log θ . Hence, or otherwise, find the mean and variance of this distribution. 2. A random variable X has a beta distribution with parameters α > 0 and β > 0 if its density function is f (x) = (α + β) α−1 x (1 − x)β−1 (α) (β) for 0 < x < 1. Show that E Xn = (α + β) (n + α) (α) (n + α + β) and hence find the mean and variance of X .
As with the logarithmic utility function, it is generally difficult to obtain closed form solutions in problems involving fractional power utility functions. 6 Notes and references A comprehensive reference on utility theory is Gerber and Pafumi (1998), which discusses applications in both risk theory and finance. For a more general discussion of the economics of insurance, including applications of utility theory, see Borch (1990). 7 Exercises 1. An insurer, whose current wealth is W , uses the utility function u(x) = x − x2 , 2β where x < β, for decision making purposes.