FINANC2

FINANC2 - Some Formulas neglected in Anderson Sweeny and...

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1 Some Formulas neglected in Anderson, Sweeny, and Williams, with a Digression on Statistics and Finance Transformations of a Single Random variable: If you have a case where a new random variable is defined as a linear transformation of a another random variable, there is an easy way to get the mean and variance of the new random variable. Assume you are given Y ab X =+ , where a and b are constants, and Y is a new random variable being defined as this linear transformation of the random variable X . If we know the mean and variance of the random variable X , we can easily find the mean and variance of the random variable Y using the following two formulas. µ σσ yx b = 22 2 . A word of warning: this shortcut only works if Y is a linear transformation of X . For instance, if 2 YX = it is not true that 2 µµ = . If 2 = (or for any non-linear transformation) we must use the equation that Egx gxf x x bg ch bgbg = . An Example of a Non-Linear Transformation Suppose the random variable X has the following distribution: x f(x) 0 .343 1 .441 2 .189 3 .027 It is easily verified (we did the example in class) that the mean of the random variable is () ( ) 3 0 0.9 x EX x fx = == . Suppose () 2 gx x = . One might expect that () () 2 0.9 0.81 Ex =   --- However, this is WRONG . The correct calculation is x x 2 f(x) x 2 f(x) 0 0 .343 0 1 1 .441 .441 2 4 .189 .756 3 9 .027 .243 2 xf x 1.440 Why is this important? It is central in the economics of uncertainty. When an agent’s utility depends on an uncertain outcome (a very common case in the real world) the utility achieved becomes random, so the agent can’t be thought of as maximizing utility. The simpliest generalization of utility theory that can cover this case is to assume that under uncertainty, agents maximize expected utility . If utility, U , depends on a
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2 random variable X – which for concreteness you can think of as wealth – then the agent maximizes () ()() EU x U xf x = . The fact that Egx gEx has important implications in economic theory. For instance, it implies that when wealth is uncertain, an expected utility maximizer may not wish to select a strategy that maximizes expected wealth.
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FINANC2 - Some Formulas neglected in Anderson Sweeny and...

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