Lecture_35_Risky_Assets

Lecture_35_Risky_Assets - Lecture 35: Risky Assets c...

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Unformatted text preview: Lecture 35: Risky Assets c & 2008 Je/rey A. Miron Outline 1. Mean-Variance Utility 2. The Tradeo/ Between Risk and Return 3. The CAPM 4. Diversi&cation 1 Introduction Last time, we developed a model for analyzing consumer decisions made under un- certainty. The key ideas in the general version of the model were states of nature and contingent commodities. The main version of the model that we considered was the expected utility model. This model said that a consumer maximizes a utility function that equals the ex- pected value of the utility the consumer attaches to the di/erent contingent com- modities (equivalently, to consumption in di/erent states of nature). This model has many applications; it is useful for modeling uncertainty generally. One speci&c application we have examined is the allocation of a &nancial portfolio to di/erent assets, some of which are risky. We now want to continue with applications in &nancial markets, but to make new progress, it is useful to further specialize the utility function that agents maximize. This version is known as the mean-variance utility model. 1 2 Mean-Variance Utility We are now going to assume that we are talking speci&cally about portfolio allo- cations between risky and non-risky &nancial asset, or between di/erent possible combinations of risky assets. We are also going to assume that a consumers preferences over di/erent possible portfolios can be summarized by exactly two portfolio features: the mean and variance of the return on the portfolio. Speci&cally, suppose that each &nancial asset has a return, r , that we can model as a random variable. That is, the return can take one of S di/erent values, r 1 ; :::; r S , each with some known probability & s . This random variable has a mean, which is given by r = S X s =1 & s r s and a variance, which is given by 2 r = S X s =1 & s ( r s & r ) 2 Now suppose the consumer has some amount of wealth, W , and the consumer invests this wealth in the various assets. For simplicity, keep the number of assets at two for now. The consumer invests a fraction x of this wealth into the &rst asset, and 1 & x in the second asset. The return on the consumers portfolio is therefore r x = xr 1 + (1 & x ) r 2 : This portfolio has an expected return given by x = S X s =1 & s r s x = S X s =1 & s ( xr s 1 + (1 & x ) r s 2 ) 2 = x S X s =1 & s r s 1 + (1 & x ) S X s =1 & s r s 2 = x r 1 + (1 & x ) r 2 and variance given by 2 x = S X s =1 & s ( r s x & x ) 2 = S X s =1 & s ( xr s 1 + (1 & x ) r s 2 & x r 1 & (1 & x ) r 2 ) 2 = S X s =1 & s [( xr s 1 & x r 1 ) + ((1 & x ) r s 2 & (1 & x ) r 2 )] 2 = S X s =1 & s [ x 2 ( r s 1 & r 1 ) 2 + (1 & x ) 2 ( r s 2 & r 2 ) 2 + 2 x (1 & x )( r s 1 & r 1 )( r s 2 & r 2 )] = 2 1 + 2 2 + 2 12 Then our assumption is that the consumer chooses the portfolio &that is, chooses a value of x & that maximizes a utility function that depends only on these two parameters: max x u ( x...
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Lecture_35_Risky_Assets - Lecture 35: Risky Assets c...

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