LectureXI - Lecture XI: Market Evaluation of Investment...

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Lecture XI: Market Evaluation of Investment Risk Part II: Capital Asset Pricing Model I. Tobin’s Model: Introduction of a Riskless Asset A. In the 1950s macroecnomics still pondered the question: why do people hold money? James Tobin developed a model to explain the phenomenon by positing a risk-free asset called money and a risky interest rate. Under this scenario consumers would hold both money and risky assets dependent on their risk preferences. Another implication of Tobin’s work is the second separation theorem. B. Theorem: In the presence of a risk-free asset, the optimal mix of risky assets is independent of the individual’s risk aversion coefficient. 1. Proof: Assume we have three assets A, B, and C. Assume further that the assets must sum to total wealth: 1 abg =++ where α , β , and γ are the percent of total wealth in assets A, B, and C respectively. Assume that assets A and B are risky investments with normal distributions and C is a risk-free asset. Thus, ( 29 ( 29 ( 29 2 2 2 2 2 22 , , 1 2 1 A AA B BB PA BC B AB rN ms m a m b m a bm s a s b s abs g ab =+ +-- =-- : : The expected utility maximization model based on normality and the negative exponential distribution function then becomes ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 ma x 12 2 2 20 2 A B C A B AB A C A AB A C A AB AC A BA z z r a m b m a b m a s b s abs r m m a s bs a mm r a s bs b s as r s r ss +- - - ++ =- - +=
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LectureXI - Lecture XI: Market Evaluation of Investment...

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