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Theory of Capital Markets
Security Markets VI Miloslav S Vosvrda Theory of Capital Markets
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The explanation of security prices
One of the principal applications of security market theory is the explanation of security prices. The static Capital Asset Pricing Model or CAPM, begin with a set Y of random variables with finite variance on some probability space. Each y in Y corresponds to the random payoff of some security.
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span(Y) The space of linear combination of elements of Y,
meaning x is in L if and only if for some scalars and some in Y. The elements of L are portfolios. Some portfolio in L denoted 1 is riskless. The utility functional of each agent i is assumed to be strictly variance averse, meaning that whenever and
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The market portfolio The total endowment M of portfolios is the market
portfolio, and is assumed to have non-zero variance. Suppose is a competitive equilibrium for this economy. The equilibrium price functional p is represented by a unique portfolio in L via the formula:
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For the equilibrium choice of agent i, consider
the least squares regression of on : where the residual term e has zero expectation and zero covariance with ; that is Since both 1 and are available portfolios, agent could have chosen the portfolio
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We have implying that is budget feasible for agent i. Since and string variance aversion implies that . Since is optimal for agent i, it follows that e=0. For any portfolio , relation implies that
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where k= and K= ,
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Defining the return on any portfolio x with non-
zero market value to be and denoting the expected return by and where which is known as the beta of portfolio x.
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The Capital Asset Pricing Model
is relation By words: The expected return on any portfolio in excess of the riskless rate of return is the beta of that portfolio multiplied by the excess expected return of the market portfolio.
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the riskless expected rate of return.
Consider the linear regression of on , where aby is any portfolio with non-zero variance. The solution is For the particular case of y=M, we have By taking expectations shows that This is a special property distinguishing the market portfolio. A portfolio whose return is uncorrelated with the market return has the riskless expected rate of return.
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