Lecture notes 3

Lecture notes 3 - MFE5120 Investment Science Unit 3...

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MFE5120: Investment Science Unit 3: Portfolio Selection under Expected Utility Maximization Framework Duan LI Department of Systems Engineering & Engineering Management The Chinese University of Hong Kong

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2 Linear Pricing We formalize the definition of a security as a random payoff variable, d . If an investment produces an immediate positive reward with no future payoff (either positive or negative), that investment is said to be a type A arbitrage . (Linear pricing) From the assumption that there is no possibility of type A arbitrage, if d 1 and d 2 are securities with prices P 1 and P 2 , the price of αd 1 + βd 2 must be equal to αP 1 + βP 2 . Recall that the CAPM formula in pricing form is linear .
3 A portfolio of n securities, d 1 , d 2 , ... , d n , is represented by an n -dimensional vector θ = ( θ 1 2 ,...,θ n ) , where θ i represents the amount of security i in the portfolio. The payoff of the portfolio d = n summationdisplay i =1 θ i d i while the total price is P = n summationdisplay i =1 θ i P i .

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4 If an investment has nonpositive cost, but has a positive probability of yielding a positive payoff and no probability of yielding a negative payoff, that investment is said to be a type B arbitrage . If x is a random variable, we write x 0 or x > 0 to indicate x is never negative or x is always positive, respectively.
5 An individual’s investment problem: max E [ U ( x )] s . t . n summationdisplay i =1 θ i d i = x x 0 n summationdisplay i =1 θ i P i W where W is the initial wealth. Portfolio choice theorem Suppose that U ( x ) is continuous and increasing toward infinity as x → ∞ . Suppose also that there is a portfolio such that n i =1 θ 0 i d i > 0. Then the optimal portfolio problem has a solution if and only if there is no arbitrage possibility.

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