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Unformatted text preview: CHAPTER 6 INTRODUCTION TO OPTIMAL INVESTMENT In this chapter we provide an introduction to the theory of optimal investment in finite oneperiod models. We assume throughout that a finite oneperiod model with k basic risky assets and interest rate r ≥ 0 is given. The reference probability measure P will now play a central role. As usual, the sample space will be denoted by Ω. Although the reference probabilities are not used directly to compute arbitrage free prices of derivative securities, they are used directly to determine how desirable each security is to include in a portfolio. Throughout this chapter the term “expected return” refers to the expected value of the return with respect the actual probability measure P . Recall that the return of a portfolio with strictly positive initial capital is defined by return = terminal capital initial capital initial capital . 6.1 Risk Aversion and Expected Returns Under the Reference Measure Webster’s Finance and Investment Dictionary defines risk as “the possibility that a negative event will occur such as the value of investments declining below what was paid for them ··· ”. The noarbitrage principle tells us that any investment (with initial capital X > 0) that has the possibility of doing better than the bank account (i.e., X 1 > X (1 + r )) must also have the possibility of doing worse than the bank account (i.e., X 1 < X (1 + r )). An investor who hopes to outperform the bank account must therefore be willing to take some risk. There is an investment principle known as the riskreward tradeoff which says that investments must offer higher expected returns (under the reference measure) as compensation for higher risk. Before attempting to quantify the riskreward tradeoff, let us look at a simple example. Example 6.1 : Consider a oneperiod binomial model with u = 2 , d = 1 2 , r = . 25 and S = 100. The riskneutral probabilities are ˜ p = ˜ P ( H ) = 1 2 , ˜ q = ˜ P ( T ) = 1 2 . Let us denote the reference (or actual) probabilities by p = P ( H ) , q = P ( T ) . Recall that the return of the stock is defined by 176 ρ S ( ω ) = S 1 ( ω ) 100 100 for all ω ∈ { H, T } , and the return of the bank account is r = . 25 (for sure). Under the riskneutral measure, the stock has the same expected return as the bank account, i.e. ˜ E ( ρ S ) = . 25 . In order for the stock to be an attractive investment to most rational investors we must have E ( ρ S ) > . 25 (1) because the bank account offers a guaranteed return of .25 and purchase of the stock involves risk. (Indeed investors who purchases the stock might lose half of their money.) Most investors will want to feel confident that (1) holds in order for them to buy any stock....
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 Spring '10
 SCHAFFER
 Utility, bank account, X1, initial capital

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