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Unformatted text preview: Mathematics in Finance Steven E. Shreve Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 USA [email protected] A Talk in the Series Probability in Science and Industry Carnegie Mellon University November 20, 2007 1 / 48 Outline MeanVariance Analysis Risk Measurement Controlling Risk by Hedging What is Going on Today? 2 / 48 1. MeanVariance Analysis 3 / 48 Asking the right question Question before 1952: How do I choose a good stock? 4 / 48 Asking the right question Question before 1952: How do I choose a good stock? Question after 1952: How do I choose a good portfolio? 5 / 48 Return of a stock If we invest $100 and leave it for a year, reinvesting any dividends, at the end the year the value of our investment will be some random amount Y which could be either more than $100 or less than $100. 6 / 48 Return of a stock If we invest $100 and leave it for a year, reinvesting any dividends, at the end the year the value of our investment will be some random amount Y which could be either more than $100 or less than $100. Define the return on the investment to be X = Y 100 100 . This is a a random variable that could be either positive nor negative. 7 / 48 Return of a stock If we invest $100 and leave it for a year, reinvesting any dividends, at the end the year the value of our investment will be some random amount Y which could be either more than $100 or less than $100. Define the return on the investment to be X = Y 100 100 . This is a a random variable that could be either positive nor negative. In his 1952 Ph.D. dissertation, Harry Markowitz assumed that X is a normal random variable with some mean μ and standard deviation σ . 8 / 48 Return on two stocks I Two stocks with returns X 1 and X 2 . I μ 1 = E X 1 and μ 2 = E X 2 . I σ 1 = q E ( X 1 μ 1 ) 2 and σ 2 = q E ( X 2 μ 2 ) 2 I μ 2 > μ 1 and σ 2 > σ 1 μ 1 μ 2 9 / 48 Return on two stocks (continued) ( σ 1 ,μ 1 ) ( σ 2 ,μ 2 ) μ σ 10 / 48 A portfolio of two stocks Suppose the returns are jointly normal with correlation ρ . Put a fraction α of your capital in the first stock and the remaining fraction 1 α in the second stock. Return on the portfolio is μ ( α ) = αμ 1 + (1 α ) μ 2 Standard deviation of the portfolio is σ ( α ) = q α 2 σ 2 1 + 2 ρα (1 α ) σ 1 σ 2 + (1 α ) 2 σ 2 2 . ( σ 1 ,μ 1 ) ( σ 2 ,μ 2 ) μ σ ρ = 0 ρ = 1 11 / 48 Efficient frontier μ σ 12 / 48 Optimal portfolio Theorem Let ( X 1 , X 2 ,..., X n ) be an ndimensional vector of random returns, and assume that this vector is jointly normal with a positive definite covariance matrix Γ. This can be written as Γ = σσ , where σ is a nonsingular matrix and σ denotes its transpose. Let μ be the vector of expected returns. Let m be a desired rate of return for a portfolio. The portfolio that achieves this rate of return with minimal standard deviation is found by solving the optimization problem Minimize √ α Γ α Subject to α e = 1 ,...
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This document was uploaded on 02/18/2011.
 Spring '11
 Probability

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