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Steve Shreve lecture

Steve Shreve lecture - Mathematics in Finance Steven E...

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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

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Outline Mean-Variance Analysis Risk Measurement Controlling Risk by Hedging What is Going on Today? 2 / 48
1. Mean-Variance Analysis 3 / 48

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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

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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

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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 Two stocks with returns X 1 and X 2 . μ 1 = E X 1 and μ 2 = E X 2 . σ 1 = E ( X 1 - μ 1 ) 2 and σ 2 = E ( X 2 - μ 2 ) 2 μ 2 > μ 1 and σ 2 > σ 1 μ 1 μ 2 9 / 48

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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 ) = 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

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Efficient frontier μ σ 12 / 48
Optimal portfolio Theorem Let ( X 1 , X 2 , . . . , X n ) be an n -dimensional 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 non-singular 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 , (1) α μ = m , (2) where e is the n -dimensional vector whose very component is 1.

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