(L08)MeanVar2(f)2

# (L08)MeanVar2(f)2 - Lecture #8: Mean-Variance Portfolio...

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Primbs/Investment Science 1 Lecture #8: Mean-Variance Portfolio Theory Reading: Luenberger Chapter 6, Sections 1 – 5

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Primbs/Investment Science 2 Mean-Variance Portfolio Theory Betting Wheel Diversification Portfolio Diagrams Basic Probability Multiple Random Variables Mean and Variance of a Sum Portfolio Mean and Variance Motivation
Primbs/Investment Science 3 Motivation We want to develop a theory of single period random cash flows. Random quantities can be characterized by their probability distributions. But, these are hard to work with… Instead, we will characterize a single period investment by its expected return and its standard deviation . These are easier quantities to work with. You should think of this as a reward/risk description where expected return = reward and standard deviation = risk .

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Primbs/Investment Science 4 Mean-Variance Portfolio Theory Betting Wheel Diversification Portfolio Diagrams Basic Probability Multiple Random Variables Mean and Variance of a Sum Portfolio Mean and Variance Motivation
Primbs/Investment Science 5 Random Variables A discrete random variable X is a variable that can take on the values: x 1 , x 2 , . ..,x n with probabilities p 1 , p 2 ,..., p n . 0 i p = = n i i p 1 1 = n x x x X 2 1 w.p. n p p p 2 1 where and

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Primbs/Investment Science 6 Probability Mass Function x 1 x 2 x 3 x 4 x 5 p 1 p 2 p 3 p 4 p 5 When we write the probabilities p i as a function of x i , this is known as the probability mass function.
Primbs/Investment Science 7 Example: Roll of a die X= 1 2 3 4 5 6 1/6 1/6 1/6 1/6 1/6 1/6 w/ prob. x 1/6 1/6 1/6 1/6 1/6 1/6 1 2 3 4 5 6

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Primbs/Investment Science 8 Expectation If X and Y are random variables, and a and b are constants. E[aX+bY]=aE[X]+bE[Y] Linearity Certain Value If y is known, then E[y]=y . Non-negativity If X can only take on positive values, then E[X]>0 . The Expectation of a random variable is defined as: [ ] = = = n i i i p x X E X 1
Primbs/Investment Science 9 Example: Roll of a die X= 1 2 3 4 5 6 1/6 1/6 1/6 1/6 1/6 1/6 w/ prob. x 1/6 1/6 1/6 1/6 1/6 1/6 1 2 3 4 5 6 = = 6 1 ] [ i i i p x X E 5 . 3 6 1 6 1 = = = i i 6 1 6 ... 6 1 2 6 1 1 + + = The expectation is where this picture would balance on your finger

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Primbs/Investment Science 10 Variance and Standard Deviation It has the same units as X and measures the spread of a random variable around its mean. The Standard Deviation is the square root of the Variance. ) ( X Var X = σ The Variance of a random variable is defined as: ] ) [( ) ( 2 X X E X Var - = 2 2 ] [ X X E - = Notation: The variance is often written as: 2 ) ( X X Var =
Primbs/Investment Science 11 Example: Roll of a die X= 1 2 3 4 5 6 1/6 1/6 1/6 1/6 1/6 1/6 w/ prob. x 1/6 1/6 1/6 1/6 1/6 1/6 1 2 3 4 5 6 92 . 2 = ] ) [( ) ( 2 X X E X Var - = 2 2 2 ) 5 . 3 6 ( 6 1 ... ) 5 . 3 2 ( 6 1 ) 5 . 3 1 ( 6 1 - + + - + - =

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Primbs/Investment Science 12 Mean-Variance Portfolio Theory Betting Wheel Diversification Portfolio Diagrams Basic Probability Multiple Random Variables Mean and Variance of a Sum Portfolio Mean and Variance Motivation
Primbs/Investment Science 13 Several Random Variables Let X 1 and X 2 be two random variables.

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## This note was uploaded on 11/15/2008 for the course MS&E 242 taught by Professor Primbs during the Fall '06 term at Stanford.

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(L08)MeanVar2(f)2 - Lecture #8: Mean-Variance Portfolio...

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