(L13)APT(f)

(L13)APT(f) - Lecture # 13 The Arbitrage Pricing Theory...

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Primbs/Investment Science 1 Lecture # 13 The Arbitrage Pricing Theory (APT) Reading: Luenberger Chapter 8, Sections 1 – 4
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Primbs/Investment Science 2 Models, Data, and APT Models, Data, and APT Factor Models Relation to CAPM The Arbitrage Pricing Theory Means and Variances Well Diversified Portfolios
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Primbs/Investment Science 3 The difficulty with mean-variance The mean-variance approach requires that you estimate the mean, variance, and covariance of all securities in your portfolio. This can be a huge number. For n stocks, we need n means, n variances, and n(n-1)/2 covariances. (for 1000 stocks, this is 501500 values to be estimated!) Clearly, we need an approach that requires less computation.
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Primbs/Investment Science 4 Factor Models Rates of return are related to factors (which are random quantities). i i i i e f b a r + + = Consider the following model: random factor f constants b i is called the factor loading e i is a random error term with E[ e i ]=0.
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Primbs/Investment Science 5 Factor Models i i i i e f b a r + + = Factors models correspond to using a linear regression to fit data. r i f x x x x x x x a i slope b i y-intercept ( factor loading ) e i x x x x x x x
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Primbs/Investment Science 6 Multi-Factor Models i i i i i e f b f b a r + + + = 2 2 1 1 j j j j j e f b f b a r + + + = 2 2 1 1 We can also consider models where rates of returns are determined by more than a single factor. These are called multi-factor models. Assumption We will assume that the factors f 1 , f 2 , . .. are uncorrelated random variables. Furthermore, we will assume that the error terms, e i , e j are uncorrelated with each other and with the factors. That is Cov(e i ,e j ) = 0 and Cov(e i ,f j ) = 0 .
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Primbs/Investment Science 7 What are possible factors? External Factors: GDP, PPI, CPI, Unemployment, Interest Rates, etc. .. Extracted Factors: Market Portfolio, Industry Averages, etc. .. (One can also include firm specific factors such as size, price/book, etc. However, in the following APT theory only factors that apply to many firms will be relevant.)
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Primbs/Investment Science 8 Models, Data, and APT Models, Data, and APT Factor Models Relation to CAPM The Arbitrage Pricing Theory Means and Variances Well Diversified Portfolios
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Primbs/Investment Science 9 Calculation of Mean and Variance (Single Factor Model) The mean of a return is given by: f b a i i + = ] [ ] [ i i i i i e f b a E r E r + + = = ] [ ] [ i i i e E f E b a + + = Known: E(e i )=0 and cov(f,e i )=0 . ) ( i i i i e f b a r + + =
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Primbs/Investment Science 10 Calculation of Mean and Variance (Single Factor Model) Known: E(e i )=0 and cov(f,e i )=0 . The variance is given by: ] )) ( ) [(( ] ) [( 2 2 2 f b a e f b a E r r E i i i i i i i + - + + = - = σ ] ) ) ( [( 2 i i e f f b E + - = ] ) ( 2 ) ( [ 2 2 2 i i i i e e f f b f f b E + - + - = ) var( ) , cov( 2 ) var( 2 i i i i e e f b f b + + = ) var( ) var( 2 i i e f b + = 2 2 2 i e f i b + = ) ( i i i i e f b a r + + =
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Primbs/Investment Science 11 Calculation of Mean and Variance
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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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(L13)APT(f) - Lecture # 13 The Arbitrage Pricing Theory...

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