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Unformatted text preview: Introduction to Financial Econometrics Chapter 3 The Constant Expected Return Model Eric Zivot Department of Economics University of Washington January 6, 2000 This version: January 23, 2001 1 The Constant Expected Return Model of Asset Returns 1.1 Assumptions Let R it denote the continuously compounded return on an asset i at time t. We make the following assumptions regarding the probability distribution of R it for i = 1 , . . . , N assets over the time horizon t = 1 , . . . , T. 1. Normality of returns: R it ∼ N ( μ i , σ 2 i ) for i = 1 , . . . , N and t = 1 , . . . , T. 2. Constant variances and covariances: cov ( R it , R jt ) = σ ij for i = 1 , . . . , N and t = 1 , . . . , T. 3. No serial correlation across assets over time: cov ( R it , R js ) = 0 for t 6 = s and i, j = 1 , . . . , N. Assumption 1 states that in every time period asset returns are normally dis tributed and that the mean and the variance of each asset return is constant over time. In particular, we have for each asset i E [ R it ] = μ i for all values of t var ( R it ) = σ 2 i for all values of t The second assumption states that the contemporaneous covariances between assets are constant over time. Given assumption 1, assumption 2 implies that the contem poraneous correlations between assets are constant over time as well. That is, for all 1 assets corr ( R it , R jt ) = ρ ij for all values of t. The third assumption stipulates that all of the asset returns are uncorrelated over time 1 . In particular, for a given asset i the returns on the asset are serially uncorre lated which implies that corr ( R it , R is ) = cov ( R it , R is ) = 0 for all t 6 = s. Additionally, the returns on all possible pairs of assets i and j are serially uncorrelated which implies that corr ( R it , R js ) = cov ( R it , R js ) = 0 for all i 6 = j and t 6 = s. Assumptions 13 indicate that all asset returns at a given point in time are jointly (multivariate) normally distributed and that this joint distribution stays constant over time. Clearly these are very strong assumptions. However, they allow us to de velopment a straightforward probabilistic model for asset returns as well as statistical tools for estimating the parameters of the model and testing hypotheses about the parameter values and assumptions. 1.2 Constant Expected Return Model Representation A convenient mathematical representation or model of asset returns can be given based on assumptions 13. This is the constant expected return (CER) model. For assets i = 1 , . . . , N and time periods t = 1 , . . . , T the CER model is represented as R it = μ i + ε it (1) ε it ∼ i.i.d. N (0 , σ 2 i ) cov ( ε it , ε jt ) = σ ij (2) where μ i is a constant and we assume that ε it is independent of ε js for all time periods t 6 = s . The notation ε it ∼ i.i.d. N (0 , σ 2 i ) stipulates that the random variable ε it is serially independent and identically distributed as a normal random variable with mean zero and variance...
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This note was uploaded on 09/13/2011 for the course ECON 503 taught by Professor Pujara during the Spring '11 term at Punjab Engineering College.
 Spring '11
 Pujara
 Econometrics

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