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constantexpectedreturn

# constantexpectedreturn - Chapter 1 The Constant Expected...

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Chapter 1 The Constant Expected Return Model The fi rst model of asset returns we consider is the very simple constant ex- pected return (CER) model. This model assumes that an asset’s return over time is normally distributed with a constant (time invariant) mean and vari- ance The model also assumes that the correlations between asset returns are constant over time. Although this model is very simple, it allows us to discuss and develop several important econometric topics such as estimation, hypothesis testing, forecasting and model evaluation. 1.0.1 Constant Expected Return Model 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 distributed and that the mean and the variance of each asset return is con- stant over time. In particular, we have for each asset i and every time period 1

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2 CHAPTER 1 THE CONSTANT EXPECTED RETURN MODEL t E [ R it ] = μ i var ( R it ) = σ 2 i The second assumption states that the contemporaneous covariances between assets are constant over time. Given assumption 1, assumption 2 implies that the contemporaneous correlations between assets are constant over time as well. That is, for all assets and time periods corr ( R it , R jt ) = ρ ij 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 uncorrelated 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 1-3 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. How- ever, they allow us to development 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. 1.0.2 Regression Model Representation A convenient mathematical representation or model of asset returns can be given based on assumptions 1-3. This is the constant expected return (CER) regression model. For assets i = 1 , . . . , N and time periods t = 1 , . . . , T the CER model is represented as R it = μ i + ε it (1.1) ε it iid. N (0 , σ 2 i ) cov ( ε it , ε jt ) = σ ij (1.2) 1 Since all assets are assumed to be normally distributed (assumption 1), uncorrelated- ness implies the stronger condition of independence.
3 where μ i is a constant and ε it

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constantexpectedreturn - Chapter 1 The Constant Expected...

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