1.2 Distributional Properties of Returns

To study asset returns, it is best to begin with their distributional properties. The objective here is to understand the behavior of the returns across assets and over time. Consider a collection of N assets held for T time periods, say, t = 1, …, T. For each asset i, let rit be its log return at time t. The log returns under study are {rit;i = 1, …, N;t = 1, …, T}. One can also consider the simple returns {Rit;i = 1, …, N;t = 1, …, T} and the log excess returns {zit;i = 1, …, N;t = 1, …, T}.

1.2.1 Review of Statistical Distributions and Their Moments

We briefly review some basic properties of statistical distributions and the moment equations of a random variable. Let Rk be the k-dimensional Euclidean space. A point in Rk is denoted by x ∈ Rk. Consider two random vectors X = (X1, …, Xk)′ and Y = (Y1, …, Yq)′. Let P(X ∈ A, Y ∈ B) be the probability that X is in the subspace A ⊂ Rk and Y is in the subspace B ⊂ Rq. For most of the cases considered in this book, both random vectors are assumed to be continuous.

Joint Distribution

The function

where x ∈ Rp, y ∈ Rq, and the inequality ≤ is a component-by-component operation, is a joint distribution function of X and Y with parameter θ. Behavior of X and Y is characterized by FX, Y(x, y;θ). If the joint probability density function fx, y(x, y;θ) of X and Y exists, then

In this case, X and Y

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