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Chapter 3 Financial Data: I.I.D. Modeling

“All models are wrong, but some are useful.”

— George E. P. Box

Under the efficient-market hypothesis (Fama, 1970), the price of a security is a good estimate of its intrinsic value. That is, any information about future prospects is already incorporated in the current price, so that the forecast is just the current price. This leads to modeling the prices as a random walk (Malkiel, 1973) and, equivalently, the returns as a sequence of independent and identically distributed (i.i.d.) random variables. In the case of multiple assets, the random variables denote the returns of all the assets, leading to a multivariate random variable. This is a simple and convenient model, which in fact was already employed in Markowitz’s seminal paper on portfolio design (Markowitz, 1952).

Under the i.i.d. model, no temporal structure is incorporated and the returns at a given time are assumed to be independent from other time instances; in addition, the distribution of the random returns over time is assumed fixed. Hence the terminology “independent and identically distributed.” This chapter explores the characterization of the multivariate i.i.d. distribution, from the simplest sample estimators to the more sophisticated robust non-Gaussian estimators that may include prior information in the form of shrinkage, factor modeling, or prior views.

This material will be published by Cambridge University Press as Portfolio Optimization: Theory and Application by Daniel P. Palomar. This pre-publication version is free to view and download for personal use only; not for re-distribution, re-sale, or use in derivative works. © Daniel P. Palomar 2025.

References

Fama, E. F. (1970). Efficient capital markets: A review of theory and empirical work. The Journal of Finance, 25(2), 383–417.

Malkiel, B. G. (1973). A Random Walk down Wall Street. New York: W. W. Norton.

Markowitz, H. M. (1952). Portfolio selection. The Journal of Finance, 7(1), 77–91.