Factor models are heavily used in finance to create variance matrices. Here’s why.<\/p>\n
Factor models:<\/p>\n
First off, what does this mean?<\/p>\n
The technical term is that you want your estimate of the variance matrix to be positive definite<\/em>.\u00a0 In practical terms what that means is that all portfolios have positive volatility according to the estimate.\u00a0 Optimizers would be very pleased<\/a> to find a portfolio with negative volatilty — we don’t want to give them the opportunity.<\/p>\n Suppose you have a universe of 1000 stocks.\u00a0 If you want to estimate the variance of their returns (note: returns not prices<\/a>), then you need more than 1000 observations to get a positive definite estimate using the sample variance.\u00a0 Probably 2000 observations would be the minimum you’d want to use.\u00a0 That is 8 years of daily data (and if your universe is global, then daily data will have asynchrony problems<\/a>).<\/p>\n Factor models always produce positive definite estimates (well, possibly they’ll require a nudge here and there).<\/p>\n A big universe means that the variance matrix is really big.\u00a0 If the universe has 1000 assets, then the variance matrix is going to take roughly 8 megabytes.\u00a0 Years ago that was a problem — as in “no way Jose”.<\/p>\n If the universe has 20,000 assets, then the variance matrix takes about 3 gigabytes.\u00a0 That need not be at all problematic now.<\/p>\n The issue of space was a key driver in factor models being adopted.\u00a0 We should reconsider the merits of factor models now that computers are massively bigger and there are alternative estimators.<\/p>\n Factor models assume the fiction that there is a set of ghosts that drive the relationship of the returns in the universe.\u00a0 Fiction is sometimes enlightening, sometimes not<\/a>.<\/p>\n There are three major classes of factor models<\/a>:<\/p>\n Statistical factor models are mute about the sources of risk because both the factors and the sensitivities to the factors are estimated.\u00a0 The ghosts are anonymous.<\/p>\n In the other two classes, the ghosts have names — names like: “Energy sector”, “Value”, “Momentum”, “Short-term interest rates”.<\/p>\n This is the unique proposition that factor models hold.\u00a0 If what you want is names on partitions of your risk, then you want a (non-statistical) factor model.<\/p>\n Conversely, if you don’t care about named partitions, then you should at least consider alternatives.\u00a0 I hypothesize that it is suboptimal to use factor models for optimization.<\/p>\n The leading alternative to factor models is shrinkage models.\u00a0 Start with the sample variance matrix and then shrink towards some target.\u00a0 The model I know of that makes the most sense to me is to shrink towards equal correlation.\u00a0 That is commonly known as the Ledoit-Wolf shrinkage<\/a> estimate.<\/p>\n There is an R implementation of the Ledoit-Wolf estimate (as well as a statistical factor model) in the BurStFin package<\/a>.<\/p>\n Today we have naming of parts. Japonica from “Lessons of the War: Naming of Parts” by Henry ReedSpace<\/h2>\n
Sources of risk<\/h2>\n
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\u00a0Alternatives<\/h2>\n
Epilogue<\/h2>\n
\nGlistens like coral in all of the neighbouring gardens,
\nAnd today we have naming of parts.<\/p><\/blockquote>\n
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