Facts about Factors Paula Cocoma Megan Czasonis Mark Kritzman David Turkington

Why use factors? Factors may be less correlated than assets and therefore provide better diversification. Factors may be more effective than assets at reducing noise. Investors may be more skilled at predicting factors than assets. Factors may have less estimation error than assets.

Diversification

Diversification – a specious argument Assets define the opportunity set. It is impossible to create a more efficient in-sample portfolio with the same constraints and of the same periodicity by regrouping assets into factors. Principal components (all factors uncorrelated) Asset classes (correlations range from to 0.82) Frontier with leverageFrontier without leverage Frontier with leverageFrontier without leverage Notes: This analysis incorporates the following asset classes: U.S. large cap, U.S. small cap, EAFE equities, emerging equities, global sovereigns, U.S. government bonds, U.S. corporate bonds, commodities, and hedge funds. It is based on monthly returns over the period Jan 1990 through Dec Excess returns represent the return over the risk-free rate. All data obtained from DataStream.

Noise Reduction

Noise reduction: Means Does consolidating toward factors reduce noise more effectively than consolidating toward assets? Portfolio returns are less noisy than the returns of the assets within them due to diversification. However, we have already shown that efficient frontiers comprised of factors are identical to efficient frontiers comprised of assets. It follows, therefore, that we cannot reduce dispersion around portfolio means more effectively by using factor groupings as opposed to asset groupings.

Noise reduction: Covariances If asset returns are normally distributed and serially independent, the average noise in covariances across a group of assets is no less than the average noise in covariances across the individual assets that form that group. For assets with non-normal distributions or serial dependence, different methods of grouping may result in different levels of noise. Nonetheless, there is no reason to expect factor groupings to produce less noise than asset class groupings.

Predictability

Superior predictability – an untestable hypothesis Some investors may be more skilled at predicting factors than assets. However, some investors may be more skilled at predicting assets than factors. It is impossible to test these hypotheses generically; they are investor specific.

Superior predictability – an untestable hypothesis Some investors may be more skilled at predicting factors than assets. However, some investors may be more skilled at predicting assets than factors. It is impossible to test these hypotheses generically; they are investor specific. Nonetheless, those who prefer to predict factors are faced with the additional challenge of predicting how these factors map onto factor mimicking portfolios.

Estimation Error

Sources of estimation error Small-Sample Error Independent-Sample Error Interval Error Mapping Error … … … … … … … … … … … … … … Vary: Small samples Hold constant: Forecasting sample Factor mapping Measurement interval Vary: Forecasting sample Hold constant: Sample size Factor mapping Measurement interval Vary: Factor mapping Hold constant: Sample size Forecasting sample Measurement interval Vary: Measurement interval Hold constant: Sample size Forecasting sample Factor mapping

Small-sample error The realization of parameters from a small sample will likely differ from the parameter values of a large sample from which it is selected. We call this small-sample error. When A and B are the same asset, this formula will measure the error in the standard deviation of that asset.

Small-sample error The realization of parameters from a small sample will likely differ from the parameter values of a large sample from which it is selected. We call this small-sample error. When A and B are the same asset, this formula will measure the error in the standard deviation of that asset. m, j indicates monthly estimates from a 36-month testing subsample m alone indicates monthly estimates from the full sample

Independent-sample error The realization of parameters from a future sample will likely differ from the parameter values of an independent historical sample. We call this independent-sample error. Notes: In rare instances, small sampler error exceeds the mean squared error between two independent samples. In these cases, we record small sample error as the mean squared error between the two samples, and record independent sample error as zero.

Independent-sample error The realization of parameters from a future sample will likely differ from the parameter values of an independent historical sample. We call this independent-sample error. Notes: In rare instances, small sampler error exceeds the mean squared error between two independent samples. In these cases, we record small sample error as the mean squared error between the two samples, and record independent sample error as zero. m, j indicates monthly estimates from a 36-month testing subsample m, j indicates monthly estimates from a 36-month independent subsample immediately preceding the testing subsample ^

Mapping error Assets define the opportunity set for investing. A desired factor exposure must be mapped onto a portfolio of assets to be investable. The mapping that best tracks a factor in the future will likely differ from the mapping of an independent historical sample. We call this mapping error. Mapping error applies only to factors. The calculations for small sample error and independent sample error for factors do not reflect mapping error.

Mapping error Mapping error can be isolated by comparing the covariance of the best-fit factor- mimicking portfolio to the covariance of a factor-mimicking portfolio estimated from an independent sample, holding the evaluation period constant.

Mapping error Mapping error can be isolated by comparing the covariance of the best-fit factor- mimicking portfolio to the covariance of a factor-mimicking portfolio estimated from an independent sample, holding the evaluation period constant. A, B indicate factor mappings derived from the independent subsample A, B indicate factor mappings derived from the testing subsample ^^

Interval error Parameters estimated from monthly or higher-frequency returns often differ from those estimated from lower-frequency returns, within the same sample. We call this interval error.

Interval error

Correlation = 0.69 U.S. and emerging markets stocks: monthly returns

Interval error Correlation = 0.44 U.S. and emerging markets stocks: annual returns

Interval error Correlation = 0.04 U.S. and emerging markets stocks: triennial returns

The volatility of the cumulative continuous returns of x over q periods is given by: Interval error: Long-horizon and short-horizon volatility

The volatility of the cumulative continuous returns of x over q periods is given by: Interval error: Long-horizon and short-horizon volatility The volatility of the cumulative continuous returns of x over q periods is given by: This term reflects annualization in the absence of lagged effects

The volatility of the cumulative continuous returns of x over q periods is given by: Interval error: Long-horizon and short-horizon volatility The volatility of the cumulative continuous returns of x over q periods is given by: This term captures the impact of auto- correlation

Interval error: Long-horizon and short-horizon correlation The correlation between the cumulative returns of x and the cumulative returns of y over q periods is given by:

Interval error: Long-horizon and short-horizon correlation The correlation between the cumulative returns of x and the cumulative returns of y over q periods is given by: This term captures the lagged cross-correlation between x and y

Interval error: Long-horizon and short-horizon correlation The correlation between the cumulative returns of x and the cumulative returns of y over q periods is given by: This term captures the auto- correlation of x This term captures the auto- correlation of y

Interval error Interval error can be isolated by comparing a covariance matrix estimated from low- frequency returns to a covariance matrix estimated from high-frequency returns.

Interval error Interval error can be isolated by comparing a covariance matrix estimated from low- frequency returns to a covariance matrix estimated from high-frequency returns. tri, j indicates implied 3-year estimates from a 36-month testing subsample m, j indicates monthly estimates from the same 36-month testing sample

Total covariance error We summarize the degree of error associated with an entire covariance matrix by averaging the squared errors across all of the elements of the matrix and then taking the square root.

Composite instability score The four sources of estimation error are independent from one another, which means we can sum the variances of each error and then take the square root of this sum to compute a composite instability score.

Classifications and Data

Asset classes, fundamental factors and principal components Notes: Our analysis uses monthly data over the period Jan 1990 through Jul We proxy U.S. large cap, U.S. small cap, U.S. government bonds, U.S. corporate bonds, commodities, and hedge funds with the following indices: S&P 500, Russell 2000, Barclays U.S. Agg Government, Barclays U.S. Agg. Corporate, S&P GSCI Commodity, and HFRI Fund of Funds Composite, respectively. We proxy inflation, growth, term premium, credit premium, small premium, and value premium with the following: U.S. CPI, U.S. GDP growth forecast, 10-2 yr U.S. Treasury, Baa Corp-10 yr U.S. Treasury, Fama-French SMB factor, and Fama-French HML factor, respectively. To analyze fundamental factors, we build six factor-mimicking portfolios by regressing each factor on the six asset classes. To analyze principal components, we build six portfolios using principal components analysis (each portfolio corresponds to an eigenvector).

Principal components

Industries, attributes and principal components Notes: Our analysis uses monthly returns over the period Jan 1989 through Jan We narrow the universe of 400 stocks (based on the constituents in the MSCI U.S. Index as of Jan 2015) to the 194 stocks that have full price, market cap, and book-to-market history over this sample. To form industry portfolios, we use GICS classifications. To form attribute portfolios, we use average market cap, book-to-market ratios, and 1-year returns, respectively. To form principal components portfolios, we run a PCA on the 194 stocks and form portfolios corresponding to the top 10 eigenvectors. * Because we run the PCA on 194 stocks using 36-month sub-samples, we restrict our universe to only the top 36 principal components.

Results

Asset classes, fundamental factors, and principal components Small-Sample ErrorIndependent-Sample Error Mapping ErrorInterval Error Composite Instability Score: Sources of Error

Asset classes, fundamental factors, and principal components Composite Instability Score: Total Covariance Error

Industries, attributes, and principal components (10 groups) Small-Sample ErrorIndependent-Sample Error Mapping ErrorInterval Error Composite Instability Score: Sources of Error

Industries, attributes, and principal components (10 groups) Composite Instability Score: Total Covariance Error

Noise reduction in covariances

Summary It has become fashionable to use factors instead of assets as the building blocks for forming portfolios. Some have argued that factors offer greater potential for diversification, but this argument is specious. Others argue that factor groupings reduce noise more effectively than asset groupings, but this too is untrue. It is also argued that it is easier to predict factors than assets, but this argument is generically untestable. Nonetheless, it may be the case that factor parameters are more stationary than asset parameters. We find no evidence that factors produce more stable results taking into account small- sample error, independent-sample error, mapping error, and interval error. Instead, we find the opposite to be true.

Conclusion Why use factors?

Conclusion Why use factors? In our view, the case is yet to be made that investors should use factors rather than assets as building blocks for forming portfolios. However, investors may be able to gather useful insights by attributing portfolio performance to factor exposures in addition to asset exposures. And understanding a portfolio’s factor exposures may help investors to identify concentration risk and to hedge unwanted exposures, which otherwise might go undetected.