As we saw in my previous post on performance attribution, the returns of a quantitative portfolio that is a the result of a factor model can be decomposed via straightforward regression analysis of the asset returns on the alpha factors and/or risk factors. If we analyze the left hand side of the regression a little more carefully, it’s easy to see that the portfolio contributions of each asset is simply the product of the asset return and the weight of that asset in the portfolio at the beginning of the return period. By regressing the contributions on the factors, we are assuming a set relationship between the weight of the asset in the portfolio and the combination of risk factors and alpha factors. Thus we see that this methodology implicitly assumes a fixed relationship between portfolio weights and the factors.

But what if we have to explain the returns of a portfolio over multiple rebalancing periods? Because the portfolio weights are adjusted during every rebalance, the relationship between the portfolio weights and the factors will change over time. Hence, we cannot assume that the total asset returns are due to a single set of portfolio weights. We must turn to something different.

**Summing Up Single Period Attributions**

The most obvious way to analyze multiperiod returns is to run a single period attribution between each portfolio rebalance and aggregate the contributions. However, this method has many moving pieces. Depending on whether the trading level is reset in every period, compounding may or may not be valid. Some practitioners like to use geometric returns for easier aggregation while others like to stick with simple returns for ease of interpretation and add various smoothing factors to combine returns. Moreover, because the single period BHB analysis doesn’t account for transactions related items, the aggregated return generally does not equal the whole period return.

The final nail in the coffin is that with this method we don’t have a sense for our overall exposure to various factors within the period. For very stable portfolios whose factor exposures are relatively stable over time, this is not such a big deal. But for portfolios that turnover quickly or are constructed from high turnover factors, this is a non-trivial problem. We could approximate the average exposures with a portfolio-value-weighted mean of the factor exposures, but in many instances a fund manager doesn’t necessarily rebalance on a fixed and uniform schedule. Thus we would also have to take the number of days into account. This gets very messy.

**Return Time-Series Attribution**

An alternative to summing up the results from the cross sectional regressions is to use a time-series regression. Instead of using a fundamental factor model paradigm where we have known factor exposures in every period, we could use a macroeconomic factor model paradigm where we have know factor returns but unknown factor exposures. Under this paradigm, we can take the time-series of overall portfolio returns and regress them on the time-series of factor returns. Factor returns are usually based on the return of the factor portfolio. We can use the simple factor portfolio return, or we could estimate the factor return by taking the top and bottom quintiles of the factor portfolio.

Once we have the exposures, the factor contributions are straightforward to compute. Just take the cumulative returns of the factor during the entire attribution window and multiply it by the estimated realized factor exposure. The noise component then is the difference between the cumulative portfolio return and the sum of all the factor contributions.

**Pitfalls**

First and foremost is the sample size issue. The Barra risk model has something like 68 factors, and if we incorporate our own alpha model, the number of factors can easily get up to 80 or 90. Even if we are doing daily return attributions, it won’t be possible to do monthly attributions. The problem is even worse if our factors generally act over longer time horizons than just a day. For daily returns, it may be possible to regress out the style factors from the alpha factors and then decompose the returns for a month (~20 observations) on the residualized alpha factors. But even that means you have to have less than 20 factors. If you have multiple strategies built on the same factors, it maybe possible to setup a panel regression that gets around this problem of sample size, but it’s unclear whether it’s sensible to add this much complexity into the process when the cross-sectional method is available to us as a simple straight-forward alternative.

Another problem with the time-series method is autocorrelation. If we use daily returns, it is fairly reasonable to assume that we maybe violating the no-autocorrelation assumption in OLS. This means we also have to perform additional testing using Durbin-Watson or Breusch-Godfrey tests. The additional testing is especially crucial here because of the sample size issue, since the presence of non-spherical disturbances can make a non-trivial impact on the efficiency of the OLS estimator.

The last problem with the time-series method is that we cannot guarantee that we can reconcile it with sub-period or single period attributions. This problem is a deal-breaker in many situations when we want to look at overall return decompositions and then drill down into sub-periods that had significant contributions.

**Conclusion**

The cross sectional return decomposition methods are more appropriate for equities portfolios because in most macro strategies the cross section tend to be very small. Decomposing time-series of returns to obtain loadings on various factors or assets is a more comprehensive view on the characteristics of your portfolio over a time window.