Performance Attribution
In addition to decomposing risks, the Scientific Portfolio platform also allows for an exact decomposition of the returns into contributions from risk factors.
We first analyze the returns associated with fundamental, or rewarded, risk factors and attribute them proportionally to the portfolio's exposures to these factors. Returns associated with fundamental risk factors are shown by academia to be the primary drivers of long-term performance. The remainder of returns, also known as residuals, are denoted by . They correspond to the part of the returns that are not driven by risk factors.
Method
The first step consists of identifying the returns associated with the portfolio's exposures to fundamental risk factors. These exposures are evaluated by regressing the returns of the portfolio to the fundamental factors over the period considered, they are denoted by . Once these exposures are estimated, the returns are decomposed into contributions from the various risk factors.
Mathematically, for a portfolio the decomposition of the cross-section of returns is particularly straightforward, as
where the columns of the matrix correspond to the returns of the Market, Value, Size, Momentum, Investment, Profitability and Low-Risk factors.
Over the total period selected, the returns that attributed to a given factor are proportional to:
when the total of the factor over the period is given by . The residual, which corresponds to the part of the returns that is unexplained by the model is obtained by taking
Relative returns
The relative returns correspond to the returns of the portfolio minus those of the reference. The absolute returns for both portofolios are calculated separately, then they are subtracted to obtain the relative returns. The relative return attributed to each instrument are proportional to their weight in the portfolio, that is:
Performance linking and the heat map
Although the decomposition of the cross-section of returns is straightforward, the attribution of performance over the full period suffers from the same issues as any other performance decomposition methodology. This is a consequence of returns being compounded rather than summed to measure performance over time. As the sum of daily returns is a poor approximation of the portfolio's long-term performance, the performance of the portfolio over the full period is obtained via the product
When point-in-time returns are very small, the sum over time might provide an approximation of the period performance, but as the number of time periods grow, this approximation becomes poor.
Several methodologies provide adjustment factors that allow the decomposition of the performance over the full period by summing daily contributions. These factors, which we denote by , are defined such that
Once adjusted, the daily contributions can be summed over time which allows for the decomposition of performance over time, characteristics and instruments.
We use a linking methodology known as the Frongello method [1] that was recently proven to mitigate the problem of amplifying small returns that occur with the well-known Cariño method [2]. Frongello adjustments are defined as
With the returns properly adjusted, we can achieve a granular, exact performance decomposition over time and across instruments.
In the heatmap, the returns shown are thus “linked” so that all returns shown sum up to the total performance over the period. To do so we link the returns of the factors by multiplying them by the appropriate and recompute the residual returns:
with the returns of the “linked” factors. At each time, the return is thus decomposed into parts attributed to either a risk driver
Notice that the total figures shown for the selected period are not affected by linking, only the returns shown in the heatmap. Also, similarly as before, in the relative case:
Confidence interval
The precision of the performance decomposition depends on the confidence interval of the exposures to the risk factors. To obtain an estimate of the standard error around the performance associated with a given factor over the period considered, we use the delta method, which yields:
where is the standard error associated with . Using the delta method again, the standard error around the residual is given by:
with .
Decomposition of the unexplained performance
Note that in the above, the weights are associated with the funds or mandates that compose portfolio. Whenever the holdings of the portfolio are available, it is possible to approximate the contribution of each stock to the unexplained risk. To do so we aggregate the holdings in each stock to form a portfolio of stocks denoted by and estimate the stocks betas . This analysis is particularly relevant when comparing two portfolios with similar holdings, in which case we might consider the relative weights between the two instead of absolute weights. The contribution of each stock to the residual risk over the period is then approximated as follows:
with
References
1 Andrew Frongello. (2002). Linking Single Period Attribution Results. Journal of Performance Measurement. 6(3).
2 Yindeng Jiang and Joseph Saenz. (2014). Comparing Performance Attribution Linking Methods: An Empirical Study. SSRN pre-print. 10.2139/ssrn.2463378.