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. We compute the contributions of the returns associated with fundamental risk factors first because they are shown by academia to be the primary drivers of long-term performance. The remainder of returns, denoted by $e_t$, are then separated from the returns driven by fundamental risk factors.

This approach to performance decomposition reflects the hierarchy between factors in explaining returns. The fundamental risk factors have been shown to provide excess returns over the long term, making the portfolio's exposure to these factors the most important driver of long-term performance, taking precedence over other factors. Exposures to other sector-based risk factors or active management decisions are related to the tactical exposure of the portfolio rather than its long-term performance and therefore are taken into account in the residual.

Method

The first step consists of identifying the returns associated with the portfolio's exposures to fundamental risk factors. These exposures are equal to the portfolio characteristics used throughout the platform. Once these returns are identified, they are decomposed into contributions from the various risk factors. The remaining returns are then categorized as 'residual'.

Mathematically, for a portfolio $\bold w\in\Bbb R^{N\times 1}$ the decomposition of the cross-section of returns is particularly straightforward, as $$\begin{align*} r_{w,t} &= \bold w^T \bold c \bold F_t^{\text{fundamental}} + e_t\\ &=\sum_{\ell}\sum_i^N (\bold r_{w,t})_{\ell,i} + e_t, \end{align*}$$ where the term $(\bold r_{w,t})_{\ell,i}$ and the columns of the matrix $\bold F_t^{\text{fundamental}}\in\Bbb R^{L_{fundamental} \times T}$ correspond to the returns of the Market, Value, Size, Momentum, Investment, Profitability and Low-Risk factors. The matrix $\bold c \in\Bbb R^{N\times L_{fundamental}}$ contains the static characteristics of the instruments associated with the fundamental characteristics. The contribution of the $i^{th}$ instrument's $\ell^{th}$ characteristic to the portfolio's return on day $t$ is given by

$$(\bold r_{w,t})_{\ell,i}= w_i c_{\ell,i} \bold F_{\ell, t}.$$ The residual is then obtained as $$e_t = r_{w,t} - \bold w^T \bold c \bold F_t^{\text{fundamental}}.$$

Performance linking

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 $$r_w = \prod_t (1+r_{w,t})-1\neq\sum_t r_{w,t}.$$ 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.

Because the decomposition of performance over several periods is difficult, 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 $\gamma_t$, are defined such that $$\begin{align*} \tilde{r}_{w,t}&= \gamma_t r_{w,t}\\ r_w &= \sum_t \tilde{r}_{w,t}. \end{align*}$$ 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 $$\gamma_t^F=\left[ \prod_{i=1}^{t-1}(1+r_{w,t})\right].$$ With the returns properly adjusted, we can achieve a granular, exact performance decomposition over time and across instruments.

References

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