Question:

In Grinold Chapter 17, Performance Attribution and Analysis deconstructs periodic returns into components ("attributes returns to components"). For example, if the portfolio return is 12% and the benchmark is 10%, then the ACTIVE RETURN is +2%. This active return is attributed to components.

(i) What are the components?
(ii) Which are systematic and which are residual, and what do these terms mean?
(iii) Which are components of active management?
(iv) Where is skill captured?
(v) Where is luck captured?

Answer:

(i)
The portfolio active return is attributed to:
1. Total Active Systemic Return,
2. Active Residual Returns attributed to common factors
3. Active Residual Returns attributed to portolio specific factors

(ii)
The key here is the analogy to the capital asset pricing model (CAPM). In the CAPM, a security has systematic returns owing to systematic exposure to the market portfolio (i.e., beta is the security’s sensitivity to the single-factor risk premium called the equity risk premium) and specific/residual returns. Specific/residual returns are those returns that are uncorrelated with the market.

Now take the same idea and replace the market portfolio with a benchmark. ACTIVE SYSTEMATIC RETURNS are those owing to holding securities in common with the benchmark; if the benchmark has an expected excess return, then holding the benchmark produces excess return but no alpha! The first term above is analogous to CAPM:

* In CAPM, the return owing to systematic exposure = beta * ERP
* In the above, the return owing to holdings in common with the benchmark = active portfolio beta * benchmark return

Residual returns, therefore, are generated by exposures that are uncorrelated to the benchmark. Residual returns include both factor exposures (i.e., exposure to factors that are not the benchmark) and manager-specific skill (i.e., alpha in the strictest sense)

(iii)
First, there is not necessarily a concrete answer to this. Grinold et al say “Performance analysis, just like attribution, is not uniquely defined. The scheme outlined here is simply a reasonable approach to distinguishing the various sources of typical strategy returns.”

Second, here is the scheme they outline. In short, they attribute or give credit to active management ALL ACTIVE PORTFOLIO RETURNS EXCEPT THE EXPECTED ACTIVE BETA RETURN. Per Table 17.4, this includes:

* Active beta surprise (inside total active systemic return, above)
* Active benchmark timing (inside total active systemic return, above)
* ALL active residual returns: Industry factors (inside second term, above); risk index factors (second term, above), and Specific (third term, above)

(iv) and (v)

Skill would be limited to the positive active returns (less expected active beta returns), HOWEVER, the identification of skill/versus luck is not found in the attribution per se (this is the key point of the beginning of the reading). For example, we can find +2% attributed to active management but unless the finding is STATISTICALLY SIGNIFICANT we cannot distinguish luck from skill (out-performance per se could be luck or skill; proof of skill requires sustained and statistically significant out-performance).

Under Grinold et al, the identification of skill requires:

* Calculation of the information ration associated with the attributed return, and
* Calculation of the t statistic to determine significance