1.3 Traditional Risk and Risk-Adjusted Return Measures .1 Volatility.1Volatility
1.3.3 Relationship of Tracking Error and Alpha
1.3.2.3 Conclusion
Tracking error isthe risk measure in benchmark-oriented portfolio management.
As a relative risk measure, it corresponds in its definition and interpretation to volatility as an absolute risk measure in benchmark-agnostic portfolio management.
Both risk measures are symmetrical, and their interpretation which is based on the mathematical concept of confidence intervals is only valid, if the underlying data (either an absolute return or a relative return time series) are normally distributed.
This especially means that tracking error (as well as volatility) treats positive and negative deviations in the same way.
in 1989. This version was then improved in 2002 by Roger Clark,26 Harindra de Silva27and Steven Thorley,28see Clarke, de Silva, and Thorley (2002).
The link between the engaged relative risk (measured by tracking error) and the relative return (measured by alpha) in benchmark-oriented portfolio management is based on the assumption that in the portfolio all desired positions can be implemented. No restrictions, for example, on short selling or portfolio weights are considered. This, of course, is unrealistic in active portfolio management practice as there are severe restrictions, like the long-only constraint which prohibits short positions in portfolios.29These restrictions were included into the fundamental law of active portfolio management by Clark, de Silva and Thorley in 2002. De Silva and Clark work for Analytic Investors, a quantitative asset management firm in Los Angeles.30Their version is what nowadays is understood under thefundamental law of active managementand this is the version that will be presented here.
Simply speaking, the basic message is “no risk no fun”. In active portfolio management, the fun part is alpha while the risk part is the engaged tracking error.
Before we can mathematically state the fundamental law of active management we need to define a few variables:
In portfolio management, the investment universe is the set of all securities a portfolio manager chooses from when creating his portfolio. For a European equity portfolio, the investment universecomprises all European equities. Let IC be the information coefficient,TCthetransfer coefficientandN thebreadthof the investment universe:
1. BreadthN:
N is the number of stocks in the investment universe adjusted by correlations between the stocks to achieve independence.31 Each stock can be seen as a
26Roger G. Clarke, Ph.D., is the chairman of Analytic Investors. Recognized as an authority with more than 20 years experience in quantitative investment research, Roger Clarke has authored numerous articles and papers including two tutorials for the CFA Institute. He also served on the faculty of Brigham Young University for 8 years where he specialized in investment and options theory and continues to lecture as a guest professor.
27Harindra de Silva, Ph.D., CFA, is the president of Analytic Investors and a portfolio manager.
De Silva has authored several articles and studies on finance-related topics including stock market anomalies, market volatility and asset valuation.
28Steven Thorley, Ph.D., CFA, is the H. Taylor Peery Professor of Finance at the Marriott School of Management at Brigham Young University in Provo, UT.
29This is most common in retail funds, i.e., funds available for public distribution.
30Analytic Investors, LLC was founded in 1970. The original firm was known for its expertise in derivatives strategies. Nowadays, Analytic is part of Old Mutual Asset Management (U.S.), a group of affiliate firms selected by Old Mutual that have complementary investment styles (non- overlapping) and are considered top-quality investment management firms.
31This is, for example, done by using risk models like BARRA. It is a highly mathematical task, so we do not go into further detail here.
“strategy”,32i.e., as the outcome of a decision on how to weight a particular stock in the portfolio. In active portfolio management, its weight is mainly expressed as the relative security weight in the portfolio versus the index. This is also called over- or underweight. For example, if a security has1:5% weight in the index but only1:0% weight in the portfolio, the security has an underweight of 0:5% versus the benchmark. This over- or underweight decision compared to the benchmark is a strategic decision of the portfolio manager. In this respect,N is the number of independent strategies the portfolio manager can invest in.
2. Information CoefficientIC:
TheICis a coefficient that measures the correlation between the predicted and the finally realized alphas, i.e., the quality of the predictions of the portfolio manager about the future alpha of each stock in the investment universe.
3. Transfer CoefficientTC:
The TC is a coefficient that measures the correlation between the portfolio manager’s alpha predictions and the implemented strategies in the portfolio, i.e., the implementation quality of a portfolio manager.
Using these definitions the ex-ante alpha of the fundamental law of active management can be written as
˛exanteIC TC TE p
N : (1.39)
Equation (1.39) is only valid when the impact of fees like management fees, administration fees, depot bank cost, etc. is neglected. In this case alpha is called gross-of-fee alpha.33 These costs only pertain to the portfolio but not to the benchmark.
It is important to repeat that the fundamental law of active management holds for all types of active asset management vs. a benchmark. The termfundamental does not refer to fundamental asset management (as opposed to quantitative asset management) but simply states that this is a general (i.e., fundamental) rule governing all types of benchmark-related active asset management, let it be fundamental or quantitative, equity or fixed income portfolio management.
Example 6
Let us consider an equity portfolio comprising European large-cap stocks managed against the MSCI Europe Index. While the index comprises roughly 450equities, the number of stocks an active portfolio manager can usually invest in, i.e., the investment universe, is almost 1;600.34 However, some of these securities are highly correlated. For example, stocks within the same industry and
32Sometimes also calledbet.
33If these costs are already subtracted, alpha is callednet-of-fee alpha.
34Assuming the portfolio manager is not restricted to invest only in benchmark securities, i.e., off-benchmark positions are allowed.
within the same country are likely to have a higher correlation than stocks from different industries and countries. BreadthN is corrected for this correlation using financial engineering software. As a result, breadth N of a European investment universe is roughly900.
Assuming the allowed tracking error as given by the client is3%D 1003 we can calculate the expected alpha of the portfolio. But whileN (via the investment universe) andTE are both specified by the client in the portfolio’s investment guidelines,35ICandTCdepend solely on the quality of the portfolio manager.
For a very good portfolio manager, we roughly have the long-runICD 101 and TC D 13. Putting these data in Eq. (1.39), we get as the expected (i.e., ex-ante) alpha:
˛exanteIC TC TE p N D 1
10 1
3 3
100 p 900
D3%: (1.40)
This means, the fundamental law of active management predicts the expected (therefore, ex-ante) p.a. alpha based on the client’s investment universe and tracking error specification in combination with the skills of the portfolio manager.
End of Example 6