Active is as Active Does: Active Share vs Tracking Error

Transcription

1 Active is as Active Does: Active Share vs Tracking Error Sebastian Ceria Chief Executive Officer Axioma, Inc. Joint work with Vishv Jeet and Dieter Vandenbussche March 2015 FACTSET 2015 SYMPOSIUM NOT FOR REDISTRIBUTION

2 Goals Active Share is increasingly being used and reported as an important characteristic of a portfolio to measure how active the portfolio is. This metric is reported by commercial performance evaluators and asset owners are asking their managers to report such metric on a regular basis: Our goal is NOT to opine in the debate about whether active share is an indicator of outperformance or not. Rather, this presentation s objective is to help quantitative portfolio managers understand how the various inputs to the quantitative portfolio construction process affect Active Share. 2

3 Agenda Quick review of Active Share Quantitative Portfolio Construction and the impact on Active Share: Diversification Alpha Process Constraints Benchmark Some other observations on Active Share How does Active Share change over time? Factor bets and Active Share The role of Optimization Conclusions 3

4 Active Share Active Share is a measure of activeness of the portfolio. Active Share: 1 w 2 i i b i w i : Portfolio weights, b i : Benchmark weights. Active share is the fraction of the portfolio different from the benchmark. Also, it s the sum of the long side of the active holdings. Active share is always between 0 and 100% for a long-only, fully invested portfolio. For fully invested portfolios: Active Share = 1 min (w i, b i ) i Benchmark Coverage 4

5 Illustrating Active Share Active Weights Portfolio Benchmark Fully invested and Long-only portfolio. Active share is sum of long active weights. In this example, Active Share of the portfolio is 30%. 5

6 Active Share and Factor Risk Models Active Share: 1 2 w i b i i Indifferent to many small versus few big active bets Does not fully capture the conviction a portfolio manager is showing A risk measure with a naïve risk model that knows nothing about variances and covariances Active Specific Risk: δ i 2 i w i b i 2 Accounts for assets with differing volatilities Weighted version of active share Encourages diversification in active bets Active Factor Risk: k,l Ω kk g k g l where g = X (w b) Can take large active bets at the asset level without taking factor bets Helps identify systematic risk 6

7 Fund Classification (Active Share vs TE) Active Share High Low Diversified Stock Pickers Closet Indexing Concentrated Stock Picks Factor Bets Pure Indexing Low Tracking Error High Cremers and Petajisto used active share and tracking error to classify funds into different categories. They observe outperformance for funds with higher Active Share. These results have been challenged in subsequent studies. 7

8 Quantitative Portfolio Construction and Active Share Proposed Questions Is Active Share a true measure of diversification? How does the Alpha Process affect Active Share? Do constraints bias Active Share? What is the role of the Benchmark? 8

9 Portfolio Diversification and Active Share Concentrated Portfolio Diversified Portfolio T PG IBM CVX GE MSFT AAPL XOM Portfolio Benchmark T PG IBM CVX GE MSFT AAPL XOM Concentrated Diversified Active Share 20.00% 20.00% Active Risk 3.16% 2.08% Active Specific Risk 2.61% 1.60% Active Factor Risk 1.78% 1.34% Active Share ignores diversification altogether! 9

10 The Alpha Process and Active Share Simulate normally distributed Alphas (expected returns), optimize to maximize alpha, subject to budget, long-only, and fixed tracking error target to the S&P 500. Distribution of Active Share The distribution of the Alpha can make a difference in the Active Share for quantitative portfolios. 10

11 Fat Tailed Alphas and Active Share Statistically significant T-stat (~ -23.0) for the negative slope Fat tails (higher kurtosis) of cross-sectional alpha distribution reduces Active Share. 11

12 Cross-sectional Alpha Distributions Some alphas encourage bigger, less diversified active bets, which leads to higher tracking error. These alphas bump into the tracking error constraint sooner. Fat tailed alpha (higher k) can reduce Active Share. Fat tailed alpha can also increase the variability of Active Share. Alpha Distributions Active Share Histograms 12

13 Alphas and the Long-only Constraint Small cap assets are harder to underweight due to long-only constraint. Optimized portfolios tend to have a small size bias. To see the effect, we generate random alphas (normally distributed). On average, alphas have no size bias. Now we maximize alpha, fully invested with a 3% tracking error bound and long-only. 13

14 Alpha and Size Correlation of Alpha and Size Size Exposure of Optimal Portfolio Mean correlation between alpha and size is zero. Mean exposure between alpha-optimized portfolio and size is negative. 14

15 Higher Correlation of Alpha and Size Leads to Lower Active Share 15

16 The Effect of Alphas With Explicit Size Tilt Active Share Histograms Negative size tilt in alpha can raise Active Share of quant portfolios and vice-versa 16

17 Kurtosis and Size Bias are not Related Effects Fat Tailed Alpha do not necessarily have size tilt. The two effects are completely independent (zero correlation). 17

18 How Does the Tracking Error Constraint Affect Active Share? 1 Active Share Vs. Tracking Error Lower Tracking Error limit can drastically reduce Active Share. 18

19 Impact of the Turnover Constraint 1 Turnover Frontier Active Share is unaffected by the turnover limit constraint. 19

20 Impact of the Long-Only Constraint 1.2 Active Share: Shorting Allowed Longonly Active Share can be improved if shorting is allowed. 20

21 Impact of Active Asset Bounds Constraints Active Share Histograms Active Asset Bounds can make a difference to Active Share, the more the merrier. 2% bounds reduced the dispersion of Active Share around the mean. 1% bounds reduced the mean Active Share. 3% bounds raised the mean Active Share. 21

22 Benchmark Selection and Active Share 1 Active Share S & P 500 Russell 1000 Russell 3000 Russell Global Broader benchmarks allow for higher Active Share in quant strategies. 22

23 Benchmarks: Active Share Distribution Active Share Histograms Cross-sectional distribution of benchmark affect the Active Share of quant portfolios exactly the same way as alpha. Concentrated benchmarks results in lower Active Share. 23

24 Active Share of a Quant Strategy Through Time Momentum Value Growth Active Share varies quite a bit thru time for a strategy with fixed Tracking Error target. During crises, due to raised market volatility and fixed TE targets, Active Share must be compromised. 24

25 Explaining Time Variation in Active Share The idiosyncratic portion of total market volatility better aligns with Active Share. 25

26 Optimization with Active Share Imposing an upper bound constraint on Active Share results in convex optimization problem: Just like other constraints, can serve as a backstop to the risk model (see DeMiguel et al). Makes the strategy more conservative. Imposing a lower bound on Active Share results in a nonconvex optimization problem (i.e a very difficult problem to solve). Arbitrarily maximizing active share is not the goal anyway, even if there is correlation with outperformance, there is no evidence of causality. Active Share is not the same thing as skill. Lower bounds may introduce perverse incentives to the optimizer Off-setting active positions in highly correlated assets. 26

27 Conclusions Active Share is an interesting measure of portfolio s activeness that complements other measures of activeness such as Tracking Error and its components. Active Share is a good auxiliary measure that may help some asset owners and investors interested in aggressive strategies to screen out closet indexers. Many factors impact Active Share of a fund, making it difficult to compare Active Share across funds. Given that Active Share does not capture the intrinsic variation in volatility across assets and across time, nor the diversification effect, one cannot use Active Share in isolation to categorize funds. We cannot identify factor betting strategies on the basis of Active Share and Tracking Error. The portion of variance coming from active specific variance may be a better alternative. 27