Should Benchmark Indices Have Alpha?
Revisiting Performance Evaluation
Martijn Cremers (Yale)
Antti Petajisto (Yale)
Eric Zitzewitz (Dartmouth)
How Would You Evaluate These Funds?
ƒ Regress 3 stock portfolios on the four-factor
Fama-French-Carhart model
ƒ 26-year period from 1980 to 2005
ƒ Portfolio A:
ƒ α = 0.82% (t = 2.95) per year
ƒ Portfolio B:
ƒ α = –2.41% (t = –3.35) per year
ƒ Portfolio C:
ƒ α = 5.24% (t = 3.97) per year
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
1
Even Passive Indices Appear to Have Skill
ƒ All portfolios are entirely passive:
ƒ A: S&P 500
ƒ B: Russell 2000
ƒ C: S&P 500 Growth – Russell 2000 Growth
ƒ Easily replicated by cheap index funds
ƒ Most common US equity benchmarks
ƒ … significant biases for performance evaluation?
ƒ Together A and B cover 85% of US equity market
ƒ … relevant for portfolios with size or size-value tilt
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
2
Implications for Active Trading Strategies
ƒ Fama-French 2x3 size-value grid
ƒ Active long-short portfolio
ƒ Long positions in Large Growth stocks
ƒ Short positions in Large Value
ƒ 1980 to 2005:
ƒ Long side underperformed short side by 1.66% p/y
ƒ Carhart model: positive alpha of 3.90% p/y
ƒ Fama-French alpha is 4.33%
ƒ Mechanical application may lead to incorrect
inferences about performance
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
4
Research Questions
1) How large a problem are these index alphas?
ƒ Effect for all common indices
2) Why do we get non-zero index alphas?
ƒ Problems in factor construction
3) What would be a better factor model?
ƒ Modified factors and index-based factors
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
5
Main Results
ƒ Benchmark ‘alphas’ for Fama-French-Carhart model
ƒ Large impact on performance evaluation
ƒ Especially across size-value grid
ƒ Benchmark ‘alphas’ misleading
ƒ Primarily from FF factor construction
ƒ Also: CRSP market index construction and (Russell 2000)
index reconstitution
ƒ Alternative size and value factors based on
benchmark indices
ƒ Improve estimation of fund alphas
ƒ Improve pricing
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
6
Data Sources
ƒ Benchmark index returns
ƒ Monthly and daily returns directly from index providers:
S&P, Russell, DJ Wilshire
ƒ Benchmark index holdings
ƒ Monthly data directly from index providers
ƒ Mutual fund returns
ƒ Monthly returns from CRSP
ƒ Daily returns from CRSP, Yale ICF, and S&P
ƒ Mutual fund holdings
ƒ Quarterly/semiannually from Thomson Financial
ƒ Stock and firm data from CRSP and Compustat
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
8
Benchmark Indices: S&P and Wilshire
Large
S&P 500
S&P500
Value
S&P500 Large
Growth
Wilshire 5000
S&P400
S&P400
S&P 400
Value
Growth
Wilshire 4500
S&P600
S&P600
S&P 600
Growth
Value
Small
Small
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
9
Benchmark Indices: Russell
Large
Large
Russell 3000
Russell 1000
Russell Midcap
Russell 2000
Small
Small
Large
R3000
Value
R3000
Growth
R1000
Value
R1000
Growth
R2000
Value
R2000
Growth
Large
R Mid
Value
Small
R Mid
Growth
Small
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
10
S&P Indices Are a Subset of the Market
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
11
Russell Indices Include Entire Market
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
12
Fama-French-Carhart vs. Indices
Benchmarks implied by FF-C model versus indices:
ƒ CRSP-VW includes assets most indices/portfolios do not
ƒ SMB is EW mix of growth and value
ƒ Indices are VW
ƒ Small value has outperformed => EW SMB > VW SMB
ƒ SMB excludes stocks with no/negative BtM (e.g., IPOs)
ƒ Including reduces SMB spread
ƒ FF-C: same value effect for Big and Small stocks
ƒ Boundaries differ
ƒ FF-C includes Midcaps in Big
ƒ Indices include both High and Med BtM stocks in Value
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
13
Index Alphas 1980-2005
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
14
Choice of the Market Portfolio:
Not just US Common Stocks in CRSP-VW
1%
5%
1%
1%
US Stocks
Non-US stocks
Closed-End
Funds
REITS
Other
92%
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
15
Choice of the Market Portfolio:
FF-C Annualized Alphas for CRSP-VW Categories
1.00%
0.00%
-1.00%
US
Non-US Closed- REITS
Stocks stocks
End
Funds
Other
-2.00%
-3.00%
-4.00%
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
16
Fama-French Component Portfolios: Market Weights
SmallGrowth
4%
SmallMedian
4%
SmallValue
2%
Big-Value
13%
BigMedian
29%
BigGrowth
48%
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
17
Fama-French Component Portfolios
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
18
Weights of SMB and HML
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
19
Size Decile 10 vs. Its 3-Factor Benchmark
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
20
Size Decile 4 vs. Its 3-Factor Benchmark
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
21
S&P 500 Alpha and Modified FF Factors: Annual Alpha
0.90%
0.80%
0.70%
0.60%
0.50%
0.40%
0.30%
0.20%
0.10%
0.00%
FF model +
Momentum
Only US
stocks in
Market
Also Use
SMB-VW
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
22
Russell 2000 Alpha and Modified FF Factors: Annual Alpha
0.00%
-0.50%
-1.00%
FF model +
Momentum
Only US
stocks in
Market
Also Use
SMB-VW
-1.50%
-2.00%
-2.50%
-3.00%
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
23
Russell 2000 Reconstitution Effect: Annualized Alpha
0.00%
June-July
-0.20%
Remainder of
the Year
-0.40%
-0.60%
-0.80%
ƒ Index reconstitution costs (e.g., Petajisto (2008))
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
24
Criteria for a Good Model
ƒ Pricing
ƒ Simplest possible explaining the crosssection of expected returns
ƒ Benchmarking money managers
ƒ Most accurate estimate of a manager’s
value added relative to a passive strategy
ƒ Benchmark model includes pricing model
ƒ Add non-priced factors
ƒ Reduce noise in alpha estimates
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
25
Selection of Alternative Models
ƒ Carhart: MktRf, SMB, HML, UMD
ƒ Carhart + S&P500 + Russell 2000
ƒ Modified Carhart:
ƒ Only US stocks, value-weighted SMB and HML
ƒ Seven-factor model:
ƒ MktRf, MMB, SMM, BHML, MidHML, SHML, UMD
ƒ Index-based models:
ƒ S5, R2-S5, R3V-R3G, UMD
ƒ S5, RM-S5, R2-RM, R3V-R3G
ƒ S5, RM-S5, R2-RM, S5V-S5G, RMV-RMG, R2V-R2G, UMD
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
26
Our Tests for Alternative Models
ƒ Explain common time series variation
ƒ Tracking error volatility of all-equity mutual funds
ƒ Explain cross-section of average returns
ƒ Mutual funds
ƒ 3 x 3 portfolio sort by size and value
ƒ 100 Fama-French size-value portfolios
ƒ 10 x 10 portfolio sort by size and value
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
27
Index-Based Models Perform the Best
ƒ Out-of-sample Tracking Error of mutual funds
ƒ Index models: lowest TE (5% - 10% lower)
ƒ Cross-section of average mutual fund returns:
ƒ Index models
ƒ close-to-zero alphas across all fund groups
ƒ Carhart model
ƒ small-cap funds underperform large-caps;
ƒ conclusion fully reversed (by 5% per year!) if control for
benchmark index
ƒ Cross-section of average return on 100 FF portfolios
ƒ Index models produce the highest R2
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
28
Mutual Fund Alphas: Fama-French factors
FFC Annualized Alpha of
Fund Return
FFC Alpha, Benchmarkadjusted Fund Return
1.00%
0.00%
-1.00%
0.00%
Large Cap
Funds
Small Cap
Funds
-1.00%
Large Cap
Funds
Small Cap
Funds
-2.00%
-2.00%
-3.00%
-3.00%
-4.00%
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
30
Mutual Fund Alphas: Index factors
4-factor Index Model
Annualized Alpha,
Fund Return
4-factor Index Model
Annualized Alpha,
Benchmark-adjusted Fund
Return
1.00%
1.00%
0.00%
0.00%
Large Cap
Funds
Small Cap
Funds
Large Cap Small Cap
Funds
Funds
-1.00%
-1.00%
-2.00%
-2.00%
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
31
Cross-Section of 100 Fama-French Portfolios
R2 of Mean Returns on Betas
70.00%
60.00%
50.00%
40.00%
30.00%
20.00%
10.00%
0.00%
4-Factor
7-Factor
FFC
+ S&P 500 + Russell
Index ShouldIndex
Midcap
B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluat
ion
+ Russell
2000
32
So What Model Do We Propose?
3 approaches:
1) Modify the Fama-French factors
ƒ
Value-weight SMB, only US stocks, etc.
2) Use benchmark-adjusted returns
ƒ
FF factors relatively harmless
3) Use index-based models
ƒ
ƒ
ƒ
7 factors: S5, RM-S5, R2-RM, S5V-S5G, RMVRMG, R2V-R2G, UMD
4 factors: S5, R2-S5, R3V-R3G, UMD, or
S5, RM-S5, R2-RM, R3V-R3G
Simplest and most general approach
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
33
Further Considerations for
Portfolio Performance Evaluation
Use benchmark factors that
ƒ … include similar assets as portfolio
ƒ E.g., both only US common stocks
ƒ … are representative of the portfolio asset class(es)
ƒ E.g., separate index for each asset class
ƒ … use similar weights
ƒ E.g., value-weighted
ƒ E.g., similar weights to value and growth, liquid and
illiquid, traded and non-traded, high and low transaction
costs, etc.
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
34
Conclusions
ƒ Common indices
ƒ Large nonzero Fama-French alphas
ƒ Affects performance evaluation
ƒ Benchmark FFC alphas are misleading
ƒ FF factor construction methodology
ƒ CRSP market and Russell 2000 reconstitution
ƒ Alternative models based on benchmark indices
ƒ Improve estimation of alphas
ƒ Improve cross-sectional pricing
Should B enchm ark Indices H ave A lpha? R evisiting Perform ance Evaluation
37