amihud and goyenko

Amihud and Goyenko

Mutual Fund’s R2 as Predictor of Performance

Fund R2 as predictor of performance

PRMIA-CIRANO- Lunch Conference December 7, 2011

Yakov Amihud Stern – New York University

Ruslan GoyenkoMcGill University

Cremers & Petajisto (2009): Active Share – the sum of absolute deviations of the fund’s portfolio from the benchmark portfolio – predicts alpha from a regression on the benchmark index.

Brand, Brown & Gallagher (2005) measure “active management” by a divergence index, the sum of squared deviations of the fund portfolio’s weights from the benchmark portfolio, using Australian data. The divergence index positively predicts fund performance

Daniel, Grinblatt, Titman and Wermers (1997): while securities that are picked by mutual funds outperform a characteristic-based benchmark, gains from stock picking approximately equal the funds’ average management fee.

Motivation: Does selectivity enhance mutual fund performance?

Christoffersen, Goyenko, Jacobs, and Karoui Illiquidity Premia in the Equity Options MarketAmihud and Goyenko Fund R2 as predictor of performance

Kacperczyk, Sialm and Zheng (2005) find that funds exhibit better performance if they have greater industry concentration of holdings compared to the weights of these industries in a diversified portfolio

Kacperczyk and Seru (2007) find that funds whose stocks holdings are related to company-specific information different from analysts’ expectations exhibit better performance

Kacperczyk, Sialm and Zheng (2008) find that the higher return gap, the difference between the reported fund return and the return on a portfolio that invests in the previously disclosed fund holdings, predicts better subsequent performance

Motivation


These measures are hard to calculate, require a lot of data which are not readily available

Problem: Indexing vs Selectivity: if a fund puts x% in S&P500 and (1-x%) in, say, Russell 2000, (stating that its benchmark is S&P500), AS (CP, 2009) will classify it as "active" while it is not, it is just indexing

Require a well specified benchmark. CP(2009) : “Since 1998, the SEC has required each fund to report a benchmark index in its rospectus; however, this information is not part of any publicly available mutual fund database, and prior to 1998, it does not exist for all funds. These self-declared benchmarks might even lead to a bias: some funds could intentionally pick a misleading benchmark to increase their chances of beating the benchmark by a large margin, as discussed in Sensoy (2009).”

For active equity funds only! What if a fund also holds corporate bonds? (split benchmark!)

Motivation


Measuring active management by tracking error (Wermers 2003), the StdDev of S&P500-adjusted fund return. It is positively related to the contemporaneous fund alpha. But others find that tracking error does not predict performance (Cremers & Petajisto (2009))

Omitted variable problem: for funds (not individual stocks), tracking error is a small % of the total volatility. The total volatility -- or systematic volatility -- is an omitted variable which is correlated with the tracking error

Motivation


Use the fund’s R2 (from a regression on common factor) as a proxy measure of selectivity.

Higher R2 means greater indexation.Lower R2 means greater selectivity better performance.

Our suggestion:


R2 = 1 – VARIANCE

RMSE 2

= 22

2

RMSERiskSystematicRiskSystematic

Lower R2 measures greater selectivity or active management and it significantly predicts higher fund performance.

Funds sorted into lowest-quintile lagged R2 and highest-quintile lagged alpha produce significant annual alpha of 2.4% or more

R2 also predicts performance of funds that hold corporate bonds

Preview of the Results


Fama-French-Carhart factors: RM-Rf, SMB, HML, UMD + Russell2000resids given the results of Cremers, Petajisto and Zitzewitz (2010) – (FFCR)

Four-factor model proposed by Cremers et al (2010) : S&P500 -Rf, Russell 2000 -S&P500, Russell 3000 value -Russell 3000 growth, and UMD – (CPZC)

Also standard four factor model (FFC)

Methodology: Benchmark Models


R2 : calculated from semi-annual regressions of daily/weekly fund returns on the FFCR (CPZC) factor returns, contemporaneous and one-day lag (Dimson, 1979) (robustness: monthly, 24-month rolling window)

Performance measures: alpha and InfRatio = alpha/RMSE (Treynor-Black, 1973).Information Ratio gives the improvement in the Sharpe Ratio due to adding an asset (fund) to an investor’s efficient portfolio.SR2

P = SR2

M + [alphaA / RMSEA]2

Greater InfRatio greater demand for the asset

Methodology:


Performancey = b*TR2y-1 + control variables

TR2 = log[(√R2+c)/(1- √R2+c)] ,

where c = 0.5/n, n being the sample size (Cox (1970, p. 33)). N=120 for the daily return estimation and 40 for weekly returns

Hypothesis: b < 0. Greater selectivity, or greater deviation from indexing better performance

Estimation model:


*Expense ratio*Fund size (TNA) at the end of the year. (We use funds with TNA > $15m (Elton, Gruber & Blake, 1996)).*Turnover*Managerial Tenure*Age of the fund*alpha or InfRatio (skill)*Style dummies: (i) Aggressive Growth, (ii) Equity Income, (iii) Growth, (iv) Long term growth, (v) Growth and Income, (vi) Mid-Cap, (vii) Micro-Cap funds, (viii) Small cap, and (ix) Maximum Capital Gains

Control variables (all from previous half-year) (standard):


Sample: actively-managed all-equity funds with identifiable objective. Excluded are funds with “Index” or abbreviation of common indexes in name. Funds with at least 70% holdings in common stocks

Data: 1990-2010. Earlier data are from Yale SOM’s database.CRSP data from 1999 2,460 funds, 42 periods (the test period is always 6 months), 37,198 fund-periods Required: at least 120 daily returns or 40 weekly returns in y-1 (estimation period)R2: We trim off the top and bottom 0.5%.Mean: 0.902, range: 0.270 to 0.995

Sample and Data


FM regressions are similar to Carhart (1997) and Chen, Hong, Huang and Kubik (2004)

Fama-Macbeth (1973) regressions: sample 1990-2010


Performancej,t = γtTR2j,t-1 + δ1tExpensesj,t-1 + δ2tlog(TNA)j,t-1 + δ3t[log(TNA)]2

j,t-1

+ δ4tTurnoverj,t-1+ δ5tlog(Fund Age)j,t-1+ δ6tlog(Manager tenure)j,t-1

+ δ7tPerformancej,t-1 +

9

11,,n

ntnjStyleDummy

Daily Returns


Explanatory variables (lagged)

Daily Returns FFCR factor model CPZC factor model (1) (2) (3) (4)

alpha InfRatio alpha InfRatio TR2

TR2 Weighted Mean

-0.914 (3.32)

Med: -0.818 Neg: 33/42+

-0.749 (3.50)

-0.015 (5.66)

Med: -0.014 Neg: 34/42+

-0.013 (5.16)

-1.046 (3.16)

Med:-1.059 Neg: 33/42+

-0.943 (4.09)

-0.014 (7.05)

Med: -0.013 Neg: 37/42+

-0.012 (6.26)

Expenses -0.629 (2.36)

-0.010 (4.56)

-0.893 (3.10)

-0.012 (5.56)

log(TNA) 0.597 (1.24)

-0.002 (0.47)

0.399 (0.86)

-0.003 (0.72)

[log(TNA)]2 -0.035 (0.96)

2*10-4 (0.85)

-0.024 (0.69)

3*10-4 (1.08)

Turnover 0.004 (1.44)

2*10-5 (1.15)

0.002 (0.54)

8*10-7 (0.04)

log(Fund Age)

-0.226 (1.54)

-0.002 (1.36)

-0.246 (1.68)

-0.003 (2.15)

log(Manager Tenure)

-0.234 (1.50)

-0.002 (1.38)

-0.211 (1.32)

-0.001 (0.78)

Dependent variable lagged

0.139 (4.84)

0.129 (5.78)

0.122 (4.39)

0.116 (6.27)

R-sqr 0.15 0.17 0.15 0.15

Weekly Returns



Weekly Returns FFCR factor model CPZC factor model (1) (2) (3) (4)

alpha InfRatio alpha InfRatio TR2

TR2 Weighted Mean

-1.219 (2.81)

Med: -0.700 Neg: 30/42+

-0.729 (2.47)

-0.035 (4.40)

Med: -0.031 Neg: 30/42+

-0.031 (4.25)

-1.435 (2.79)

Med: -1.123 Neg: 35/42+

-0.966 (2.78)

-0.034 (5.48)

Med: -0.035 Neg: 35/42+

-0.032 (5.09)

Expenses -0.778 (3.53)

-0.022 (4.18)

-1.133 (4.48)

-0.030 (6.27)

log(TNA) 0.386 (0.96)

-0.002 (0.22)

0.216 (0.58)

-0.006 (0.73)

[log(TNA)]2 -0.025 (0.79)

3*10-4 (0.47)

-0.016 (0.54)

6*10-4 (0.91)

Turnover 0.004 (1.43)

4*10-5 (0.80)

9*10-4

(0.26) -3*10-5 (0.72)

log(Fund Age) -0.229 (1.35)

-0.003 (0.82)

-0.202 (1.36)

-0.005 (1.33)

log(Manager Tenure)

-0.193 (1.33)

-0.004 (1.25)

-0.181 (1.09)

-0.003 (0.90)

Dependent variable lagged

0.177 (4.37)

0.158 (4.74)

0.166 (4.81)

0.164 (6.51)

R-sqr 0.16 0.17 0.15 0.15

Results for Untransformed R2

daily returns, FFCR, coeff (R2t-1 )= 6.03 with ‑ t = 2.06, the

weighted mean= 5.60 with ‑ t = 2.98

Economic magnitudes: decline in R2 from 0.9 (the mean) to 0.8 leads to increase in alpha by an annual 0.60% (and 0.56% for the weighted mean). Using TR2 coefficient, the same decline in R 2 would raise the annualized alpha by 0.68%

If only explanatory variables are TR2t-1 Alphat-1 and style dummies

(TNA>$15 mln), then coeff (TR2t-1 )= -0.775 with t = 3.12

Robustness and Economic Significance


Panel regressions: style and time dummies, and standard errors are clustered by both time and fund. For Alpha equation - the coefficient of TR2 is –1.163 (t = 2.77) for the FFCR model and –1.060 (t = 2.52) for the CPZC model.

FFC model: Alpha equation- the mean coefficient of TR2t-1 is 0.801 (‑ t = 2.33) and

the weighted mean is 0.749 (‑ t = 2.90).

Monthly returns, 24 months rolling window, 216 monthly test periods, for 1993-2010, 2,438 funds, 227,726 fund test-period months: FFC model - the mean coefficient of TR2

t-1 is 1.077 with ‑ t = 2.23, the weighted mean coefficient is 0.858 ‑with t = 3.21

CPZC model: the mean coefficient of TR2t-1 is 1.328 with ‑ t = 3.52, the

weighted mean is 1.328 with ‑ t = 4.32

Robustness:


Active Share (AS), the sum of absolute deviations of the fund’s stock holdings (weights) from those of its benchmark index portfolio, predicts fund performance. More recently, Cremers, Ferreira, Matos and Starks (2011) show that AS positively predicts the performance of international mutual funds.

R2 and AS are negatively correlated (data 1990-2006 from CP) with the mean of -0.36 (ranges between 0.07 to -0.68). ‑

AS measures the deviations from a single benchmark index while R2 measures deviations from multiple benchmark indexes

Active Share: Cremers and Petajisto (2009)


TR2 and TAS (weekly returns)



FFCR factor model CPZC factor model alpha InfRatio alpha InfRatio

TR2 TR2 Weighted Mean

-1.110 (2.99)

Med:-0.672 Neg:26/35+

-0.654 (2.18)

-0.023 (3.16)

Med: -0.016 Neg:25/35+

-0.022 (3.15)

-1.484 (3.08)

Med: -0.623 Neg: 28/35+

-0.834 (2.84)

-0.025 (3.72)

Med: -0.021 Neg: 28/35+

-0.023 (3.59)

TAS TAS Weighted Mean

0.396 (1.27)

Med:0.062 Pos:19/35

0.168 (0.71)

0.015 (2.34)

Med: 0.014 Pos:22/35^

0.013 (2.48)

0.542 (2.36)

Med: 0.392 Pos: 23/35+

0.315 (1.79)

0.018 (3.17)

Med: 0.015 Pos: 26/35+

0.014 (2.89)

Other variables (see Table 2)

Yes

Yes

Yes

Yes

ICI: the sum of the squared deviation between the fund’s weights in various industries and the weights of these industries in the market portfolio

Rgap: the difference between the reported fund return and the return on a portfolio that invests in the previously disclosed fund holdings gauges the effects of trades by skilled fund managers who use their informational advantage to trade individual stocks optimally.

Industry Concentration (ICI) and Return Gap (RGap) (Kacperczyk, Sialm and Zheng, 2005, 2008)


Panel Regressions as in Kacperczyk et al.



Alpha (1) (2) (3) (4)

TICI 0.932 (1.95)

0.511 (1.03)

RGap 0.060 (2.10)

0.060 (2.06)

TR2 -1.183 (1.92)

-1.267 (2.77)

Other variables

Yes Yes Yes Yes


InfRatio (5) (6) (7) (8)

TICI 0.011 (3.15)

0.006 (1.88)

RGap 0.0003 (1.39)

0.0003 (1.44)

TR2 -0.014 (2.77)

-0.017 (4.12)

Other variables

Yes Yes Yes Yes

In FM regressions, Alpha equation: both TR2 and TICI are significant, the coeff are -0.905 (t = 2.12) and 0.737 (t = 2.00) respectively; both TR2 and RGAP significant , the coeff are -1.267 (t = 2.77) and 0.060 (t = 2.06) respectively. InfRatio equation: while TR2 is negative and significant, TICI and RGAP are not

"Characteristic Selectivity" (CS), is the difference between the fund return and the weighted average return on one of 125 passive portfolios of stocks that match each of the fund’s stocks on the basis of market capitalization, book-to-market and prior-year return, the weights being equal to the fund’s stock holdings

“Characteristic Timing” (CT), is the weighted annual return on each of the 125 characteristics portfolios, where the weights are those of the stocks that match these characteristics

Following Daniel et al. (1997), both CS return and CT return are “Carhart-adjusted”

Characteristics-based excess return (Daniel, Grinblatt, Titman and Wermers, 1997)


Fama-MacBeth regressions


Explanatory variables (lagged) CS CT (1) (2)

TR2

(t-statistic, corrected for AR(1))

-0.423 (1.47) (2.17)

Med: -0.278 Neg: 23/34+

0.049 (0.14) (0.17)

Med: -0.200 Neg: 13/21

Expenses -0.148

(0.63) -0.079 (0.57)

log(TNA) -0.813 (2.21)

0.349 (1.64)

[log(TNA)]2 0.072 (2.37)

-0.026 (1.67)

Turnover 0.0005 (0.18)

0.002 (1.30)

log(Fund Age) -0.235 (1.62)

-0.076 (0.52)

log(Manager Tenure) 0.115 (1.06)

0.126 (0.71)

Dependent variable, Lagged 0.064 (2.02)

0.054 (0.85)

R-sqr 0.12 0.13

AR(1) coefficient of the series of coefficients of TR2 in the CS regression is -0.389 with t = 2.39

Passive stock portfolios: Fama-French 100 portfolios sorted on size and on book-to-market (10x10)

“styles”: D-micro cap = 1 for the smallest two size decile portfolios, D Small cap‑ = 1 for size portfolios 3 and 4, D growth‑ = 1 for the lowest three book-to-market portfolios, and D value‑ = 1 for the highest three book-to-market portfolios (zero otherwise)

Also repeated the analysis for Fama-French 48 industry portfolios which can be viewed as passive “sector” funds

Is the R2 effect due to selectivity or to pricing of volatility?


Fama-French 100 portfolios



Performance measure

Alpha InfRatio

TR2 0.509 (0.93)

0.003 (0.81)

Dependent variable, lagged

0.085 (2.13)

0.103 (3.57)

D-micro cap -1.694 (0.95)

-0.010 (0.85)

D-small cap -1.270 (1.62)

-0.007 (1.33)

D-growth (low book/mkt)

-0.767 (0.74)

-0.005 (0.77)

D-value (high book/mkt)

-0.483 (0.55)

-1*10-4 (0.02)

R-sqr 0.16 0.18

Portfolio Sorting: FFCR Alpha (sample 1990-2010, Net Returns)


R2t-1

Alphat-1 Low 2 3 4 High All Low-High

Low -2.057* (1.81)

-2.246*** (2.94)

-1.991*** (2.78)

-2.132*** (3.67)

-2.968*** (5.82)

-2.264*** (3.78)

0.911 (0.85)

2 -0.951 (1.19)

-0.940 (1.51)

-1.253** (2.49)

-1.649*** (3.55)

-2.260*** (5.46)

-1.412*** (3.22)

1.310 (1.60)

3 1.273 (1.65)

-0.473 (0.80)

-0.689 (1.43)

-1.081** (2.24)

-2.118*** (5.62)

-0.620 (1.53)

3.392*** (4.42)

4 2.071*** (2.59)

0.652 (0.99)

-0.505 (0.82)

-0.358 (0.74)

-2.023*** (4.84)

-0.051 (0.12)

4.094*** (4.87)

High 2.419*** (2.45)

2.725*** (3.35)

1.196 (1.42)

-0.121 (0.13)

-0.825 (1.38)

1.082 (1.62)

3.245*** (3.28)

All 0.555 (0.83)

-0.032 (0.06)

-0.643 (1.41)

-1.066** (2.40)

-2.044*** (5.66)

-0.624 (1.63)

2.600*** (3.82)

High-Low

4.476*** (2.95)

4.971*** (4.67)

3.188*** (2.05)

2.012** (2.05)

2.143*** (3.20)

3.349*** (3.94)

Portfolio Sorting: CPZC Alpha (sample 1990-2010, Net Returns)


R2t-1

Alphat-1 Low 2 3 4 High All Low-High

Low -1.166 (1.04)

-1.947** (2.31)

-2.313*** (3.31)

-2.482*** (3.99)

-2.365*** (5.70)

-2.048*** (3.35)

1.199 (1.16)

2 -0.538 (0.64)

-0.407 (0.57)

-1.001* (1.76)

-1.251*** (2.80)

-1.655*** (5.38)

-0.972** (2.01)

1.117 (1.44)

3 1.908** (2.27)

0.172 (0.28)

-0.951 (1.39)

-0.463 (1.04)

-1.309*** (3.94)

-0.132 (0.29)

3.217*** (4.10)

4 2.988*** (3.55)

1.434** (2.06)

0.118 (0.19)

-0.473 (0.89)

-1.016*** (2.87)

0.603 (1.28)

4.004*** (5.01)

High 4.220*** (3.73)

2.729*** (3.10)

1.901** (2.34)

0.796 (1.07)

-1.290* (1.67)

1.666* (2.37)

5.510*** (4.70)

All 1.484** (2.00)

0.406 (0.68)

-0.454 (0.84)

-0.771* (1.74)

-1.514*** (4.62)

-0.156 (0.38)

2.995*** (4.37)

High-Low

5.386*** (3.55)

4.676*** (4.55)

4.214*** (4.91)

3.278*** (4.19)

1.075 (1.43)

3.713*** (4.79)

Determinants of R2


Variables lagged half-year FFCR CPZC Expenses -0.273

(9.56) -0.290 (9.82)

log(TNA) 0.149 (4.57)

0.163 (4.55)

[log(TNA)]2 -0.007 (2.79)

-0.009 (2.99)

Turnover -0.0002 (1.33)

-0.0001 (1.21)

log(Fund Age) -0.035 (2.41)

-0.030 (1.84)

log(Manager Tenure) -0.098 (7.13)

-0.102 (7.13)

Style dummy variables: Aggressive Growth - - Equity Income 0.195

(2.00) 0.344 (3.49)

Growth 0.255 (3.24)

0.307 (3.69)

Long term growth 0.199 (1.33)

0.236 (1.58)

Growth and Income 0.427 (4.96)

0.504 (5.65)

Mid-Cap 0.042 (0.54)

-0.006 (0.07)

Micro-Cap -0.277 (2.36)

-0.338 (2.79)

Small Cap 0.175 (1.98)

0.144 (1.57)

Maximum Capital Gains 0.211 (1.28)

0.271 (1.70)

R-sqr 0.48 0.47

Funds that invest in corporate bonds more than 35% of their net asset value, which accommodates “Balanced Funds” (average holdings of corporate bonds - 69.5% across all funds)

Exclude Treasury, government or municipal bond funds

Benchmark models:

Bessembinder et al. (2008) (Fama-French+DEF+TERM) [average R2 =0.53]

Elton et al. (1995): MKT-RF, Bond Index-Rf, DEF and OPTION (Barclays GNMA index – Barclays Gov Intermediate index) [average R2 =0.46]

Sample: 2001-2010 (on average 67 funds per period)

Corporate bond funds: predicting performance with R2


The effect of R2 on Bond fund performance


Variables lagged Bessembinder et al.’s (2008) model Elton et al.’s (1995) model alpha InfRatio alpha InfRatio

TR2

TR2 Weighted Mean

-0.621 (1.74)

Med: -0.966 Neg: 16/20+

-0.885 (3.86)

-0.015 (3.21)

Med: -0.012 Neg: 15/20+

-0.014 (3.36)

-0.745 (-3.30)

Med: -0.466 Neg: 16/20+

-1.027 (5.60)

-0.014 (3.02)

Med: -0.016 Neg: 15/20+

-0.015 (4.38)

Expenses -1.066 (1.32)

-0.019 (2.71)

-0.689 (0.60)

-0.012 (1.79)

log(TNA) -0.210 (0.24)

0.0003 (0.04)

0.141 (0.16)

0.001 (0.16)

[log(TNA)]2 0.010 (0.17)

-0.0001 (0.24)

-0.022 (0.35)

-0.0003 (0.45)

Turnover 0.002 (1.18)

-1*10-5 (0.61)

0.002 (1.61)

-9*10-6 (0.48)

log(Fund Age) -0.327 (1.05)

-0.004 (0.91)

-0.255 (0.78)

-0.002 (0.47)

log(Manager Tenure) -0.097 (0.33)

-0.002 (0.42)

0.115 (0.35)

0.001 (0.42)

Dependent variable, Lagged

0.205 (2.92)

0.182 (3.49)

0.269 (5.08)

0.240 (4.41)

R-sqr 0.55 0.56 0.53 0.53

Conclusion:


Lower R 2

Higher Selectivity

Better Performance

A new convenient way to predict mutual fund performance using only their return data : the R2 from a regression of the fund return on a multi-factor model that is considered a common benchmark for fund performance

The lower R2 the greater selectivity, the better the subsequent performance, after controlling for fund characteristics and past performance

the highest-alpha and lowest-R2 portfolio produces an average annual excess return of 2.4% or more (depending on the benchmark factor model)

R2 is related to fund characteristics such as fund size, expenses, manager tenure, and style. These characteristics explain nearly 50% of the cross-fund variation in R2.

R2 also predicts performance for mutual funds that hold corporate bonds

Conclusion:


amihud and goyenko

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