Risk, Return, and the Optimal Exploitation of Stock ...lamfin.arizona.edu/rsrch/RROEslides.pdf ·...

Risk, Return, andthe Optimal

Exploitation ofStock

Characteristics

Introduction

Characteristics andthe Cross Section

Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio

Certainty Equivalent

Factor exposures

Month-by-Month

What next?

Risk, Return, and the Optimal Exploitationof Stock Characteristics

Christopher G. Lamoureux&

Huacheng Zhang

May 29, 2014



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Backdrop

1. We know that the cross-section of stock returns has astrong factor structure, and we also know that stockcharacteristics are linked to factor structure.

2. Furthermore characteristics can predict the cross-sectionof expected returns out-of-sample: Lewellen (2013).Although Nagel (2012) notes that this is a pseudoout-of-sample test.

3. Lewellen, Nagel and Shanken (2010) and Kogan andTian (2013) (Nagel 2012) note that what this factorstructure implies for asset pricing models is unclear.

4. Recent addition to variables that explain cross-sectionalpredictability of expected returns: lagged same monthreturns (Heston and Sadka 2008).



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

This paper’s contributions

1. We use characteristics to form optimal portfolios usingthe Brandt, Santa-Clara, and Valkanov (2009)algorithm.

I Maximize expected utility by a fully-invested portfoliowith weights linear in measurable characteristics.

2. Question is whether a risk inverse investor can usecharacteristics to significantly increase expected utilityout-of-sample.

I Addresses to some extent the pseudo out-of-samplecritique with a different loss function.

I Provides a convenient and natural way to combinecharacteristics, and assess each one’s marginalcontribution.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?


3. By including lagged same-month returns ascharacteristics we explore this in the context of theother characteristics.

4. Because we have a relatively long sample we isolatefactor loadings conditional on month of the year.

5. Period includes 2009, and the momentum crash:I We analyze effect of characteristics on portfolio weights

on a yearly basis.I We consider the (small sample) bias in the effects of

characteristics on portfolio weights.

6. We compare a 15-year rolling “training period” withupdating (over our 38 year out-of-sample period).



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?


7. Since we have a CRRA (isoelastic) utility function, weexamine the effect of changing relative risk aversion onthe desired characteristic exposure.

E (U) = 1T ·

T−1∑t=0

(1+rp,t+1)1−γ

1−γ .

8. We bootstrap over the optimization to providestatistical context to all results.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Results

1. The loss function matters.I Optimal portfolio for γ 3 investor with all

characteristics, using rolling protocol has 45%Fama-French-Carhart alpha, and CE of -23%.

2. The only characteristics that generate statisticallysignificant improvement in EU are book-to-market andbeta.

I The dynamic characteristics do not help–even beforetransactions costs.

3. Portfolios formed based on lagged same-month returnsdo have higher (lower) exposures to known factors inmonths when the factors have high (low) expectedreturns. This is not evident in unconditional betas(Keloharju, Linnainmaa and Nyberg, 2013).

4. There is material bias and imprecision in the sample θestimates.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Results

5. There is an in-sample complementarity betweenmultiple factors; ex.: size and beta.

6. While there is weak in-sample substitution betweenlagged same-month returns and momentum, theycomplement size and book-to-market.

7. θ coefficients are lower in later years–notably onmomentum after 2009. But all are significant; (McLeanand Pontiff, 2012).

8. The updating protocol dominates the rolling protocolout-of-sample.

9. Investors’ desired exposure to characteristics diminishingin γ, with the exception of beta.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Results

10. New – not yet in paper: Note that this optimizingalgorithm asks the investor to put a dogmatic prior on amodel wherein the relationships between returnmoments (and co-moments) between laggedcharacteristics and next month returns are robust andstable over time.

I We find significant improvement in Certainty Equivalentby using the optimal portfolio selected for an investorwho is more risk averse.

I Although, there is no explcit model of the returndistribution.

I This overturns the results on the dynamiccharacteristics.

I Raises the important question of why this does so well.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Portfolio Formation

maxθ

T−1∑t=0

(1 + rp,t+1)1−γ

1− γ

(1

T

)

rp,t+1 =Nt∑i=1

(ωi ,t +

1

Ntθ′xi ,t

)· ri ,t+1

I θ has dimension K , the number of characteristics.

I BSV use size, book-to-market, and momentum, andfind that the optimal portfolio outperforms the market,and the equally-weighted market.

I Extremely parsimonious relative to mean-varianceoptimization, which suffers notoriously from estimationrisk.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Data

I Start forming optimal portfolios on December 31, 1959.

I Sample consists of all stocks with no missing returnsfrom January 1955 through December 1959, and validCompustat data.

I On every month-end date, the sample contains Nt

K + 1 vectors Υi ,t .

I Υi ,t = (xi ,t , ri ,t+1). (No look-ahead bias.)

I First in-sample period: returns from January 1960 -December 1975.

I Use the θ vector just obtained to form out-of-sampleportfolio for each of the next 12 months.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Data (continued)

I Rolling Protocol: Drop 12 months in 1960, so thatthere are always 180 months used to estimate θ.

I Updating Protocol: Add next 12 months. Sample sizedused in estimating θ grows from 180 months to 624months.

I Note that stocks can come into the sample at anytime–in both the in- and out-of-sample sets.

I Exclude smallest 10% of eligible stocks pre-1975 and20% post-1975.

I Sample size : 290 in January 1960, 3,033 in May 1999.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Data – Characteristics

I Size: Number of outstanding shares (CRSP) × closingprice–both month-end before portfolio formation.

I Book-to-market: Fiscal year-end book value fromCompustat – lagged 6 - 18 months. Time-matched tomarket cap.

I Book value: Total assets + deferred taxes + ITC - totalliabilities - preferred equity.

I Momentum: Compounded return from month -12 to -1.

I Beta: OLS coefficient of excess stock return on excessmarket return in months -59 - 0.

I Lagged same-month return:I rt−11

I5∑

i=1

rt−(12i−1) · 15



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Data – Characteristics

I One of the brilliant insights of BSV is to standardizeand normalize the coefficients.

I Makes the portfolio weight constraint natural.I Allows direct comparison of characteristics’ effects on

weights.I Also means that the characteristics are not different

from shrunken characteristics.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Number of stocks in sample–by month

0

500

1000

1500

2000

2500

3000

3500

12/1959 06/1965 12/1970 06/1976 11/1981 05/1987 11/1992 04/1998 10/2003 04/2009



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Bootstrap

I An interesting aspect of the BSV portfolio selectionalgorithm is that there is no likelihood function (orerrors).

I Construct sampling distributions by drawing (withreplacement) from Υi ,t to construct a pseudo-sample ofthe same term and size as in the data. We then selectthe optimal portfolios (to obtain θ) at the beginning ofeach oos year. Repeat 10,000 times to construct asampling distribution of θ and functions of interest of θand the data (e.g., alpha, Sharpe ratio, and CE).

I I believe that this approach requires that the investorhave a dogmatic prior (not exactly clear what that is).



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

θ Coefficients

Exploit our empirical design

I Average θ coefficients on each characteristic in isolation.

I Time series behavior of these coefficients.

I Substitutability / Complementarity betweencharacteristics when used in combination.

I Difference between rolling and updating on coefficients.

I Effects of γ on coefficients.

I Small sample properties.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

θ on size

−14

−12

−10

−8−6

−4

Year 1975 − 2012 −− Model 5

Thet

a(3)

xx

x

x

xx x x

x x x xx

x xx

x x x x x xx x

xx

xx x x x x x

x x

x x x

−5−4

−3−2

−10

Year 1975 − 2012 −− Model 8

Thet

a(1)

This plot shows the bootstrap distributions for the coefficients on the stock's size from Model 5 (top panel) and Model 8 (lower panel). The box and whiskers plot shows the 95%ile range (whiskers), the interquartile range (box), and median (bar inside box) for the coefficient in each year. The sample estimate is shown as a red X.x

x

x

x xx

xx

xx

xx

x

x xx

x x x x x x x xx x x

x xx x x x

xx x x x



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Summary of Size θ

I By itself and in conjunction with mom, btm, and laggedsame-month returns:

I Significantly negative for γ = 3.I Significantly positive for γ ≥ 15.I Not much difference between rolling and updating

protocols.

I When beta is in characteristic set:I Significantly negative for all γ.I Rolling shrinks θ for γ = 3.I Bootstrap standard deviations are twice as high than

when beta is not.I Sample bias much larger (∼ 1 σ) than when beta is not.

I Overall:I Appeal of small stocks declines monotonically in γ.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

θ on book-to-market ratio

46

810

Year 1975 − 2012 −− Model 5

Thet

a(2)

This plot shows the bootstrap distributions for the coefficients on the stock's book− to−market ratio from Model 5 (top panel) and Model 9 (lower panel). The box and whiskers plot shows the 95%ile range (whiskers), the interquartile range (box), and median (bar inside box) for the coefficient in each year. The sample estimate is shown as a red X.

x xx

xx

x

xx x

x x x x xx x

x xx

x x x x x x x x x xx x x x

xx x x x

45

67

8

Year 1975 − 2012 −− Model 9

Thet

a(1)

x x

xx x

x

x

x

xx x x x x

x x

xx

xx x x x x

x

xx x x x x x x

xx

x x x



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Summary of Book-to-Market θ

I Significantly positive for all γ, all characteristic sets.

I Substitutability with beta (Significantly lower in Model2 than in Model 1, for all γ.)

I Complementarity with lagged same-month returns(Significantly higher in Model 4 than Model 1, for all γ.)

I No complementarity or substitutability with momentumand size.

I Rolling and updating very similar.

I Exposure to value monotonically declining in γ–althoughit flattens out at high γ at a significantly positive value.

I (Recalling that characteristics are standardized), theeffect is larger than size, momentum and beta.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

θ on momentum

34

56

Year 1975 − 2012 −− Model 6

The

ta(1

)

This plot shows the bootstrap distributions for the coefficient on the stock's momentum from Model 6 (top panel) and Model 7 (bottom panel). The box and whiskers plot shows the 95%ile range (whiskers), the interquartile range (box), and median (bar inside box) for the coefficient in each year. The sample estimate is shown as a red X.

x

x x

x x

xx

x

x

x xx

xx x

xx x x x x

x x x xx

xx x

x x x xx

x

x x x

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Year 1975 − 2012 −− Model 7

The

ta(1

)

x

xx

x xx

xx

x

xx

x xx

x

x

x x x x xx x x

xx

x

x

xx x x x

xx

x x x



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Momentum θ: γ 3 and γ 5

34

56

Year 1975 − 2012 −− Model 6, gamma 3

The

ta(1

)

This plot shows the bootstrap distributions for the coefficient on the stock's momentum from Model 6, gamma 3 (top panel) and Model 6, gamma 5 (bottom panel). The box and whiskers plot shows the 95%ile range (whiskers), the interquartile range (box), and median (bar inside box) for the coefficient in each year.

The sample estimate is shown as a red X.

x

x x

x x

xx

x

x

x xx

xx x

xx x x x x

x x x xx

xx x

x x x xx

x

x x x

1.5

2.0

2.5

3.0

3.5

Year 1975 − 2012 −− Model 6, gamma 5

The

ta(1

)

x

x x

xx

x

x

x

x

x xx x

x xx

x x x x x x x x xx

xx x

x x x xx x

x x x



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Summary of Momentum θ

I Weakly complementary to size, book-to-market, andbeta.

I Weakly substitutable with lagged same-month returns.

I Model 2 significantly higher than Model 7, for all γ,rolling and updating.

I Exposure to momentum monotonically declining in γ.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

θ on beta

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

Year 1975 − 2012 −− Model 10

Thet

a(1)

x

xx x

xx

x

xx x

xx

x x x x xx

x x x x xx x

xx x

xx x x x x

xx x x

−7−6

−5−4

−3−2

−1

Year 1975 − 2012 −− Model 2

Thet

a(4)

This plot shows the bootstrap distributions for the coefficient on the stock's monthly beta from Model 10 (top panel) and Model 2 (bottom panel). The box and whiskers plot shows the 95%ile range (whiskers), the interquartile range (box), and median (bar inside box) for the coefficient in each year. The sample estimate is shown as a red X.

x xx

xx

x xx x x x x x

x xx x

x x x x x x x

x

xx

x

x x x x x x xx x x



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Summary of Beta θ

I Remarkably constant across γ.

I Always significantly negative (all γ, characteristic sets,and Rolling / Updating).

I Strong complementarity with size.

I The complementarities shrink in γ.

I (As we see on the previous slide), Complementaritiesvary widely over time.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

θ on last-year’s same-month return

−2

02

46

810

Year 1975 − 2012 −− Model 3

The

ta(1

)

This plot shows the bootstrap distributions for the coefficient on the stock's same−month return last year from Model 3 (top panel) and Model 4 (bottom panel). The box and whiskers plot shows the 95%ile range (whiskers), the interquartile range (box), and median (bar inside box) for the coefficient in each year. The sample estimate is shown as a red X.

x x x

xx

x

xx

x x xx

x x x xx x x

x x xx x x x

x

x

x x x x x x x x x x

23

45

Year 1975 − 2012 −− Model 4

The

ta(4

)

x

xx

xx

x

x

x

xx x

x x xx

x

x x xx x

x xx

xx

x x

x x x x xx x

x x x



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

θ on 5-year MA same-month return

02

46

8

Year 1975 − 2012 −− Model 3

The

ta(1

)

This plot shows the bootstrap distributions for the coefficient on the stock's same−month return averaged over the previous 5 years from Model 3 (top panel) and Model 4 (bottom panel). The box and whiskers plot shows the 95%ile range (whiskers), the interquartile range (box), and median (bar inside box) for the coefficient in each year. The sample estimate is shown as a red X.

x x xx x

x x

x

x x x

x

x xx

x

xx x x

x x xx

xx

x x

x xx x x x

x x x x

78

910

1112

13

Year 1975 − 2012 −− Model 4

The

ta(4

)

x x

x x x x

x x xx x x

x xx x

x xx

x x x xx

xx

x x x x x x xx

x

x x x



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

θ on both same-month return characteristics

−20

−15

−10

−50

510

Year 1975 − 2012 −− Model 5−−Rolling protocol

Thet

a(6)

This plot shows the bootstrap distributions for the coefficient on the stock's same−month return from last year from Model 5−−Rolling protocol (top panel) and average same−month return over the preceding 5 years from Model 5−−Rolling protocol (bottom panel). The box and whiskers plot shows the 95%ile range (whiskers), the interquartile range (box), and median (bar inside box) for the coefficient in each year. The sample estimate is shown as a red X.

xx x x

x x x x x xx

xx

x xx

xx x

x x x

x

x

xx

x

x

x x x x x x x x x x

05

1015

2025

3035

Year 1975 − 2012 −− Model 5−−Rolling protocol

Thet

a(5)

x x x x x x x

x x xx

x xx

x x xx x x x x

x

x

xx

xx

x x x xx x

xx x x



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Sum of θ on both same-month returncharacteristics: Updating vs. Rolling

510

1520

Year 1975 − 2012 −− Model 5 (Updating Protocol)

Thet

a(5)

+The

ta(6

)

This plot shows the bootstrap distributions for the sum of the coefficient on the stock's lagged same−month return and the moving average of the same−month return over the previous 5 years from Model 5, with coefficient of relativerisk aversion = 3. The top panel is from the updating protocol and the bottom panel is from the rolling protocol.The box and whiskers plot shows the 95%ile range (whiskers), the interquartile range (box),and median (bar inside box) for the coefficient in each year. The sample estimate is shown as a red X.

x x x xx x

x

x x x x x x x x x x x x x x x x x x x

x xx x x x x x x x x x

010

2030

Year 1975 − 2012 −− Model 5 (Rolling Protocol)

Thet

a(5)

+The

ta(6

)

x x x x x xx

x x x x x x xx x x x x

x x x

x

x

x x

xx

x x x xx x

xx x x



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Summary of lagged same-month return θs

I Effect is very large for low γ.

I Appeal of predictability diminishes sharply in γ.

I Substitutability with Momentum, and complemantaritywith: size, BtM, and beta.

I (As seen on the last slide), Rolling is much more volatilethan Updating.

I (As seen on the last slide), Rolling subject to muchlarger sample bias than Updating.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Short positions by month, Updating protocol

Model 1

1.0

2.0

3.0

4.0

Model 2

12

34

5

Model 3

01

23

4 Color Code: gamma = 3 gamma = 5 gamma = 10 gamma = 15 gamma = 20

Model 4

12

34

56

78

Model 5 Year

1980 1990 2000 201002

46

8



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Short positions by month, Updating protocol

Model 6

0.0

0.5

1.0

1.5

Model 7

01

23

4

Model 8

0.0

0.4

0.8

1.2

Color Code: gamma = 3 gamma = 5 gamma = 10 gamma = 15gamma = 20

Model 9

0.5

1.0

1.5

2.0

2.5

Model 10 Year

1980 1990 2000 2010

0.3

0.5

0.7

0.9



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Discussion

I Leverage generally declines in γ (as expected).I Interesting Exception: Size, post 1987.

I Individual Characteristics:I Momentum: Leverage peaks in late 1998, declines in

last three years.I BtM: Highest of 4 traditional characteristics–declining

over time.I Size: Largest temporal changes.I Lagged same-month return: Higher than other 4

characteristics; declines significantly over time.I More stable for persistent characteristics: mom, BtM,

size; large month-to-month variation for more volatilecharacteristics.

I Combinations:I From Models 4 and 5 reaches high of 8 in first three

years, declines over time.



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Unconditional FFC alpha

I As BSV note, not surprising that conditioning on size,BtM, and momentum do not produce significantlypositive FFC alpha.

I Therefore surprising that conditioning on beta does.I By itself, 8 bps (significantly > 0).I Case of γ = 3, Updating, adding to: size, BtM, and

mom increases alpha from 6 (insig.) to 60 bps (sig.)I Same case, adding beta adds 60 bps to all other

characteristics (not sig.)

I Points of Reference:I Value-Weighted Index: 3 (−2 , 8).I Value-Weighted Index: 7 (5 , 10).



Characteristics

Introduction


Contributions

Results

BSV

Data

Bootstrap

Detailed Results

θ

Leverage

Out-of-sample

Alpha

Sharpe Ratio


Factor exposures

Month-by-Month

What next?

Unconditional FFC alpha

I Lagged same month returns produce significant oosalpha.

I Case of γ = 3, Updating, alpha is 162 (119 , 208) bpsper month.

I (Same case): Adding size, mom, and BtM to rt−11 andr t−11 increases alpha to 210 (147 , 281) bps.

I Maximum bootstrap mean of the oos alpha across 100cases: γ = 3, Rolling format, all 6 characteristics: 334(251 , 428) bps. (Sample alpha: 45% annual.)

I Not surprisingly, alpha declines monotonically in γ:γ = 15, Rolling, all 6 characteristics: 78 (58 , 101) bps.(Sample alpha: 10.5% annual.)



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Sharpe Ratio

I Unlike alpha, SR does not diminish in γ.

I In all 10 cases, maximized using all 6 characteristics(Model 5). 1 exception: γ = 3, Updating: excludingbeta does just as well.

I Maximized: γ = 5, Updating, (All 6 characteristics):Sample: 1.25; B-S: 1.19 (1.10 , 1.29), (annualized).

I Rolling, γ = 5, All 6 characteristics: Sample: 1.16; B-S:1.12 (1.01 , 1.23).

I Reference:I Value-Weighted Index: 0.49 (0.45 , 0.53).I Equally-Weighted Index: 0.57 (0.56 , 0.59).



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I Interesting that the out-of-sample alpha and Sharperatios of the optimized portfolios are so large, sinceneither was maximized in-sample.

I Here, we have 456 months of out-of-sample returns,that we use to measure the average utility. CE return isthe constant rate of return on the portfolio that wouldprovide the same average utility.

I This is what was maximized in-sample.



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I Benchmarks for γ = 3:I Value-Weighted Index: Sample: 75, B-S: 75 (70 , 80)

bps per month.I Equally-Weighted Index: Sample: 86, B-S: 86 (83 , 88)

bps per month.

I Benchmarks for γ = 5:I Value-Weighted Index: Sample: 53, B-S: 53 (47 , 58)

bps per month.I Equally-Weighted Index: Sample: 54, B-S: 54 (51 , 56)

bps per month.

I Benchmarks for γ = 10:I Value-Weighted Index: Sample: -10, B-S: -12

(−18 ,−5) bps per month.I Equally-Weighted Index: Sample: -45, B-S: -46

(−50 ,−41) bps per month.



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Certainty Equivalent: Inference

I Inference:I Which characteristics lead to a significant increase in

CE?I Is this effect robust across γ ?I Do these effects parallel effects on alpha and SR?

I We confirm the BSV findings: Model 1:I γ = 3:

I Updating: B-S 2.5%ile: 98 (EWI 97.5%ile: 88).I Rolling: B-S 2.5%ile: 51.

I γ = 5:I Updating: B-S 2.5%ile: 66 (VWI 97.5%ile: 58).I Rolling: B-S 2.5%ile: 48.

I γ = 10:I Updating: B-S 2.5%ile: -8 (VWI 97.5%ile: -5).I Rolling: B-S 2.5%ile: -15.



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Certainty Equivalent: Using a higher γ

Model 1, Updating Protocol, γ 3 investor.

γ B-S B-S B-SUsed 2.5%ile Median 97.5%ile

EWI 83.38 85.79 88.27

3 97.72 120.93 144.385 135.11 150.94 168.64



VWI 47.37 52.47 57.60

5 66.09 81.77 97.4210 86.96 97.99 109.89



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CE: Size, BtM, mom individually

I Momentum in isolation:I For all γ values and both protocols, B-S mean CE is less

than preferred index (difference not significant).I Ex: 95% bounds for γ = 3, Updating: 71 (57 , 83),

(recall EWI: 86 (83 , 88).)

I Size in isolation:I For γ = 3, 5, 10, both protocols, B-S mean CE is

significantly less than preferred index.I Ex: 95% bounds for γ = 3, Updating: (68 , 78), (recall

EWI 2.5%ile: 83).I For γ = 15, 20, both protocols, not worse than index,

but not significantly better.



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CE: Size, BtM, mom separately (Cont’d.)

I Book-to-Market in isolation:I For γ = 3, 5, 10, both protocols, CE is significantly

larger than preferred index.I Ex: γ = 3, Updating: 107 (93 , 122), Rolling: 113

(100 , 128).I For γ = 15, Rolling protocol significantly better than

EWI, but not updating.I For γ = 20, Not sig. different from VWI.I Additional example: γ = 10: Updating: 5 (−6 , 16),

Rolling: 16 (6 , 25), (VWI: -12 (−18 ,−5).

I Using a higher γ does not have much effect on the CEfrom size, book-to-market, and momentum individually.



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CE: Size, BtM, mom

I Does adding mom and size to BtM enhance CE–beyondusing only BtM?

I No. The difference between CE from Models 1 and 9 isnot significantly different in any of the 10 cases.

I Example: γ = 3: Updating: BtM only: 107 (93 , 122),size, BtM, and mom: 124 (98, , 144).

I Example: γ = 3: Rolling: BtM only: 113 (100 , 128),size, BtM, and mom: 85 (51, , 114).

I Example: γ = 10: Updating: BtM only: 5 (−6 , 16),size, BtM, and mom: 5 (−8, , 19).

I Example: γ = 10: Rolling: BtM only: 16 (6 , 25), size,BtM, and mom: 0 (−15, , 14).

I BtM only does as well with rolling as with updating.Rolling much worse when size included.

I Transactions costs and leverage are much higher whenmom is included in characteristic set.



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CE: Size, BtM, mom

I Does adding mom and size to BtM enhance CE–beyondusing only BtM?

I Using a higher γ adds to momentum’s appeal.I Adding size and momentum to book-to-market affords

significantly higher CE for the γ 3 investor who uses theγ 5 optimal portfolio.



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Beta

I In isolation:I Not much difference between updating and rolling

protocols.I For γ = 3, CE significantly less than EWI.I For γ = 5, CE not significantly different from VWI.I For γ = 10, CE significantly higher than VWI.I For γ > 10, CE not significantly different from VWI.I Similarly the γ 5 investor using the optimal portfolio

conditioned only on beta for the γ 10 investordominates the indices.

I Complementing BtM, size, mom:I In 9 of 10 cases, adding beta to these three does not

significantly change the CE.I The one case where it does: γ = 10, Rolling Protocol:

size, BtM, and mom: 0 (−15, , 14), add beta: 30(14 , 43). (This is the maximum CE for the γ = 10investor.)



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Lagged same-month return

I In isolation:I Big difference between rolling and updating protocols.I Ex: γ = 3: Updating: 50 (−13 , 88) (not sig. different

from EWI).Rolling: -336 (−1984 ,−3).

I Updating: Not sig. different from higher index forγ = 3, 5.Sig. less than higher index for γ = 10, 15, 20.

I Complementarities and substitutabilitiesI Rolling protocol, Adding to other characteristics lowers

CE:I Significantly for γ ≤ 10.I Insignificantly for γ > 10.

I Updating protocol: For all γ no sig. change. Alwaysreduces precision.

I Ex.: γ = 3, Updating: mom only: 71 (57 , 83);add lsmr: 11 (−137 , 76).



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I This characteristic is most positively affected by using ahigher γ.

I –Both individually and in combination.I For both the γ 3 investor using γ of 5 to optimize, and

the γ 5 investor using γ of 10 to optimize, individuallythe same-month return characteristics can be used tosignificantly increase CE above the indices.

I Additionally, this characteristic–while deleterious to CEwhen using your own γ–is very efficacious incombination with other characteristics when usinghigher γ.



EWI 83.38 85.79 88.27

3 2.58 138.63 206.275 192.40 225.35 259.61



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VWI 47.37 52.47 57.60

5 -11.48 87.10 131.4610 120.87 145.03 166.13



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Unconditional market loading

I {Mom, size, BtM}: Insig. different from 1 for all γ, andboth protocols.

I Conditioning on beta:I In isolation:

I Sig. < 0 for all γ and both protocols.

I Adding to other characteristics:I All cases: lowers market loading significantly.I Ex.: γ = 3, Updating:{Mom, size, BtM}: 1.07 (1.00 , 1.14){Mom, size, BtM, beta}: 0.30 (0.13 , 0.47)

I Conditioning on lagged same-month returns:I In isolation:

I Sig. > 1 for all γ and both protocols, except γ = 20,Updating.

I Shrinking in γ.



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I Conditioning on lagged same-month returns:I Adding to other characteristics:

I Adding to {mom} and {mom, size, BtM}: increasessignificantly for γ ≤ 15No sig. effect for γ = 20–Both protocols.

I Adding to {mom, size, BtM, beta}: No sig. effect (allγ, both protocols).

I Conditioning on {mom}:I Slightly > 1 for γ ≤ 10, both protocols.I Insig. different from 1 for γ = 15, 20.

I Conditioning on {size}:I Updating:

I All insig. different from 1.

I Rolling:I All sig. > 1.



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I Conditioning on {BtM}:I Sig < 1 in all 10 cases.I Ex.: γ = 3, Updating protocol:

0.93 (0.88 , 0.98)



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Unconditional SMB loading

I Conditioning on {beta}:I Sig. < 0 in all 10 cases.I Ex.: γ = 3, Updating protocol:

-0.69 (−0.80 ,−0.59).

I Conditioning on {size}:I Sig. > 0, for γ = 3.I Ex.: Updating Protocol:

0.62 (0.38 , 0.85).I Sig. < 0 for γ ≥ 10.I Ex.: γ = 10, Updating Protocol:

-0.18 (−0.25 ,−0.10).

I Conditioning on {BtM}:I Sig. > 0 for γ ≥ 5I Ex.: Updating Protocol, γ = 3:

0.25 (0.15 , 0.35).I Insig. different from 0 for γ ≥ 10.



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I Conditioning on {mom}:I Sig. < 0 for γ ≥ 10.I Sig. > 0 for γ ≤ 5, Rolling protocol

Ex.: -0.69 (−0.80 ,−0.59).



-0.18 (−0.25 ,−0.10).





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I Conditioning on {rt−11, r t−11}:I Sig. < 0 for γ ≥ 10.I Sig. > 0 for γ ≤ 5, Rolling protocol

Ex.: -0.69 (−0.80 ,−0.59).



-0.18 (−0.25 ,−0.10).





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Unconditional HML loading

I Conditioning on {rt−11, r t−11}:I Sig. < 0 for γ ≤ 10.I Insig. different from 0 for γ ≥ 15

I Both protocols.I Ex.: γ = 3, Updating: -0.60 (−0.76 ,−0.46).

I Conditioning on {size}:I Sig. > 0, for γ ≤ 5.I Ex.: Updating Protocol, γ = 3:

0.17 (0.10 , 0.23).

I Conditioning on {BtM}:I Sig. > 0 for all γ, both protocols.I Large drop from γ = 3: 2.4, (2.1 , 2.7) to γ = 10: 1.1,

(1.0 , 1.2).I Flat as γ increases above 10.I Protocols very similar.



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Focus on rt−11, r t−11

I Summarizing, optimal portfolios conditioned on laggedsame-month returns for γ = 3, have:

I 1.5 loading on market risk premium.I -0.3 loading on SMB.I -0.6 loading on HML.I 0.1 loading on MOM.

I So, it appears that the positive alpha on this strategy isdistinct from the traditional sources of risk.

I Keloharju, Linnainmaa, and Nyberg (2013) suggest thatthe lagged same-month effect has to harness known riskfactors.

I Intuition: stocks that did well last January are likelysmall stocks, and because they are small, they are alsolikely to do well next January.

I We examine this by analyzing the FFC regressions foreach month of the year.

FFC factors by month

Factor All Jan Feb Mar Apr May Jun Jul Aug Sep Octmonths

Rm − rf 0.63** 1.33 0.19 0.92 1.42* 0.80 0.42 -0.02 0.20 -0.87 -0.05(0.21) (0.86) (0.67) (0.65) (0.62) (0.63) (0.57) (0.69) (0.85) (0.77) (1.11)

SMB 0.29 1.50* 1.04 0.32 0.07 0.39 0.92 -0.54 -0.16 0.16 -1.39*(0.14) (0.52) (0.68) (0.59) (0.48) (0.44) (0.48) (0.46) (0.45) (0.37) (0.56)

HML 0.36* 1.09 0.70 1.04* 0.66 0.02 -0.16 0.78 0.61 0.24 -0.57(0.14) (0.65) (0.69) (0.36) (0.46) (0.34) (0.44) (0.49) (0.41) (0.41) (0.48)

MOM 0.64** -1.47 1.74* 0.62 -1.04 -0.27 2.14** 1.06 -0.34 1.96** 0.59(0.21) (0.97) (0.70) (0.58) (1.10) (0.64) (0.65) (0.47) (0.49) (0.56) (0.69)

Factor Nov DecRm − rf 1.65* 1.55*

(0.74) (0.55)

SMB 0.40 0.82*(0.45) (0.36)

HML -0.45 0.30(0.54) (0.46)

MOM 0.75 1.95*(0.80) (0.68)



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Factor exposures by month

I So, there is significant variation across the months inFFC factors.

I Look at FFC regressions separately in each of the 12months (38 observations).

I Conditioning on {rt−11 , r t−11} γ = 3, Updating:I HML loading:

I Lowest loading in Nov: -2.5 (−3.16 ,−1.91).I Highest loading in Mar: 1.2 (0.65 , 2.50).

I We have 2 alphas:I From the full 456 month FFC regression.I Aggregating over the 12 month-of-the year regressions.

I The latter minus the former is the measure of bias dueto ignoring month-dependent loadings.

I This can also be decomposed into the bias due to eachof the four factors.

I We have bootstrap distributions on all of thesefunctions.



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Continuing with {rt−11 , r t−11} γ = 3, Updating

I Unconditional alpha in this case: 162.3 (119.5 , 207.9)(bps per month).

I Biases in unconditional alpha from:I Rm − rf : 7.6 (1.2 , 14.2).I SMB: 22.4 (13.4 , 32.1).I HML: 21.1 (14.2 , 28.4).I MOM: 10.6 (14.2 , 28.4).I Total: 61.8 (43.3 , 81.5).



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Biases in unconditional alphas

I Largest bias: γ = 3, Rolling,x ∈ {mom, BtM, size, rt−11, r t−11}

I Biases in unconditional alpha from:I Rm − rf : 20.0 (5.1 , 35.4).I SMB: 47.8 (26.2 , 71.0).I HML: 23.4 (7.6 , 40.0).I MOM: -7.2 (−32.01 , 18.2).I Total: 84.0 (40.5 , 129.3).



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So?

I The information in lagged same-month returns can beused to increase expected returns.

I Adding it to traditional characteristics increasesexpected returns and Sharpe ratios.

I This information is useful to increase expected utility ifthe investor takes account of model uncertainty.

I The alternative loss functions is another response topseudo-out-of-sample evidence.

I Keloharju, Linnainmaa, and Nyberg (2013) arecorrect–conditioning on this information increasesexposure to traditional sources of systematic risk.



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Decision Theory

The sampling variation–in both θ coefficients and CE–varieswidely across various x .

I This paper has used statistical comparisons.

I What are the normative implications for a risk-averseinvestor?

I Two approaches using machine learning (still nolikelihood):

I Bootstrap Aggregation (Bagging / Bragging).I Data Augmentation.

I Generally the DA approach is better than BA.

I Largest effect is on larger models (almost no effect onthose with just one characteristic).

I Model 4 does beat Model 9 statistically using DA: M4CE: 183 (127 , 231); std: 146 (36 , 213).



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Decision Theory

I But even these gains in certainty equivalent pale incomparison to the gains achieved by using a higher γ.

I So the next step might be to combine these.

I Also since using a higher γ works so well, one possibilityis a lack of stationarity. But updating generallydominates rolling. This motivates using the machinelearning / Data augmentation with the rollingprocedure. (Trade-off estimation noise andnon-stationarity.)

Risk, Return, and the Optimal Exploitation of Stock ...lamfin.arizona.edu/rsrch/RROEslides.pdf ·...

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