Current draft: November 18, 2016First draft: February 1, 2012

An Empirical Assessment of Characteristics and Optimal Portfolios

Christopher G. Lamoureux∗

andHuacheng Zhang†

Measurable stock characteristics can be used to generate positive expected returns that areorthogonal to the known factors. The cross-section of equity returns also exhibits a strongmonth-of-the-year seasonal pattern. Since returns on long-short portfolios and Fama-Macbethcharacteristic-based portfolios have relatively high variances, fat tails, and negative skew, thispaper examines the efficacy of stock characteristics to maximize a convex objective function (suchas a power utility function).

The Brandt, Santa-Clara, and Valkanov (2009) algorithm exhibits large estimation risk whichmust be solved before it can provide insight into our research question. We address this by usinghigher values of risk-aversion to select portfolios. We examine complementarity and substitutabil-ity across characteristics. We find that the book-to-market ratio is redundant in the context ofsize and market model residual standard deviation. In isolation beta appears to be unambigu-ously bad, but in combination with the market model residual standard deviation this effectvanishes. Momentum and average same-month return are independent of the other characteris-tics. The bootstrap shows that there is a high level of imprecision in the relationships betweenvectors of characteristics and the distribution of (future) portfolio returns. While beta and av-erage same-month return are effective in increasing expected utility for relatively risk-tolerantinvestors, they are dropped from the optimal set of characteristics for more risk-averse investors.

Optimal portfolios tend to be positively skewed and fat-tailed. Their exposure to the themarket factor is significantly less than unity, and they have generally positive exposure to themomentum and value factors. Optimal portfolios have high (but not maximal) Sharpe ratios,and significantly positive Fama, French, Carhart alphas. Alpha and our objective functions arequite disparate however, as those portfolio with the highest alphas are severely penalized by theobjective function’s convexity.

Key Words: Stock characteristics; optimal portfolios∗Department of Finance, The University of Arizona, Eller College of Management, Tucson, 85721, 520–621–

7488, lamoureu@email.arizona.edu.†Department of Finance, Southwest University of Finance and Economics, Changdu, China,

zhanghuacheng1@gmail.com.We are grateful to Scott Cederburg, Kei Hirano, and Michael Weber. The current version of this paper can bedownloaded from lamfin.arizona.edu/rsch.html .

1. Introduction

Firm characteristics can predict future stock returns in the cross-section (Lewellen 2015). The

cross-section of expected stock returns also has a strong seasonal component (Keloharju, Linnain-

maa, and Nyberg 2016). This paper analyzes these facts from the perspective of a risk-averse

expected utility maximizing investor. To what extent does this cross-sectional predictability

cause such an investor to tilt her optimal portfolio weights away from the market portfolio? This

is an open question since many alpha generating strategies entail high volatility, fat tails, and/or

negative skew (Barroso and Santa-Clara 2015). To this end, we use Brandt, Santa-Clara, and

Valkanov’s (2009) normative algorithm that allows optimal portfolio weights to depend on stock

characteristics. We consider estimation risk and optimal characteristic usage by investors with

varying degrees of risk aversion. We document the properties of these optimal portfolios in terms

of return moments and factor loadings. All of our statistical analysis is conducted out-of-sample,

using the bootstrap to conduct inference.

Lewellen (2015) notes that much of the empirical literature on characteristics and returns

uses portfolios that are formed by sorting on one, and possibly two characteristics. As such, we

have limited information about characteristics’ complementarity and substitutability–especially

as these may affect characteristics’ efficacy for optimizing investors. Because characteristics are

not independent these effects are important. For example, investors prefer low beta to high beta

stocks when beta is the only characteristic available.1 However, when we add residual standard

deviation to beta–allowing weights to depend on both–then beta becomes attractive (to risk-

averse investors with relatively high risk tolerance), and investors prefer low residual standard

deviation stocks along with higher beta stocks. We also find that the book-to-market ratio is

redundant in the context of the other characteristics that we consider: momentum, size, beta,

residual standard deviation, and lagged same-month returns. Since there is significant estimation

risk, which is mitigated by parsimony, it is never optimal to include the book-to-market ratio in

the optimal set of characteristics.

We consider nine different objective functions, or “investors”–indexed by the coefficient of

relative risk aversion (γ)–and examine how the set of optimal characteristics varies as we increase

γ. Investors with γ of 2, 3, and 4 optimally use: momentum, size, beta, residual standard

deviation, and five-year average same-month return. Investors with γ of 5, 6, and 7 drop beta

from the optimal set of characteristics. Investors with γ of 8, 9, and 10 also drop the average

same-month return–leaving only three characteristics to optimally tilt portfolio weights.

We examine in depth the estimation risk inherent in the selection algorithm. Estimation risk

1Such a result would be consistent with Frazzini and Pedersen (2015) who argue that leverage-constrainedinvestors bid up the prices of high beta stocks, which therefore have lower expected returns, ceteris paribus.

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is severe. We find that an updating protocol dominates a rolling protocol, so estimation risk in

this context is not due to slowly evolving relations between (cross-)moments and characteristics.

Instead the estimation risk that plagues this algorithm is conventional overfitting–relying too

heavily on in-sample patterns. We analyze the nature of estimation risk as a statistical problem

by comparing the out-of-sample expected utility of all portfolios selected using an investor’s own

gamma with those selected using higher values of gamma.

This effectively means that our analysis is not normative since we make statistical choices

based on “out-of-sample” results. Instead we are using portfolio selection as a descriptive ana-

lytical tool to evaluate the relationships between characteristics and portfolio returns. In what

follows we will use the word gamma to reference a statistical loss function (or “an investor”),

and the symbol γ to reference the parameter used to select the portfolio. As such we will have

a certainty equivalent for each “investor” (indexed gamma 2, . . . , 10, using all values of γ in the

integer set: {2, . . . , 13, 16, 22} for γ values greater than or equal to the “investor’s” gamma. We

find that out-of-sample, all loss functions are maximized by using a value for γ that is larger than

(the investor’s actual) gamma.

We examine the properties of the optimal portfolios for each loss function: their factor

loadings, alphas, Sharpe ratios, means, variances, skewness, and kurtosis, and how the various

characteristics affect these portfolio measures. For example, the gamma 2 investor’s optimal

portfolio, obtained using γ = 3, has a monthly alpha of 296 basis points per month, significantly

less exposure to the market, and significantly more exposure to SMB, HML, and MOM than

both the value-weighted and equally-weighted indices. The portfolio’s mean Sharpe ratio is 1.24,

almost twice as high as that of the equally-weighted index (whose mean Sharpe ratio is 0.63). This

portfolio’s skewness and kurtosis are not significantly different from those of the two benchmark

portfolios.2

Similarly the optimal portfolio for the gamma 6 investor, obtained using γ = 11, has signifi-

cantly less exposure to the market and significantly more exposure to HML and MOM than the

benchmarks. However, this portfolio has insignificant exposure to SMB. This portfolio’s mean

Sharpe ratio is 1.27, and it has significantly more positive skewness and kurtosis than the bench-

marks. The optimal portfolio for the gamma 10 investor also has significantly less exposure to

the market and significantly more exposure to HML and MOM than the benchmarks. It has

significantly negative loading on SMB. Its mean Sharpe ratio is 1.07, and it also has significantly

more positive skewness and kurtosis than the benchmarks. This portfolio also has a statistically

significant alpha of 49 basis points per month. While this portfolio’s return standard deviation

2The equally-weighted and value-weighted benchmarks are portfolios that are formed from the (eligible sample)set of securities. We avoid using the term index to characterize these portfolios because they are different fromthe Fama French value-weighted index which we use as a factor.

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is larger than that of the value-weighted index, its interquartile range is statistically significantly

smaller.

Bootstrapping highlights the lack of precision in the role for characteristics. Much of this is

due to the complementarity between the characteristics as we show that adding characteristics

generally decreases the precision of the coefficients that relate portfolio weights to characteris-

tics. Furthermore estimation risk increases in the dimensionality of the set of characteristics

used. Lewellen, Nagel, and Shanken (2010) stress the importance of examining this precision.

Nevertheless the benefits of using characteristics to tilt portfolio weights transcend this impreci-

sion. The approach in this paper addresses several of Lewellen, Nagel, and Shanken’s concerns

about traditional tests. In particular we use individual stock returns and we only evaluate the

out-of-sample performance of the optimal portfolios.

We examine the month-of-the-year properties of the optimal portfolios. Lagged same-month

return is largely independent of the other characteristics, and calls attention to the possibility

that optimal portfolios’ exposures to risk factors may vary across the 12 months of the year. We

summarize this by looking at the bias in alpha from the Fama, French, Carhart regressions that

is due to assuming that the factor loadings are the same across the 12 months. We decompose

this bias into risk-premium exposure and volatility exposure. Referring to the optimal portfolio

for the gamma 2 investor (with mean alpha of 296 basis points per month), the mean calendar

bias is 46 basis points per month. The largest portion of this bias is due to volatility timing of

the SMB factor.3 That is the portfolio loads more heavily on SMB in months when SMB has

higher volatility than in other months.4

Ultimately we add to what we know about the predictive content of characteristics for con-

structing optimal portfolios. Just as they are useful in predicting expected returns, they can

be used to form portfolios with significantly higher certainty equivalent returns than the value-

weighted and equally-weighted indices.5 Complementarities are extremely important to under-

stand the optimal role of beta and size. Substitution effects are important to understand the role

of the book-to-market ratio, which we find to be redundant. Risk averse investors want exposure

to SMB, HML, and the momentum factor. In particular exposure to the momentum factor re-

3We use the measures of volatility timing and factor exposure timing from Boguth, Carlson, Fisher, and Simutin(2011).

4This is artifactual as is the portfolio’s alpha itself (that is the algorithm is not trying to maximize alpha, perse). In the sense that we do not know a priori whether alpha in the Fama, French, Carhart regressions is desirableto the CRRA investors. We find that the alphas of those optimal portfolios selected using only lagged same-monthreturns have a very large month-of-the-year timing bias component. For example, when γ = 2 is used to select theoptimal portfolio, and the only characteristic is 5-year average same-month return, the portfolio’s mean alpha is211 basis points per month, of which 92 basis points is timing bias, and 77 basis points of that is due to volatilitytiming biases which are statistically significant on the market, SMB, and HML.

5This provides contrast to the optimality of equal weighting as a robust rule for portfolio formation, as shownby DeMiguel, Garlappi, and Uppal (2009).

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duces the out-of-sample occurrence of very large negative returns that occur when momentum is

excluded from the characteristic set. The only characteristic that allows the investor to control

portfolio exposure to the momentum factor is the stock’s momentum. Similarly the only way

investors can regulate their exposure to SMB is with the size characteristic. However, investors

do not need the book-to-market characteristic to achieve optimal exposure to HML. This can be

obtained using log size and residual standard deviation.

The remainder of the paper is organized as follows. We describe the data and empirical ap-

proach in Section 2. Summary results concerning the algorithm and estimation risk are provided

in Section 3. Section 4 contains a discussion of the paper’s empirical findings. Section 5 concludes

the paper.

2. Portfolio Selection

2.1 Algorithm

In Brandt, Santa-Clara, and Valkanov’s (2009) algorithm, investors choose the vector θ in

order to maximize average utility over T periods:

maxθ

T−1∑t=0

(1 + rp,t+1)1−γ

1− γ

(1T

)(1)

by allowing the weights to depend on observable stock characteristics:

rp,t+1 =Nt∑i=1

(ωi,t +

1Ntθ′xi,t

)· ri,t+1 (2)

Where: xi,t is the K-vector of characteristics on firm i, measurable at time t; ωi,t is the weight

of stock i in the (value-weighted) market portfolio at time t; and Nt is the number of stocks in

the sample at time t.6 An alternative perspective on this loss function is that it is asymmetric

in the portfolio’s return. It would be relevant for an institutional money manager who wants to

avoid large losses in a single month.

2.2 Data

An observation of stock i at time t consists of the return in month t + 1 and the set of

characteristics that are measurable at time t. We use the following characteristics: momentum

(M), book-to-market (V), size (S), beta (β), market model residual standard deviation (σε), and

rt−12, and rt−12j , for j = 1, . . . , 5. Momentum is measured as the compounded return from

month t − 13 though t − 2. Market capitalization is the market value of all of a company’s

6We condition only on information that is available to investors at the time the portfolios are formed. Thisavoids the overconditioning bias analyzed by Boguth, Carlson, Fisher, and Simutin (2011).

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outstanding shares (aggregated across all classes for companies with multiple share classes) at

time t− 2. Book value is obtained from the Compustat database for the most recent fiscal year-

end between t − 6 and t − 18. The book-to-market ratio is the ratio of book value to market

capitalization. Size is the natural logarithm of market capitalization. Beta and the residual

standard deviation are obtained by regressing monthly returns from months t− 60 through t− 1

on the CRSP value-weighted index. All characteristic and return data is drawn from the merged

CRSP-Compustat file on WRDS. To be eligible for inclusion in the sample in month t, the stock

must have no missing returns in the CRSP database for the previous 60 months, and it must

have book value information in the Compustat database for a fiscal year-end between t− 6 and

t − 18. We obtain the US GDP deflator from the Federal Reserve (FRED) and use this to

construct a minimum size criterion of $50 million in January 1990 dollars. Stocks whose market

capitalization is less than this inflation-adjusted size criterion are excluded from the sample. This

excludes stocks with market capitalization less than $11.5 million in January 1960, and $93.3

million in December 2015. We next exclude the smallest 10% of stocks that meet all inclusion

criteria prior to February 1978, when the first Nasdaq stocks enter the sample, and the smallest

20% afterwards. If the stock return is missing in month t, we look to the CRSP delisting return.

If that is missing, we substitute -30% for NYSE- and AMEX-listed stocks and -50% for Nasdaq

stocks.

Figure 1 shows the sample size used each month. There are 395 (exclusively New York Stock

Exchange-listed) stocks in the sample in January 1960. There is a jump in the sample size

in August, 1967 (from 675 to 875) when the American Stock Exchange stocks are eligible for

inclusion in our sample. The largest jump is in January 1978 (from 1,000 to 1,419 stocks) when

Nasdaq stocks enter our sample. The maximum number of stocks is 2,291 in April, 2006. There

are 1,728 stocks in our sample in the last month, December 2015.

As noted, we normalize and standardize the characteristics–ensuring that the weights will sum

to unity for any value of θ. This also means that the characteristics are observationally equivalent

to shrinkage values. For example, let β be a stock’s OLS beta, were we to use shrinkage betas,

such as βS = .5·β+.5·1, the normalized βS are identical to the normalized β. A single observation

(Ψi,t) comprises stock i’s return in month t, ri,t, as well as the vector of characteristics, measurable

at month t− 1, for stock i, i = 1, . . . , Nt.

2.3 Specifications

A model is defined by the (sub-)set of characteristics used in portfolio selection. We consider

41 models which include each of the six characteristics as a singleton, and all seven variables

together. We consider 14 values of γ, which means that we evaluate 574 unique portfolios.

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Empirical analysis of the portfolios selected using the expected utility optimizing algorithm must

be conducted out-of-sample. We use both a rolling and updating protocol. The first out-of-

sample month is January 1975. At this point we have estimated the θ-vector using 180 months

of data. As in Brandt, Santa-Clara, and Valkanov (2009), we use this θ-vector construct the

optimal portfolios in each of the next 12 months. We add the 12 months of 1975 to the original

180 months under the updating protocol, whereas we drop the first 12 months in the last sample

(i.e., the 12 months of 1960) under the rolling protocol to estimate the θ-vector to use in forming

the optimal portfolios in 1976. The last out-of-sample year is 2015, so the last optimization/θ

estimation (applied to the 12 months of 2015, out-of-sample) uses 660 months under the updating

protocol (and the 180 months of 2000 – 2014 under the rolling protocol). This means that we

optimize utility (and estimate the θ vector) from each of the 574 cases 41 times–at the end of

each in-sample period. The out-of-sample period comprises the 41 years 1975 – 2015.

2.4 The bootstrap

Our interest is in statistical comparisons across various models. We use the bootstrap to

construct sampling distributions of the functions of interest in this paper, such as certainty

equivalent, portfolio loading on factors, portfolio skew, etc. The bootstrap also shows that

there is a small sample bias in the θ estimates, and other functions of θ. Our bootstrap is

designed as follows. As noted the data in month t in our sample consist of the Nt vectors

Ψi,t, for i = 1, . . . , Nt and t = 1, . . . , 672. A bootstrap draw resamples (with replacement) Nt

vectors from Ψi,t in all months. Thus each bootstrapped sample consists of the same number

of observations in each period as the original sample, and the calendar structure of the original

data is preserved. The latter is important because we consider the possibility that the 12 months

of the year are important characteristics. We take 10,000 bootstrap samples to estimate the

sampling distributions for all 574 unique portfolios.7

3. Algortithm efficacy and estimation risk

Table 1 reports the model with the highest bootstrap 2.5%ile certainty equivalent (across

the 41 specifications) for each of the nine investors, for her own and all higher γ values used to

estimate θ. The first row under each γ value is the result of using the updating protocol and

the second row is for the rolling protocol. This table also reports the bootstrap and sample

values of the certainty equivalent for both the equal- and value-weighted indices for all investors.

This table is excerpted from Tables IA-1 through IA-9,8 which report the bootstrap distributions

7The computational burden is non-trivial. A typical bootstrap takes 26 hours of CPU time on a 3.7 GHzXeon processor. However, since the bootstrap draws are independent (unlike in Markov Chain Monte Carlo, forexample), the algorithm is easily parallelizable.

8Tables starting with IA are collected in this paper’s Internet appendix.

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and sample estimates of the certainty equivalent for each of the nine investors, respectively,

using each value of γ and all 41 models under the updating protocol. That combination (of γ

and characteristic set) that produces the maximum 2.5%ile value of the certainty equivalent is

highlighted in bold face.

For example, the gamma 2 investor’s median certainty equivalent for the value- (equally-

weighted) index is 106 (86) basis points per month. When this investor uses her own γ, the model

(i.e., the set of characteristics whose θ is not restricted to equal 0 in all periods) with the highest

2.5%ile certainty equivalent is that comprising momentum, book-to-market, and size, using the

updating protocol. The gamma 2 investor does not prefer this portfolio to the equally-weighted

index, on a statistical basis. Such a preference requires that the 2.5%ile certainty equivalent

value of the preferred portfolio exceed the alternative portfolio’s 97.5% certainty equivalent. A

glance at Table IA-1 shows the nature and severity of the estimation risk problem in this case.

All 11 portfolios selected using five or more characteristics have a 2.5%ile certainty equivalent

of -100%. This means that in at least 250 of the bootstrap samples, the portfolio lost at least

100% of its value in at least one month in the out-of-sample period. This is also the case with all

models that include last year’s same-month return regardless of the number of characteristics.

Table 1 also shows that estimation risk is extenuated by using the rolling protocol. Using

γ = 2, the 2.5%ile certainty equivalent return (as well as the 25%ile certainty equivalent) of all

41 models is -100% for the gamma 2 investor.

By contrast, when the gamma 2 investor uses γ = 3 along with the characteristics: momen-

tum, log size, beta, residual standard deviation, and (lagged) 5-year average same-month return;

and the updating protocol to estimate θ, the median certainty equivalent is more than three

times higher than that of the equally-weighted index, and the 2.5%ile is 288 basis points per

month. Table IA-1 makes clear that using γ = 3 is not a panacea for the gamma 2 investor,

as nine of the 41 models produce a 2.5%ile certainty equivalent of -100%, including most that

contain last year’s same-month return. Table IA-1 also demonstrates the imprecision with which

the coefficients interact with portfolio moments. There are five additional models that are not

statistically dominated by the optimal. That is the 2.5%ile certainty equivalent exceeds the

97.5%ile of the equally-weighted index and the 97.5%ile exceeds the 2.5%ile of the the optimal

portfolio’s certainty equivalent. For example, adding book-to-market to the characteristic set

lowers the 2.5%ile certainty equivalent by 25 basis points per month, but increases the median

and 97.5%ile. The optimal model significantly dominates all (19) of the sets of characteristics

involving three or fewer characteristics. Table IA-1 also shows that using the optimal set of

characteristics and higher values of γ produces portfolios that are not statistically worse for the

gamma 2 investor. Increasing γ by a factor of one reduces the mean certainty equivalent return

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by 20 basis points. For γ values of 8 and higher, the optimal portfolio is statistically dominated

by the optimal portfolio selected using γ = 3.

Table IA-1 shows that including momentum in the set of characteristics is important to

reduce estimation risk. For the gamma 2 investor using γ = 3, and all sets of four, five, and six

characteristics that do not include momentum, the optimal portfolio’s 2.5%ile certainty equivalent

return is -100%. For example when the book-to-market ratio is substituted for momentum in

the characteristic set, the mean certainty equivalent drops from 336 to -1,230 basis points per

month. Adding last year’s same-month return to this set of characteristics further exacerbates

estimation risk. Even dropping momentum from the set of optimal characteristics, leaving log

size, beta, residual standard deviation and lagged average same-month returns results in an

optimal portfolio whose 2.5%ile certainty equivalent return is -100% when γ = 3 is used to

estimate θ. The optimal portfolio using all (six) characteristic variables excluding momentum

has a mean certainty equivalent of -6,045 basis points per month, when γ = 3. The mean certainty

equivalent using this set of characteristics is also negative for the gamma 2 investor, using a γ

value as high as 5 to select the optimal portfolio. As we will show below, adding momentum

to the other four characteristics increases the mean minimum (monthly) return on the optimal

portfolio using γ = 3 from -79% to -52% by significantly increasing the portfolio’s exposure to

the momentum factor. This is the reason that adding the momentum characteristic to the other

four characteristics has such a large effect on the optimal portfolio’s certainty equivalent return.

Using γ = 3 with the rolling protocol does not produce any optimal portfolios that allow the

gamma 2 investor to dominate the benchmarks. In fact, Table 1 shows that for this investor the

optimal result under the rolling protocol requires the use of γ = 5, and all 7 characteristics. This

portfolio also statistically dominates the benchmarks, and is not statistically dominated by the

global optimum portfolio for the gamma 2 investor (from the updating protocol, as its 95%ile

sampling interval certainty equivalent return is [248 , 366] (basis points per month). Table 1

shows that the optimal γ to use is also not precise, as the gamma 2 investor’s optimal portfolio

is not statistically dominated by the global optimum for γ values of 4, 5, 6, and 7. Table IA-1

also shows that in these cases, adding book-to-market and last year’s same-month return to the

five variables in the original optimal set has a trivial (and statistically insignificant) effect on the

optimal certainty equivalent returns.

The nature of the optimal portfolios and estimation risk for the gamma 3 and gamma 4 in-

vestors are very similar to that of the gamma 2 investor. The equally-weighted index statistically

dominates the value-weighted index for these investors. The gamma 3 investor’s optimal portfolio

is achieved using γ = 5 and the gamma 4 investor optimally uses γ = 7. The optimal character-

istic set for these two investors is the same as for the gamma 2 investor: momentum, log size,

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beta, residual standard deviation, and (lagged) average same-month return. The proportional

gains in certainty equivalent returns for these investors are also similar to that of the gamma 2

investor. The gamma 4 investor increases certainty equivalent mean return from 77 basis points

per month to 199 basis points per month using these five characteristics to tilt portfolio weights

away from the market portfolio–versus the 1/N rule. The rolling protocol has more estimation

risk. The gamma 4 investor for example has to increase γ to 9 to maximally reduce estimation

risk under the rolling protocol. The optimal portfolio in this case also significantly dominates

the equally-weighted index and is not dominated by the global optimal portfolio. For all three

of these investors the optimal portfolio under the rolling protocol is obtained using all seven

characteristic variables, and that under the rolling protocol drops the book-to-market ratio and

last-year’s same-month return from this set.

The gamma 5 and gamma 6 investors are statistically indifferent between the equally-weighted

and value-weighted indices and the gamma 7 investor prefers the value-weighted index. These

three investors optimally use the same characteristic set under the updating protocol: momentum,

log size, residual standard deviation, and average same-month return, so that beta is dropped

from the set used by more risk-tolerant investors. For these three investors the rolling protocol

is strictly dominated by the updating protocol, since all of the optimal portfolios generated by

the former are significantly dominated by those from the latter. The magnitude of the utility

gain for these investors is similar to that of the less risk-averse group. For example the gamma

7 investor’s mean certainty equivalent increases from 34 to 116 basis points per month switching

from the value-weighted index to the optimal portfolio, which is obtained using γ = 13.

The gamma 8, gamma 9, and gamma 10 investors comprise the third set of investors. All

three of these investors optimally use γ = 16 (since the grid of γ used to obtain portfolios is not

continuous), and the optimal set of characteristics is: momentum, log size, and residual standard

deviation. For these investors the rolling protocol is also optimized using these three character-

istics and γ = 16 (with very similar results using γ = 13). These optimal rolling portfolios are

not significantly dominated by the optimal portfolios produced under the updating protocol.

Overall the relationship between the updating and rolling protocols means that slowly evolv-

ing relationships between the (multivariate) return distribution and the characteristics is not

driving estimation risk–in the context of the optimal set of characteristics. Nor are there im-

portant structural breaks in these relationships. If there were, then the rolling protocol would

yield portfolios that outperform those selected using the updating protocol. The rolling proto-

col’s underperformance–relative to updating–also suggests that in the multivariate context there

is no evidence of a diminution of the predictive content of the characteristics over time, as in

Mclean and Pontiff (2016). Although Mclean and Pontiff evaluate long-short portfolios using one

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characteristic at a time. We see below that the evidence is different for singleton characteristic

sets than for multivariate sets of characteristics.

Estimation risk increases in the dimensionality of the characteristic set, with the exception

that adding momentum always mitigates estimation risk. In what follows we will consider how

momentum affects the factor exposure and moments of the optimal portfolios to consider its

effect on estimation risk in more detail.

It is never optimal to include last-year’s same-month return in the characteristic set. This is

because in the context of the five-year average same-month return this is very noisy and results

in increased estimation risk. The optimal portfolio also never conditions on the book-to-market

ratio. We will show below that this is not because these portfolios are not exposed to the HML

factor, but instead that size and residual standard deviation provide enough flexibility to allow

optimal exposure to the value factor. In other words the book-to-market ratio is redundant and

adds estimation risk.

The estimation risk in this context then is conventional overfitting. This inference is also

consistent with the fact that the dimensionality of the optimal model shrinks as investor risk

aversion increases. Those investors with gamma values higher than 5 eschew beta, and those

with gamma values higher than seven also drop the average same-month return from the optimal

characteristic set. The benefits delivered by these characteristics to lower gamma investors are

offset by estimation risk for the more risk-averse investors.

These results suggest that θ shrinkage or even holding a larger portion of assets in cash and/or

the market is also not optimal. In most cases, the adjustment involved with using a higher γ

than in the actual utility function is to lower the exposure to the characteristic (i.e., θ shrinks in

absolute value), but this is not the case when characteristics are optimally removed from the set

used to estimate θ. We will revisit this as well when we evaluate the θ coefficients below.

4. Characteristics and portfolios

4.1 Characteristics in isolation

Tables IA-1 – IA-9 show that momentum, book-to-market, and log size never serve as a single-

ton characteristic to produce significantly higher certainty equivalent–for any investor, using any

γ value. The most risk-tolerant investors (those whose coefficient of relative risk aversion is two

and three) can use the average same-month return by itself to create portfolios with significantly

higher certainty equivalent than the equally-weighted benchmark. Investors with midlevels of risk

aversion (those with coefficients of relative risk aversion between five and eight) can significantly

increase certainty equivalent above the benchmark by using the residual standard deviation in

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isolation. Investors with high risk aversion (those with coefficients of relative risk aversion be-

tween seven and nine) can significantly increase certainty equivalent above the benchmark by

using beta in isolation. Therefore, most–and in some cases all– of the (out-of-sample) utility

gains from using the characteristics in combination.

4.2 Substitutions and complementarities

Whereas momentum and log size never generate such a portfolio in isolation, both of these

characteristics–along with the residual standard deviation–are used in the optimal characteris-

tic set for all nine investors. Therefore the salutary effects of the characteristics for risk-averse

utility optimizing investors is primarily due to substitution and complementary effects between

the characteristics. This fact belies traditional analysis of the role of characteristics and returns,

which has focused mainly on the relationship between characteristics and expected returns, and

has generally analyzed these effects one characteristic at a time (Lewellen 2016). Another im-

portant difference between this approach and much of the literature is that this analysis has to

be out-of-sample (since the model places no testable restrictions on the data generating process),

(Lewellen, Nagel, and Shanken 2010).

Tables IA-10 through IA-16 report bootstrap properties for θ coefficients on each of the seven

characteristic variables for all the optimal portfolios produced under the updating protocol, from

all of the combinations of characteristic sets that contain that variable, for all 14 γ values. Table

2 shows the sampling distributions of the average of the 41 annual θ coefficients on the (seven)

portfolios that are optimal for the nine investors. Figures 2 – 6 show bootstrap properties of the

estimated θ coefficients by year (41 years: 1974 through 2014, used to form the out-of-sample

portfolios in the following year). We show each of the five characteristic variables that enter

the optimal set for the gamma 2, 3, and 4 investors, when they are used in isolation(top panel),

and in the optimal portfolio using γ = 3 (lower panel). Figure 2 shows the bootstrap properties

of the θ coefficients on momentum. When momentum is used in isolation, there appears to be

a structural break after year 25, so we split the results temporally at this point in the tables.

Comparing the two panels in Figure 2 it is clear that the sampling distribution of the estimated

θ is much tighter when momentum is the only characteristic. This is also evident in Table IA-10

as the standard deviation of the momentum θ in the second period (Years 26 - 41) is 0.18, when

momentum is the only characteristic used, and 0.40 when momentum is used along with: log size,

beta, residual standard deviation, and average same-month returns. Also the imprecision in the

latter case is such that there is no evidence of a structural break in the momentum θ coefficient

in the optimal model for the gamma 2 investor.

The figures also report the sample estimate of θ in the context of the parameter’s bootstrap

11

distribution. These coefficients are biased (away from zero). This bias is extenuated by dimen-

sionality of the sample space, and in most cases appears to get worse over time.9 These biases

support our use of bootstrap means instead of the sample estimates for our statistical analyses.

These biases also explain the differences between the bootstrap mean certainty equivalent and

the sample value, shown in Table 1 as well as Tables IA-1 – IA-9. The sample certainty equivalent

for the gamma 2 investor’s optimal portfolio is 376.4 basis points per month, above the bootstrap

75%ile value.

Momentum enters the optimal characteristic set for all nine investors. For the gamma 8, 9,

and 10 investors the mean θ coefficient on momentum is 1.54 in 1975 - 1999 and 0.91 in 2000

- 2015. The difference between these is statistically significant. These momentum θ coefficients

are not significantly different from the optimal θ values for the gamma 6 and 7 investors, but

they are significantly smaller than the optimal θ for those investors whose gamma is less than 6.

Figure 2 shows that momentum is largely independent of the other characteristics since the

levels of θ are similar when momentum is used in isolation, as when it is part of the optimal

characteristic set. There appears to be some evidence of complementarity in the latter part of

the sample. In this period, the 95% sampling interval of the momentum θ used in isolation is

[2.62 , 3.32], whereas in the optimal set this interval is [3.87 , 5.44].

Figure 3 and Table IA-12 report properties of the θ coefficient on log size. As with momentum,

the figure shows that the sampling distributions are much less precise in the higher dimensional

case. The sample bias is also worse in this context, and at the end of the sample. This figure and

table show that size is complementary to the other characteristics. In isolation, the 95% sampling

interval of the θ coefficient on log size in the two periods are: [−3.0 , −1.7] and [−2.5 , −1.6],

respectively. By contrast, in the optimal model for the gamma 2 investor, these ranges are:

[−13.7 , −9.6] and [−10.7 , −8.3], respectively. We can look at all of the specifications reported

in Table IA-12 to isolate which characteristic(s) are most complementary to log size. Adding the

book-to-market, momentum, and same-month returns, individually or in combination have no

effect on the log size θ. Beta, like residual standard deviation, tends to complement log size, and

the effects of these two are distinct.

Table IA-13 and Figure 4 show properties of the θ coefficient on beta. When beta is the

only characteristic (top panel in both Figure 4 and Table IA-13) the θ coefficient is significantly

negative for all gamma values used to select the optimal portfolio in both subperiods. Further,

there is no statistical difference between the coefficients as γ ranges from 2 through 22. Such9This suggests that a bias adjustment technique such as bootstrap aggregation (or “bagging”) might improve

matters in terms of selecting optimal portfolios. Experimentation with both bagging and bragging suggests thatthe methods produce small gains in certainty equivalent that are orders of magnitude smaller than the gainsachieved with simply using a higher γ to select the optimal portfolio. The numerical demands of such techniquesare also overwhelming.

12

a result is consistent with Frazzini and Pedersen (2016), who argue that high beta stocks are

bid-up by borrowing-constrained investors, so that they offer lower expected rates of return after

adjusting for risk. If this were indeed the case, then the appeal of low beta stocks in this setting

would be independent of the investor’s risk aversion, since we assume away borrowing constraints.

This result is not robust. All of the portfolios formed by conditioning only on beta (and which

have an average negative θ coefficient on beta) are dominated by the optimal portfolio for all

investors. For the gamma 2, 3, and 4 investors these involve a statistically insignificant θ in the

first 25 years and a significantly positive θ coefficient on beta in the 2000 – 2015 period. For the

more risk-averse investors with gamma values 5 and higher, these optimal portfolios place a 0

weight on beta. Table 2 shows that the average coefficient on beta is not statistically different

from 0 over the full 41 year period. This is also evident in Figure 4. Despite this, investors with

gamma values of 4 and lower are better off including beta in the set of characteristics, which

speaks to the time-series properties of beta. Clearly the coefficient on beta is significantly positive

in the latter 16 years of the sample (Table IA-13).

Table IA-13 shows the effects of adding the other characteristics to beta: the residual standard

deviation, momentum, and log size are the sources of complementarity with beta. In particular,

adding residual standard deviation to beta leads to a positive θ on beta in both periods, using

γ = 2, and in the 2000–2015 period using γ ≤ 5. This suggests that beta is correlated with

something that all investors do not like. And the residual standard deviation is also correlated

with this latent factor. Kogan and Papanikolaou (2013) argue that firms with higher idiosyncratic

volatility have higher growth opportunities, and lower risk premia. When relatively risk-tolerant

investors condition on both beta and the residual standard deviation, the residual standard

deviation allows them to reduce their exposure to this factor, and then higher beta stocks are

more attractive than low beta stocks. The fact that beta is dropped from the optimal set of

characteristics when investor gamma is 5 and higher suggests that it has a lot of estimation risk,

and that its appeal in-sample does not produce benefits out-of-sample.

Figure 4 shows one reason for isolating the period 2000-2015, as the θ coefficient on beta–

when beta is used in isolation–behaves differently over this period than in the preceding period.

The figure also shows that the bias in the sample θ coefficient is larger when beta is combined

with the four other characteristics than when it is used in isolation.

Table IA-14 and Figure 5 report the properties of the θ coefficient on the residual standard

deviation. The figure shows that for the gamma 2 investor, the coefficient on the residual standard

deviation is uniformly negative, and that this is complemented by the other characteristics,

especially size and beta. For example, the average θ coefficient on the residual standard deviation

for the gamma 2 investor using γ = 3 when residual standard deviation is the only characteristic

13

is -2.0. In the optimal model, this average bootstrap coefficient is -11.89. The residual standard

deviation, like momentum and log size is optimally used by all investors, and has a significantly

negative θ coefficient in all optimal portfolios. Figure 5 shows that as the case with the other

characteristics the sample bias is extenuated by the model dimensionality and the increased

sample size. Table IA-14 shows that the θ coefficient on this characteristic is much more sensitive

to investor risk aversion when it is used in combination with size and beta, but it is significantly

negative in all optimal portfolios.

Table IA-15 and Figure 6 show properties of the θ coefficient on lagged five-year average

same-month return. Table IA-15 shows that the θ coefficient on this characteristic is significantly

higher in the second subperiod in the optimal models for the gamma 3, 4, 5, 6, and 7 investors

than in the model in which it is the only characteristic. The 95% sampling bands on the θ

coefficient on rt−12j in the two subperiods for the optimal portfolio for the gamma 6 investor

(using γ = 11) are: [1.7 , 3.3] and [1.9 , 3.0], respectively. When rt−12j is the only coefficient in

the characteristic set, these bands are [1.2 , 2.1] and [0.6 , 1.2], respectively, when γ = 11. This

means that in the second subperiod, for investors whose gamma exceeds 2, momentum, log size,

and residual standard deviation are complementary to lagged same-month return. Figure 6 shows

that when rt−12j is used in isolation the θ coefficient varies significantly from year to year (which

is also apparent in significant differences between the two subperiods for this model). The lower

panel of Figure 6 shows that this is not the case for the optimal model for the gamma 2 investor,

largely because the sampling variances on these coefficients are so large.

Two of the seven characteristic variables, last-year’s same-month return (whose θ coefficients

are reported in Table IA-16) and the book-to-market ratio are not used in any of the optimal

portfolios. Tables IA-1 – IA-9 show that for the most part this is because of redundancy and

estimation risk, rather than direct harm. Adding both characteristic variables to the optimal set

for the gamma 4 investor results in an insignificant drop in the certainty equivalent; the 95%ile

bands on the certainty equivalent decrease from [173 , 226] to [157 , 223] (basis points per month).

Adding a characteristic which is largely spanned by the other characteristics adds estimation risk,

seen in the drop in the 2.5%ile of the sampling distribution in these cases.

5. Portfolios and Factors

5.1 Factor loadings

In this subsection we consider what these optimal portfolios’ exposures are to the traditional

risk factors as well as their alphas. Table 3 contains the results of these regressions for the

benchmarks and the optimal portfolios for the nine investors. The results for all portfolios are

14

contained in Tables IA-17 – IA-30, for each of the 14 γ values used in portfolio selection. The

exposure to the market portfolio is constant across levels of investor risk-aversion. All seven of

the optimal portfolios have loadings on the market that are significantly less than the equally-

weighted and value-weighted benchmarks.

The optimal portfolios’ exposure to SMB decreases in risk-aversion. The gamma 2 investor

has the largest exposure to this factor, with a 95% sampling band of [0.6 , 1.5]. This is significantly

larger than the equally-weighted benchmark. The gamma 3 investor’s optimal portfolio also has

significant positive loading on SMB, however this is not significantly different from the exposure

of the equally-weighted benchmark to this factor. The gamma 4 investor’s optimal portfolio also

has significant positive loading on SMB, but this is significantly less than the exposure of the

equally-weighted benchmark to SMB. The optimal portfolios of the gamma 5, 6, and 7 investors

have insignificant exposure to SMB, and none of these are significantly different from the value-

weighted benchmark’s SMB loading. The optimal portfolio for the most risk-averse investors

with gamma values of 8, 9, and 10 are significantly negative, although not significantly different

from the value-weighted benchmark.

All of the seven optimal portfolios have significant positive exposure to HML, as does the

equally-weighted benchmark. All of the portfolios’ loadings on the HML factor exceed that

of the equally-weighted benchmark, and decline only slightly in risk-aversion. The gamma 2

investor’s optimal portfolio has a significantly higher loading on HML than the optimal portfo-

lios of investors with gamma of 4 or higher. The optimal portfolio of the gamma 4 investor’s

loading is not significantly different from that of investors with gamma between 5 and 8. The

value-weighted benchmark has no exposure to the momentum factor and the equally-weighted

benchmark has significantly negative exposure to this factor. All of the optimal portfolios have

significant exposure to this factor. The momentum loading decreases in investor risk aversion.

Tables IA-17 – IA-30 provide evidence about the linkages between the characteristics and

factor loadings. Consider the 41 portfolios used by setting γ = 3 (which include the optimal

portfolio for the gamma 2 investor) shown in Table IA-18. When beta is the only characteristic,

the optimal portfolio has significantly less exposure to the market factor than the two benchmarks.

This portfolio has a significant negative loading on SMB, a significant positive loading on HML, as

well as a significantly positive loading on momentum. Conditioning on only the residual standard

deviation similarly reduces the exposure to the market portfolio below that of the benchmarks.

The negative weight on this characteristic implies a large negative loading on SMB, as large

stocks tend to have lower residual standard deviations. This characteristic also affords exposure

to HML. Conditioning only on the average same-month return induces a significant increase in

exposure to the market factor, and significant negative loadings on SMB and HML. None of the

15

optimal portfolios are created by conditioning on the book-to-market ratio. When we replace

momentum in this investor’s characteristic set with the book-to-market ratio, the portfolio’s

exposure to HMB is unchanged. The 95%ile sampling band of the HML loading with the book-

to-market ratio in the characteristic set is [1.4 , 2.2]; when we replace the book-to-market ratio

with momentum in the characteristic set this band is [1.4 , 2.2]. Adding the book-to-market ratio

to the five optimal characteristics does not produce a significant difference in the factor loadings,

which is not surprising as this portfolio’s certainty equivalent is not significantly different from

the optimal portfolio’s.

The most risk-averse investors optimally include only momentum, log size, and residual stan-

dard deviation in their optimal characteristic set, and use γ = 16. Table IA-29 shows that

substituting the book-to-market ratio for size in this optimal characteristic set, portfolio expo-

sure to the market factor increases significantly: 95%ile sampling bands go from [0.56 , 0.65] to

[0.76 , 0.82]. The exposure to SMB decreases from [−0.25 , −0.06] to [−0.65 , −0.52]. The two

portfolios’ exposures to HML are not significantly different. Since momentum is largely indepen-

dent of the other characteristics these results identify the complementarity between log size and

residual standard deviation.

The gamma 5 investor drops beta from the optimal characteristic set. Comparing the optimal

portfolio factor loadings for the gamma 4 investor with those of the gamma 5 investor’s optimal

portfolio shows that the effect of removing beta is virtually nil. The mean loading on the market

factor is identical after removing beta, and the sampling variance of this exposure is lower.

Comparing the optimal portfolio of the gamma 7 investor with that of the gamma 8 investor

shows the effect of removing the average same-month return from the optimal characteristic set.

The mean loadings between these two portfolios are virtually identical, and the sampling variance

is lower on all.

Table IA-20 shows that using average same-month return in isolation along with γ = 5, loads

more heavily on the market than the equally-weighted benchmark. In Section 4.1 we saw that this

portfolio has a significantly higher certainty equivalent than this higher benchmark for the gamma

2 and gamma 3 investors. The 95% sampling band of this portfolio’s loading on the market factor

is [1.16 , 1.27] . This portfolio also has significant negative exposure to SMB [−0.31 , −0.14] and

HML: [−0.35 , −0.19]. The equally-weighted index has significant positive loadings on these two

factors. The optimal portfolio selected using γ = 10 and the residual standard deviation in

isolation (Table IA-25) has significantly lower exposure to the market factor [0.75 , 0.80] than

the market benchmarks. It also has significantly negative exposure to SMB [−0.70 , −0.61]. Its

exposure to HML [0.42 , 0.50] is significantly higher than the value-weighted benchmark.

The desired exposure to HML is achieved with momentum, size, and the residual variance.

16

Table IA-24 shows the effect of adding the book-to-market ratio to the optimal set of character-

istics for the gamma 5 investor (using γ = 9). The optimal portfolio conditions on momentum,

size, residual variance, and average same-month return. Its mean exposure to the momentum

factor is 0.63 ([0.54 , 0.72]) and to HML, 1.17 ([1.03 , 1.31]). Adding the book-to-market ratio

does not significantly affect the portfolio’s HML exposure: 1.33 ([1.16 , 1.51]), but it significantly

decreases its exposure to the momentum factor: 0.41 ([0.32 , 0.51]). Kogan and Papanikolaou

(2013) provide the rationale for the residual standard deviation to regulate portfolio exposure

to the value factor, HML. In their model firms with higher growth options have higher market

model idiosyncratic volatility.

5.2 Alpha

Table 3 and Tables IA-17 – IA-30 also report the sampling values and bootstrap properties

of the intercepts from the Fama, French, Carhart regressions in basis points per month. The

maximum 2.5%ile alpha value of 371 is realized when the characteristic set comprises the book-

to-market ratio, log size, beta, residual standard deviation, last year’s same-month return, and

the average same-month return; that is all characteristic variables except momentum. This

portfolio’s mean alpha is 480 basis points per month. Table IA-1 shows that this portfolio’s

mean certainty equivalent for the gamma 2 investor is -9,305 basis points per month–highlighting

the fact that maximizing expected utility is not pari-passu with maximizing alpha. Indeed, in

general adding momentum to a characteristic set reduces the portfolio alpha, however all optimal

portfolios include momentum in the characteristic set.

Focusing on the optimal portfolios, Table 3 shows that alpha declines monotonically in in-

vestor gamma. This means that while these returns are orthogonal to the four Fama, French,

Carhart factors they are not a free lunch. It also provides an example of Ferson’s (2013) observa-

tion that a positive alpha strategy may be increasingly leveraged to further increase alpha (unlike

the Sharpe ratio, which is largely unchanged in leverage). The large drop from the gamma 7 to

the gamma 8 investor also highlights the usefulness of lagged same-month returns in producing

alpha (Heston and Sadka 2009). This is the only case in Table 3 where an adjacent portfolio

alpha is statistically different from its predecessor. While the characteristic same-month return

is related to future expected returns, this relationship is noisy enough to deter those investors

with gamma 8 and higher from exploiting this predictive content. This is manifest in the fact

that alpha’s sampling standard deviation drops from 10 to 6 basis points per month when the

average same-month return is removed from the optimal characteristic set.

Table IA-20 shows that the γ = 5, singleton mean same-month return portfolio’s mean alpha

is 78 basis points per month [57.15 , 101.72] . The γ = 10 singleton residual standard deviation

17

portfolio has a mean alpha of 29 basis points per month [21 , 37].

This evidence provides another reminder of the danger of using alpha as a performance

measure, since the investor’s objective function is not linear. This fact is well known; for example

Leland (1999) and Ferson (2013) discuss the failings of alpha in evaluating equity portfolios.

Broadie, Chernov, and Johannes (2009) provide a reminder in the context of option returns.

6. Return Distributions and Moments

Figures 7 and 8 show the return distributions for four of the optimal portfolios–each contrasted

with the return distribution of the investor’s preferred benchmark.10 Table 4 shows sampling

distributions of portfolio moments for the benchmark portfolios and the nine optimal portfolios.

Tables IA-31–IA-44 provide this information for all 41 portfolios using each of the 14 gamma

values, respectively. Figure 7 shows that the risk tolerance of the gamma 2 investor generates

an optimal portfolio with a much greater scale than the equally-weighted index. The mean (95%

sampling band) of this optimal portfolio’s mean return is 549 ([465 , 645]) basis points per month.

The analogous moments for the equally-weighted benchmark are: 133 ([131 , 135]) basis points

per month. The Sharpe ratio shows that whereas the standard deviation is also larger than

the benchmark’s, the increase in the second moment is proportionately less. The Sharpe ratios

reported in these tables are annualized (i.e., the monthly numbers are multiplied by√

12). In

addition to mean, standard deviation, and Sharpe ratio, the tables provide bootstrap properties

of the return median, interquartile range, minimum return, and robust measures of skewness

(K3) and kurtosis (K4):

K3 =r+.95 − r

−.05

r+.5 − r−.5

− 2.63 (3)

and

K4 =r.95 − r.05

r.75 − r.25− 2.90 (4)

Where: r+.95 is the mean of the highest 5% of returns, r−.05 is the mean of the smallest 5% of

returns, r+.5 is the mean of the top half of returns, and r−.5 is the mean of the bottom half of

returns; and rx is the observation corresponding to the x%ile of the return data.11

Whereas Figure 7 shows that the return distribution of the optimal portfolio for gamma

2 investor is very different from that of the equally-weighted benchmark, it is neither more

skewed nor leptokurtic than this benchmark. The optimal portfolios for all more risk-averse10The figures aggregate the 492 months (in the out-of-sample period) over the 10,000 bootstrap draws, so that

the plotted empirical distributions have 4,920,000 observations.11The measures of skewness and kurtosis are recommended and discussed by Kim and White (2003).

18

investors are significantly more right-skewed than the benchmark. Tables IA-31 – IA-44 show

that some portfolios have significantly higher skews than the optimal one–using the same γ.

For example when γ = 3 is used to generate optimal portfolios the maximal skew is obtained

by conditioning on the book-to-market ratio, mean same-month returns, and last-year same-

month return. This portfolio’s median sampling skewness measure (95% sampling band) is

1.00 ([0.87 , 1.14])–significantly higher than the skewness measure of the returns on the optimal

portfolio for the gamma 2 investor: 0.44 ([0.31 , 0.57]). The portfolio with higher skew also has

fatter tails. The kurtosis median sampling kurtosis measure (95% sampling band) for the higher

skew portfolio is 1.29 ([0.85 , 1.80]), whereas that for the gamma 2 investor’s optimal portfolio

is 0.51 ([0.15 , 0.90]). The gamma 2 investor is the only one of the nine investors whose optimal

portfolio does not exhibit significantly more (positive) skewness than the preferred benchmark.

Table IA-32 shows that when used in isolation with γ = 3, momentum, book-to-market ratio,

residual standard deviation, and same-month return produce optimal portfolios that are more

positively skewed than the benchmark returns. Size, and beta do not.

The upper panel of Figure 8 shows the return distribution of the optimal portfolio for

the gamma 4 investor. The distribution is significantly more skewed and leptokurtic than the

benchmark–both facts are evident in this figure. The mean return of the most risk-averse in-

vestors’ optimal portfolio is 74 basis points higher (per month) than that of the value-weighted

benchmark. The standard deviation is not significantly higher than that of the benchmark.

Furthermore, the with a 95%ile sampling band of ([4.37 , 5.11]) (basis points per month), this

portfolio’s interquartile range is significantly lower than that of the benchmark ([6.03 , 6.33]).

Especially since these portfolios are more leptokurtic than the normal distribution the standard

deviation may give a distorted sense of the distribution’s scale. The interquartile range is a

nonparametric measure of scale. The lower panel of Figure 8 shows this portfolio along with the

value-weighted benchmark.

Table IA-34 shows that the γ = 5, singleton average same-month return optimal portfolio

has a significantly higher Sharpe ratio than that of the equally-weighted benchmark; its 95%

sampling band is ([0.67 , 0.81]) (the 97.5%ile of the equally-weighted benchmark’s Sharpe ratio

is 0.65). Both the portfolio mean return ([1.64 , 2.08]) and interquartile range ([6.42 , 7.69]) are

significantly higher than those of the benchmark. The use of same-month return to weight stocks

creates a portfolio that is significantly more positively skewed than the benchmark. It is also

more leptokurtic. This explains why the same-month return can be used in isolation by the most

risk-averse investors to improve utility relative to the equally-weighted benchmark. The variance

is lowered by reduced by lowered exposure to SMB and HML. This portfolio is far from optimal

for these risk-tolerant investors for this same fact. In contrast, the optimal portfolios for these

19

investors (Table 3) have lowered exposure to the market and increased exposure to SMB and

HML, relative to the equally-weighted benchmark. In addition they have significant exposure to

the momentum factor, which can only be achieved when momentum is included in the available

set of characteristics.

Table IA-40 that the γ = 11, singleton residual standard deviation has a significantly higher

Sharpe ratio than the value-weighted benchmark. This is primarily because the portfolio has a

lower standard deviation than the benchmark, as these portfolios’ mean returns are not statis-

tically different. Not only is this portfolio’s standard deviation lower than the value-weighted

benchmark’s, the interquartile range is proportionally much lower (with 95% band: [4.07 , 4.56]

compared to the value-weighted benchmark [5.20 , 5.59]). This is because this portfolio has sig-

nificantly higher kurtosis than the benchmark. Its skewness is not significantly different from

that of the value-weighted benchmark.

The optimal portfolio for the gamma 6 investor is constructed using γ = 11 as well, however

this portfolio has significantly higher mean and standard deviation than the benchmark. This

highlights another aspect of the complementarity between the characteristics. Midlevel risk-

averse investors optimally use the residual standard deviation to reduce portfolio variance, even

though they tolerate a higher variance than the market benchmark. The other characteristics,

especially the same-month return and momentum work to increase the expected return.

With γ = 9 Table IA-38 shows that using residual variance in isolation yields an optimal

portfolio with significantly higher mean return than using beta in isolation: [1.10 , 1.25] versus

[0.96 , 1.07]. Table IA-24 shows that these two portfolios have similar loadings on the Fama,

French, Carhart factors. The difference in mean returns is due entirely to alpha–using residual

variance in isolation produces a significantly higher alpha: 29 versus 13 basis points per month.

7. Monthly seasonal patterns

Since the average same-month return enters into the optimal characteristic set for all but the

most risk-averse investors, we turn to the question of how optimal portfolios vary across the 12

months of the year. Referring to the Fama, French, Carhart regressions, we examine the extent

to which there are timing biases in alpha that result from not conditioning on the month. To

this end, we compare the sampling distribution of the (unconditional) alpha in Tables IA-17 –

IA-30 with the (conditional) alpha from the 12 regressions that use only the 41 observations

from one month (e.g., all 41 Januaries, etc.). We follow Boguth, Carlson, Fisher, and Simutin

(2011) in defining the volatility timing bias which reflects correlation between the monthly factor

loading and the monthly factor variance; and risk-premium timing bias which reflects correlation

between the monthly factor loading and the monthly factor expected return.

20

In particular, define buj to be the unconditional loading of the portfolio returns on factor j

(estimated using the 492 out-of-sample months). Let bm,j be the loading of the portfolio return

in month m on fj,m, the return on factor j in month m, and bj be 112

12∑m=1

bm,j . The volatility

timing bias associated with factor j is then: fuj (bj − buj ), where fuj is the unconditional mean of

the return on factor j (over the 492 out-of-sample months). Defining f j,m as the mean return

on factor fj in month m, the total bias in α attributable to factor j is: 112

12∑m=1

bm,jf j,m − buj · f

uj .

The α− bias attributable to timing the risk premium is then defined as the total bias minus the

volatility timing bias.

We introduce alpha as that amount of the mean portfolio return that is orthogonal to the

Fama, French, Carhart factors, and do not suggest that it is devoid of risk. Indeed, since portfolio

alpha declines monotonically in γ there is no reason to identify it as a “free lunch.” Therefore we

use these biases simply as a metric to measure the importance of varying exposure to the Fama,

French, Carhart factors across the 12 months of the year.

For this decomposition to be meaningful, we need significant differences across the 12 months

in factor means and variances. Panel A of Table 5 provides the unconditional factor means and

standard errors over the 41 years in the out-of-sample period (1975 – 2015), by month of the year.

All four factors exhibit monthly seasonal patterns in both mean returns and standard deviations.

For example, the highest return on the market is in November (168 basis points) and the lowest

is in September (-84 basis points) – a statistically significant difference. The return on SMB in

January (141 basis points) is significantly higher than it is in October (-128 basis points), April

(-18 basis points), July (-66 basis points), and August (-13 basis points). The standard deviation

of momentum returns is more than twice as large in April as it is in August.

Tables IA-45 – IA-58 show the bootstrap sampling distributions for the volatility-, risk-

premium-, and total timing biases in alpha for the optimal portfolios selected using all 41 models,

for each of the 14 γ values, respectively. Panels B - H of Table 5 cull from these tables the optimal

portfolios for the 9 investors, respectively. Another reason to examine timing in the metric of the

alpha timing biases is that conceptually it can explain the disconnect between (out-of-sample)

alpha and expected utility.

The largest bias in Table IA-46, where γ = 3, is on the portfolio that conditions on momen-

tum, book-to-market, log size, and average same-month return. As reported in Table IA-18, the

average alpha on this portfolio is 154 basis points per month. This table shows that over half of

this alpha is due to non-constant loadings across the 12 months. With this set of characteristics

the portfolio invests more in the momentum factor in months when this factor has a higher risk

premium. This timing bias on momentum accounts for 21 of the 88 basis point bias. The two

largest components of the bias are volatility timing on SMB and HML. In all volatility timing

21

accounts for 69% of the total timing bias. SMB volatility timing accounts for 31 of the 88 basis

point bias, and HML volatility timing accounts for 25 basis points.

The timing biases in the nine optimal portfolios reported in Table 5 are much smaller. Of

these, the largest is also using γ = 3, which is optimal for the gamma 2 investor. In this case

varying factor loadings across the 12 months of the year means that the portfolio has significantly

less exposure to the market and to HML in those months when the market risk premium is higher

and the return on HML is higher. The portfolio has significantly more exposure to SMB and

MOM in those months when the returns on these two factors are higher. This portfolio also has

significantly more exposure to the market and SMB in those months when the returns on these

factors are more volatile, and significantly less exposure to MOM in the months when this factor

is more volatile.

Studies examining the ability of same-month return to produce alpha have not explored the

possibility that some of this alpha may be spurious, since the means and variances of the four

factors exhibit significant month-by-month variation.12 Table IA-49 shows the timing biases of

the optimal portfolio obtained using γ = 6. The portfolio that is formed by conditioning only

on the two same-month return characteristics has a mean monthly alpha of 63 basis points per

month–close to the 68 basis points reported in Heston and Sadka (2008).13 The table shows that

this alpha includes a 45% bias. Most of this bias (90% of the total timing bias) is the result

of increased exposure to the three Fama and French factors in months when they have higher

volatility. There is significantly positive volatility timing bias on the market risk premium, SMB,

and HML. This portfolio also significantly positively times the risk premium of momentum and

SMB– loading more heavily on these factors in months when they have higher expected returns.

It also negatively times the market and momentum–loading more heavily on these factors in

months when their expected returns are lower. While all four of these risk-premium timing

biases are statistically significant, their magnitudes are small relative to the volatility timing

biases. This is entirely consistent with Keloharju, Linnainmaa, and Nyberg.

Table IA-46 shows that conditioning on momentum by itself, using γ = 3, results in a portfolio

whose unconditional factor loadings significantly understate exposure to the volatility in SMB.

This results in overstating alpha by 8 basis points per month. The converse of this is also shown

in this table: a portfolio formed using γ = 3 and conditioning only on size has significantly lower

12This is a motivation of Keloharju, Linnainmaa, and Nyberg (2016). Indeed they note that a portfolio of stocksranked by January returns is likely to look like SMB in January. Their point is that same-month returns proxyfor systematic factor exposure. They posit that these factors are latent, and many may be relatively small. Herewe measure the extent to which this time-variation in factor loadings applies to the four Fama, French, Carhartfactors.

13See Heston and Sadka (2008) Table 3, p. 426. A portfolio that is long the top 10% same-month returns inyears 2 – 5 and short the bottom 10% same-month returns in years 2 – 5 earns a Fama and French alpha of 68basis points per month (with a 3.7 t−statistic).

22

exposure to the momentum factor in those months when this factor is less volatile. The table also

shows that in isolation, beta generates an optimal portfolio that successfully times the volatility

in the market, SMB, and HML. The portfolio’s loadings on all three of these factors are lower in

those months when they are more volatile.

Table IA-46 shows that risk-tolerant investors using average same-month return in isolation

produces a portfolio that has higher exposures to: the market factor, SMB, and HML in the

months when those factors have higher variances. In contrast, Table IA-52 shows that midlevel

risk-averse investors conditioning only residual standard deviation reduce portfolio variance by

loading more heavily on SMB and HML in those months when these factor variances are low.

Table IA-52 also shows that conditioning only on beta has a similar effect.

Table IA-57 shows that the most risk-averse loss functions in our analysis use (γ = 16, and)

the characteristics: momentum, size, and residual standard deviation to reduce the portfolio’s

exposure to SMB, HML, and MOM, the momentum factor, in those months when their variances

are high. This results in understating the portfolio alpha by 15 basis points per month on average.

Although momentum is the only “dynamic” characteristic among these three–used optimally by

the gamma 8, 9, and 10 investors–this table suggests that momentum in isolation does not

successfully time the volatility exposure to any of the four factors. Conditioning on the residual

standard deviation and/or beta, which are both persistent by construction, successfully times

the exposures to SMB and HML, in the sense that factor exposure is lower in the months of the

year when the factor is more volatile.

8. Conclusion

This paper complements studies that examine the usefulness of observable stock character-

istics to predict returns in the cross-section. To what extent is this documented predictability

useful for risk-averse investors? The first step in answering this question is addressing estimation

risk. Estimation risk has plagued portfolio selection algorithms over the years, manifest in the

popularity of norms such as equal-weighting (as emphasized by DeMiguel, Garlappi, and Uppal

2009). Although the Brandt, Santa-Clara, and Valkanov (2009) algorithm estimates a small

number of coefficients it also suffers from estimation risk. We address that risk by inflating the

investor’s risk aversion coefficient to select the portfolios. For all the investors we consider the

optimal portfolio represents a large and statistically significant improvement in expected utility.

Momentum, log size, and the market model residual standard deviation enter the optimal

set of characteristics of all investors. As investors become more risk-tolerant lagged same-month

return is added to the optimal characteristic set. Increasing risk-tolerance further, makes the

market model beta a useful characteristic. The book-to-market ratio never enters the optimal

23

characteristic set. The information in this characteristic is spanned by the other characteristics

and since all of our analysis is out-of-sample parsimony is rewarded.

All of the optimal portfolios’ loading on the market factor is significantly less than one,

and are insensitive to investor risk-aversion. The portfolios’ exposure to the size factor (SMB),

value factor (HML) and momentum factor (MOM) all decrease in risk-aversion. All portfolios

have a significant four-factor alpha which also decreases monotonically in risk-aversion. With

the exception of the most risk-averse investor’s, these optimal portfolios’ return distributions

are significantly more positively skewed and have fatter tails than market index portfolios. The

Sharpe ratios of the optimal portfolios are significantly larger than the market benchmarks–

roughly twice as large. The efficacy of measurable characteristic-based investing is not diminished

(and perhaps it is even expanded) by taking into consideration risk aversion and portfolio return

variance, skewness, and kurtosis.

While Brandt, Santa-Clara, and Valkanov (2009) is structured as a normative algorithm we

use it to test and measure the relationships between characteristic and future portfolio returns

in our data. There are two reasons that limit the normative implications of this study. First,

all of our tests are out-of-sample in the sense that data used to evaluate portfolio properties is

different from the data estimate portfolio weights. However, we use the “out-of-sample” periods

to manage estimation risk (i.e., overfitting), which effectively brings it “in-sample.” Second, as

noted by Brandt, Santa-Clara, and Valkanov, we use data whose properties have already been

studied–informing our choice of eligible characteristics. Finally, we do not include a riskless asset

or other asset classes, whose presence might also mitigate estimation risk.

24

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Boguth, Oliver, Murray Carlson, Adlai Fisher, and Mikhail Simutin, 2011, Conditional risk and

performance evaluation: Volatility timing, overconditioning, and new estimates of momentum

alphas, Journal of Financial Economics 102, 363–389.

Brandt, Michael W., Pedro Santa-Clara, and Rossen Valkanov, 2009, Parametric portfolio policies:

Exploiting characteristics in the cross-section of equity returns, Review of Financial Studies 22,

3411–3447.

Broadie, Mark, Mikhail Chernov, and Michael Johannes, 2009, Understanding index option returns,

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DeMiguel, Victor , Lorenzo Garlappi, and Raman Uppal, 2009, Optimal versus naive diversification:

How inefficient is the 1/N portfolio strategy?, Review of Financial Studies 22, 1915–1953.

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Frazzini, Andrea and Lasse Heje Pedersen, 2013, Betting against beta, Journal of Financial Economics

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Finance 71, 1557–1589.

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Simulation and application to the S&P 500 Index, Working Paper, University of California, San

Diego.

Kogan, Leonid and Dimitris Papanikolaou, 2013, Firm characteristics and stock returns: The role of

investment-specific shocks, Review of Financial Studies 26, 2718–2759.

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Leland, Hayne, 1999, Beyond mean-variance: Performance measurement in a nonsymmetrical world,

Financial Analysts Journal, 27–36.

Lewellen, Jonathan, 2013, The cross section of expected stock returns, Critical Finance Review 4,

1–44.

Lewellen, Jonathan, Stefan Nagel, and Jay Shanken, 2010, A skeptical appraisal of asset pricing tests,

Journal of Financial Economics 96, 175–194.

McLean, R. David and Jeffrey Pontiff, 2016, Does academic research destroy stock return predictabil-

ity? Journal of Finance 71, 5–31.

26

0

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This panel shows the bootstrap distributions for the coefficient onmomentum for the case when momentum is the only characteristic.The box and whiskers show the 95%ile range (whiskers), the interquartilerange (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

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x x xx

xx x x x x x x x

x x x x x x

45

67

Year 1974 − 2014 Using gamma = 3

The

ta C

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t on

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entu

m

This panel shows the bootstrap distributions for the coefficient on momentumwhen the characteristics are: momentum, size, beta, residual standarddeviation, and average same−month return.

The box and whiskers show the 95%ile(whiskers), the interquartile range (box), and median (bar inside box) for thecoefficient in each year.The sample estimate is shown as a red X.

xx x

xx x

xx

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

xx

x

x xx x x x x x

x x x x x x

Figure 2. Annual coefficient on stocks’ momentum This graph depicts properties ofthe sampling distributions of the coefficient on (standardized and normalized) momentum foreach year in the out-of-sample period (1975 – 2015). Both panels are for the case where γ (thecoefficient of relative risk-aversion) is 3, and the updating protocol is used. This is the θ coefficientthat governs the effect of momentum on the weight placed on each stock in the optimal portfolioin the following year. The set of characteristics used in the lower panel generates the optimalportfolio for the gamma 2 investor.

−4

−3

−2

−1

0T

heta

Coe

ffici

ent o

n si

ze

This panel shows the bootstrap distributions for the coefficient on size for the case when size is the onlycharacteristic. The box and whiskers show 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

x xx x

xx

x x xx

x xx

xx x x x x x x

x x x

x xx x x x

x x x x x x x x

−14

−12

−10

−8

Year 1974 − 2014 Using gamma = 3

The

ta C

oeffi

cien

t on

size

This panel shows the bootstrap distributionsfor the coefficient on size when the characteristicsare: momentum, size, beta, residual standarddeviation, and average same−month returns.The box and whiskers show the 95%ile range(whiskers), the interquartile range (box),and median (bar inside box) for the coefficientin each year.

The sample estimate is shown as a red X.

x x

x

xx

x

x

x xx x x x

x xx

x x x x xx x x

xx

x x x x x x x

x xx x x x x

x

Figure 3. Annual coefficient on stocks’ log size This graph depicts properties of thesampling distributions of the coefficient on (standardized and normalized) log size for each yearin the out-of-sample period (1975 – 2015). Both panels are for the case where γ (the coefficientof relative risk-aversion) is 3, and the updating protocol is used. This is the θ coefficient thatgoverns the effect of log size on the weight placed on each stock in the optimal portfolio in thefollowing year. The set of characteristics used in the lower panel generates the optimal portfoliofor the gamma 2 investor.

−2.

5−

1.5

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5T

heta

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ta

This panel shows the bootstrap distributions for the coefficient on beta for the case when beta is the only characteristic. The box and whiskers show 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

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x x x x xx

x x x x xx x

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−4

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02

46

8

Year 1974 − 2014 Using gamma = 3

The

ta C

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beta

This panel shows the bootstrap distributionsfor the coefficient on beta when the characteristicsare: momentum, size, beta, residual standarddeviation, and average same−month returns.The box and whiskers show the 95%ile range(whiskers), the interquartile range (box),and median (bar inside box) for the coefficientin each year.

The sample estimate is shown as a red X.

x x x x

x

xx

x x x x x x x x x xx

x x x x x xx

x x x xx x x x x x

x x x x x x

Figure 4. Annual coefficient on stocks’ beta This graph depicts properties of the samplingdistributions of the coefficient on (standardized and normalized) beta for each year in the out-of-sample period (1975 – 2015). Both panels are for the case where γ (the coefficient of relativerisk-aversion) is 3, and the updating protocol is used. This is the θ coefficient that governs theeffect of beta on the weight placed on each stock in the optimal portfolio in the following year.The set of characteristics used in the lower panel generates the optimal portfolio for the gamma2 investor.

−3.

0−

2.0

−1.

0T

heta

Coe

ffici

ent o

n si

gma

This panel shows the bootstrap distributions for the coefficient on residual standard deviation forthe case where it is the only characteristic. The box and whiskers show the 95%ile range(whiskers), the interquartile range (box), and median (bar inside box) for the coefficient in eachyear.

The sample estimate is shown as a red X.

x xx x xx

x

x

x

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−18

−14

−10

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−4

Year 1974 − 2014 Using gamma = 3

The

ta C

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t on

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a

This panel shows the bootstrap distributions for the coefficient on residual standard deviationwhen the characteristics are: momentum, size, beta, residual standard deviation, and averagesame−month returns. The box and whiskers show 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

xx x x x x x x x x x x

xx x x x x x x x x x x

x x x xx x

x x x x x x

Figure 5. Annual coefficient on stocks’ residual standard deviation This graph depictsproperties of the sampling distributions of the coefficient on (standardized and normalized) resid-ual standard deviation for each year in the out-of-sample period (1975 – 2015). Both panels arefor the case where γ (the coefficient of relative risk-aversion) is 3, and the updating protocol isused. This is the θ coefficient that governs the effect of residual standard deviation on the weightplaced on each stock in the optimal portfolio in the following year. The set of characteristicsused in the lower panel generates the optimal portfolio for the gamma 2 investor.

68

1012

The

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This panel shows the bootstrap distributions for the coefficient on average same−month return forthe case where it is the only characteristic. The box and whiskers show the 95%ile range(whiskers), the interquartile range (box), and median (bar inside box) for the coefficient in eachyear.

The sample estimate is shown as a red X.

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Year 1974 − 2014 Using gamma = 3

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This panel shows the bootstrap distributions for the coefficient on average same−month returnwhen the characteristics are: momentum, size, beta, residual standard deviation, and averagesame−month returns. The box and whiskers show the 95%ile range (whiskers), the interquartile

range (box), and median (bar insidebox) for the coefficient in each year.

The sample estimate is shown as a red X.

xx x x

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Figure 6. Annual coefficient on stocks’ 5-year average same-month return This graphdepicts properties of the sampling distributions of the coefficient on (standardized and normal-ized) the 5-year average same-month return for each year in the out-of-sample period (1975 –2015). Both panels are for the case where γ (the coefficient of relative risk-aversion) is 3, and theupdating protocol is used. This is the θ coefficient that governs the effect of the 5-year averagesame-month return on the weight placed on each stock in the optimal portfolio in the followingyear. The set of characteristics used in the lower panel generates the optimal portfolio for thegamma 2 investor.

dens

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Figure 7. Densities of monthly returns on optimal portfolio and indices The panels inthis figure compare the density function of the monthly returns from an optimal portfolio withthe relevant benchmark portfolio.

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Figure 8. Densities of monthly returns on optimal portfolio and indices The panels inthis figure compare the density function of the monthly returns from an optimal portfolio withthe relevant benchmark portfolio.

Tab

le1

Max

imin

cert

aint

yeq

uiva

lent

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sis

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tspe

rm

onth

Por

tfol

ioB

oots

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tstr

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oots

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ean

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dard

2.5%

ile25

%ile

50%

ile75

%ile

97.5

%ile

Cha

ract

eris

tics

Dev

iati

onG

amm

a2

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I:10

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910

3.6

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106.

610

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387

.983

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.387

.889

.492

.3U

sing

γ=

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9.9

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154.

117

8.6

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-376

0.3

4706

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74.1

-100

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00.0

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3.0

Usi

ngγ

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,σε,rt−

12j

335.

810

6.8

376.

428

7.6

318.

433

6.2

354.

639

0.3

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134.

443

4.6

157.

531

.914

2.2

163.

417

9.4

203.

5U

sing

γ=

4M

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12j

320.

722

.935

0.0

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930

4.9

319.

833

5.7

367.

3M

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190.

815

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3.9

158.

918

1.6

191.

420

1.0

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9U

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γ=

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12j

297.

720

.132

1.0

260.

028

3.8

297.

131

0.9

339.

1M

,V

,S,β

,σε,r t−

12,r t−

12j

298.

831

2.0

342.

424

7.5

291.

230

9.8

328.

436

5.5

Usi

ngγ

=6

M,

V,

S,β

,σε,r t−

12,r t−

12j

277.

918

.829

7.7

242.

526

5.2

277.

429

0.2

316.

9M

,V

,S,β

,σε,r t−

12,r t−

12j

291.

015

1.2

323.

024

4.0

277.

729

3.6

309.

934

2.0

Usi

ngγ

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M,

V,

S,β

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12,r t−

12j

260.

017

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7.4

228.

324

8.3

259.

327

1.1

294.

8M

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12,r t−

12j

277.

121

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2.6

236.

326

2.7

276.

929

1.1

320.

2U

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γ=

8M

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241.

413

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1.8

216.

623

2.4

241.

025

0.0

268.

1M

,V

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12,r t−

12j

261.

219

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3.9

224.

224

7.9

261.

327

4.3

300.

0U

sing

γ=

9M

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12j

230.

412

.123

9.6

207.

722

2.1

230.

023

8.2

255.

5M

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,S,β

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12,r t−

12j

247.

517

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7.5

214.

023

5.4

247.

125

9.4

283.

0

Tab

le1

Con

tinu

ed

Max

imin

cert

aint

yeq

uiva

lent

:ba

sis

poin

tspe

rm

onth

Por

tfol

ioB

oots

trap

Boo

tstr

apSa

mpl

eB

oots

trap

Boo

tstr

apB

oots

trap

Boo

tstr

apB

oots

trap

/M

ean

Stan

dard

2.5%

ile25

%ile

50%

ile75

%ile

97.5

%ile

Cha

ract

eris

tics

Dev

iati

onG

amm

a2

inve

stor

,co

nti

nu

ed:

Usi

ngγ

=10

M,

S,σε,r t−

12j

220.

911

.622

9.3

199.

821

2.9

220.

522

8.5

244.

7M

,V

,S,β

,σε,r t−

12,r t−

12j

235.

016

.425

3.0

203.

722

3.7

234.

724

6.0

267.

8U

sing

γ=

11M

,S,σε,r t−

12j

212.

510

.822

0.3

192.

520

5.2

212.

021

9.6

235.

0M

,V

,S,β

,σε,r t−

12,r t−

12j

224.

415

.424

0.3

195.

521

3.7

224.

123

4.7

255.

8U

sing

γ=

12M

,S,σε,r t−

12j

205.

710

.421

2.5

186.

519

8.4

205.

421

2.4

226.

7M

,V

,S,β

,σε,r t−

12,r t−

12j

214.

414

.322

9.1

187.

620

4.7

214.

222

3.6

243.

5U

sing

γ=

13M

,S,σε,r t−

12j

199.

49.

820

5.7

181.

119

2.5

199.

020

5.9

219.

3M

,V

,S,β

,σε,r t−

12,r t−

12j

206.

113

.621

9.1

180.

319

6.7

205.

721

5.0

233.

7U

sing

γ=

16M

,S,σε,r t−

12j

184.

58.

918

9.4

167.

617

8.4

184.

319

0.3

202.

8M

,S,σε,r t−

12j

182.

89.

818

8.8

163.

517

6.2

182.

718

9.3

202.

5U

sing

γ=

22M

,S,σε,r t−

12j

164.

68.

416

7.5

148.

815

8.9

164.

417

0.1

181.

5M

,S,σε,r t−

12j

161.

69.

716

5.4

143.

115

5.1

161.

416

7.9

180.

7

Gam

ma

3in

vest

or:

EW

I:91

.71.

191

.889

.490

.991

.792

.593

.9V

WI:

77.6

2.3

77.9

73.1

76.1

77.7

79.2

82.1

Usi

ngγ

=3

M,

S,β

,σε,r t−

12j

209.

111

1.4

256.

013

6.4

191.

621

3.1

233.

326

9.3

M,

V,

S,σε

-61.

954

9.0

-13.

2-6

73.7

-41.

528

.066

.710

8.6

Usi

ngγ

=4

M,

S,β

,σε,r t−

12j

248.

820

.527

9.8

208.

923

5.2

248.

326

2.2

290.

0M

,V

,S,σε

122.

133

.712

1.5

45.8

111.

412

8.5

141.

016

1.0

Tab

le1

Con

tinu

ed

Max

imin

cert

aint

yeq

uiva

lent

:ba

sis

poin

tspe

rm

onth

Por

tfol

ioB

oots

trap

Boo

tstr

apSa

mpl

eB

oots

trap

Boo

tstr

apB

oots

trap

Boo

tstr

apB

oots

trap

/M

ean

Stan

dard

2.5%

ile25

%ile

50%

ile75

%ile

97.5

%ile

Cha

ract

eris

tics

Dev

iati

onG

amm

a3

inve

stor

,co

nti

nu

ed:

Usi

ngγ

=5

M,

S,β

,σε,rt−

12j

249.

117

.627

3.0

215.

523

7.1

248.

926

0.9

284.

4M

,V

,S,σε

151.

314

.215

2.3

119.

714

3.6

152.

516

0.7

175.

4U

sing

γ=

6M

,S,β

,σε,r t−

12j

240.

016

.025

9.9

209.

422

9.1

239.

625

0.5

272.

9M

,V

,S,β

,σε,r t−

12,r t−

12j

228.

818

6.6

259.

715

9.2

220.

923

7.7

253.

228

2.8

Usi

ngγ

=7

M,

S,β

,σε,r t−

12j

229.

214

.524

6.2

202.

321

9.2

228.

923

8.7

258.

8M

,V

,S,β

,σε,r t−

12,r t−

12j

232.

526

.625

4.6

182.

721

9.5

234.

324

8.1

274.

7U

sing

γ=

8M

,S,σε,r t−

12j

217.

011

.922

7.2

194.

420

8.8

216.

722

4.9

240.

9M

,V

,S,β

,σε,r t−

12,r t−

12j

225.

025

.924

5.2

181.

921

3.0

226.

523

9.0

263.

0U

sing

γ=

9M

,S,σε,r t−

12j

209.

311

.121

8.4

188.

520

1.6

209.

021

6.4

232.

3M

,V

,S,β

,σε,r t−

12,r t−

12j

217.

122

.423

4.7

179.

620

5.7

217.

522

9.7

252.

1U

sing

γ=

10M

,S,σε,r t−

12j

202.

210

.721

0.4

182.

419

4.8

201.

820

9.2

224.

3M

,V

,S,β

,σε,r t−

12,r t−

12j

208.

617

.122

4.3

174.

719

7.7

208.

922

0.1

241.

1U

sing

γ=

11M

,S,σε,r t−

12j

195.

610

.120

3.2

176.

918

8.7

195.

120

2.3

216.

4M

,V

,S,β

,σε,r t−

12,r t−

12j

200.

816

.021

4.6

170.

319

0.4

200.

921

1.3

231.

6U

sing

γ=

12M

,S,σε,r t−

12j

190.

09.

719

6.8

171.

918

3.1

189.

819

6.3

209.

7M

,V

,S,β

,σε,r t−

12,r t−

12j

192.

814

.820

5.5

163.

918

3.0

192.

720

2.4

221.

8U

sing

γ=

13M

,S,σε,r t−

12j

184.

79.

319

1.0

167.

317

8.3

184.

319

0.9

203.

5M

,V

,S,β

,σε,r t−

12,r t−

12j

185.

914

.219

7.1

158.

317

6.5

185.

919

5.1

213.

5

Tab

le1

Con

tinu

ed

Max

imin

cert

aint

yeq

uiva

lent

:ba

sis

poin

tspe

rm

onth

Por

tfol

ioB

oots

trap

Boo

tstr

apSa

mpl

eB

oots

trap

Boo

tstr

apB

oots

trap

Boo

tstr

apB

oots

trap

/M

ean

Stan

dard

2.5%

ile25

%ile

50%

ile75

%ile

97.5

%ile

Cha

ract

eris

tics

Dev

iati

onG

amm

a3

inve

stor

,co

nti

nu

ed:

Usi

ngγ

=16

M,

S,σε,r t−

12j

171.

68.

617

6.5

155.

016

5.8

171.

417

7.2

189.

2M

,S,σε,r t−

12j

167.

010

.517

2.5

145.

716

0.1

167.

117

3.9

187.

3U

sing

γ=

22M

,S,σε,r t−

12j

153.

18.

315

6.2

137.

514

7.5

153.

015

8.7

169.

7M

,S,σε,r t−

12j

146.

910

.615

0.6

125.

513

9.9

147.

015

4.0

167.

4

Gam

ma

4in

vest

or:

EW

I:76

.91.

277

.074

.676

.176

.977

.779

.2V

WI:

67.2

2.3

67.6

62.5

65.6

67.2

68.7

71.7

Usi

ngγ

=4

V,

S,σε

101.

18.

210

4.2

84.6

95.8

101.

210

6.6

116.

8M

,V

,S,σε

28.6

94.8

24.9

-166

.812

.750

.573

.410

2.0

Usi

ngγ

=5

M,

S,β

,σε,r t−

12j

196.

718

.622

2.0

159.

118

5.3

197.

120

9.0

230.

6M

,V

,S,σε

101.

225

.998

.539

.791

.110

6.0

117.

313

4.0

Usi

ngγ

=6

M,

S,σε,r t−

12j

192.

914

.420

6.5

164.

118

4.0

193.

120

2.2

219.

8M

,V

,S,σε

124.

613

.912

4.0

91.9

117.

412

6.2

133.

914

6.5

Usi

ngγ

=7

M,

S,β

,σε,rt−

12j

198.

813

.721

6.0

172.

718

9.5

198.

820

7.8

226.

2M

,S,σε

134.

110

.413

4.4

111.

212

8.2

134.

814

1.0

152.

1U

sing

γ=

8M

,S,σε,r t−

12j

191.

211

.420

1.2

169.

218

3.6

191.

119

8.7

213.

8M

,S,σε

137.

78.

513

9.1

120.

013

2.5

138.

014

3.4

153.

5

Tab

le1

Con

tinu

ed

Max

imin

cert

aint

yeq

uiva

lent

:ba

sis

poin

tspe

rm

onth

Por

tfol

ioB

oots

trap

Boo

tstr

apSa

mpl

eB

oots

trap

Boo

tstr

apB

oots

trap

Boo

tstr

apB

oots

trap

/M

ean

Stan

dard

2.5%

ile25

%ile

50%

ile75

%ile

97.5

%ile

Cha

ract

eris

tics

Dev

iati

on

Gam

ma

4in

vest

or,

conti

nu

ed:

Usi

ngγ

=9

M,

S,σε,r t−

12j

187.

110

.619

6.0

167.

017

9.9

187.

019

4.0

208.

5M

,V

,S,β

,σε,r t−

12,r t−

12j

183.

648

.919

9.9

128.

817

3.6

187.

119

9.4

221.

6U

sing

γ=

10M

,S,σε,r t−

12j

182.

510

.219

0.6

163.

217

5.6

182.

318

9.2

203.

4M

,V

,S,β

,σε,r t−

12,r t−

12j

180.

022

.519

3.9

134.

216

9.4

182.

219

3.6

214.

4U

sing

γ=

11M

,S,σε,r t−

12j

177.

89.

718

5.3

159.

417

1.2

177.

518

4.2

197.

5M

,V

,S,β

,σε,r t−

12,r t−

12j

175.

320

.518

7.2

135.

016

5.0

176.

518

7.5

207.

8U

sing

γ=

12M

,S,σε,r t−

12j

173.

49.

418

0.2

155.

416

6.9

173.

317

9.6

192.

4M

,S,σε,r t−

12j

164.

314

.717

0.5

133.

915

6.3

165.

417

3.7

189.

0U

sing

γ=

13M

,S,σε,r t−

12j

169.

29.

117

5.4

152.

116

3.1

169.

017

5.3

187.

7M

,S,σε,r t−

12j

160.

813

.616

6.4

131.

615

3.1

161.

816

9.5

184.

6U

sing

γ=

16M

,S,σε,r t−

12j

158.

18.

616

3.0

141.

315

2.4

158.

016

3.7

175.

1M

,S,σε,r t−

12j

149.

612

.615

4.5

122.

414

2.3

150.

515

8.0

171.

5U

sing

γ=

22M

,S,σε,r t−

12j

140.

98.

514

4.1

124.

613

5.3

140.

814

6.6

157.

8M

,S,σε,r t−

12j

130.

612

.813

4.1

103.

812

2.8

131.

313

9.2

153.

4

Tab

le1

Con

tinu

ed

Max

imin

cert

aint

yeq

uiva

lent

:ba

sis

poin

tspe

rm

onth

Por

tfol

ioB

oots

trap

Boo

tstr

apSa

mpl

eB

oots

trap

Boo

tstr

apB

oots

trap

Boo

tstr

apB

oots

trap

/M

ean

Stan

dard

2.5%

ile25

%ile

50%

ile75

%ile

97.5

%ile

Cha

ract

eris

tics

Dev

iati

on

Gam

ma

5in

vest

or:

EW

I:61

.41.

261

.659

.060

.661

.462

.263

.8V

WI:

56.4

2.4

56.9

51.6

54.7

56.4

58.0

61.1

Usi

ngγ

=5

V,

S,σε

91.1

7.5

94.3

75.9

86.3

91.4

96.1

105.

4M

,V

,S,σε

38.5

52.8

31.2

-87.

023

.151

.069

.292

.2U

sing

γ=

6M

,S,β

,σε,r t−

12j

160.

317

.518

1.7

123.

614

9.9

161.

217

1.8

192.

2M

,V

,S,σε

84.4

23.0

80.5

25.6

74.9

89.1

99.4

114.

5U

sing

γ=

7M

,S,σε,r t−

12j

160.

514

.317

2.1

130.

715

2.3

161.

116

9.9

186.

0M

,V

,S,σε

102.

915

.110

1.6

68.5

95.4

105.

211

2.8

125.

1U

sing

γ=

8M

,S,β

,σε,r t−

12j

166.

513

.318

1.8

140.

215

8.1

166.

817

5.2

192.

0M

,V

,S,σε

111.

810

.811

1.3

87.1

106.

011

2.8

118.

812

9.9

Usi

ngγ

=9

M,

S,σε,rt−

12j

163.

511

.217

2.3

141.

115

6.3

163.

717

0.9

184.

6M

,V

,S,σε

115.

59.

011

5.6

96.4

110.

211

6.0

121.

513

1.5

Usi

ngγ

=10

M,

S,σε,r t−

12j

161.

610

.516

9.6

140.

515

4.8

161.

616

8.6

182.

1M

,S,σε

118.

47.

811

9.6

102.

511

3.5

118.

612

3.6

132.

9U

sing

γ=

11M

,S,σε,r t−

12j

158.

910

.016

6.2

139.

415

2.4

158.

916

5.5

178.

2M

,S,σε

118.

47.

111

9.8

104.

111

3.8

118.

612

3.2

132.

0U

sing

γ=

12M

,S,σε,r t−

12j

155.

99.

616

2.6

137.

014

9.5

156.

016

2.3

174.

6M

,S,σε

117.

46.

811

9.0

103.

811

2.9

117.

412

2.0

130.

6

Tab

le1

Con

tinu

ed

Max

imin

cert

aint

yeq

uiva

lent

:ba

sis

poin

tspe

rm

onth

Por

tfol

ioB

oots

trap

Boo

tstr

apSa

mpl

eB

oots

trap

Boo

tstr

apB

oots

trap

Boo

tstr

apB

oots

trap

/M

ean

Stan

dard

2.5%

ile25

%ile

50%

ile75

%ile

97.5

%ile

Cha

ract

eris

tics

Dev

iati

on

Gam

ma

5in

vest

or,

conti

nu

ed:

Usi

ngγ

=13

M,

S,σε,r t−

12j

152.

89.

315

8.9

134.

814

6.7

152.

815

9.1

171.

4M

,S,σε

115.

76.

711

7.4

102.

211

1.2

115.

712

0.2

129.

0U

sing

γ=

16M

,S,σε,r t−

12j

143.

68.

914

8.5

125.

513

8.0

143.

714

9.5

161.

1M

,S,σε

109.

16.

711

1.0

95.9

104.

610

9.1

113.

712

2.3

Usi

ngγ

=22

M,

S,σε,r t−

12j

127.

89.

113

1.0

109.

512

1.8

127.

813

4.0

145.

6M

,S,σε

94.7

7.8

96.6

79.2

89.6

94.7

99.9

109.

7

Gam

ma

6in

vest

or:

EW

I:45

.11.

345

.342

.644

.245

.146

.047

.6V

WI:

45.1

2.5

45.9

40.3

43.5

45.1

46.8

50.0

Usi

ngγ

=6

V,

S,σε

81.4

7.4

84.5

65.8

76.8

81.7

86.4

95.0

M,

V,

S,σε

36.5

40.1

28.1

-67.

422

.346

.261

.982

.2U

sing

γ=

7M

,S,β

,σε,r t−

12j

166.

414

.715

0.3

136.

415

7.4

166.

817

6.1

193.

3M

,V

,S,σε

68.4

24.2

64.3

11.6

58.6

73.2

83.7

98.5

Usi

ngγ

=8

M,

S,β

,σε,r t−

12j

137.

615

.815

3.6

103.

312

8.8

138.

914

7.9

165.

1M

,V

,S,σε

84.6

15.5

81.9

47.0

77.7

87.1

94.8

106.

8U

sing

γ=

9M

,S,β

,σε,r t−

12j

139.

014

.515

2.9

109.

213

1.0

139.

814

8.1

164.

0M

,S,σε

93.8

12.0

92.3

65.8

87.9

95.5

101.

711

1.9

Usi

ngγ

=10

M,

S,σε,r t−

12j

139.

212

.214

7.0

113.

113

2.2

139.

914

7.3

160.

8M

,S,σε

98.0

9.7

97.8

76.8

92.7

98.9

104.

511

4.5

Tab

le1

Con

tinu

ed

Max

imin

cert

aint

yeq

uiva

lent

:ba

sis

poin

tspe

rm

onth

Por

tfol

ioB

oots

trap

Boo

tstr

apSa

mpl

eB

oots

trap

Boo

tstr

apB

oots

trap

Boo

tstr

apB

oots

trap

/M

ean

Stan

dard

2.5%

ile25

%ile

50%

ile75

%ile

97.5

%ile

Cha

ract

eris

tics

Dev

iati

on

Gam

ma

6in

vest

or,

conti

nu

ed:

Usi

ngγ

=11

M,

S,σε,rt−

12j

138.

611

.414

5.8

114.

913

2.0

139.

214

6.1

158.

8M

,S,σε

100.

08.

310

0.3

82.1

95.1

100.

610

5.6

114.

8U

sing

γ=

12M

,S,σε,r t−

12j

137.

110

.914

3.7

114.

113

0.5

137.

614

4.4

156.

8M

,S,σε

100.

37.

710

0.9

84.2

95.4

100.

610

5.5

114.

4U

sing

γ=

13M

,S,σε,r t−

12j

135.

310

.514

1.2

113.

212

8.9

135.

714

2.2

154.

9M

,S,σε

99.4

7.4

100.

484

.194

.799

.710

4.4

113.

7U

sing

γ=

16M

,S,σε,r t−

12j

128.

110

.113

2.9

106.

812

2.1

128.

613

4.6

146.

6M

,S,σε

94.0

7.3

95.4

79.4

89.1

94.1

99.1

107.

8U

sing

γ=

22M

,S,σε,r t−

12j

113.

410

.511

6.6

91.6

106.

811

3.9

120.

613

2.8

M,

S,σε

79.5

8.8

81.4

61.5

73.8

79.6

85.4

96.1

Gam

ma

7in

vest

or:

EW

I:27

.91.

428

.125

.226

.927

.928

.830

.6V

WI:

33.5

2.6

34.3

28.4

31.8

33.5

35.3

38.5

Usi

ngγ

=7

V,

S,σε

71.0

7.8

74.5

55.2

66.0

71.2

76.3

85.0

M,

V,

S,σε

28.0

39.3

20.3

-65.

914

.337

.052

.371

.2U

sing

γ=

8V

,S,σε

74.6

7.2

78.1

59.7

70.0

74.8

79.5

88.3

M,

V,

S,σε

53.9

23.5

48.4

-5.5

44.7

58.7

69.1

83.4

Tab

le1

Con

tinu

ed

Max

imin

cert

aint

yeq

uiva

lent

:ba

sis

poin

tspe

rm

onth

Por

tfol

ioB

oots

trap

Boo

tstr

apSa

mpl

eB

oots

trap

Boo

tstr

apB

oots

trap

Boo

tstr

apB

oots

trap

/M

ean

Stan

dard

2.5%

ile25

%ile

50%

ile75

%ile

97.5

%ile

Cha

ract

eris

tics

Dev

iati

on

Gam

ma

7in

vest

or,

conti

nu

ed:

Usi

ngγ

=9

M,

S,β

,σε,r t−

12j

112.

019

.012

6.8

70.1

103.

711

4.2

123.

714

0.6

M,

V,

S,σε

67.1

17.2

63.4

28.4

59.8

70.0

77.9

90.3

Usi

ngγ

=10

M,

S,β

,σε,r t−

12j

114.

615

.812

7.0

78.7

106.

411

6.1

124.

914

0.2

M,

S,σε

75.7

13.2

73.7

45.0

69.5

77.6

84.4

95.5

Usi

ngγ

=11

M,

S,σε,r t−

12j

116.

714

.712

3.7

84.3

109.

611

8.3

126.

013

9.5

M,

V,

S,σε

77.6

11.1

75.7

51.9

71.9

78.9

84.9

95.4

Usi

ngγ

=12

M,

S,σε,r t−

12j

116.

813

.512

3.3

85.8

109.

711

8.2

125.

613

8.9

M,

S,σε

81.8

9.4

81.3

60.9

76.6

82.8

88.2

97.9

Usi

ngγ

=13

M,

S,σε,rt−

12j

116.

213

.012

2.1

86.7

109.

311

7.5

124.

713

8.0

M,

S,σε

82.0

8.8

82.0

62.7

76.8

82.6

87.8

97.8

Usi

ngγ

=16

M,

S,σε,r t−

12j

111.

112

.411

5.9

83.5

104.

511

2.3

119.

113

1.8

M,

S,σε

77.9

8.4

78.8

60.6

72.6

78.2

83.7

92.9

Usi

ngγ

=22

M,

S,σε,r t−

12j

97.5

13.0

100.

567

.989

.998

.710

6.4

119.

5M

,S,σε

63.0

10.4

65.0

40.7

56.6

63.6

70.0

81.9

Gam

ma

8in

vest

or:

EW

I:9.

51.

59.

76.

58.

59.

510

.512

.5V

WI:

21.3

2.7

22.3

15.9

19.4

21.3

23.1

26.5

Usi

ngγ

=8

V,

S,σε

60.2

8.5

64.1

42.2

55.1

60.8

66.0

75.4

M,

V,

S,σε

18.7

35.9

9.6

-72.

06.

326

.941

.459

.1

Tab

le1

Con

tinu

ed

Max

imin

cert

aint

yeq

uiva

lent

:ba

sis

poin

tspe

rm

onth

Por

tfol

ioB

oots

trap

Boo

tstr

apSa

mpl

eB

oots

trap

Boo

tstr

apB

oots

trap

Boo

tstr

apB

oots

trap

/M

ean

Stan

dard

2.5%

ile25

%ile

50%

ile75

%ile

97.5

%ile

Cha

ract

eris

tics

Dev

iati

on

Gam

ma

8in

vest

or,

conti

nu

ed:

Usi

ngγ

=9

V,

S,σε

62.8

8.1

66.8

45.4

57.8

63.4

68.4

77.4

M,

V,

S,σε

38.6

25.3

32.1

-18.

829

.243

.754

.069

.0U

sing

γ=

10V

,S,σε

64.4

8.0

68.2

47.5

59.4

64.8

69.8

78.7

M,

S,σε

50.9

18.8

46.6

5.7

43.0

54.4

63.2

75.9

Usi

ngγ

=11

V,

S,σε

65.2

7.9

68.8

48.4

60.3

65.5

70.6

79.5

M,

S,σε

58.3

14.3

55.3

23.9

51.5

60.6

67.8

79.3

Usi

ngγ

=12

M,

S,σε,r t−

12j

94.5

18.0

101.

150

.486

.497

.310

6.1

120.

7M

,S,σε

61.9

12.4

59.9

33.0

56.0

63.6

70.0

80.8

Usi

ngγ

=13

M,

S,σε

76.4

8.7

78.5

57.8

71.3

76.8

82.3

91.6

M,

S,σε

63.2

11.0

62.1

37.8

57.4

64.4

70.5

81.6

Usi

ngγ

=16

M,

S,σε

79.3

7.6

82.0

63.8

74.3

79.5

84.5

93.6

M,

S,σε

60.6

10.1

60.9

38.8

54.6

61.4

67.4

77.8

Usi

ngγ

=22

M,

S,σε

72.8

8.5

75.6

54.8

67.5

73.4

78.8

88.0

M,

S,σε

45.0

12.7

47.2

16.5

37.5

46.1

53.6

66.9

Gam

ma

9in

vest

or:

EW

I:-1

0.2

1.7

-9.9

-13.

5-1

1.3

-10.

2-9

.1-6

.9V

WI:

8.5

2.9

9.7

2.9

6.5

8.5

10.5

14.1

Usi

ngγ

=9

V,

S,σε

48.5

9.8

52.9

26.7

42.8

49.5

55.2

65.1

M,

V,

S,σε

6.3

37.0

-3.7

-78.

2-6

.114

.628

.446

.8

Tab

le1

Con

tinu

ed

Max

imin

cert

aint

yeq

uiva

lent

:ba

sis

poin

tspe

rm

onth

Por

tfol

ioB

oots

trap

Boo

tstr

apSa

mpl

eB

oots

trap

Boo

tstr

apB

oots

trap

Boo

tstr

apB

oots

trap

/M

ean

Stan

dard

2.5%

ile25

%ile

50%

ile75

%ile

97.5

%ile

Cha

ract

eris

tics

Dev

iati

on

Gam

ma

9in

vest

or,

conti

nu

ed:

Usi

ngγ

=10

V,

S,σε

50.6

9.6

54.8

29.4

45.0

51.3

57.1

66.9

M,

S,σε

23.1

26.9

16.0

-42.

613

.028

.940

.555

.9U

sing

γ=

11V

,S,σε

51.7

9.4

55.8

30.8

46.3

52.4

58.2

68.2

M,

S,σε

34.3

19.7

29.0

-14.

925

.838

.247

.260

.6U

sing

γ=

12V

,S,σε

51.9

9.4

56.0

31.1

46.5

52.7

58.3

68.1

M,

S,σε

40.1

16.4

36.5

0.0

33.1

43.0

50.8

63.1

Usi

ngγ

=13

M,

S,σε

56.8

10.9

58.3

32.6

50.9

57.8

64.2

75.0

M,

S,σε

42.9

14.3

40.4

8.0

36.2

45.0

52.3

64.7

Usi

ngγ

=16

M,

S,σε

62.7

9.0

65.1

44.0

57.1

63.3

69.0

79.1

M,

S,σε

41.8

12.5

41.6

13.7

34.9

43.2

50.3

62.0

Usi

ngγ

=22

M,

S,σε

57.6

10.0

60.3

35.8

51.6

58.5

64.5

74.9

M,

S,σε

25.1

16.0

27.7

-12.

016

.026

.936

.051

.6

Gam

ma

10in

vest

or:

EW

I:-3

1.5

2.0

-31.

1-3

5.4

-32.

8-3

1.5

-30.

1-2

7.7

VW

I:-5

.03.

1-3

.6-1

1.2

-7.0

-4.9

-2.8

1.1

Usi

ngγ

=10

V,

S,σε

35.8

11.8

40.7

8.7

29.4

37.1

43.9

54.9

M,

V,

S,σε

-8.6

37.0

-19.

4-1

01.0

-21.

20.

014

.633

.5U

sing

γ=

11V

,S,σε

37.4

11.5

42.0

10.7

31.1

38.5

45.3

56.5

M,

V,

S,σε

7.6

27.2

-0.4

-61.

4-3

.213

.725

.141

.2

Tab

le1

Con

tinu

ed

Max

imin

cert

aint

yeq

uiva

lent

:ba

sis

poin

tspe

rm

onth

Por

tfol

ioB

oots

trap

Boo

tstr

apSa

mpl

eB

oots

trap

Boo

tstr

apB

oots

trap

Boo

tstr

apB

oots

trap

/M

ean

Stan

dard

2.5%

ile25

%ile

50%

ile75

%ile

97.5

%ile

Cha

ract

eris

tics

Dev

iati

on

Gam

ma

10in

vest

orC

onti

nu

ed:

Usi

ngγ

=12

V,

S,σε

37.8

11.5

42.4

11.4

31.5

39.0

45.6

56.5

M,

S,σε

16.1

22.2

10.5

-39.

67.

420

.630

.345

.0U

sing

γ=

13V

,S,σε

37.4

11.7

42.2

10.5

31.1

38.8

45.5

56.2

M,

S,σε

20.7

19.0

16.7

-26.

812

.624

.032

.947

.1U

sin

gγ

=16

M,

S,σε

44.9

10.9

46.9

21.1

38.3

45.8

52.6

64.0

M,

S,σε

21.4

15.9

20.6

-15.

513

.323

.632

.145

.6U

sing

γ=

22M

,S,σε

41.1

12.1

43.8

13.9

34.2

42.4

49.5

61.3

M,

S,σε

3.0

20.5

6.2

-45.

2-7

.95.

817

.035

.2

For

each

of9

inve

stor

s,in

dexe

dby

γi

=2,...,

10w

eev

alua

teth

esa

mpl

ing

dist

ribu

tion

ofth

ece

rtai

nty

equi

vale

ntre

turn

s(a

cros

s10

,000

boot

stra

psa

mpl

es),

for

41di

ffere

ntm

odel

s(o

rse

tsof

char

acte

rist

ics)

andγ∈{γ

i,...,

13,1

6,22}.

The

retu

rns

inal

lca

ses

are

from

the

out-

of-s

ampl

epe

riod

Janu

ary

1975

thro

ugh

Dec

embe

r20

15(4

92m

onth

s).

The

upda

ting

prot

ocol

uses

all

hist

oric

alda

tapr

ior

toth

est

art

ofth

ene

xtou

t-of

-sam

ple

year

,th

ero

lling

prot

ocol

uses

the

prec

edin

g18

0m

onth

spr

ior

toth

est

art

ofth

ene

xtou

t-of

-sam

ple

year

.T

his

tabl

ere

port

sth

epr

oper

ties

ofth

ece

rtai

nty

equi

vale

ntre

turn

from

mod

elth

atpr

oduc

esth

ehi

ghes

t2.

5%ile

,fo

rea

chva

lue

ofγ

,un

der

each

prot

ocol

.

The

upda

ting

resu

lts

are

repo

rted

first

wit

hth

ero

lling

resu

lts

repo

rted

belo

wth

em.

Glo

bal

max

ima

acro

ssal

lγ

valu

esan

dpr

otoc

ols

are

high

light

edin

bold

face

.

The

upda

ting

resu

lts

inth

ista

ble

are

extr

acte

dfr

omT

able

sIA

-1–

IA-9

inth

ein

tern

etap

pend

ix.

Table 2

Means of annual θ coefficients from optimal models

Characteristics used in portfolio selection: M, S, β σε, rt−12j using γ = 3Optimal portfolio for the gamma 2 investor

Bootstrap BootstrapCharacteristic Mean σ Sample 2.5%ile 25%ile 50%ile 25%ile 97.5%ile

M 5.40 0.92 5.54 3.73 4.66 5.49 6.08 7.06S -10.80 1.64 -11.66 -14.11 -12.00 -10.66 -9.49 -8.13β 3.50 2.51 4.63 -1.69 1.68 3.84 5.46 7.52σε -11.89 2.34 -13.53 -16.05 -13.47 -12.11 -10.50 -6.72

rt−12j 9.16 1.48 10.03 6.11 8.26 9.19 10.10 12.10

Characteristics used in portfolio selection: M, S, β σε, rt−12j using γ = 5Optimal portfolio for the gamma 3 investor

Bootstrap BootstrapCharacteristic Mean σ Sample 2.5%ile 25%ile 50%ile 25%ile 97.5%ile

M 3.48 0.58 3.60 2.42 3.01 3.53 3.91 4.53S -7.10 1.08 -7.64 -9.16 -7.93 -7.07 -6.21 -5.30β 1.60 1.55 2.33 -1.49 0.44 1.74 2.84 4.16σε -7.63 1.38 -8.72 -10.17 -8.55 -7.72 -6.81 -4.63

rt−12j 5.66 0.91 6.21 3.78 5.09 5.66 6.23 7.47

Characteristics used in portfolio selection: M, S, β σε, rt−12j using γ = 7Optimal portfolio for the gamma 4 investor

Bootstrap BootstrapCharacteristic Mean σ Sample 2.5%ile 25%ile 50%ile 25%ile 97.5%ile

M 2.55 0.43 2.65 1.79 2.22 2.57 2.86 3.36S -5.44 0.85 -5.83 -6.99 -6.11 -5.47 -4.73 -3.97β 0.73 1.14 1.27 -1.50 -0.13 0.78 1.65 2.64σε -5.63 0.97 -6.45 -7.46 -6.27 -5.67 -5.04 -3.57

rt−12j 4.01 0.68 4.41 2.63 3.59 4.01 4.44 5.35

Characteristics used in portfolio selection: M, S, σε, rt−12j using γ = 9Optimal portfolio for the gamma 5 investor

Bootstrap BootstrapCharacteristic Mean σ Sample 2.5%ile 25%ile 50%ile 25%ile 97.5%ile

M 1.98 0.39 2.02 1.28 1.66 2.01 2.27 2.68S -4.43 0.71 -4.77 -5.67 -5.00 -4.49 -3.86 -3.16σε -4.18 0.80 -4.39 -5.42 -4.85 -4.37 -3.36 -2.89

rt−12j 3.05 0.54 3.35 1.98 2.70 3.05 3.40 4.14

Table 2 Continued

Means of annual θ coefficients from optimal models

Characteristics used in portfolio selection: M, S, σε, rt−12j using γ = 11Optimal portfolio for the gamma 6 investor

Bootstrap BootstrapCharacteristic Mean σ Sample 2.5%ile 25%ile 50%ile 25%ile 97.5%ile

M 1.65 0.32 1.69 1.11 1.39 1.65 1.89 2.29S -3.78 0.63 -4.06 -4.86 -4.28 -3.86 -3.28 -2.61σε -3.76 0.73 -3.94 -4.87 -4.37 -3.94 -2.99 -2.57

rt−12j 2.46 0.47 2.69 1.54 2.15 2.46 2.77 3.40

Characteristics used in portfolio selection: M, S, σε, rt−12j using γ = 13Optimal portfolio for the gamma 7 investor

Bootstrap BootstrapCharacteristic Mean σ Sample 2.5%ile 25%ile 50%ile 25%ile 97.5%ile

M 1.42 0.28 1.45 0.98 1.20 1.39 1.62 2.01S -3.32 0.59 -3.56 -4.30 -3.79 -3.42 -2.85 -2.20σε -3.47 0.69 -3.62 -4.50 -4.05 -3.66 -2.73 -2.34

rt−12j 2.07 0.43 2.25 1.25 1.78 2.06 2.35 2.91

Characteristics used in portfolio selection: M, S, σε using γ = 16Optimal portfolio for the gamma 8, 9, and 10 investors

Bootstrap BootstrapCharacteristic Mean σ Sample 2.5%ile 25%ile 50%ile 25%ile 97.5%ile

M 1.29 0.39 1.35 0.69 0.88 1.37 1.60 1.96S -2.56 0.62 -2.72 -3.49 -3.05 -2.73 -1.98 -1.44σε -3.00 0.69 -3.14 -3.94 -3.58 -3.26 -2.19 -1.91

This table reports the sampling distribution of the time-series means of the θ coefficientson each characteristic from the optimal models for investors with γ values ranging from 2

through 10 in the objective function:T−1∑t=0

(1+rp,t+1)1−γ

1−γ(

1T

),

where: rp,t+1 =Nt∑i=1

(ωi,t + 1

Ntθ′xi,t

)· ri,t+1.

The objective function is optimized at the end of 1974 and every year thereafter through2014 (41 years), using an updating protocol. The characteristic abbreviations are: momentum(M), log size (S), market model β (estimated on the preceding 60 months of returns), theresidual standard deviation from that market model (σε), and the five-year average of nextmonth’s return (rt−12j).

Tab

le3

Fam

a-Fr

ench

-Car

hart

regr

essi

onco

effici

ents

ofbe

nchm

ark

port

folio

s

Reg

ress

ion

Boo

tstr

apB

oots

trap

Sam

ple

Boo

tstr

apB

oots

trap

Boo

tstr

apB

oots

trap

Boo

tstr

apC

oeffi

cien

tM

ean

σ2.

5%ile

25%

ile50

%ile

75%

ile97

.5%

ileV

alu

e-w

eigh

ted

ind

ex(o

fel

igib

lest

ock

s)

α(b

ps/

mon

th)

2.82

2.41

2.78

-1.9

61.

212.

804.

447.

58M

kt0.

990.

010.

990.

980.

990.

991.

001.

01SM

B-0

.09

0.01

-0.1

0-0

.12

-0.1

0-0

.09

-0.0

9-0

.07

HM

L0.

050.

010.

050.

020.

040.

040.

060.

07M

OM

0.00

0.01

0.00

-0.0

2-0

.01

0.00

0.00

0.01

Equ

ally

-wei

ghte

din

dex

(of

elig

ible

stock

s)

α(b

ps/

mon

th)

8.90

1.23

8.93

6.48

8.08

8.89

9.70

11.3

1M

kt1.

030.

001.

031.

021.

031.

031.

031.

04SM

B0.

480.

010.

480.

470.

480.

480.

480.

49H

ML

0.26

0.01

0.26

0.25

0.26

0.26

0.27

0.27

MO

M-0

.08

0.00

-0.0

8-0

.09

-0.0

8-0

.08

-0.0

8-0

.07

Op

tim

alp

ortf

olio

for

gam

ma

2in

vest

or(u

sin

gγ

=3)

α(b

ps/

mon

th)

296.

4040

.51

325.

9922

3.25

268.

2229

4.32

322.

3938

1.41

Mkt

0.53

0.12

0.57

0.31

0.45

0.53

0.61

0.76

SMB

1.05

0.22

1.19

0.61

0.90

1.05

1.20

1.49

HM

L1.

870.

211.

961.

491.

731.

872.

012.

28M

OM

1.43

0.14

1.38

1.15

1.33

1.43

1.53

1.72

Tab

le3

Con

tinu

edFa

ma-

Fren

ch-C

arha

rtre

gres

sion

coeffi

cien

tsof

benc

hmar

kpo

rtfo

lios

Reg

ress

ion

Boo

tstr

apB

oots

trap

Sam

ple

Boo

tstr

apB

oots

trap

Boo

tstr

apB

oots

trap

Boo

tstr

apC

oeffi

cien

tM

ean

σ2.

5%ile

25%

ile50

%ile

75%

ile97

.5%

ileO

pti

mal

por

tfol

iofo

rga

mm

a3

inve

stor

(usi

ngγ

=5)

α(b

ps/

mon

th)

192.

5624

.94

211.

2814

6.42

175.

2319

1.39

208.

5824

4.57

Mkt

0.55

0.07

0.57

0.42

0.50

0.55

0.60

0.69

SMB

0.45

0.13

0.52

0.19

0.37

0.45

0.54

0.71

HM

L1.

460.

121.

531.

231.

381.

461.

541.

71M

OM

0.97

0.09

0.95

0.80

0.91

0.97

1.03

1.14

Op

tim

alp

ortf

olio

for

gam

ma

4in

vest

or(u

sin

gγ

=7)

α(b

ps/

mon

th)

143.

6118

.49

157.

2910

9.95

130.

6914

2.93

155.

6818

1.95

Mkt

0.57

0.05

0.58

0.47

0.53

0.56

0.60

0.67

SMB

0.20

0.10

0.25

0.02

0.14

0.20

0.27

0.39

HM

L1.

270.

091.

331.

101.

211.

271.

331.

45M

OM

0.74

0.06

0.73

0.61

0.70

0.74

0.78

0.86

Op

tim

alp

ortf

olio

for

gam

ma

5in

vest

or(u

sin

gγ

=9)

α(b

ps/

mon

th)

115.

4913

.48

123.

0490

.19

106.

2011

4.90

124.

0514

3.39

Mkt

0.57

0.03

0.55

0.50

0.55

0.57

0.59

0.64

SMB

0.05

0.07

0.08

-0.1

00.

000.

050.

100.

20H

ML

1.17

0.07

1.22

1.03

1.11

1.16

1.21

1.31

MO

M0.

630.

050.

640.

540.

600.

630.

660.

72

Tab

le3

Con

tinu

edFa

ma-

Fren

ch-C

arha

rtre

gres

sion

coeffi

cien

tsof

benc

hmar

kpo

rtfo

lios

Reg

ress

ion

Boo

tstr

apB

oots

trap

Sam

ple

Boo

tstr

apB

oots

trap

Boo

tstr

apB

oots

trap

Boo

tstr

apC

oeffi

cien

tM

ean

σ2.

5%ile

25%

ile50

%ile

75%

ile97

.5%

ileO

pti

mal

por

tfol

iofo

rga

mm

a6

inve

stor

(usi

ngγ

=11

)

α(b

ps/

mon

th)

99.3

911

.52

105.

6078

.20

91.3

298

.86

106.

8512

3.38

Mkt

0.60

0.03

0.58

0.54

0.58

0.60

0.62

0.66

SMB

-0.0

40.

06-0

.02

-0.1

7-0

.09

-0.0

40.

000.

08H

ML

1.08

0.06

1.13

0.96

1.04

1.08

1.12

1.20

MO

M0.

530.

040.

540.

450.

500.

530.

560.

61

Op

tim

alp

ortf

olio

for

gam

ma

7in

vest

or(u

sin

gγ

=13

)

α(b

ps/

mon

th)

88.7

010

.20

93.5

469

.85

81.6

388

.25

95.3

610

9.44

Mkt

0.62

0.03

0.60

0.56

0.60

0.62

0.64

0.67

SMB

-0.1

10.

06-0

.10

-0.2

3-0

.15

-0.1

1-0

.07

0.00

HM

L1.

020.

061.

070.

920.

981.

021.

061.

13M

OM

0.46

0.04

0.47

0.39

0.44

0.46

0.49

0.54

Op

tim

alp

ortf

olio

for

gam

ma

8,9,

and

10in

vest

ors

(usi

ngγ

=16

)

α(b

ps/

mon

th)

49.1

15.

7251

.72

38.2

145

.19

48.9

452

.93

60.6

5M

kt0.

600.

020.

590.

560.

590.

600.

620.

65SM

B-0

.15

0.05

-0.1

4-0

.25

-0.1

9-0

.15

-0.1

2-0

.06

HM

L1.

000.

041.

060.

910.

971.

001.

031.

09M

OM

0.41

0.03

0.43

0.35

0.39

0.41

0.43

0.47

The

(out

-of-

sam

ple)

retu

rns

ofth

eop

tim

alpo

rtfo

lios

for

each

inve

stor

inde

xed

byco

effici

ent

ofre

lati

veri

skav

ersi

onar

epr

ojec

ted

linea

rly

onth

efo

urFa

ma,

Fren

ch,

Car

hart

fact

ors

inth

em

onth

sJa

nuar

y19

75th

roug

hD

ecem

ber

2015

(492

mon

ths)

.T

his

tabl

ere

port

sth

esa

mpl

ing

dist

ribu

tion

sof

the

regr

essi

onco

effici

ents

.T

hein

form

atio

nin

this

tabl

eis

extr

acte

dfr

omT

able

sIA

-17

thro

ugh

IA-3

0of

the

Inte

rnet

App

endi

x.

Table 4

Portfolio moments of benchmark portfolios formed using eligible stocks

Return BootstrapProperty Mean σ 2.5%ile 25%ile 50%ile 75%ile 97.5%ile

VWIMean 1.08 0.02 1.03 1.06 1.08 1.09 1.12

Std Dev 4.38 0.03 4.33 4.36 4.38 4.40 4.43Median 1.35 0.06 1.23 1.31 1.35 1.39 1.47

Intq Rng 5.40 0.10 5.20 5.33 5.40 5.46 5.59Minimum -22.28 0.51 -23.27 -22.62 -22.29 -21.93 -21.25Skewness 0.32 0.02 0.27 0.31 0.32 0.34 0.37Kurtosis 0.30 0.09 0.13 0.24 0.30 0.36 0.48

SR (ann.) 0.54 0.02 0.50 0.53 0.54 0.55 0.57

EWIMean 1.33 0.01 1.31 1.32 1.33 1.34 1.35

Std Dev 5.13 0.01 5.10 5.12 5.13 5.13 5.15Median 1.59 0.05 1.48 1.55 1.59 1.63 1.69

Intq Rng 6.18 0.08 6.03 6.13 6.18 6.23 6.33Minimum -25.93 0.30 -26.51 -26.13 -25.93 -25.72 -25.35Skewness 0.33 0.01 0.30 0.32 0.33 0.33 0.35Kurtosis 0.31 0.06 0.20 0.27 0.31 0.35 0.42

SR (ann.) 0.63 0.01 0.62 0.63 0.63 0.64 0.65

M, S, β, σε, rt−12j Optimal for gamma 2 investorMean 5.49 0.46 4.65 5.17 5.46 5.79 6.45

Std Dev 14.19 0.98 12.41 13.51 14.13 14.83 16.25Median 5.66 0.53 4.68 5.30 5.65 6.01 6.76

Intq Rng 16.48 1.27 14.16 15.57 16.41 17.30 19.17Minimum -53.15 6.37 -67.83 -56.77 -52.41 -48.73 -42.86Skewness 0.44 0.07 0.31 0.39 0.44 0.49 0.57Kurtosis 0.51 0.19 0.15 0.38 0.51 0.64 0.90

SR (ann.) 1.24 0.06 1.14 1.21 1.24 1.28 1.35

M, S, β, σε, rt−12j Optimal for gamma 3 investorMean 3.88 0.28 3.37 3.69 3.87 4.06 4.47

Std Dev 9.42 0.58 8.35 9.02 9.39 9.79 10.64Median 4.09 0.33 3.48 3.86 4.08 4.30 4.75

Intq Rng 10.62 0.78 9.14 10.08 10.58 11.12 12.24Minimum -36.14 4.50 -46.83 -38.50 -35.51 -33.07 -29.22Skewness 0.51 0.07 0.37 0.46 0.51 0.55 0.64Kurtosis 0.62 0.20 0.24 0.47 0.61 0.75 1.04

SR (ann.) 1.28 0.05 1.18 1.25 1.28 1.32 1.39

Table 4 Continued

Portfolio moments of benchmark portfolios formed using eligible stocks

Return BootstrapProperty Mean σ 2.5%ile 25%ile 50%ile 75%ile 97.5%ile

M, S, β, σε, rt−12j Optimal for gamma 4 investorMean 3.13 0.21 2.75 2.99 3.12 3.27 3.56

Std Dev 7.33 0.42 6.57 7.04 7.32 7.61 8.19Median 3.34 0.24 2.89 3.17 3.33 3.50 3.83

Intq Rng 8.03 0.57 7.00 7.63 8.01 8.40 9.19Minimum -29.52 4.38 -40.29 -31.76 -28.79 -26.43 -23.17Skewness 0.57 0.07 0.44 0.53 0.57 0.62 0.71Kurtosis 0.73 0.21 0.34 0.59 0.73 0.87 1.17

SR (ann.) 1.30 0.05 1.19 1.26 1.29 1.33 1.40

M, S, σε, rt−12j Optimal for gamma 5 investorMean 2.71 0.15 2.43 2.61 2.70 2.80 3.01

Std Dev 6.26 0.29 5.72 6.06 6.25 6.45 6.87Median 2.94 0.18 2.60 2.82 2.94 3.07 3.32

Intq Rng 6.63 0.39 5.90 6.36 6.62 6.90 7.44Minimum -27.96 4.69 -38.74 -30.82 -27.27 -24.48 -20.79Skewness 0.66 0.06 0.54 0.62 0.66 0.70 0.78Kurtosis 0.84 0.21 0.46 0.70 0.83 0.97 1.26

SR (ann.) 1.28 0.05 1.19 1.25 1.28 1.32 1.38

M, S, σε, rt−12j Optimal for gamma 6 investorMean 2.45 0.13 2.22 2.36 2.44 2.53 2.72

Std Dev 5.62 0.24 5.17 5.45 5.61 5.77 6.11Median 2.67 0.16 2.36 2.56 2.67 2.77 2.99

Intq Rng 5.86 0.33 5.24 5.64 5.85 6.08 6.53Minimum -26.87 4.53 -36.72 -29.75 -26.45 -23.50 -19.48Skewness 0.67 0.06 0.55 0.63 0.67 0.71 0.79Kurtosis 0.87 0.20 0.50 0.73 0.86 1.00 1.28

SR (ann.) 1.27 0.05 1.18 1.24 1.27 1.31 1.37

M, S, σε, rt−12j Optimal for gamma 7 investorMean 2.28 0.11 2.07 2.20 2.27 2.35 2.50

Std Dev 5.22 0.20 4.85 5.08 5.21 5.35 5.64Median 2.47 0.14 2.21 2.37 2.47 2.57 2.77

Intq Rng 5.38 0.28 4.87 5.18 5.37 5.56 5.96Minimum -26.69 4.39 -36.05 -29.50 -26.38 -23.48 -19.16Skewness 0.68 0.06 0.56 0.64 0.68 0.72 0.80Kurtosis 0.88 0.19 0.52 0.75 0.87 1.01 1.27

SR (ann.) 1.25 0.05 1.16 1.22 1.25 1.29 1.35

Table 4 Continued

Portfolio moments of benchmark portfolios formed using eligible stocks

Return BootstrapProperty Mean σ 2.5%ile 25%ile 50%ile 75%ile 97.5%ile

M, S, σε Optimal for gamma 8, 9, and 10 investorsMean 1.82 0.07 1.69 1.77 1.82 1.87 1.96

Std Dev 4.64 0.12 4.41 4.56 4.63 4.72 4.87Median 1.96 0.10 1.76 1.89 1.96 2.03 2.17

Intq Rng 4.73 0.19 4.37 4.60 4.72 4.85 5.11Minimum -22.24 2.30 -27.33 -23.65 -21.99 -20.59 -18.45Skewness 0.67 0.05 0.57 0.63 0.67 0.71 0.77Kurtosis 1.07 0.19 0.72 0.94 1.07 1.20 1.46

SR (ann.) 1.07 0.04 0.99 1.04 1.07 1.10 1.15

This table provides sampling distributions (formed from 10,000 bootstrapsamples) of the (out-of-sample) return distribution moments of the optimalportfolios for investors indexed by their coefficient of relative risk aversion(gamma). There are 492 monthly returns, over the period January 1975through December 2015.

Intq Rng refers to the interquartile range (the difference between the 75%ilereturn and the 25%ile return).

Skewness is the estimate (K3) of the third moment:

K3 =r+.95 − r

−.05

r+.5 − r−.5

− 2.63

Kurtosis is the estimate (K4) of the fourth moment:

K4 =r.95 − r.05

r.75 − r.25− 2.90

Where:r+.95 is the mean of the highest 5% of returns,r−.05 is the mean of the smallest 5% of returns,r+.5 is the mean of the top half of returns,and r−.5 is the mean of the bottom half of returns;and rx is the observation corresponding to the x%ile of the return data.

SR (ann.) is the monthly Sharpe ratio ×√

12.

Tab

le5

Mon

th-b

y-M

onth

Fac

tor

Exp

osu

reP

anel

A.

Mon

th-b

y-m

onth

FF

Cfa

ctor

retu

rns

Fact

orA

llJa

nFe

bM

arA

prM

ayJu

nJu

lA

ugSe

pO

ctN

ovD

ecm

onth

sRm−r f

0.67∗∗

1.21

0.47

0.93

1.37∗

0.89

0.39

0.10

0.08

-0.8

40.

311.

68∗

1.46∗

(0.

20)

(0.

81)

(0.

64)

(0.

60)

(0.

56)

(0.

57)

(0.

53)

(0.

65)

(0.

80)

(0.

71)

(1.

03)

(0.

68)

(0.

51)

SMB

0.26

1.41∗

0.98

0.35

-0.1

80.

381.

03-0

.66

-0.1

30.

06−

1.28∗

0.45

0.75∗

(0.

14)

(0.

48)

(0.

63)

(0.

55)

(0.

46)

(0.

41)

(0.

44)

(0.

44)

(0.

42)

(0.

36)

(0.

53)

(0.

43)

(0.

35)

HM

L0.

30∗

0.92

0.58

1.05∗∗

0.70

0.00

-0.1

90.

640.

580.

16-0

.54

-0.5

50.

26(

0.13

)(

0.60

)(

0.65

)(

0.35

)(

0.42

)(

0.31

)(

0.41

)(

0.47

)(

0.39

)(

0.38

)(

0.44

)(

0.50

)(

0.43

)M

OM

0.65∗∗

-1.2

81.

620.

60-1

.23

-0.1

32.

081.

27-0

.36

2.02

0.45

0.79

1.91

(0.

20)

(0.

90)

(0.

65)

(0.

54)

(1.

01)

(0.

60)

(0.

60)

(0.

48)

(0.

46)

(0.

52)

(0.

64)

(0.

73)

(0.

62)

Pan

elB

.T

imin

gbi

ases

inFa

ma

Fren

chC

arha

rtre

gres

sion

alph

asin

the

opti

mal

port

folio

ofth

ega

mm

a2

inve

stor

form

edus

ingγ

=3

wit

hch

arac

teri

stic

s:M

,S,β

,σε,r t−

12j

Bia

sdu

eto

:M

arke

tSM

BH

ML

MO

MT

otal

Bia

sα

Tot

alB

ias:

Mea

n5.

3031

.04

9.50

0.41

46.2

529

6.40

Std

Dev

5.24

8.18

5.26

11.0

117

.07

40.5

12.

5%ile

-4.8

716

.06

-0.7

3-2

1.45

13.8

022

3.25

Med

ian

5.23

30.7

59.

440.

4145

.89

294.

3297

.5%

ile15

.72

47.9

920

.19

22.0

880

.46

381.

41

Vol

atilit

yT

imin

gB

ias:

Mea

n16

.36

22.6

013

.74

-15.

6037

.10

Std

Dev

4.56

8.08

5.47

8.77

15.6

42.

5%ile

7.67

7.62

3.21

-32.

907.

83M

edia

n16

.30

22.2

913

.65

-15.

5936

.62

97.5

%ile

25.6

339

.43

24.8

61.

5968

.77

Mea

nT

imin

gB

ias:

Mea

n-1

1.06

8.43

-4.2

416

.01

9.15

Std

Dev

2.80

1.95

2.11

3.94

5.89

2.5%

ile-1

6.79

4.72

-8.5

68.

29-2

.47

Med

ian

-10.

978.

38-4

.14

16.0

19.

1897

.5%

ile-5

.81

12.3

9-0

.30

23.7

620

.56

Table 5 Month-by-Month Factor Exposure Continued

Panel C. Timing biases in Fama French Carhart regression alphas in the optimal portfolio of thegamma 3 investor formed using γ = 5 with characteristics: M, S, β, σε, rt−12j

Bias due to:Market SMB HML MOM Total Bias α

Total Bias: Mean 1.52 16.16 3.47 -1.18 19.97 192.56Std Dev 3.27 5.12 3.33 6.90 10.64 24.942.5%ile -4.86 6.70 -2.95 -14.86 0.07 146.42Median 1.53 16.03 3.42 -1.20 19.63 191.39

97.5%ile 7.96 26.67 10.27 12.47 41.50 244.57

Volatility Timing Bias: Mean 9.47 10.32 6.39 -10.91 15.27Std Dev 2.82 5.07 3.39 5.47 9.642.5%ile 4.03 0.89 -0.01 -21.64 -2.82Median 9.41 10.17 6.33 -10.98 14.94

97.5%ile 15.14 20.69 13.33 -0.18 34.78

Mean Timing Bias: Mean -7.95 5.84 -2.92 9.73 4.70Std Dev 1.73 1.23 1.31 2.48 3.762.5%ile -11.47 3.50 -5.60 4.85 -2.82Median -7.90 5.80 -2.89 9.73 4.69

97.5%ile -4.64 8.31 -0.51 14.65 11.97

Panel D. Timing biases in Fama French Carhart regression alphas in the optimal portfolio of thegamma 4 investor formed using γ = 7 with characteristics: M, S, β, σε, rt−12j

Bias due to:Market SMB HML MOM Total Bias α

Total Bias: Mean -0.26 9.20 0.67 -2.21 7.42 143.61Std Dev 2.49 3.78 2.49 5.16 8.01 18.492.5%ile -5.02 2.20 -4.11 -12.50 -7.71 109.95Median -0.27 9.09 0.69 -2.15 7.24 142.93

97.5%ile 4.63 16.94 5.63 7.82 23.93 181.95

Volatility Timing Bias: Mean 6.31 4.54 3.01 -8.99 4.87Std Dev 2.13 3.73 2.53 4.10 7.242.5%ile 2.31 -2.38 -1.83 -17.17 -8.76Median 6.25 4.43 2.98 -8.98 4.69

97.5%ile 10.59 12.15 8.12 -0.99 19.93

Mean Timing Bias: Mean -6.57 4.66 -2.34 6.78 2.54Std Dev 1.32 0.93 0.99 1.84 2.842.5%ile -9.23 2.89 -4.32 3.12 -2.98Median -6.52 4.65 -2.31 6.78 2.55

97.5%ile -4.09 6.53 -0.45 10.34 8.08

Table 5 Month-by-Month Factor Exposure Continued

Panel E. Timing biases in Fama French Carhart regression alphas in the optimal portfolio of thegamma 5 investor formed using γ = 9 with characteristics: M, S, σε, rt−12j

Bias due to:Market SMB HML MOM Total Bias α

Total Bias: Mean -1.04 5.32 -1.74 -4.50 -1.97 115.49Std Dev 1.92 2.87 1.87 3.67 5.72 13.482.5%ile -4.81 -0.13 -5.37 -11.84 -12.85 90.19Median -1.04 5.24 -1.75 -4.50 -2.04 114.90

97.5%ile 2.77 11.17 1.99 2.74 9.47 143.39

Volatility Timing Bias: Mean 5.20 1.42 0.78 -9.21 -1.82Std Dev 1.55 2.83 1.88 2.95 5.032.5%ile 2.20 -3.97 -2.82 -15.07 -11.38Median 5.19 1.34 0.75 -9.21 -1.97

97.5%ile 8.33 7.06 4.61 -3.46 8.34

Mean Timing Bias: Mean -6.24 3.90 -2.53 4.71 -0.15Std Dev 0.97 0.73 0.74 1.33 2.042.5%ile -8.23 2.52 -4.00 2.15 -4.10Median -6.24 3.90 -2.51 4.69 -0.17

97.5%ile -4.36 5.35 -1.13 7.37 3.88

Panel F. Timing biases in Fama French Carhart regression alphas in the optimal portfolio of thegamma 6 investor formed using γ = 11 with characteristics: M, S, σε, rt−12j

Bias due to:Market SMB HML MOM Total Bias α

Total Bias: Mean -1.55 3.15 -2.30 -4.27 -4.97 99.39Std Dev 1.69 2.48 1.59 3.18 4.97 11.522.5%ile -4.90 -1.55 -5.41 -10.54 -14.23 78.20Median -1.57 3.10 -2.32 -4.29 -5.13 98.86

97.5%ile 1.77 8.13 0.87 2.00 5.23 123.38

Volatility Timing Bias: Mean 4.12 -0.26 -0.09 -7.93 -4.16Std Dev 1.36 2.44 1.60 2.58 4.372.5%ile 1.47 -4.95 -3.13 -12.96 -12.19Median 4.11 -0.30 -0.12 -7.94 -4.31

97.5%ile 6.81 4.63 3.13 -2.90 4.89

Mean Timing Bias: Mean -5.67 3.41 -2.21 3.66 -0.80Std Dev 0.85 0.63 0.64 1.14 1.762.5%ile -7.37 2.21 -3.50 1.44 -4.22Median -5.65 3.40 -2.21 3.66 -0.83

97.5%ile -4.05 4.67 -0.99 5.92 2.67

Table 5 Month-by-Month Factor Exposure Continued

Panel G. Timing biases in Fama French Carhart regression alphas in the optimal portfolio of thegamma 7 investor formed using γ = 13 with characteristics: M, S, σε, rt−12j

Bias due to:Market SMB HML MOM Total Bias α

Total Bias: Mean -1.95 1.59 -2.69 -4.29 -7.34 88.70Std Dev 1.55 2.23 1.44 2.86 4.50 10.202.5%ile -5.08 -2.72 -5.47 -9.85 -15.95 69.85Median -1.96 1.55 -2.72 -4.28 -7.40 88.25

97.5%ile 1.05 6.03 0.20 1.28 1.79 109.44

Volatility Timing Bias: Mean 3.35 -1.51 -0.68 -7.20 -6.04Std Dev 1.23 2.20 1.45 2.32 3.972.5%ile 0.97 -5.76 -3.49 -11.79 -13.48Median 3.34 -1.54 -0.71 -7.18 -6.13

97.5%ile 5.77 2.87 2.23 -2.67 1.94

Mean Timing Bias: Mean -5.30 3.09 -2.01 2.91 -1.30Std Dev 0.77 0.57 0.58 1.02 1.602.5%ile -6.82 2.01 -3.18 0.93 -4.38Median -5.30 3.08 -2.00 2.91 -1.32

97.5%ile -3.82 4.22 -0.91 4.94 1.91

Panel H. Timing biases in Fama French Carhart regression alphas in the optimal portfolio of thegamma 8, 9, and 10 investors formed using γ = 16 with characteristics: M, S, σε

Bias due to:Market SMB HML MOM Total Bias α

Total Bias: Mean -3.44 -5.59 -5.13 -4.52 -18.68 49.11Std Dev 1.10 1.68 1.15 2.19 3.01 5.722.5%ile -5.56 -8.86 -7.38 -8.83 -24.53 38.21Median -3.45 -5.63 -5.13 -4.51 -18.70 48.94

97.5%ile -1.30 -2.22 -2.87 -0.24 -12.85 60.65

Volatility Timing Bias: Mean 0.41 -7.15 -4.21 -4.25 -15.20Std Dev 0.89 1.66 1.15 1.81 2.732.5%ile -1.32 -10.36 -6.42 -7.82 -20.54Median 0.40 -7.17 -4.21 -4.23 -15.22

97.5%ile 2.17 -3.87 -1.98 -0.73 -9.95

Mean Timing Bias: Mean -3.85 1.56 -0.92 -0.27 -3.48Std Dev 0.56 0.40 0.42 0.74 1.242.5%ile -4.94 0.76 -1.75 -1.73 -5.89Median -3.85 1.56 -0.93 -0.27 -3.48

97.5%ile -2.73 2.35 -0.10 1.18 -0.96

Table 5 ContinuedMonth-by-month factor exposure

The numbers in parentheses below the mean returns in Panel A are standard errors.

Total bias is defined:

112

12∑m=1

bm,jf j,m − buj · f

uj

where:buj is the unconditional loading of the portfolio returns on factor j.fj,m is the return on factor j in month m.bm,j is the loading of the portfolio return in month m on fj,m.

bj =112

12∑m=1

bm,j

Volatility bias is defined:

fuj (bj − buj )

where:fuj is the unconditional mean of the return on factor j.

Risk-premium (or mean-) timing bias is defined as the difference between total bias and volatility-timing bias.