Equal-weighted strategy: Why it outperforms value-weighted strategies? Theory and evidence

Abstract

Recent academic papers and practitioner publications suggest that equal-weighted portfolios (or 1/N portfolios) appear to outperform various other portfolio strategies. In addition, as the equal-weighted portfolio does not rely on expected average returns, it is therefore assumed to be more robust compared to other price-weighted or value-weighted strategies. In this paper, we provide a theoretical framework to the equal-weighed versus value-weighted equity portfolio model, and demonstrate using simulation as well as real-world data from 1926 to 2014 that an equal-weighted strategy indeed outperforms value-weighted strategies. Moreover, we demonstrate that a significant portion of the excess return is attributable to portfolio rebalancing. Finally, we show that because of equal-weighting, the excess returns are higher than the higher costs incurred due to higher portfolio turnover. Therefore, even after accounting for higher portfolio turnover costs, equal-weighting makes economic sense.

Full text

Equal-weighted Strategy: Why it outperforms value-weighted
strategies? Theory and evidence.
Rama Malladi
Kubera Investments LLC, Irvine, CA, USA
rmalladi@gmail.com
Frank J. Fabozzi
EDHEC Business School, Nice, France
1

Abstract
Recent academic papers and practitioner publications suggest that equal-weighted portfolios (or
1/N portfolios) appear to outperform various other portfolio strategies. In addition, as the equal-
weighted portfolio does not rely on expected average returns, it is therefore assumed to be more
robust compared to other price-weighted or value-weighted strategies. In this paper we provide
a theoretical framework to the equal-weighed versus value-weighted equity portfolio model, and
demonstrate using simulation as well as real-world data from 1926 to 2014 that an equal-weighted
strategy indeed outperforms value-weighted strategies. Moreover, we demonstrate that a significant
portion of the excess return is attributable to portfolio rebalancing. Finally, we show that because
of equal-weighting the excess returns are higher than the higher costs incurred due to higher port-
folio turnover. Therefore, even after accounting for higher portfolio turnover costs, equal-weighting
makes economic sense.
JEL classification: G11
Keywords: Equal-weighted, cap-weighted, 1/N, asset allocation, portfolio optimization
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1 Introduction
Since the introduction of the S&P 500 index in 1957, most indices have been weighted by market
capitalization (or value-weighted, abbreviated as VW). By the end of 2014, S&P Dow Jones Indices
estimates that over $7.8 trillion was benchmarked to the S&P 500 alone, with indexed assets making
up $2.2 trillion of this total.1The theoretical foundation for VW indices as a benchmark for
investors is provided by the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner
(1965), and the Efficient Market Hypothesis (EMH) of Fama (1970). According to the CAPM
model, the expected return implicit in the price of a stock should be commensurate with the risk of
that stock. Based on the CAPM and EMH theory, the most efficient portfolio would be the entire
market and a broad VW index would represent the optimal mean-variance efficient investment.
However, following the critique by Roll (1977), there has been considerable debate (examples of
which are available in Gruber and Ross (1978) and Gibbons (1982)) as to how efficient the market
portfolio is in practice. Thus, there are countless different strategies2as documented by Arnott
et al. (2005) to beat the market. This has led to indices created based on alternative factors that
measure different strategies, sometimes referred to as Smart Beta strategies, as explained by Amenc
et al. (2011b) and Amenc and Goltz (2013). Investors have been attracted by the performance of
these indices compared to traditional cap-weighted indices. Some of the popular alternative indices
are FTSE EDHEC-Risk Efficient Index, Intech’s Diversity-Weighted Index, Research Affiliates
Fundamental Index, QS Investors’ Diversification Based Investing (DBI), and TOBAM’s Maximum
Diversification Index.
Recent studies by DeMiguel et al. (2009) and Plyakha et al. (2015) suggest that equal-weighted
portfolios (also known as 1/N, abbreviated as EWP) appear to outperform 14 different portfolio
weighting strategies. The out-of-sample performance of an EWP of stocks is significantly better
than that of a value-weighted portfolio (abbreviated as VWP), and no worse than that of portfolios
from a number of optimal portfolio selection models. Plyakha et al. (2014) report that for the
14 models that they studied using seven empirical datasets, none were consistently better than
the EWP in terms of Sharpe ratio, certainty-equivalent return, or turnover. Because the EWP
1http://www.spindices.com/documents/index-policies/spdji- indexed-assets- survey-2014.pdf
2http://faculty-research.edhec.com/_medias/fichier/edhec- position-paper- smart-beta- 2-0_1378195044229- pdf
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does not rely on expected average returns, it is therefore assumed to be more robust compared to
other price-weighted or VW strategies. Moreover an EW strategy minimizes the risk of a portfolio
deviating from the investor’s target allocation objectives.
In this paper our objective is not to support one of the several alternative beta strategies in
the market, as they all seem to outperform VW indices according to Arnott et al. (2005), Chow
et al. (2011), and Amenc et al. (2011a). Rather, we focus on the nature and source of EWP
returns. Given the positive excess returns of EWPs relative to VWPs in the equity asset class, two
natural questions arise: (1) Can this excess return be realized under various market scenarios and
time horizons? and (2) What is the source of these positive excess returns? Plyakha et al. (2015)
identified rebalancing as the main source of excess returns by using two experiments with simulated
data. We develop a model to explain the source of excess returns, and test it with both simulated
and real-world data. In this paper we provide a theoretical framework for the EWP model, and
demonstrate using simulation and historical data that an EW strategy indeed outperforms VW
strategies. Then we show that rebalancing is a key driver behind the positive excess retun of the
EWPs
To undertake our analysis, a two-period equity portfolio model with two assets is developed.
This model can be extended to multiple periods as well as multiple assets. This equity portfolio
model proves that after portfolio rebalancing, if smaller-cap stocks outperform larger-cap stocks,
then the EWP will produce higher returns than the VWP. Given that smaller-cap stocks are riskier
than larger-cap stocks, it is quite natural to expect a higher return from smaller-cap stocks. So it
is plausible for an EWP to outperfom a VWP. We validate the model with five tests to see if EWPs
produce higher returns and Sharpe ratios than VWPs.
The flow of this paper is as follows. Section 2 describes the portfolio model for VWP and
EWP. In Section 3 we describe the data used. The results are presented and discussed in Section
4, followed by our conclusions in Section 5.
2 Portfolio Model
In this section, an equity portfolio is built with two (n=2) stocks A and B. Either A or B can be
a large cap stock. At t=0, an investor’s wealth is invested in either a VW or EW portfolio. At
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t=1, single-period returns of value-weighted portfolio (denoted as V) and equal-weighted portfolio
(denoted as E) are computed. The main difference between the two portfolios (V and E) is that
equal-weighted portfolio has to be rebalanced after t=1, whereas the value-weighted portfolio does
not require rebalancing. At t=2, the returns denoted by R2Vand R2E, the standard deviation of
returns denoted by σ2Vand σ2E, and Sharpe ratios denoted by S2Vand S2Eare computed. In the
end, the following five metrics are computed:
1. Excess return of EWP over VWP: R2E−R2V
2. Excess risk of EWP over VWP: σ2E−σ2V
3. Excess Sharpe ratio of EWP over VWP: S2E−S2V
4. Decomposed excess return due to rebalancing effect
5. Decomposed excess return due to size effect.
2.1 Value-weighted portfolio return
Let M0denote the investable wealth (or portfolio value) at t=0 and then
Marketcap of firm A at t=0 is: VA0=PA0QA, where PA0is price of A, QAis number of shares outstanding
Marketcap of firm B at t=0 is: VB0=PB0QB, where PB0is price of B, QBis number of shares outstanding
Portfolio value invested in firm A, at t=0, according to value - weight is: MA0V=M0PA0QA
PA0QA+PB0QB
Portfolio value invested in firm B, at t=0, according to value - weight is: MB0V=M0PB0QB
PA0QA+PB0QB
where M0=MA0V+MB0V
Number of firm A stocks in portfolio at t=0: NA0V=MA0V
PA0
=M0QA
PA0QA+PB0QB
Number of firm B stocks in portfolio at t=0: NB0V=MB0V
PB0
=M0QB
PA0QA+PB0QB
Similarly at t=1: VA1=PA1QA, VB1=PB1QB, MA1V=NA0VPA1, MB1V=NB0VPB1
Value-weighted portfolio return between t=0 and 1 = R1V=MA1V+MB1V
MA0V+MB0V
−1
⇒R1V=QA(PA1 −PA0) + QB(PB1 −PB0 )
QAPA0 +QBPB0
(1)
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Since the VWP is not rebalanced after t=1, NA1V=NA0V, NB1V=NB0V
Similarly at t=2: VA2=PA2QA, VB2=PB2QB, MA2V=NA1VPA2, MB2V=NB1VPB2
Value-weighted portfolio return between t=1 and 2 = R2V=MA2V+MB2V
MA1V+MB1V
−1
⇒R2V=QA(PA2 −PA1) + QB(PB2 −PB1 )
QAPA1 +QBPB1
(2)
2.2 Equal-weighted portfolio return
Again, letting M0denote the investable wealth (or portfolio value) at t=0 and then
Marketcap of firm A at t=0 is: VA0=PA0QA, where PA0is price of A, QAis number of shares outstanding
Marketcap of firm B at t=0 is: VB0=PB0QB, where PB0is price of B, QBis number of shares outstanding
Given that n = 2, 50% of the portfolio will be invested in A and B each.
⇒MA0E=MB0E=M0
2, NA0E=M0
2PA0
, NB0E=M0
2PB0
, MA1E=NA0EPA1, MB1E=NB0EPB1
Equal-weighted portfolio return between t=0 and 1 = R1E=MA1E+MB1E
MA0E+MB0E
−1
⇒R1E=PA1
2PA0
+PB1
2PB0−1 (3)
Unlike the value-weighted portfolio, the equal-weighted portfolio weights have to be rebalanced
after t=1. Given the portfolio value of MA1E+MB1Eat t=1 before rebalancing, the portfolio value of
A and B after rebalancing will be MA1ER =MB1ER =1
2(MA1E+MB1E).
As a result, NA1ER =MA1ER
PA1
, and NB1ER =MB1ER
PB1
.
So at t=2: VA2=PA2QA, VB2=PB2QB, MA2E=NA1ER PA2, MB2E=NB1ER PB2
Equal-weighted portfolio return between t=1 and 2 = R2E=MA2E+MB2E
MA1E+MB1E
−1
⇒R2E=PA2
2PA1
+PB2
2PB1−1 (4)
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2.3 Excess return for the portfolio
We define excess return, R2E−R2V, as the EWP return minus the VWP return at t=2.
R2E−R2V=PA2
2PA1
+PB2
2PB1−1−QA(PA2 −PA1) + QB(PB2 −PB1 )
QAPA1 +QBPB1
⇒R2E−R2V=QAP2
A1PB2 +PA2QBP2
B1 −PA1PB1 (QAPA2 +QBPB2)
2PA1PB1 (QAPA1 +QBPB1)(5)
⇒R2E−R2V=(PA1PB2−PB1PA2) (PA1QA−PB1 QB)
2PA1PB1 (QAPA1 +QBPB1)(6)
Based on equation (6), at t=2, the EWP will produce a higher return than a VWP if and only
if R2E−R2V>0. This is possible under the two scenarios discussed next.
Scenario (a): PA1PB2−PB1PA2>0⇒PB2/PB1 > PA2/PA1⇒B has a higher return than A.
and PA1QA−PB1QB>0⇒PA1QA> PB1 QB⇒B has a smaller cap compared to A.
Together the above two conditions imply that if the return of the small cap stock is greater
than that of the large cap stock, then the excess return will be positive.
Scenario (b): PA1PB2−PB1PA2<0⇒PB2/PB1 < PA2/PA1⇒A has higher return than B.
and PA1QA−PB1QB<0⇒PA1QA< PB1 QB⇒A has a smaller cap compared to B.
As in scenario (a), the above two conditions imply that if the return of the small cap stock
exceeds that of the large cap stock, then the excess return will be positive.
In summary, in both scenarios (a) and (b) after rebalancing the portfolio at t=1, if the smaller-
cap stock outperforms the larger-cap stock, then the EWP will produce higher returns than a VWP.
Given that the smaller-cap stocks are riskier than the larger-cap stocks, it is quite natural to expect
a higher return from the smaller-cap stock in the long-run, thus it follows that because of portfolio
rebalancing, an EWP will produce higher returns than a VWP. The same logic can be extended to
the multi-period and multi-asset cases.
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2.4 Excess risk (Standard Deviation)
In this section, excess risk is defined as the difference between the EWP standard deviation and
the VWP standard deviation, denoted by σE-σV. If at t=2, the standard deviation for the large
cap (L) and the small cap (S) returns are denoted by σLand σS, and the correlation between those
returns is denoted by ρ, then the portfolio variance
σ2
P=W2
Lσ2
L+W2
Sσ2
S+ 2WLWSσLσSρ
For EWP, WL=WS= 0.5. Letting σS-σL=d, then
σ2
E= 0.25 σ2
L+σ2
S+ 0.5σLσSρ= 0.25 σ2
L+ (σL+d)2+ 0.5σLρ(σL+d) (7)
For VWP, WL>0.5.Letting WL= 0.5 + e, where 0 <e<0.5⇒WS= 0.5−e, then
σ2
V= (0.5 + e)2σ2
L+ (0.5−e)2(σL+d)2+ 2 (0.5 + e) (0.5−e)σL(σL+d)ρ(8)
⇒σ2
E−σ2
V= 0.5d2(1 −e)e+de(1 + e(ρ−1))σL+e2(ρ−1)σ2
L(9)
The EWP will have a higher variance than the VWP if σ2
E−σ2
V= (σE−σV)(σE+σV)>0.
Since (σE+σV)>0 always, the excess risk will be positive when σE-σV>0. Excess risk is
therefore a function of f(d, e, ρ, σL). EWP will have a higher variance depending on the value of
d(difference in standard deviation of small cap and large cap), e(large cap weight minus equal
weight), and ρ(correlation between large cap and small cap). Given that equation (9) is a function
of d, e, ρ, and σL, when excess risk is positive, the following parameter values are realized.
d >
σLh1 + e(ρ−1) ±p1 + e2(ρ2−1)i
e−1(10)
e > d2+ 2dσL
d2+ 2σL(1 −ρ)(d+σL)(11)
σL>
−dh1 + e(ρ−1) ±p1 + e2(ρ2−1)i
2e(ρ−1) (12)
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ρ > (e+ 1)σL2+ (e−1)(d+σL)2
2σL(d+σL)e(13)
2.5 Excess Sharpe Ratio
The excess Sharpe ratio is defined as EWP Sharpe ratio minus VWP Sharpe ratio, denoted by
SE−SV. If at t=2, the realized annual returns of small cap and large cap are RS,RLrespectively,
annual risk-free rate is RF, standard deviations of large cap and small cap returns are σL,σS, and
the correlation between large cap and small cap returns is ρ,
EWP return, RE= 0.5(RL+RS)
VWP return, RV=RL(0.5 + e) + RS(0.5−e)
Then the Sharpe ratios of EWP and VWPs are
SE=0.5(RL+RS)−RF
q0.25σL2+ 0.5ρσL(d+σL)+0.25(d+σL)2(14)
SV=RL(0.5 + e) + RS(0.5−e)−RF
q(0.5 + e)2σL2+ 2 (0.5−e) (0.5 + e)ρσL(d+σL) + (0.5−e)2(d+σL)2(15)
SE−V= Excess Sharpe Ratio = SE−SV(16)
2.6 Rebalancing and Size effect
If the EWP has two stocks consisting of a large cap (A) and small cap (B), and produces a positive
excess return, a natural question arises about the source of these excess returns. EWP and VWPs
differ only in terms of weight of the small cap stocks and rebalancing. So the excess return can be
attributed to one of these two. In this section, the excess return is decomposed into excess return
attributable to the rebalancing effect and excess return attributable to the size (small-cap) effect.
In the Section 2.2, the EWP return is computed using equation (4), when the portfolio is
rebalanced after t=1. If the EWP is not rebalanced after t=1, then the number of stocks in the
non-rebalanced EWP will continue to be NA1E, and NB1E, instead of NA1ER , and NB1ER as in
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Section 2.2. So without rebalancing, the portfolio value of the non-rebalanced EWP at t=2 will be
M2E NR =MA2E N R +MB2E N R ,where MA2E NR =NA1EPA2, MB2E NR =NB1EPB2.
The non-rebalanced EWP return between t=1 and 2 = R2E NR =M2E N R
M1E
−1
⇒R2ENR =MA2E N R +MB2E NR −M1E
M1E
=PB0(PA2−PA1) + PA0 (PB2−PB1)
PA1PB0 +PA0PB1
(17)
The excess return, R2E−R2V, can be decomposed into excess return due to rebalancing effect and
excess return due to size (small-cap) effect as shown below.
Excess Return = R2E−R2V= (R2E−R2E N R )
| {z }
Rebalancing ef fect
+ (R2E N R −R2V)
| {z }
Siz e eff ect
(18)
Rebalancing Effect = R2E−R2E N R =PA0PA2P2
B1+PB0PB2P2
A1−PA1PB1(PA2PB0+PA0PB2)
2PA1PB1(PA1PB0+PA0PB1)
(19)
Size Effect = R2E NR −R2V=(PA2PB1−PA1PB2) (PB0QB−PA0QA)
(PA1PB0+PA0PB1) (QAPA1 +QBPB1)(20)
As shown in equation (19), the rebalancing effect does not depend on the market cap of stocks
in the portfolio, and dependents solely on price movements. The return due to rebalancing is
positive, i.e., R2E−R2E NR >0, when PA0PA2P2
B1+PB0PB2P2
A1> PA1PB1(PA2PB0+PA0PB2).
By further simplification, it can be shown that R2E−R2E N R >0 when RB1> RA1. In other
words when the small cap return is more than the large cap return at t=1, the rebalancing effect
for the EWP is going to be positive. Given that small cap stocks are riskier than the large cap
stocks, it is quite natural to expect a higher return from the small cap stock in the long-run, thus
it follows that because of portfolio rebalancing, an EWP will produce higher returns than a VWP.
As shown in equation (20), the size effect depends on both price fluctuations and the size
of stocks. The return due to size effect is positive, i.e. R2E N R −R2V>0, when PA2PB1<
PA1PB2and PB0QB< PA0QA⇒PA2
PA1 <PB2
PB1and PB0QB< PA0 QA. This implies that size effect
does not depend on the returns at t=1, but instead depends on the returns at t=2. In addition,
given that B is a small cap stock, if the return on stock B is more than the return on the large cap
stock A at t=2, then the size effect is going to be positive.
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2.7 Excess portfolio turnover and transaction costs
Since the EWP gets rebalanced after every time period, it will have a higher portfolio turnover
compared to that of the VWP. In this section we analyze the excess turnover, denoted by T OEX ,
defined as the percentage of stocks rebalanced (or traded) to maintain equal weights of stocks in
each time period. Since EWP is rebalanced before t=2,
T OEX =((NA1ER +NB1ER )−(NA1E+NB1E))
(NA1E+NB1E)=(PA1PB0−PA0PB1) (PA1 −PB1)
2PA1PB1 (PA0 +PB0)(21)
Next, to compute the impact of excess portfolio turnover on portfolio return, we use Aggregate
Trading Costs (ATC) based on the methodology and findings of Edelen et al. (2013). ATC captures
invisible costs (commission, bid-ask spread, price impact, and the volume of trades) and should be
added to the visible cost (expense ratio). ATC is computed by first calculating the per unit cost of
a trade and then multiplying the per unit cost of each trade (portfolio change) by the dollar value
of the trade and summing across all trades for the time period. As a result of portfolio turnover,
the net impact on portfolio return is going to be a negative T OEX ∗AT C.
3 Data
Historical stock-level data are not required in Sections 4.1, and 4.2 because prices are simulated.
When individual portfolios are not required as in Section 4.3, and 4.4, VW and EW index returns
(including dividends) are used from the CRSP (ASI) dataset for years from 1926 to 2014. To create
VWP and EWPs in Section 4.5, stock-level data are obtained from CRSP monthly stock dataset
from January 1926 to December 2014. A total of 4,150,448 monthly stock returns were obtained
for the analysis.3
When investable indexes are required for comparison purposes as in Section 4.7, the Russell
1000 index (which contains 1000 large cap firms with 90% of the total market capitalization of
Russell 3000 index) is used as a proxy for large cap returns, and the Russell 2000 index (which
contains 2000 small cap firms with 10% of the total market capitalization of Russell 3000 index) is
3Since the same ticker symbol is used for multiple companies, and same company can have multiple classes of
shares during the window of analysis, PERMCO is used as a unique key for analysis.
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used as a proxy for small cap returns. Russell indices monthly data are available between October
1992 and December 2014. Summary statistics of data is shown in Table 1.
Insert Table 1 here.
To ensure that there is enough liquidity in the underlying stocks and to compare performance
with the S&P500, the maximum number of stocks in the EWP is limited to the largest 500 stocks
by market value. If any company’s stock is delisted during the year, that stock position is liquidated
using the last available price at the beginning of the year, and those proceeds are allocated either
according to the weights (in case of VWP) or equally (in case of EWP). For the VWP, the portfolio
is rebalanced at the beginning of every year to account for any new listings and delistings. For the
EWP, the portfolio is rebalanced at the beginning of every year to account for any new listings,
delistings, and change in portfolio weights during the year.
4 Results
The VW and EW equity portfolio model developed in Section 2 is validated based on the following
five tests. For analysis and discussion purposes, the VW return R2Vis computed with equation
(2), EW return R2Ewith equation (4), and excess return R2E−R2Vwith equation (6). Excess
risk σE-σVis computed as described in Section 2.4, and excess Sharpe ratio SE−Vwith equation
(16). The five tests are
Test 1: Random prices, or random returns for small cap and large cap stocks.
Test 2: Normally distributed returns with higher mean and standard deviation for small cap
stocks.
Test 3: Bootstrapping simulation using historical data from 1926 to 2014.
Test 4: Curve fitting using historical data from 1926 to 2014.
Test 5: Constructing EWP and VWPs using historical data from 1926 to 2014.
4.1 Random prices
In this test, it is assumed that the initial market cap at t=0 for A is larger than that of B.
All the prices PA0, PA1, PA2, PB0, PB1, PB2are randomly chosen between $0 and $100 with equal
12

probabilities. Although this type of price pattern is unrealistic in real-world, the purpose of this
test is to demonstrate that the excess return R2E−R2Vcan be positive when stock prices are
randomly-generated.
As shown in Figure 1, after 10,000 such simulations, the EWP produces positive returns 59% of
the time, whereas VWP produces positive returns 49% of the time. The excess return R2E−R2V
is positive 66% of the time, with a mean of 15.5%. The Sharpe ratio is also higher for the EWP.
VWP has a mean return of -0.55%, standard deviation of 36.94%, and Sharpe ratio of -0.096. EWP
has a mean return of 14.92%, standard deviation of 48.90%, and Sharpe ratio of 0.24. Although
the results reported here assume a risk-free rate of 3%, the results are unchanged for any risk-free
rate between 0% and 5%. In repeated simulations it is found that the EWP has a higher mean
return, higher standard deviation, and higher Sharpe ratio. This case demonstrates that even with
totally random prices, a EWP produces superior returns compared to a VWP.
Insert Figure 1 here.
4.2 Normally distributed returns
In this test, returns RA1=PA1−PA0
PA0, and RA2=PA2−PA1
PA1are normally distributed for a large cap
stock, with a mean and standard deviation of (µL, σL). Similarly, returns RB1=PB1−PB0
PB0, RB2=
PB2−PB1
PB1are normally distributed for a small cap stock, with a mean and standard deviation of
(µS, σS). Given that small cap returns are larger and more volatile than the same for large cap, it
is quite natural to have µL< µS, and σL< σS. For the purpose of our illustration, the results of
10,000 such excess return simulations for µL= 5%, µS= 8%, σL= 10%, σS= 15% are shown in
Figures 2, 3, and 4. However, the results are tested for robustness for any µL< µS, and σL< σS.
The EWP produces positive returns 76% of the time, whereas VWP produces positive returns
75% of the time. The mean excess return R2E−R2Vis 0.79%, and it is positive in 56.4% of the
simulations. The mean, and median excess returns are also positive. This pattern is consistent in
repeated simulations for any µL< µS, and σL< σS.
The Sharpe ratio is also higher for the EWP. VWP has a mean return of 5.8% and standard
deviation of 8.46%, whereas EWP has a mean return of 6.6% and standard deviation of 9.0%.
Assuming a risk-free rate of 3%, the Sharpe ratio is 0.33 for VWP and 0.40 for EWP.
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Insert Figures 2, 3, 4 here.
In summary, both the random prices in Section 4.1, and normally distributed returns in Section
4.2 illustrate that EWP outperforms VWP in both mean return and Sharpe ratio. In the next
section historical data are used rather than simulated random prices and simulated normal returns.
4.3 Bootstrapping historical data
In this test, VW and EW annual returns (including dividends) are obtained from CRSP (ASI)
dataset for the years from 1926 to 2014. A summary of these two returns are shown in Figures 5,
6, and Table 1.(2).
Insert Figures 5 and 6 here.
To compute a 2-period return, any two years are randomly selected using replacement from
1926 to 2014. Then for each selected year, the corresponding VW and EW returns are used from
the dataset. By doing so, the RA1,RA2,RB1,RB2used in Section 4.2 are no longer selected from
a normal distribution. Instead they are picked from the past historical data. This process was
repeated 10,000 times.
The results are summarized in Figures 7, 8, and 9. After 10,000 such simulations, EWP produces
positive returns 75.2% of the time, whereas VWP produces positive returns 73.9% of the time. The
excess return R2E−R2Vis positive 54.1% of the time, has mean of 1.01%, and similar to the
1.8% annual excess return found by the S&P Indices global research team over a 20-year period
between 1998 and 2008.4In addition, the Sharpe ratio of the EWP is also slightly higher than the
VWP (0.46 vs 0.45), again assuming a risk-free rate of 3%. Given the similarity of EWP and VWP
returns, we conducted a pair-wise t-test on EWP and VWP to check the statistical significance
of results. The results are shown in Table 2. A p-value of 0.00 suggests that EW returns are
statistically different to VW returns, and the 95% confidence interval shows that excess returns are
positive. As in the previous two test, in this test the EWP produces statistically significant and
better excess return and Sharpe ratios.
Insert Figures 7, 8, 9, and Table 2 here.
4https://us.spindices.com/documents/research/EqualWeightIndexing_7YearsLater.pdf
14

4.4 Curve fitting historical data
In this test, VW and EW annual returns (including dividends) are used from CRSP (ASI) dataset
for years from 1926 to 2014. The actual return distribution is fitted to the best possible matching
distribution using the Anderson-Darling (A-D) method. Using this approach, the best match for
large cap stocks with an A-D of 0.1741 is a Weibull distribution with a location of -101.10%, scale
of 120.97%, and shape of 6.672. Similarly, the best match for small cap stocks with an A-D of
0.2377 is a logistic distribution with a mean of 15.56%, and scale of 15.29.
Next, using these matching parameters, returns are computed for EWP and VWPs using 10,000
simulations. As shown in Figure 10, the EWP produces positive returns 79.4% of the time, whereas
VWP produces positive returns 78.1% of the time. The annual excess return R2E−R2Vis positive
54.8% of the time with a mean of 1.03%. In addition, the Sharpe ratio of the EWP is also higher
than the VWP (0.63 vs 0.58).
Insert Figure 10 here.
4.5 Constructed portfolios
In this section, EWP and VWPs are constructed consisting of all the publicly traded stocks using
monthly historical data from 1926 to 2014. To ensure that there is enough liquidity in the underlying
stocks and to compare performance with the S&P500, the maximum number of stocks in the EWP
is limited to the largest 500 stocks by market value. If any company’s stock is delisted during the
year, that stock position is liquidated using the last available price at the beginning of the year,
and those proceeds are allocated either according to the weights (in case of VWP) or equally (in
case of EWP). For the VWP, the portfolio is rebalanced at the beginning of every year to account
for any new listings and delistings. For the EWP, the portfolio is rebalanced at the beginning of
every year to account for any new listings, delistings, and change in portfolio weights during the
year.
After rebalancing every year, returns are computed for EWP and VWPs. The results of a pair-
wise t-test on EW and VW constructed portfolio returns are shown in Tables 3 and 4. Between the
years 1926 and 2014, EWP produced positive returns 60.2% of the time, whereas VWP produced
positive returns 56.8% of the time. The annual excess return R2E−R2Vis positive 69.3% of the
15

time, with a mean of 4.1%. In addition, the Sharpe ratio of the EWP is also higher than the VWP
(0.11 vs -0.07).
Insert Tables 3 and 4 here.
In conclusion, in the absence of trading costs, in all the five tests (i.e.,: theoretical random
prices, normally distributed returns, matching historical index prices using bootstrapping simula-
tion, simulated returns using historical rates using curve fitting, and actual portfolio construction),
EWP returns are higher and positive more number of times when compared to the VWP returns.
In addition, Sharpe ratios are higher for EWPs. The results from five tests are summarized in
Table 4.
4.6 Excess risk
In this section, we switch our attention from return to risk in order to determine if the excess
return is obtained by taking excess risk, and, if so, what happens to the risk-adjusted returns. In
the context of EWP and VWPs, excess risk is defined for a given time period as the EWP standard
deviation minus the VWP standard deviation. Excess risk is computed as explained in Section 2.4
using equation (9).
The distribution of d,e,ρ,σLin this section is based on the historcial VW and EW annual
return data (including dividends) obtained from the CRSP (ASI) dataset for the years from 1926
to 2014. The following parameters characterize the underlying data: dis between 5%, and 15%
with equal probability, eis between 5% and 45% with equal probability, ρis between -0.9 and 0.9
with a likely value of 0.6 using a triangular distribution, and σLis between 10% and 25% with
equal probability. All these parameters, small cap return RS, and large cap return RLare selected
using bootstrap by replacement method as described in Section 4.3.
After 10,000 such simulations, the excess risk is found to be positive 99% of the time. It simply
means that EWP produces excess return over VWP, but as one would expect, at the cost of higher
risk. However, as shown in Table 4, and explained in Section 4.7, the Sharpe ratio is always higher
for the EWPs. As shown in Figure 11, the excess risk varies from -2.18% to 4.37%, with a mean of
1.15%, a median of 1.01%, and a standard error of 0.01%. The excess risk is negative only when
the large cap weight approaches 100% of the portfolio.
16

Insert Figure 11 here.
4.7 Excess Sharpe Ratio
In the context of EWP and VWPs, the excess Sharpe ratio is defined for a given time period as
the EWP Sharpe ratio minus the VWP Sharpe ratio. As explained in Section 2.5, excess Sharpe
ratio is computed using equation (16).
The distribution of d,e,ρ,σLin this section is the same as the one described in Section
4.6. Although the results reported here assume a risk-free rate of 3%, the results do not change
significantly for any risk-free rate between 0% and 5%.
As shown in Figure 12, after 10,000 simulations the excess Sharpe ratio of EWP over VWP is
found to be positive 63% of the time, and the average varies from 0.009 for small equal-weighting
effect to 0.06 for large equal-weighting effect. The average excess Sharpe ratio for the entire dataset
is 0.03. This is close to the observed excess Sharpe ratio of 0.04 between two of the largest exchange-
traded funds, Guggenheim S&P 500 Equal Weight ETF (RSP)5, and SPDR S&P 500 ETF (SPY).6
As of July 2015, the Sharpe ratio for RSP is 1.98 and 1.94 for SPY.
For robustness, we computed the 95% confidence interval of the mean excess Sharpe ratio. It
is between 0.0272 and 0.0331 with a t-statistic of 20. This clearly validates that the risk-adjusted
return of the EW portfolios is higher than that of the VW portfolios. Though in aggregate the
excess Sharpe ratio is positive, as one would expect, during some periods it can be negative. As an
example during the credit crisis (years 2007 and 2008) as well as during the dot-com bubble (years
1999, 2000, 2001) excess Sharpe ratio is positive. However post credit crisis (years 2009 to 2014)
excess Sharpe ratio is negative. Parameter values for these sub-periods can be found in Table 1.
Insert Figure 12 here.
4.8 Decomposition of excess return into rebalance and size effect
After showing that excess returns are positive, the next question to answer is about the source
of these excess returns. As described in Section 2.6, the excess return is decomposed into return
due to rebalancing, and return due to size (small-cap). Although Perold and Sharpe (1995) did
5http://finance.yahoo.com/q/rk?s=RSP
6http://finance.yahoo.com/q/rk?s=SPY
17

not use the word equal-weighting, they recognized that periodic rebalancing of a portfolio to its
target allocation as one of the dynamic strategies for asset allocation. What they called Constant
Proportion Portfolio Insurance (CPPI) can be thought of as a variation of EWP. CPPI was found
to outperform other dynamic strategies in a bull market and simpler to implement than an option-
based portfolio insurance.
In this section, excess return R2E−R2Vis decomposed into return due to rebalancing R2E−
R2E NR using equation (19), and return due to size effect R2E N R −R2Vusing equation (20). We
present here the results from the Section 4.1 (random prices), as they are most generic and not
dependent on any index or market condition, to illustrate the point that significant portion of excess
returns are due to rebalancing of the portfolio.
In each of the 10,000 simulation runs, the excess return is decomposed into return due to
rebalancing, and return due to size effect. The results are summarized in Figure 13. In 19 out of 20
categories, rebalancing is the main contributor to the excess return. On average, 85% of the excess
returns are due to rebalancing effect. In addition, it can be seen in Figure 14 that almost all the
average excess return R2E−R2Vof 15.64% is due to the return from the equal-weighting (R2Eof
15.97%), with very little attributable to the return from value-weighting (R2Vof 0.34%). Almost
all the excess return is due to the rebalancing effect of 15.67%.
Insert Figures 13, and 14 here.
4.9 Excess portfolio turnover and transaction costs
As explained in Section 2.7, excess turnover (T OEX ) is defined as the percentage of stocks traded
to rebalance and maintain equal weights of stocks in each time period.
As the portfolio becomes more equal-weighted, portfolio turnover increases and excess return
also increases. As shown in Figure 15, for every 1% increase in portfolio return the portfolio
turnover increases by 0.4669%. The mean of T OE X is 13.61% with a standard deviation of 25.35%.
Next, to compute the impact of excess portfolio turnover on portfolio return, we use ATC based
on the methodology and findings of Edelen et al. (2013). They calculated ATC as 1.69% for large
funds (with an average of $2.88 billion assets) and 1.19% for small funds (with an average of $164
million assets), based on the 3,799 open-end domestic equity mutual funds data using quarterly
18

portfolio holdings data from Morningstar from 1995 to 2006. Fund size plays a role in trading
costs because estimated per unit trading costs are more than 30 bps higher for large funds than
for small funds. As a result of portfolio turnover, the net impact on annual portfolio return is
going to be a negative T OEX ∗AT C , or -0.23% for large funds and -0.16% for small funds. As
shown in Table 4, in all five tests the mean of excess returns is higher than 0.23%. Because the
benefit of equal-weighting is higher than the cost, equal-weighting makes economic sense even after
accounting for higher portfolio turnover costs.
Insert Figure 15here.
5 Conclusion
Recent studies by DeMiguel et al. (2009) and Plyakha et al. (2015) suggest that equal-weighted
portfolios (also known as 1/N, sometimes abbreviated as EWP) appear to outperform 14 different
portfolio weighting strategies. Several portfolio weighting strategies, including alternative beta
strategies7have emerged in the marketplace, as they all seem to outperform VW indices according
to Arnott et al. (2005), Chow et al. (2011), and Amenc et al. (2011a). Given the positive excess
returns of EWPs over VWPs in the equity asset class, two natural questions arise. First, Can this
excess return be realized under various market scenarios and time horizons? Second, what is the
source of these positive excess returns? In this paper we provide a theoretical framework for the
EWP model, and demonstrate using simulation and historical data that an EW strategy indeed
outperforms VW strategies. We then show that rebalancing is a key driver behind the positive
excess retun of the EWPs.
To undertake our analysis, we first develop a EWP and VWP model that allows for portfolio
rebalancing. This model demonstrates that after rebalancing a portfolio, if the smaller-cap stocks
outperform the larger-cap stocks, then the EWP will produce higher returns than a VWP. We use
five tests to confirm that the EWP outperforms the VWP in terms of return and Sharpe ratio.
In order to understand the source of the positive excess return of EWP over VWP, we decompose
the excess return into return due to rebalancing effect, and return due to size effect. The rebalancing
7http://www.edhec-risk.com/edhec_publications/all_publications/RISKReview.2015- 03-26.2929/attachments/
EDHEC_Publication_Alternative_Equity_Beta_Investing_Survey.pdf
19

effect does not depend on the market cap of stocks in the portfolio, and dependents solely on price
movements. The size effect depends on both price fluctuations and the market cap of stocks. We
show that in 19 out of 20 categories, rebalancing is the main contributor to the excess return. On
average, 85% of the excess returns are due to rebalancing effect, and the remaining 15% are due to
size effect.
Finally, we show that because of equal-weighting the excess returns are higher than the higher
costs incurred due to higher portfolio turnover. Therefore, even after accounting for higher portfolio
turnover costs, equal-weighting makes economic sense.
References
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Amenc, N., F. Goltz, and L. Martellini. 2011a. A Survey of alternative equity index strategies: A
comment. Financial Analysts Journal 67:14–16.
Amenc, N., F. Goltz, L. Martellini, and P. Retkowsky. 2011b. Efficient indexation: An alternative
to cap-weighted indices. Journal of Investment Management, Fourth Quarter .
Arnott, R. D., J. Hsu, and P. Moore. 2005. Fundamental indexation. Financial Analysts Journal
61:83–99.
Chow, T.-m., J. Hsu, V. Kalesnik, and B. Little. 2011. A survey of alternative equity index
strategies. Financial Analysts Journal 67:37–57.
DeMiguel, V., L. Garlappi, and R. 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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mutual fund performance. Financial Analysts Journal 69.
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Finance 25:383–417.
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Gibbons, M. R. 1982. Multivariate tests of financial models: A new approach. Journal of Financial
Economics 10:3–27.
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portfolios and capital budgets. Review of Economics and Statistics pp. 13–37.
Perold, A. F., and W. F. Sharpe. 1995. Dynamic strategies for asset allocation. Financial Analysts
Journal 51:149–160.
Plyakha, Y., R. Uppal, and G. Vilkov. 2014. Equal or Value Weighting? Implications for Asset-
Pricing Tests. SSRN, http://ssrn.com/abstract=1787045 .
Plyakha, Y., R. Uppal, and G. Vilkov. 2015. Why Do Equal-Weighted Portfolios Outperform
Value-Weighted Portfolios? SSRN, http://ssrn.com/abstract=2724535 .
Roll, R. 1977. A critique of the asset pricing theory’s tests Part I: On past and potential testability
of the theory. Journal of Financial Economics 4:129–176.
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21

Figure 1. Simulated excess return R2E−R2Vusing random prices PA0, PA1, PA2, PB0, PB1, PB2.
After 10,000 simulations, EWP produces positive returns 59.2% of the time, whereas VWP produces
positive returns 49.2% of the time. This pattern is consistent in repeated simulations. The excess
return R2E−R2Vis positive 65.5% of the time. The Gaussian fitted curve shows a mean return of
19% for EWP, and -1% for VWP. More detailed statistics can be seen in Table 4
.
Figure 2. Simulated VW return R2Vusing normal returns µL= 5%, µS= 8%, σL= 10%, σS=
15%.
22

Figure 3. Simulated EW return R2Eusing normal returns µL= 5%, µS= 8%, σL= 10%, σS=
15%.
Figure 4. Simulated excess return R2E−R2Vusing normal returns µL= 5%, µS= 8%, σL=
10%, σS= 15%. After 10,000 such simulations, the excess return is positive 56.4% of the time. The
mean, and median excess returns are also positive. More detailed statistics can be seen in Table 4.
23

Figure 5. Historical returns of Value-weighted (top 500 VW) and Equal-weighted (bottom 2000
by EW) index.
Figure 6. Excess returns (equal-weight minus value-weight) using historical data (1926-2014).
24

Figure 7. Value-weighted returns R2Vusing historical data (1926-2014) and Bootstrapping sim-
ulation. Annualized VW mean return for the period is 13%, with a standard deviation of 21%.
Figure 8. Equal-weighted returns R2Eusing historical data (1926-2014) and Bootstrapping sim-
ulation. Annualized EW mean return for the period is 14%, with a standard deviation of 23%.
25

Figure 9. Excess returns R2E−R2Vusing historical data (1926-2014) and Bootstrapping sim-
ulation. After 10,000 simulations, annualized EW mean return for the period is 1.01%, with a
standard deviation of 3.41%.. In addition, the Sharpe ratio of the EWP is also higher than the
VWP (0.461 vs 0.457), and a positive skewness of 2.23.
Figure 10. Excess returns R2E−R2Vusing historical data (1926-2014) and curve fitting. After
10,000 runs, the EWP produced 1.03% more annual return than the VWP. The excess return has
a positive skewness of 0.14. In addition, the Sharpe ratio of the EWP is also higher than the VWP
(0.63 vs 0.58).
26

Figure 11. Excess risk of EWP over VWP as dand echange. The excess risk is positive 99% of
the time and it varies from -2.18% to 4.37%, and has a mean of 1.15%, a median of 1.01%, and
a standard error of 0.01%. The excess risk is negative only when the large cap weight approaches
100% of the portfolio (shown in lower left part of the chart).
Figure 12. Excess Sharpe ratio of EWP over VWP. The excess Sharpe ratio is positive 63% of
the time and the average varies from 0.009 for small equal-weighting effect to 0.06 for large equal-
weighting effect. When e increases (move rightward on the x-axis), it means large cap weight is
increasing in the portfolio. In other words, the portfolio is deviating more from an equal-weight
portfolio. That means equal-weighting effect is going to be more noticeable. The average excess
Sharpe ratio for all the 10,000 simulations is 0.03. The 95% confidence interval of the mean excess
Sharpe ratio is between 0.0272 and 0.0331 with a t-stat of 20.
27

Figure 13. Decomposition of excess returns R2E−R2V(x-axis) into return due to rebalancing
effect R2E−R2E NR (shown in lower bars) using equation (19), and return due to size effect
R2E NR −R2V(shown in upper bars) using equation (20). As shown in this figure, 85% of the
excess returns are due to rebalancing effect.
Figure 14. In aggregate, almost all the excess return R2E−R2Vof 15.64% is due to the EWP
return R2Eof 15.97% over VWP return R2Vof 0.34%. In addition, almost all the excess return is
due to the rebalancing effect of 15.67%.
28

Figure 15. Portfolio excess return results versus excess portfolio turnover using random prices test
as explained in section 4.1. As portfolio turns more EW from VW, portfolio turnover increases and
excess return also goes up. For every 1% increase in portfolio return, portfolio turnover increases
by 0.4669%. The mean portfolio turnover is 13.61% with a standard deviation of 25.35%.
29

(1) Small cap and large cap returns NMean(µ) Median Stdev (σ) L95 U95
Small cap monthly return (Russell 2000). 12/1992 to 12/2014 264 0.80% 1.60% 5.52% 0.13% 1.46%
Large cap monthly return (Russell 1000). 12/1992 to 12/2014 264 0.70% 1.17% 4.28% 0.18% 1.21%
(2) Equal-weight and Value-weight index returns
Equal-weighted CRSP monthly returns (01/1926 to 12/2014) 1,068 1.15% 1.41% 6.85% 0.74% 1.56%
Value-weighted CRSP monthly returns (01/1926 to 12/2014) 1,068 0.94% 1.30% 5.49% 0.61% 1.27%
Stock prices used in EW-VW portfolios (monthly data, 12/1926 to 12/2014)) 4,150,448
Relationship between datasets P-value F-stat R-Square Corr. (ρ)RLRS
Relationship between Russell 2000 and Russell 1000 0.00 603.73 0.70 0.84
Relationship between CRSP VW and CRSP EW 0.00 11,424 0.91 0.96
(3) Scenarios for large cap and small cap (based on Russell 2000 and 1000) NσLσSdCorr. (ρ)RLRS
All months from 10/1992 to 12/2014 264 4.27% 5.52% 1.24% 0.84 0.70% 0.80%
Years 2007 and 2008 24 5.24% 6.38% 1.14% 0.94 -1.74% -1.67%
All positive months 161 2.70% 3.03% 3.33% 0.55 2.95% 4.21%
All negative months 103 3.81% 3.86% 4.91% 0.78 -2.82% -4.54%
Years 2000 and 2001 24 5.36% 7.41% 2.05% 0.63 -0.85% 0.13%
Years 2009 to 2014 72 3.83% 5.19% 1.35% 0.93 1.12% 1.23%
Table 1. Summary statistics include the following.
(1) Russell 2000 and 1000 monthly stock returns from 12/1992 to 12/2014, as proxies for small cap and large cap.
(2) CRSP equal-weight and value-weight index returns from 01/1926 to 12/2014.
(3) Different scenarios tested on Russell 2000 and 1000 monthly stock returns from 12/1992 to 12/2014, as proxies for small
cap and large cap.
30

Paired T-test for EWP and VWP
N Mean StDev SE Mean
R2E10,000 0.13883 0.23613 0.00236
R2V10,000 0.12869 0.21582 0.00216
R2E−R2V10,000 0.01014 0.03405 0.00034
95% confidence interval for R2E−R2V(0.00947, 0.01081)
T-Test of mean of difference = 0 (vs. #0)
P-value: 0.000 T-value: 29.78
Table 2. Results of matching historical data using Bootstrapping simulation. Pair-wise summary
statistics of EW return R2E, VW return R2V, and excess return of EWP R2E−R2Vshow a positive
excess return. A p-value of 0.000 suggests that EW returns are statistically different to VW returns
using the data from years 1926 to 2014.
Paired T-test for EWP and VWP
N Mean StDev SE Mean
R2E88 0.0572 0.2502 0.0267
R2V88 0.0165 0.1931 0.0206
R2E−R2V88 0.0407 0.0955 0.0102
95% confidence interval for R2E-R2V (0.0204, 0.0609)
T-Test of mean of difference = 0 (vs. #0)
P-value: 0.000 T-value: 3.99
Table 3. Constructed portfolio statistics of EW return R2E, VW return R2V, and excess return
of EWP R2E−R2Vusing the data from years 1926 to 2014. During this period EWPs produced
statistically significant excess annual return of 4.07%.
31

Table 4. Summary of EW R2E, VW R2E, and excess returns R2E−R2Vcomputed using all five
methods (i.e.: theoretical random prices, normally distributed returns, matching historical index
prices using Bootstrapping simulation, simulated returns using historical rates using curve fitting,
and actual portfolio construction). In all five methods, when compared to VWP returns, EWP
returns are higher and positive more number of times. In addition, Sharpe ratios are higher for
EWPs.
32