Abstract
The efficient market hypothesis purports that markets are efficient and modern portfolio theory (CAPM)
builds on that hypothesis and the mean variance theory of Markowitz (1952) to develop a market
portfolio, which is regarded as the optimal portfolio of all risky assets, generating the highest return for
given risks levels. However the theoretical failings of the CAPM, mostly due to the assumptions on which it
is founded, have resulted in the market portfolio – a capitalization weighted portfolio – generating returns
that are suboptimal. Other weighting methodologies and investment styles have occasionally
outperformed the market proxy. This research employs an alternative weighting methodology, first
proposed by Arnott, Hsu and Moore (2005), based on the fundamental variables of the firm. The
methodology has proven to be superior to the cap-weighted technique in terms of being mean variance
efficient, as well as resilient. In this research paper, portfolios of the top 50 and mid-100 stocks of the
Taiwanese equity market are constructed. The performance of the varying fundamental indexes are
compared to that of the cap-weighted portfolio and also are regressed against the CAPM and Fama and
French3-factor model (1993). The results indicate that only sales and the composite index outperform the
cap-weighted portfolio on a statistically significant basis after regressing against the Fama and French 3factor model. While size and value are statistically significant in explaining the alphas of portfolios
constructed from the top 50 stocks, stocks constructed from the mid-100 stocks present a unique
investment style, with alphas unsatisfactorily explained by value and size effect.
Keywords: Efficient market hypothesis, Fundamental indexation, Capitalisation weighted index, Mean
variance efficient, Size effect, Value effect, noisy market hypothesis, Capital asset pricing model, Fama and
French 3-factor model.
Introduction
The market portfolio is not only regarded as a convenient guide for capitalisation-weighted investing but is
also held by the CAPM as the de factor standard for investing. Modern portfolio theory advocates the
efficiency of markets whereby the market value of stocks is a reflection of its intrinsic value. Based on the
efficient market hypothesis (Fama, 1970), modern portfolio theory holds only in efficient market conditions,
without which stock prices become noisy and the market portfolio is stripped of its mean variance efficient
attribute. By being noisy, market prices move independently of the underlying variables of the firm (Hsu
and Campollo, 2005). This mispricing leads to a systematic overweighting of overvalued stocks and a
corresponding underweighting of undervalued stocks within a cap-weighted index, thereby introducing a
return drag in cap-weighted indexes (Arnott, Hsu and Moore, 2005). The degree of this return drag depends
on the level of noise in the prices (Treynor, 2005; Hsu, 2004). A new form of indexing – fundamental
indexation – proposed by Arnott et al. (2005) has proven to be more efficient at limiting the noise inherent
in cap-weighted indices by using fundamental variables of the firm., as opposed to their market
capitalisation values.
This research investigates the relative performance of fundamental indexation and cap-weighting on the
Taiwanese equity market. Upon constructing fundamental indexes for the top 50 and mid-100 stocks by
varying fundamental metrics of size, the performance of the top 50 and mid-100 stocks of Taiwan
Capitalisation Weighted Stock Index (TAIEX), weighted by market capitalisation, is compared with that of
individual fundamental indices, a composite fundamental index over the period 1st January 2001 to 30th
June 2014 The results should shed more light on the relative performance of the respective weighting
methodolodogies and the degree of mean reversion on the Taiwanese equity market, and upon regressing
the results against the CAPM model and Fama an French 3-factor model, determine what factors are
statistically important in explaining the performance of the fundamental indexes.
Theoretical overview
Despite its poor empirical record, a long memory of the Capital asset pricing model (CAPM) has been
retained not only by advocates of the modern portfolio theory but the finance community as a whole.
Stemming from Markowitz‘s efficient frontier of risky assets and gaining momentum from Fama’s efficient
market hypothesis, the CAPM was first unveiled by William Sharpe (1964).
According to Markowitz (1952), investors are risks averse and choose their investable portfolios taking into
consideration only the mean and variance of the portfolio. He eventually constructed the efficient frontier
of risky assets and the mean variance tangency portfolio of risky assets, which consist of all valuable risky
assets. Combined with the risk free asset, this portfolio provides the highest return for given risk and
minimum risk for given return. It is this property that makes it a theoretically optimal portfolio. In what is
described as the separation theorem (Tobin, 1958), Investors assume positions along the tangency portfolio
in accordance with their risk appetite. The mean variance portfolio is also described as the market portfolio.
Therefore, no other portfolio of risky assets, with similar risk attribution, can generate returns superior to
the market portfolio. The market portfolio is capitalization weighted; meaning, available and marketable
risky assets are included in this portfolio in proportion to the market capitalization of the asset.
The CAPM builds on the mean variance model and attempts to model the return characteristics of an asset
or portfolio of risky assets as a linear function of market related risk, known as the beta. The CAPM
assumes that other than the market risk premium, which is the beta of the asset compounded by the
premium per unit of beta, no other factor accounts for the expected return characteristics of an asset.
However, recent finance literature and empirical evidence has alluded to and shown that the return of an
asset is fitted by other factors. Fama and French (1993) developed a three-factor model of asset pricing
after demonstrating the existence of both size and value as significant factors in explaining cross-sectional
differences in asset returns. Carhart (1997) further extended the Fama and French three-factor model into
a four-factor model after revealing that the momentum effect is equally prevalent in stock pricing.
Amongst other valid justifications aimed at discrediting the mean variance efficiency assumption of the
CAPM, the theoretical pitfalls of the CAPM arise from the numerous simplifying assumptions on which the
model is founded.
First amongst these assumptions is the efficient market hypothesis (Fama, 1970). The efficient market
hypothesis (EMH) states that security markets are efficient and that prices of assets invariably reflect all
information about the asset (with the nature of the information dependent on the degree of market
efficiency). This therefore, means that the asset price unbiasedly reflects the true value of the asset, known
as the fair value. Therefore if markets are as efficient as upheld by EMH, with the market price of stocks
reflecting their fair value at each point in time or rapidly returning to true value after interim periods of
mispricing (mean reversion), any attempt by individual investors to generate sustained values for alphas
would be futile. Alpha is the return specific to any asset, which is unrelated or unattributable to the market
premium. The implications of EMH extend beyond stock pricing in that it upholds the mean variance
efficiency of the market portfolio because, as a capitalization weighted portfolio, containing all valuable
risky assets, if the market price of all assets are fairly reflected, then investing in the market portfolio
should generate the highest return for given levels of risks.
Arguments against the CAPM
Evidence against the CAPM cuts across its theoretical failings, inability to be reliably implemented as an
asset pricing model, simplifying assumptions and, above all, its misguided hailing of the market portfolio as
mean variance efficient.
With respect to its assumptions (risk-free borrowing and lending, uniform expectations amongst investors,
investor rationality, no short selling, et ceteras), CAPM assumptions are not only overly simplistic but also
more or less unrealistic. For instance, its assumption of markets being efficient is flawed, as markets often
display upheavals, with asset prices drifting far away, over sustained periods of time, from their intrinsic
values. Under these circumstances, the paradigm of the CAPM and its associated theory of efficient markets
may need to be replaced with a paradigm of markets as vulnerable to fickle behavior (Dempsey, 2013). The
adrift movement of market prices from their intrinsic or true values, as a result of observed prolonged
periods of market inefficiencies, implies that the market portfolio - which is capitalization weighted - is
weighted on misplaced prices, resulting in a weighting scheme that is ineffective and non-optimal. This
invariably results in sub-optimal returns on investments tracking the market portfolio or weighted on
market capitalization basis. Furthermore, Rolls (1977; 1978) criticizes the market portfolio as unobservable,
making it difficult to track its performance without employing a market proxy.
The theoretical basis of CAPM is also flawed as recent research has revealed that other factors, other than
just the market risk premium, have the power of explaining at least a part of the return characteristics of
stocks. Banz (1981) while performing research on NYSE stocks from 1936-1975 found that stocks/firms with
small market capitalization outperformed stocks/firms with larger market capitalization by risk-adjusted
returns of about 19.8%. This effect was described as the size effect. In 1978, Ball also showed that CAPM is
violated by the value effect, whereby stocks with high book-to-market (BTM) ratios generate higher riskadjusted returns than their low BTM counterparts. Fama and French (1992) also investigated the effect of
size in determining stock returns and discovered a positive correlation between small stocks and relatively
higher returns. Fama and French (1993) combined both the size and value effect to the original market risk
premium CAPM to introduce a 3-factor model of asset pricing. Jegadeesh and Titman (1993) exposed
evidence to the fact that prior winner stocks have the ability to maintain their winning momentum into the
near future. While performing research on NYSE and AMEX stocks over the period 1965-1989, they found
that prior winners outperformed prior losers by an average of 1.31% per month. Other effects have been
found in stock markets that account for the return dynamics of stock prices; some effects more persistent
than others but all supporting the fact that the market risk premium is hardly the only factor capable of
explaining the return characteristics of stock returns.
Despite being heavily criticized, capitalization weighted indexes do offer considerable benefits. Because
investing in the market portfolio is a buy and hold strategy, no transaction costs are incurred in investing in
this portfolio as the index automatically rebalances itself. The only costs incurred are stock buyback and
replacement expenses. Capitalization weighted indexes also allow for broad market participation and are
highly correlated with trading liquidity.
The logic of fundamental indexation and supporting literature
The market portfolio’s optimal and mean variance efficiency status is only justified in an efficient market
setting. Because should prices deviate from their true value, and some stocks become either overvalued or
undervalued, this results to the assigned weights being amiss. Deviation of stock prices from their intrinsic
values is described as noise. Siegel (2006) advanced the noisy market hypothesis, whereby he argues that
prices of securities are likely to be influenced by speculators and momentum traders, and that the changes
in share prices are not often related to their fundamental value but as a result of investors buying and
holding stocks for diversification or tax-related reasons. Owing to the weighting methodology of the market
portfolio, which is capitalization weighted, noise in stock prices results in overvalued stocks being overrepresented in the market portfolio while undervalued stocks are under-represented (Arnott, Hsu and
Moore, 2005). This results in the capitalization weighted market portfolio being biased towards
large/growth stocks. It is this weighting flaw that results in a return drag.
Arnott et al. (2005) propose a more rationale way of weighting stocks in portfolios. They recommended the
use of fundamental metrics of size such as book value, sales, dividends and the like to weight stocks. Their
argument in favour of this size metrics is founded on the notion that fundamental variables are noise
resistant and unlike share prices, which are unobservable (Perold, 2007) and sometimes fluctuate for
alternative reasons other than reflecting the fundamental value of the share (Sharpe, 1974), provide a fairly
reasonable reflection of the intrinsic value of the firm. DeBondt and Thaler (1985) also support the volatile
nature of the share price by revealing how changes in share prices are influenced by human behavior and
investors’ overreaction to new information; unfairly inflating or decreasing the share price, with
subsequent reversal to their mean prices.
Therefore weighting portfolio stocks using fundamental metrics of size generate alphas, which are
statistically significant, especially during periods of stock market price turmoil, with the size of the alpha
closely related to the level of noise in stock prices (Treynor, 2005).
As a result of the constant rebalancing required, Fundamental indexation (FI) has been affiliated to being an
active investment strategy with associated transaction cost; unlike the buy and hold passive strategy of capweighted indexes. FI has also been criticized as being a repackaging of value investing (Asness, 2006), with
greater emphasis being placed on value stocks. Skeptics of FI claim that the alpha generated can be
associated with the value effect; that is, the excess risk-adjusted return inherent in value investing as
opposed to being a superior weighting methodology to capitalization weighting. If capitalization weighted
indexes are growth biased, then fundamental indexes are value biased (Kaplan, 2008). If FI are value biased
and because value stocks are usually small sized stocks, with exposure to distress risks, this means FI alphas
are compensation for the additional risk borne for investing in fundamental indexes. However, on
regressing the results of Fundamentally indexed portfolios over the Fama and French 3-factor model,
Arnott et al (2005) found that fundamental indexes did not have any significant loading on the size factor.
Contrarily, Chow, Hsu, Kalesnik and Little (2011) found that fundamental index alphas, when regressed
using the Carhart (1997) 4-factor model, are not significant and are primarily ascribed to size and value
factor loadings.
Preliminary work on FI was performed by Arnott et al. (2005). Using data from the CRPS database, their
research covered the period 1962-2004 and focused on the top 1000 companies by fundamental metrics.
Fundamental indexes constructed book value, and trailing five-year average of sales, dividends and cash
flow. Rebalancing was done on the first day of the year using previous end of year values. A cap-weighted
reference portfolio was also constructed under the FI methodology, as well as a composite index of four
fundamental metrics of size, and the S&P 500 proxied the market. Their findings revealed that fundamental
indexes, on average, generate excess market returns of 1.97 percentage points and excess reference
portfolio returns of 2.15 percentage points. Sales index was the best performing index with an excess
return over the reference portfolio of 2.56 percentage points. Fundamental indices did not only outperform
the reference portfolio but did so with lower betas. The performance of fundamental indexes also exhibited
resilience across time, across phases of the business cycle, bull and bear markets and rising and falling
interest rate regimes. Hsieh, Hodnett and Rensburg (2012) also investigate the performance of
fundamental indexes against cap-weighted indexes on the US market, as well as the effect of portfolio
concentration on their respective performances. Their results reveal similar findings to those of Arnott et al
(2005) and also highlight the fact that unlike fundamental portfolios, cap-weighted portfolios are exposed
to portfolio concentration, with greater concentration having a return drag on the returns of cap-weighted
portfolios.
Hemminki and Puttonen (2008) also performed test on European stocks using DJ Stoxx index 50 as the
market proxy and constructing fundamental indexes similar to that of Arnott et al. (2005), with the
exception of end of year rebalancing and three year moving averages (both differences having little or
no effect on the outcome). Fundamental indexes outperform their comparative cap-weighted portfolio
by an average of 1.76%
Arnott and Shepherd (2009) show that fundamental indexation has even greater potential to add value in
developing markets where mispricing and market inefficiency is more pronounced. Because emerging
markets are much less efficient than their developed markets counterparts, the return drag on capweighted indices is much greater, creating more opportunities for generating excess returns through
fundamental indexation. They highlight the fact that higher growth prospects in developing markets also
allow for greater diversification. By analysing the growth of a dollar across the period 1994 through
December 2009, they realised that the emerging market RAFI index adds more value than the cap-weighted
counterpart (FTSE AW index) by an excess of 9.0%. The value added increases with the noisiness of the
market. However, Lobe and Walkshäusl (2008), on performing research on 50 countries, found that Taiwan
– an emerging market – produced negative excess returns on a FI basis
Estrada (2008) investigated the presence of international diversification with fundamental indexation and
found no evidence of added diversification.
Data and Methodology
This research covers the period January 2001 to June 2014. The period is chosen in line with the availability
of consistent data such that average-trailing values for fundamental indexes can be calculated. Data is
drawn from the Taiwanese Economic Journal database that encompasses data for stocks listed both on the
primary stock market and over the counter market called the GreTai securities market (GTSM), where
stocks with lower market capitalization are listed. The market proxy used for this research is Taiwan
capitalization weighted stock index (TAIEX). As at June 2014, 841 stocks formed part of this index, weighted
according to their market capitalization.
Weight of each stock included in the index is given by the following formula:
PiNi
Wi = ∑𝑛
𝑖=1 𝑃𝑁
1.1
Whereby Wi symbolizes the weight of stock i, the numerator PiNi represents the price of stock i multiplied
by its number of shares N. While the denominator represents the sum of all stock prices in the index
multiplied by their respective number of shares. The risk free rate is the interest rate on short-term bills;
that is, the money market interest rates on commercial paper borrowings (31-90days) in the secondary
market.
The fundamental metrics of size used are similar to four of the fundamental metrics employed by Arnott et
al. (2005); book value of equity, earnings, dividends and sales. However the accumulated sales for the last
12 months of trading is used. The outcome of using the accumulated last 12 trading months of sales as
opposed to monthly sales values is not materially different. End of month values for all the other
fundamental metrics of size are obtained. Dividends needed special attention in terms of adjustments since
dividends are paid months after they are due. This eliminates any look-ahead bias and ensures that monthly
dividends figures correctly reflect corresponding values.
Contrary to research performed by Arnott et al. (2005) but in line with Hemminki and Puttonen (2008),
three year trailing averages of the fundamental metrics are derived and the values are logged.
In order to match the benchmark index construction on the Taiwanese market, whereby indices of top 50
stocks and mid-100 stocks by market capitalisation are constructed, all fundamental indexes are also
constructed in tranches of top 50 and mid-100 stocks by fundamental metric weights. The mid-100 stocks
are the next largest 100 stocks subsequent to the top 50 stocks of the varying weighting metrics of size.
Rebalancing of fundamental indexes is done on a monthly basis. This is suggestive of higher transaction
costs but because only the top 50 and mid-100 stocks are used, and because FI emulates cap-weighted
indexes in terms of broad market participation and liquidity, most of the elite stocks by market
capitalization are also represented by the fundamental indexes. The Taiwanese market is also fraught with
electronic traded funds (EFTs), which are highly liquid and not subject to high rebalancing costs. Frequent
rebalancing in this research was merely for more accurate representation of fundamental weights in the
respective indexes.
A composite index and a reference portfolio are also constructed. The composite index equally weights the
four fundamental metrics of size and when, for whatever reason, one of the fundamental metrics is not
available, the average of the remaining metrics is used. The reference portfolio is the capitalizationweighted index constructed under the fundamental index methodology. The construction of the reference
portfolio provides a more rationale benchmark to evaluate the performance of the fundamental indexes, as
like is compared with like.
In order to assess whether or not fundamental indexes are have a loading on size and value factors, using
the Fama and French 3-factor model, portfolios of size and value are constructed. In constructing the size
portfolio, the monthly return of large size portfolios is deducted from that of the small size portfolios. Small
size stocks are considered stocks that constitute the bottom twenty percentile by market capitalization
while large stocks are those constituting the top twenty percentile by market capitalization. For value
portfolios, the average monthly return of small value stocks is subtracted from the average return of large
value stocks. Large value stocks are stocks in the top twenty percentile by BTM ratio while low value stocks
are stocks in the bottom twenty percentile by BTM ratio. Below is the Fama and French 3-factor model used
for determining the expected return of a stock;
E(ri) - rf = αff + βm(E(rm- rf) + βHMLrHML + βSMBrSMB + εi
1.2
Where,
βm
signifies the sensitivity of the stock to market returns
E(rm- rf) represents the excess return of the market over the risk free rate
βHML
signifies the sensitivity of the stock to excess returns of value stocks
βSMB signifies the sensitivity of the stock to excess return of small stocks
rHML
represents the excess return generated by stocks with high book to market value
rSMB
represents the excess return generated by small capitalisation stocks relative to
big stocks
αff
represents the Fama and French alpha
Results and Analysis
In analyzing the test results of the different portfolio concentrations; that is, the top 50 and mid-100
portfolios, a break down of the statistics is presented in the order of basic return statistics, basic risks
statistics and risk-adjusted returns. Statistics for regression against the CAPM and Fama and French 3-factor
model are also presented.
The cumulative return and draw down graphs for the top 50 stocks are shown in Figure 1.1 and 1.2. The
results are supportive of the arithmetic and geometric results discussed above.
Figure 1.1: Index Cumulative Return for Top 50
Mkt Proxy
Dividends
Composite
Book Value
SalesL12M
Earnings
Reference
3.5
3
2.5
Index Cum. 2
Return (%) 1.5
1
0.5
Date
1/1/2014
1/1/2013
1/1/2012
1/1/2011
1/1/2010
1/1/2009
1/1/2008
1/1/2007
1/1/2006
1/1/2005
1/1/2004
1/1/2003
1/1/2002
1/1/2001
0
Figure 1.2: Index Draw Down for Top 50 Stocks
Market proxy
Dividends
Fund. Composite
Book Value
Sales L12M
Earnings
Reference
0
-0.1
-0.2
-0.3
Index Draw
Down
-0.4
-0.5
1/1/2014
1/1/2013
1/1/2012
1/1/2011
1/1/2010
1/1/2009
1/1/2008
1/1/2007
1/1/2006
1/1/2005
1/1/2004
1/1/2003
1/1/2002
1/1/2001
-0.6
Date
Table 1 presents the performance statistics for the Top 50 stocks. In terms of the arithmetic returns of the
indexes, the highest performing index is the sales index with an annual arithmetic return of 12.23%,
followed by the composite index with an annual return of 10.70%. All fundamental indexes outperform the
reference (capitalization-weighted) portfolio, which garnered a soft return of 5.76%. All fundamental
indexes, with the exception of dividends, outperform the market proxy.
On average, fundamental indexes outperform the market proxy and reference portfolio by 1.35% and
3.75% respectively. The geometric return, which is a better reflection of the average index returns, confirms
the results of the arithmetic return and even better, as the dividend index also outperforms the market
proxy. The large chasm between the arithmetic and geometric returns is triggered by the high levels of
standard deviation in the returns of the respective indexes. The higher the standard deviation of the index
monthly returns, the greater the discrepancy in the arithmetic and geometric returns will be.
Fundamental indexes, however, achieve greater returns at slightly higher risk. Measured in terms of
standard deviation, all fundamental indexes show higher standard deviations with the Book value reflecting
the highest value of 29.92% relative to the market proxy standard deviation of 23.68%. Sales index
produces the lowest standard deviation for fundamental indexes. The higher standard deviation exhibited
by the fundamental indexes relative to the market proxy is attributable to the lover diversification inherent
in the portfolios, since the market proxy contains over a thousand stocks listed on the Taiwanese market
while the fundamental indexes only contain the top 50 stocks.
Table 1: Performance Statistics for Top 50 Stocks
Market
Proxy
Risk
Free
rate
Reference
Portfolio
Book
Value
Index
Earnings
Index
Dividends
Index
Sales
Index
Fundamental
Composite Index
Arithmetic Return
8.13%
1.30%
5.76%
8.69%
8.59%
7.35%
12.23%
10.70%
Geometric Return
2.77%
0.01%
2.17%
4.07%
4.56%
3.60%
8.96%
5.20%
Cumulative Return
1.446
1.001
1.335
1.713
1.825
1.613
3.184
1.982
23.68%
0.03%
26.53%
29.92%
27.96%
26.89%
24.68%
27.15%
1
N/A
1.085
1.152
1.112
1.076
1.005
1.060
-56,49%
-54,35%
-57,06%
-52,94%
-56,26%
Basic Return statistics
Basic Risks Statistics
Standard Deviation
Beta
Max. Draw Down
-56.26%
-
-56,84%
Risk-adj. returns
Sharpe Ratio
0.289
N/A
0.168
0.247
0.261
0.225
0.443
0.346
Treynor Ratio
0.068
N/A
0.041
0.064
0.066
0.056
0.109
0.089
Jensen’s alpha
-
N/A
-2.95%
-0.48%
-0.31%
-1.30%
4.06%
2.16%
Information ratio
0.345
N/A
N/A
0.302
0.428
0.257
0.852
0.604
M-square
0.081
N/A
0.053
0.071
0.075
0.066
0.118
0.095
Beta, which is a measure of how movements in the market return affect index returns, also corroborates
the risk measure of standard deviation, as all fundamental indexes generate slightly higher betas than the
market proxy. However, only the betas of book value and dividends indexes outweigh that of the reference
portfolio. Whilst the sequence of the performance of fundamental indexes is similar to that performed by
Arnott et al. (2005), the higher figures for risks measures are a sharp contrast. The maximum draw down
values for the fundamental indexes are comparable with that of the market proxy and reference portfolio.
Because the evaluation of returns on a comparative basis is only valid when risks of the different elements
are taken into consideration, return-risk measures of performance are employed. The Sharpe ratio (excess
index return over risk free rate to standard deviation), Treynor ratio (excess return on risk free rate to beta)
and the M-squared indicate that fundamental indexes perform poorly relative to the market proxy. This
translates to the inadequate compensation for higher risks inherent in the fundamental portfolios relative
to the market proxy. Only sales and composite indexes outperform the market on those risk-adjusted basis.
All fundamental indexes nonetheless outperform the reference portfolio on a Sharpe ratio, Treynor ratio
and M-squared basis. All fundamental indexes project negative Jensen’s alphas except for sales and the
fundamental composite index. However, despite the negative alphas shown by Fundamental indexes, they
still outperform the cap-weighted reference portfolio.
Fig 1.3: Security Market (CAPM) Line Top 50 Stocks
Index Return
(%)
16
14
Sales L12M
12
Composite
10
Book Value
Earnings
Dividends
Market Proxy
8
6
Reference
4
2
Rf: 1.30
0
0
0.5
1
1.5
2
2.5
Beta
The information ratio (IR), which is an active performance measurement ratio and signals the excess return
per unit of standard deviation of the excess return of the index on the reference portfolio, supports prior
evidence of the superior performance of fundamental indexes to the reference portfolio. All fundamental
indexes, including the market proxy, exhibit positive IR because they all earn higher returns than the capweighted reference portfolio and therefore have higher active returns. However, when the active returns
are standardized by active risks, only earnings, sales and the composite indexes show higher IR compared
to the market proxy. Overall, sales and the composite indexes are the best performing indexes of the
fundamental indexes.
Figure 2.1: Index Cumulative Return for Mid 100 Stocks
Mkt Proxy
Book Value
Earnings
SalesL12M
Reference
Composite
Dividends
6
5
4
3
Index Cum. Return
2
(%)
1
0
Date
Figure 2.2: Index Draw Down for Mid-100 Stocks
Market Proxy
Dividends
Composite
Book Value
Sales L12M
Earnings
Reference
0
Index Draw
Down
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
Date
Table 2 presents the performance statistics of mid-100 stocks. Although generating slightly higher standard
deviations than the top 50 stocks, the excess return on the top 50 stocks is much greater. All fundamental
indexes outperform the market proxy and reference portfolio on both the arithmetic return and geometric.
Figure 2.1 and 2.2 present the cumulative return and draw down values for the varying indexes constructed
from the mid-100 stocks. The maximum draw down values of the fundamental indexes are higher than that
of the market proxy, as opposed to comparative as was observed for the top 50 stocks.
Table 2: Performance Statistics for Mid-100 Stocks
Market
Proxy
Risk
Free
rate
Reference
Portfolio
Book
Value
Index
Earnings
Index
Dividends
Index
Sales
Index
Fundamental
Composite
index
Arithmetic Return
8.13%
1.30%
9.50%
16.13%
12.53%
13.45%
19.48%
17.85%
Geometric Return
2.77%
0.01%
4.97%
10.84%
8.01%
8.77%
14.90%
12.91%
Cumulative Return
1.446
1.001
1.926
4.011
2.829
3.113
6.520
5.152
23.68%
0.03%
29.36%
31.10%
29.02%
29.41%
28.48%
29.86%
1
N/A
1.157
1.171
1.114
1.119
1.084
1.096
-56.26%
-
-63,96%
-62,37%
-59,59%
-60,19%
-57,26%
-59,30%
Sharpe Ratio
0.289
N/A
0.279
0.477
0.387
0.413
0.638
0.554
Treynor Ratio
0.068
N/A
0.071
0.125
0.098
0.105
0.162
0.141
Jensen’s alpha
-
N/A
0.30%
6.73%
3.38%
4.23%
10.51%
8.52%
Information ratio
-0.123
N/A
N/A
0.664
0,506
0.728
1.537
1.029
M-square
0.081
N/A
0.079
0.126
0.105
0.111
0.164
0.144
Basic Returns statistics
Basic Risks Statistics
Standard Deviation
Beta
Max. Draw Down
Risk-adj. returns
On average, fundamental indexes of mid-100 stocks generate an excess return of 7.76% and 6.38% over the
market proxy and reference portfolio respectively. Higher returns are generated at higher risk with respect
to the market but comparative risk with respect to the cap-weighted reference portfolio. All fundamental
indexes, despite the higher betas and standard deviations, produce better performance on a risk-adjusted
basis than the reference index and market proxy. In terms of Sharpe ratio, Treynor ratio and M-squared, all
fundamental indexes outperform the market proxy and reference portfolio. Figure 2.3 below demonstrates
how all the fundamental indexes generated positive abnormal returns, measured in terms of Jensen’s alpha
and also reveal that all fundamental indexes outperformed the reference portfolio. Plotting above the SML
line is indicative of an undervalued portfolio and vice versa.
Fig 2.3: Security Market (CAPM) Line Mid-100 Stocks
Index Return
(%)
25
20
Sales L12M
Composite
Book Value
15
Dividends
Earnings
Reference
10
SML: Market Proxy
5
Rf: 1.30
0
0
0.5
1
1.5
2
2.5
Beta
Albeit maximum draw downs of fundamental indexes for mid-100 stocks are much larger than the market
proxy, fundamental index maximum draw downs are lower than the reference index, and the ability of
fundamental indexes to outperform the market and the reference by such huge arithmetic and riskadjusted return margins after incurring such large draw downs, is indicative of the recovery or bounce back
potential of fundamental indexes after bear markets or periods of market crisis. It can also be deduced that
the degree of index concentration affects the performance of the respective indexes. The higher the index
concentration, the greater the return drag on performance. Because top 50 stocks have a higher degree of
index concentration than mid-100 stocks, the performance of mid-100 stock indexes is more glorious than
its counterpart top 50 stocks. This indicates that the benefits of diversification are more inherent in
fundamental indexes than capitalization weighted indexes.
Based on active risk, the market proxy earns a negative IR, which is in direct contrast to the positive IR
displayed by all fundamental indexes.
Performance Attribution
In order to determine the significance of the alphas of fundamental indexes and what factors account for
such outperformance, CAPM regressions and Fama and French 3-factor model regressions are performed.
Table 3, split into panel A and B, displays the comparative results of the CAPM regression of the top 50 and
mid-100 stocks respectively.
Table 3: CAPM Regression Statistics
PANEL A: Top 50 Stocks
Reference
Book Value
Earnings Index
Dividends
Sales Index
Portfolio
Index
R-Squared
93.85%
83.24%
88.89%
89.89%
93.07%
85.66%
[P-value]
0.000
0.000
0.000
0.000
0.000
0.000
Intercept
-0.002
-0.000
-0.000
-0.001
0.003
0.002
t-statistics
-1.546
-0.141
-0.120
-0.523
2.088
0.701
[P-value]
0.124
0.888
0.905
0.601
0.038
0.485
b_Market risk premium
1.085
1.151
1.112
1.075
1.005
1.061
t-statistics
49.573
28.282
35.897
37.838
46.513
31.007
[P-value]
0.000
0.000
0.000
0.000
0.000
0.000
Reference
Book Value
Earnings Index
Dividends
Sales Index
Fundamental
Portfolio
Index
R-Squared
81.68%
87.97%
87.39%
87.24%
87.22%
86.98%
[P-value]
0.000
0.000
0.000
0.000
0.000
0.000
Intercept
0.006
0.003
0.003
0.008
0.000
0.006
t-statistics
1.641
1.104
1.328
3.330
0.084
2.557
[P-value]
0.103
0.271
0.186
0.001
0.933
0.012
b_Market risk premium
1.186
1.149
1.160
1.123
1.157
1.175
t-statistics
26.791
34.305
33.409
33.173
33.146
32.800
[P-value]
0.000
0.000
0.000
0.000
0.000
0.000
Index
Fundamental
Composite Index
PANEL B: Mid-100 Stocks
Index
Composite Index
The results indicate that, for the top 50 stocks, the excess returns of all portfolios are well explained by the
market risk premium, with statistically significant beta coefficients and coefficients of determination (R 2).
However, only the sales index generates a statistically significant alpha of 0.3% with t-statistics of 2.088 at a
5% significance level.
This observation is reprised in the mid-100 stocks CAPM regression but the
composite index of the mid-100 stocks also generates a statistically significant alpha of 0.6%, with a tstatistics of 2.557. The factor loading on the market risk premium is confirmed by the high values of Rsquared for the fundamental indexes, indicative of the fact that market movements account for more than
85% of the returns of the fundamental indexes.
Table 4, with Panel A and B representing top 50 and mid-100 stocks respectively, reveals that when style
risk is incorporated into the analyses of portfolio performance using the Fama and French 3-factor model,
all fundamental indexes of the top 50 stocks exhibit significant positive coefficients to the size risk
premium, thereby evidencing a small cap bias. This implies that size is also significant in fitting the returns
of all fundamental indexes. This provides further support to the argument that fundamental indexes
generate alphas because of the possible size loading in their stock selection. Thus, fundamental indexes
tend to focus on small size stocks by market capitalization. The reference portfolio has no statistically
significant loading on either the size or value factor and the alpha is entirely explained by the market risk
premium. In addition, the book value, earnings and composite indexes exhibit significant value bias.
However, only the sales and composite indexes generate statistically significant alphas, measured by the
regression intercept, after style risks are controlled for.
After portfolio concentration is lowered, size is significant in explaining only the returns of the fundamental
indexes of book value and the composite for the mid-100 stocks. With the exception of earnings index
alone, the market risk premium is equally significant in explaining index returns. Value effect fails to
satisfactorily explain the alphas of fundamental indexes of mid-100 stocks. Only the sales index retains a
statistically significant alpha after controlling for factor loadings of style. This result is in line with that of
Arnott et al. (2005) and Hsieh (2013), where they found that sales index was superior to other indexes due
to the relative predictability associated with this index in comparison with other fundamental variables like
earnings and dividends.
These findings may also be suggestive of the fact that most of the alphas of the sales and composite
indexes of the top 50 stocks are accounted for by size and/or value since the fundamental composite of the
mid-100 stocks exhibits no statistically significant alpha with size reduction. All in all, fundamental indexes
formed from mid-100 stocks have a unique investment style inexplicable by known anomalies.
Table 4: Fama and French 3-factor Regression Statistics
PANEL A: Top 50 Stocks
Reference
Book Value
Earnings Index
Dividends
Sales Index
Portfolio
Index
R-Squared
83.66%
88.48%
87.79%
87.92%
87.32%
88.16%
[P-value]
0.000
0.000
0.000
0.000
0.000
0.000
Intercept
0.007
0.003
0.004
0.009
0.001
0.008
t-statistics
2.371
1.273
1.629
3.765
0.365
3.142
[P-value]
0.019
0.205
0.105
0.000
0.715
0.002
b_Market risk premium
1.293
1.188
1.205
1.184
1.184
1.253
t-statistics
24.796
29.093
28.277
28.855
27.313
29.415
[P-value
0.000
0.000
0.000
0.000
0.000
0.000
b_SMB (Size effect)
0.325
0.130
0.137
0.184
0.078
0.239
t-statistics
3.872
1.982
1.992
2.780
1.113
3.483
[P-value
0.000
0.049
0.048
0.006
0.268
0.001
b_HML (Value effect)
0.075
0.052
0.033
0.032
0.009
0.057
t-statistics
1.890
1.675
1.001
1.031
0.261
1.756
[P-value
0.061
0.096
0.318
0.304
0.794
0.081
Reference
Book Value
Earnings Index
Dividends
Sales Index
Fundamental
Portfolio
Index
R-Squared
83.58%
88.95%
90.09%
93.13%
94.01%
86.31%
[P-value]
0.000
0.000
0.000
0.000
0.000
0.000
Intercept
0.003
-0.000
0.000
0.003
-0.003
0.004
t-statistics
0.831
-0.120
0.022
2.013
-2.107
1.570
[P-value]
0.407
0.905
0.983
0.046
0.037
0.119
b_Market risk premium
1.210
1.120
1.111
1.016
1.053
1.128
t-statistics
24.078
0.474
31.650
37.863
39.086
27.070
[P-value
0.000
0.636
0.000
0.000
0.000
0.000
b_SMB (Size effect)
0.167
0.029
0.100
0.036
-0.088
0.185
t-statistics
2.067
0.474
1.763
0.831
-2.035
2.751
[P-value
0.040
0.636
0.080
0.407
0.044
0.007
b_HML (Value effect)
0.007
0.021
0.005
0.015
0.004
-0.008
t-statistics
0.185
0.705
0.201
0.720
0.186
-0.250
[P-value
0.853
0.482
0.841
0.473
0.853
0.803
Index
Fundamental
Composite Index
PANEL B: Mid-100 Stocks
Index
Composite Index
Conclusion
Modern portfolio theory argues that investors are rational and that the returns of a security are solely
accounted for by the market risk premium. Moreover, it upholds the market portfolio as mean variance
efficient, under the assumption that markets are efficient, at least in one of the forms. Evidence of
outperformance the market portfolio, which is capitalization weighted, by other weighting and investment
strategies have cast doubt on the market portfolio’s mean variance efficiency status and Arnott et al. (2005)
propose an alternative weighting strategy based on price-insensitive metrics size called FI. Despite
overwhelming evidence of superior performance by FI, skeptics of the model have attributed the
performance to it being a rehash of value investing, noise in the market or its undue focus on small size
stocks.
This research investigated the relative performance of fundamental indexes, constructed at different
portfolio concentration levels, on the Taiwanese equity market. The results reveal that fundamental
indexes from mid-100 stocks display better performance than those of the top 50 stocks. The performance
of the top 50 stocks can largely be explained by size, value and the market risk premium. Thus, with the
exception of sales and the composite indexes, which are the best performing fundamental indexes,
fundamental indexes of top 50 stocks do not generate statistically significant alphas after controlling for
size and value. For the mid-100 stocks, size and value could not adequately explain the performance of the
fundamental indexes and therefore present unique investment styles. Sales index of the mid-100 stocks,
nonetheless, produced statistically significant alphas after controlling for known risk factors.
Concentration levels noticeably influence the performance of FI. As concentration levels decrease from top
50 to mid-100 stocks, fundamental indexes become undervalued, generating higher returns for
comparative market risk levels. Results also reveal that although fundamental indexes experienced large
drawdowns in period of market crisis, they are quick to recover and possibly outperform their
capitalization-weighted counterparts.
Bibliography
1) Arnott, R.D., Hsu, J.C. & Moore, P. (2005). Fundamental Indexation. Financial Analysts Journal, 61(2),
83-99.
2) Arnott, R. D., & Shepherd, S. D. (2009). The Fundamental Index® Concept in Emerging Markets .
Research Affiiliates, 1-10.
3) Asness, C. (2006). The Value of Fundamental Indexing. Institutional Investor. Journal of Finance, 67-71.
4) Banz, R. (1981). The Relationship Between Return and Market Value of Common Stock. The Journal of
Financial Economics, Vol. 9, 3-18.
5) Carhart, M. M. (1997). On Persistence in Mutual Fund Performance. Journal of Finance 52, 57-82.
6) Chow, T., Hsu, J., Kalesnik, V. and Little, B. (2011). “A Survey of Alternative Equity Index
Strategies”, Financial Analysts Journal, 67 (5), pp. 37-57.
7) Dempsey, M. (2013). The Capital Asset Pricing Model (CAPM): The History of a Failed
Revolutionary Idea in Finance? Abacus, Vol. 49, Supplement.
8) De Bondt, W., & Thaler, R. (July 1985). Does the Stock Market Overreact? The Journal of Finance, Vol.
XL, No. 3 , 793-804.
9) Estrada, J. (2008). Fundamental Indexation and International Diversification. Journal of Portfolio
Management, Vol. 34, No.3, 93-109.
10) Fama, E. (1970). Efficient Capital Markets: A Review of Theory and Empirical Work. Journal of Finance 25,
383-417.
11) Fama, E., & French, K. (1992). The cross section of expected stock returns. Journal of Finance 47, 427–
465.
12) Fama, E., & French, K. (1993). Common risk factors in the returns on stocks and bonds. Journal of
Financial Economics, 33(1), 3-56.
13) Hemminki, J., & Puttonen, V. (2008). Fundamental indexation in Europe. Journal of Asset Management,
8, 401–405.
14) Hsieh, H.H. (2013) Unlocking the Secrets of Fundamental Indexes: Size effect or value effect? Evidence
from emerging markets. Investment Management and Financial Innovation Journal, 10(4), 48-62
15) Hsieh, H.H., Hodnett, K., & Rensburg P.V. (2012). Application of Tactical Style Allocation for Global
Equity Portfolios. Journal of Business and Economics Research Journal 11(7), 745-752.
16) Hsu, J.C. (2004). Cap-Weighted Portfolios Are Sub-optimal Portfolios. Research White Paper, WP5401
17) Hsu, J., & Campollo, C. (2006). New Frontiers in Index Investing: An Examination of Fundamental
Indexation. Journal of Indexes, 32-38.
18) Jegadeesh, N., & Titman, S. (1993). Returns to Buying Winners and Selling Losers: Implications for Stock
Market Efficiency. The Journal of Finance 48, 65-91.
19) Kaplan, P.D. (2008). Why Fundamental Indexation Might - or Might Not - Work. Financial Analysts
Journal, 64(5), 32-39.
20) Lobe, S., & WALKSHÄUSL, C. (2008). Fundamental indexation around the world.
21) Markowitz, H. (1952). Portfolio Selection. Journal of Finance, 7(1), 77-91.
22) Perold, A. F. (2007). Fundamentally flawed indexing. Financial Analysis Journal, 63, 31–37.
23) Roll, R. (1977). A Critique of the Asset Pricing Theory’s Tests. Journal of Financial Economics, 4(2), 129176.
24) Siegel, J. (2006) The ‘noisy market’ hypothesis. Wall Street Journal, June 14, 2006, A14.
25) Tobin, J. (1958). Liquidity Preference as Behaviour Towards Risk. Review of Economic Studies, 65-86.
26) Treynor, J. (2005) Why market-valuation-indifferent indexing works. Financial Analysts Journal, 61, 6569.