Portfolio Performance Evaluation - Infoserver |...
Transcript of Portfolio Performance Evaluation - Infoserver |...
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Diplomarbeit
zur Erlangung des akademischen Grades
Magister rerum socialium oeconomicarumque
(Mag. rer. soc. oec.)
Portfolio Performance Evaluation
Institut für Betriebswirtschaftslehre
Universität Wien
Studienrichtung: Internationale Betriebswirtschaft
o. Univ.-Prof. Dr. Josef Zechner
eingereicht von
Johann Aldrian
(Matr.nr.: 9501942)
Wien, 8. September 2000
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Eidesstattliche Erklärung
Ich erkläre hiermit an Eides statt, daß ich die vorliegende Arbeit selbständig und
ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe. Die
aus fremden Quellen direkt oder indirekt übernommenen Gedanken sind als
solche kenntlich gemacht.
Die Arbeit wurde bisher in gleicher oder ähnlicher Form keiner anderen
Prüfungsbehörde vorgelegt und auch nicht veröffentlicht.
Wien, 8. September 2000 ……………………….
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PortfolioPerformanceEvaluation
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TABLE OF CONTENT
1. INTRODUCTION 1
1.1. THE RELEVANCE OF PORTFOLIO-MANAGEMENT-EVALUATION 11.2. STRUCTURE OF THIS MASTER'S THESIS 2
2. TRADITIONAL MEASURES OF PORTFOLIO PERFORMANCEEVALUATION AND ITS IMPLICATIONS. 4
2.1. FUNDAMENTALS 42.1.1. THE CONCEPT OF EFFICIENT MARKETS 42.1.2. RETURN AND RISK AS DETERMINANTS OF THE MARKET 62.2. PORTFOLIO MANAGEMENT 82.2.1. ACTIVE PORTFOLIO MANAGEMENT 92.2.2. PASSIVE PORTFOLIO MANAGEMENT 122.2.3. WHAT INDEX TO USE 132.3. TRADITIONAL MEASURES OF PERFORMANCE 152.3.1. SECURITY-MARKET-LINE BASED PERFORMANCE MEASURES 152.3.2. CAPITAL-MARKET-LINE BASED PERFORMANCE MEASURES 182.4. WEAKNESSES OF TRADITIONAL MEASURES OF PERFORMANCE 21
3. ALTERNATIVE MEASURES OF PORTFOLIO PERFORMANCE 24
3.1. THE FAMA AND FRENCH THREE & FIVE FACTOR APT-MODEL 243.2. THE GRINBLATT & TITMAN NO BENCHMARK MODEL 273.3. THE SHARPE APPROACH: ASSET ALLOCATION AND STYLE ANALYSIS 313.3.1. DETERMINANTS OF THE MODEL 323.3.2. THE PROCEDURE 353.3.3. CRITICISMS AND IMPROVEMENTS 37
4. APPLIED STYLE ANALYSIS 40
4.1. THE DATA 404.1.1. AUSTRIAN INVESTMENT FUNDS 404.1.2. ASSET CLASSES 454.1.2.1. Equity Asset Classes 454.1.2.2. Fixed Income Asset Classes 474.1.2.3. Statistical Properties of the Employed Asset Classes 48
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4.2. DETERMINING THE FUNDS STYLE AND SELECTION RETURN 504.2.1. THE FUNDS AVERAGE COMPOSITION 504.2.2. ROLLING A WINDOW 554.2.3. COMPARISON OF REAL AND ESTIMATED STYLE WEIGHTS 614.2.4. CONTRIBUTION THROUGH SELECTION 634.2.5. SUMMARY OF FINDINGS 704.3. SOME ADDITIONAL INSIGHT USING US MUTUAL FUNDS 71
5. CONCLUSION AND FINAL REMARKS 79
DATA APPENDIX
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Abbreviations
Con: Constantia Privat Invest Fund
A 4: Appollo 4 Fund
Gen: Generali Mixfund
Rai: Raiffeisen Global Mix Fund
Ers: SparInvest Fund
Spa: Global Securities Trust Fund
EVALUE: European Value Stock Index Net Dividends Reinvested
EGROWTH: European Growth Stock Index Net Dividends Reinvested
ESTAND: European Composite Stock Index Net Dividends Reinvested
NAVALUE: North American Value Stock Index Net Dividends Reinvested
NAGROWTH: North American Growth Stock Index Net Dividends Reinvested
NASTAND: North American Composite Stock Index Net Dividends Reinvested
JPSTAND: Japanese Composite Stock Index Net Dividends Reinvested
ATX: Austrian Trading Index (Composite Stock Index)
G7GOV: Government Bond Index of the 7 Largest European Countries
API: Austrian Performance Index (Government Bond Index Interest Reinvested)
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1. Introduction
1.1. The Relevance of Portfolio-Management-Evaluation
Whenever an investor employs resources, be it in the form of hiring employees
for his company, establishing a charitable fund or investing money in an
investment fund he will want to measure the performance of his investment. In
any of the above named cases the investor will establish an evaluation system
that provides him with the feedback needed to determine whether the investment
generates the predetermined utility. In the case of the employee the investor will
demand from him the accomplishment of the agreed on work objectives. From
the manager of the charity fund he will demand evidence that the money was not
spent lavishly. Both times he will bind the executing subjects to some kind of
charta which was defined in advance. In the very same manner he will consider
the evaluation of the investment manager. The investment manager will be
bound to the investment policy and subject to a constant evaluation of his
achievements. His achievement will be the return on the capital the investor
provided.
At this point one will have to determine whether the achieved return was good or
poor and whether it was skill or luck?
This is the punchline investors are are always facing when entrusting their money
to an investment manager. The evaluation now boils down to two main
questions. The first question the investor will want to address is the question of
performance. What is good and what is poor performance and where is the line
in between - the benchmark - and what to take as the benchmark. Should we
employ the performance of a riskless asset e.g. a T-bond, a generic like the S&P
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500 or other portfolio manager's performance as the benchmark? Unfortunately,
these simplistic measures of performance generally do not produce the desired
degree of specification. The investor will also want to find out whether his
investment manager is skillful of fortunate through an evaluation process, which
can be applied to his manager and thereby finding what kind of constranints may
help to get the investment manager to achieve the goal set by the investor. In
answering how to destinct between a skilled and unskilled portfolio manager and
what is good and poor performance, I will address the question central to this
master's thesis. Can Sharpe's asset allocation model and resulting style analysis
be a useful tool in assessing an investment portfolio's performance and the level
of skill of its investment manager?
1.2. Structure of this master's thesis
The first chapter is devoted to the definition of the problem and its justification in
order to give the reader a general overview of this works content.
In chapter 2 CAPM implications on performance measurement are being
elaborated and conventional measures of performance are being discussed in a
critical context. The last part of Chapter 2 will emphasize on weaknesses and
critiques of traditional measures of performance. In Chapter 3 I will introduce
alternative measures of portfolio performance. The Fama & French Model, the
Grinblatt & Titman Model and Sharpe's Asset Allocation and Style Analysis
Model will be described. The Sharpe Model will then be explained in further
detail, as it will be the core subject of this master's thesis.
Chapter 4 will comprehend a regression analysis according to Sharpe's Model. It
will be performed on 6 Austrian investment funds. The investment funds will be:
Raiffeisen Global Mix Fund
Appollo 4 Fund
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SparInvest Fund
Generali Mixfund
Global Securities Trust Fund
Constantia Privat Invest Fund
Through constrained quadratic programming the composition of the specific
funds will be determined and the performance of each of them evaluated. In the
end of this chapter the findings will be compared to traditional measures of
performance and its influence on rankings illustrated.
In Chapter 5 I will conclude the findings of this work and critically evaluate the
initially addressed question, whether Sharpe's portfolio evaluation model is a
good and useful model in assessing a portfolio's performance based on evidence
from Austrian and US investment funds.
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2. Traditional measures of portfolio performanceevaluation and its implications.
2.1. Fundamentals
The traditional evaluation of investment management is based on a few key
concepts. In many cases the framework therefore is provided by the CAPM. In
some other cases it is the risk return relationship of an individual portfolio, its total
risk, that provides the environment for portfolio performance evaluation. On this
basis the essential concepts will be explained in this chapter, as they will be
indirectly relevant in applying and explaining some investment management
evaluation tools.
2.1.1. The Concept of Efficient Markets
The efficient market concept assumes that all investors have free access to
currently available information about the future. All investors are capable of
processing the information as well as adjusting their holdings according to the
information appropriately.1 This concept guarantees that security prices fully
reflect the investment value of the security. This further implies that there exists
no possibility to generate abnormal return - in a systematic way - with generally
available information. Eugene Fama2 classified the efficient market hypothesis
into 3 forms:
The weak form of market efficiency is defined by Fama as reflecting all
historical prices in the value of a security. According to this definition it should be
impossible for a technical analyst to systematically make profits by looking at
past prices.
1 SHARPE, ALEXANDER, BAILEY (1998), p. 932 FAMA (1970), p. 383 - 417
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The semi-strong form of market efficiency is defined as incorporating all
publicly available information. This is the form currently assumed to hold,
although there is a discussion if maybe only the weak form of market efficiency
may hold.
The strong form of market efficiency is defined as including all publicly and
privately available information. If this form of market efficiency held true one
could in no circumstances make abnormal profits by using either of the three
above mentioned sources of information.
Market efficiency is of importance to CAPM, because one of its underlying
assumptions is the competitive investor. This means that prices of securities are
in equilibrium and the expected security return tomorrow based on the
information today will be zero. Security price changes are assumed to follow a
random walk, as positive "surprises" are assumed to be as likely as negative
"surprises". If a pattern can be found to detect mispriced securities on a
systematic basis, it would mean that returns are not random walk any more and
that CAPM would not hold and therefore evaluation measures based on CAPM
would be inaccurate.
The paradox that arises with the efficient markets hypothesis is that if there aren't
investors that do not believe in the efficient market hypothesis, efficient markets
can not exist. If information is free for all participants in the market than none of
the participants has an incentive to gather information. But if no one gathers
information, the market price can not reflect the information. This problem can be
overcome if the cost of gathering information (supporting a squad of analysts) is
the same as the excess return generated through their analysis.3 The short
discussion above indicated the importance of efficient markets on portfolio
management and in the same way on its evaluation.
3 SHARPE, ALEXANDER, BAILEY (1998), p. 96
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2.1.2. Return and Risk as Determinants of the Market
Return can be defined as the rate of change in the value of an asset in a defined
time interval. The mean return, which is interesting if one looks at the prices of an
investment at the beginning and the end of the investment horizon, covers
several time periods and can be measured geometrically or arithmetically. Using
geometric mean calculation is preferable when the "calculation basis" is changing
and has the additional advantage of being additive in every case. Arithmetic
mean calculation is useful when the "calculation basis" remains constant during
the observation period. Arithmetic mean computation returns the average
increase in wealth of a constant investment and does not regard reinvestments of
its proceeds. When analyzing financial time series the basis often varies and
proceeds are reinvested and thus making geometrical mean calculation more
suitable.
Risk is the uncertainty in what a security price - and in consequence the return -
will be at a certain point in the future. Another term would be volatility. Volatility is
equal to the statistical measure of standard deviation. Generally one uses
historical volatility when introducing risk into a financial model. There is also an
alternative way to determine volatility - calculating it implicitly by using the Black-
Scholes Formula.4 The Chicago board of trade provides several implied volatility
indexes for different commodity futures and options. This should help market
participants formulating their trading strategies.
The entire CAPM universe is described by risk and return where risk is
characterized through variance. Through different combinations of risk and
return, the combination of securities with different risk - return characteristics, an
investor can reach every point on the security market line.5 These two
determinants are positively correlated in the CAPM-world. The more risk one
takes the more "reward" he should expect. The linear relationship between
systematic risk and return is at the core of the CAPM. The graph on the next 4 HULL (1997), p. 2465 REILLY, BROWN (1997), p. 24
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page shows the relationship between risk-return and the derivation of the security
market line. The formula for the CAPM is:
[ ] jfMfj rrErrE β⋅−+= )()(
E(rj) = Average expected return of security (j)
r(f) = Average risk free rate (f)
E(rM) = Average return of the market (m)
β (j) = Sensitivity of the expected return of security (j) to changes in the expected
return of the market (m)
Figure 1: Capital Market Line, Security Market Line and the linear risk return relationship
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The relationship between beta and the expected return is known as the SML.
The slope of the line is given by (Rm-Rf), in other words the units of return over
the risk-free rate per unit of systematic risk.
This linear relationship shows that an investor can increase his expected return
by increasing the risk as according to CAPM securities with higher risk must have
a higher return in order to compensate the investor for the risk. This goes along
with the risk aversion assumption put forth in the CAPM. The question of utility
functions of investors will not be treated here but it should be mentioned that
investors are assumed to have convex indifference curves. This means that for
the more risk they take they demand an even higher return.6 (The marginal rate
of substitution, return for risk, increases as risk increases.)
Another important outcome of CAPM for the risk return relationship is that the
risk for which the investor can demand to be rewarded is the systematic risk of a
security as the unsystematic risk can be diversified away. This systematic risk is
reflected in a securities beta i.e. a securities co-movements with the market. The
beta reflects the systematic risk for which the investor can expected to be
rewarded for through return. Questions concerning the validity and the testability
of CAPM shall not be addressed in this work as they are of minor importance to
the central object of this work - the evaluation of portfolio management through
Sharpes' asset allocation and style analysis framework.
2.2. Portfolio Management
Portfolio management or in other words investment management is the process
by which money is managed.7 The way portfolios are managed has severely
changed over the last 100 years. Traditionally portfolio management was strongly
based on fundamental analysis of securities or assets which were to be included
6 FISCHER (1996), p. 40 - 437 SHARPE, ALEXANDER, BAILEY (1998), p. 792
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in the portfolio.8 Fundamental analysis researches the capabilities of a company
to generate future cash flows. This system was by far not as elaborate in terms of
mathematical analysis as it is performed in modern portfolio management. Also
the belief in the possibility of "beating the market" had more acceptance than
today. With the tremendous rise in the US equity market in the nineties the issue
of beating the index (e.g. S&P 500) has become more and more difficult.9 The
controversy over the possibility to outperform the market through active portfolio
management has been reinforced.
2.2.1. Active Portfolio Management
Active portfolio management aims to beat an index by detecting securities that
are under-priced. Securities are under-priced to a certain investor who takes an
active position because his view about the future, his forecast of the securities
price in the future, differs from that of the market. This in turn implies that an
investor or portfolio manager of this sort disregards the conclusion of CAPM that
securities are priced accurately. Active portfolio management is only worthwhile if
the additional return realized through active management is higher than the cost
of maintaining the necessary staff. Costs incurred through active management
are manager fees, analyst reimbursement and higher turnover of securities held
in the portfolio. Manager fees are typically in a range from 0.2 - 1.5 % of the
assets under management.10 Another cost an active fund is more prone of is the
potentially higher turnover of investment managers who, if not reaching their
predetermined returns, are fired quickly. Different styles and beliefs of different
managers will cause (conditioned by high "manager turnover) additional turnover
cost. The cost of turnover depends on the size of the trade and the liquidity of a
title.
8 comp. GRAHAM (1949)9 SORENSON, MILLER, SAMAK (1998), p. 1810 SORENSON, MILLER, SAMAK (1998), p 18
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Size of Trade
Number of $100 $300 $500Portfolio Universe Stocks Million MillionMillion
Salornon Smith Barney large-cap /growth 50 36 bps 53 bps 64 bpsSalomon Smith Barney large-cap/value 50 26 37 44Salomon Smith Barney small-cap /growth 50 131 196 246Salomon Smith Barney small-cap /value 50 113 183 239S&P large-cap /growth 162 27 38 45S&P large-cap/value 338 27 34 44S&P small-cap /growth 234 136 187 226S&P small-cap /value 366 132 189 234Note: Costs estimated at a point in time using Salomon Smith Barney's impact-cost model.
Figure 2: Typical turnover cost for different trade sizes and different asset classes
Active managers can be categorized in three groups: market timers, sector
selectors and security selectors.
Market timers change the beta of their portfolio according to their forecast on how
the market will do.11 Market timers will increase the beta on their portfolio above
the beta of the market portfolio if their forecast is bullish. Securities with a higher
beta than the market will result in the higher appreciation of the specific security
than the appreciation of the market. The reverse will be true if their forecast is
bearish. There were multiple tests on market timing ability. Treynor and Mazuy
conducted the first study on market timing.12 They found that the management of
mutual funds did not exhibit any market timing ability.
11 ELTON, GRUBER (1992), p. 70812 TREYNOR , MAZUY (1966), p. 131 - 136
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Figure 3: Characteristic line for a mutual fund that has outguessed the market.
Mutual fund managers with market timing ability show above than average
performance through detecting when the market will be bullish and when it will be
bearish. This is essentially what the graph above shows.
Further studies on market timing abilities of mutual fund managers were
conducted, showing little evidence of successful market timing.13
Sector Selectors increase their exposure to a certain sector when they believe it
will perform above average in the future and decrease their exposure to a sector
when their belief is that it will under-perform. Sectors can be classified by
industries, products, or particular perceived characteristics like size, cyclical,
growth etc. The sector selection idea is very prominent in the investment
industry. Investment managers often specialize in sectors. The investor in turn
can choose from different "specialists" and from a portfolio of managers that he
13 IPPOLITO, (1993), p. 46
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considers most appropriate for his investment strategy. Sector selection
additionally exerts influence on the later on of discussed style analysis.
The third type of active manager is the security selector. Security selection is the
most traditional form of active portfolio management. By security selection the
investment manager tries to identify securities with higher expected returns than
suggested by the market. By identifying and getting exposure to them the active
manager will realize a higher than market performance if his judgment was right.
Security selection, like all active strategies, neglects the concept of equilibrium
prices on CAPM. There are numerous tests on the ability of active managers to
detect mispriced securities and through that generating excess returns. Excess
return is the return realized above the one with the same risk predicted by
CAPM. An early and notable study on the performance of mutual funds was
conducted by William Sharpe.14 He concluded that mutual funds did not show
better performance than the Dow Jones Industrial Index and that corollary mutual
fund managers did not have stock picking ability. Jensen15 also conducted a
study on mutual fund performance and confirmed the findings of Sharpe. There
was positive evidence found in favor of stock picking by Grinblatt & Titman.16
After all it remains still an open issue if stock-picking ability exists.
2.2.2. Passive Portfolio Management
Index funds have seen a remarkable rise in the past five to seven years.17 Elton
& Gruber also aknowledged: "One of the major companies evaluating manager
performance estimated in 1989 that during the past 20 years the S&P 500 has
outperformed more than 80 % of active managers."18
Portfolio managers who try to replicate the return pattern of a predetermined
index are said to pursue passive portfolio management. The simplest way to
14 SHARPE, (1966), p. 119 - 13815 JENSEN, (1968), p. 389 - 41616 GRINBLATT, TITMAN (1989), p. 393 - 41617 SORENSON, MILLER, SAMAK (1998), p. 1818 ELTON, GRUBER (1992), p. 705
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follow passive portfolio management is to exactly replicate the index or
benchmark. Replicating an index can be very tricky and expensive. Replicating
the S&P 500 may still be feasible without incurring excessive cost but replicating
a Russell 3000 may almost be unfeasible due to excessive turnover cost and
little liquidity in small stocks. This highlights the tradeoff between accuracy and
turnover cost in duplicating an index for a passively managed portfolio.
There are two alternative ways to reproduce an index. By finding a
predetermined number of stock which best tracked the index historically or by
finding a set of stocks that represents all the industry segments in the portfolio in
the same portion as present in the index. A mixture of the three approaches may
very well be found as well as the benefits of the different methods can be
realized. The main benefit of exactly replicating the index is that the tracking error
will be relatively low compared to the other measures. In that sense an index
fund may hold exactly the same weight of large stocks in its fund as represented
in the index. Applying one of the alternative measures presented above,
therefore realizing the benefit of lower transaction cost can solve the problem
with small and illiquid stocks. Cash holdings caused by dividend payments and
cash inflows from investors will also make it harder to track an index due to the
different risk-return characteristics of cash compared to the index.
2.2.3. What Index to use
Portfolio performance evaluation traditionally involves the application of a
benchmark or index to which the portfolios return is compared. If indices are
used as benchmarks the method used to measure the market return needs to be
considered. Friend, Blume and Crockett found in their study that the average
performance of an equally weighted NYSE index differed from the one obtained
when applying a value weighted NYSE index by 2.5 %. The equal weighed
NYSE index yielded 12.4 % whereas the value weighed index yielded only 9.9 %
on average.19 The difference may be attributed to the size effect. The size effect
19 FRIEND, BLUME, CROCKETT (1970) in IPPOLITO (1993), p. 44
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or small firm effect states that small firms stocks tend to have higher returns than
large firms.
There are three commonly used weighting methods in computing a market index
the price weighting method, the value weighting method and the equal weighting
method.
A price-weighted index is computed by summing up the prices of the securities
that are included in the index and dividing them by a constant. This returns the
average price of the securities at time t and when divided by the average price at
time 0 and added to the base of the index, it will return the value of the index at
time t. In the case of stock splits, the constant is adjusted in order to reflect the
price changes due to the stock split. The prestigious Dow Jones Industrial
Average is a price weighted index.
The value weighting method is the most common. Indices like the S&P 500,
Russell 1000, Russell 3000 and the ATX are value weighted. In calculating the
index one simply takes the market value of the securities included in the index at
time t and divides it by the market value of the securities at time 0 and adds the
value to the index base at time 0.
An equal-weighted index is calculated by multiplying the level of the index at time
t-1 with the price relatives at time t. The price relatives are calculated by dividing
the price of every single security in the index at time t by its price at t-1 and then
dividing the sum these price relatives by the number of securities included
herein. An example for an equal weighted index would be the Value Line
Composite Index.
When evaluating the performance of a portfolio and applying an index as the
benchmark one has to make sure that the return measurement method for the
index is the same as for the portfolio under evaluation. Using general market
indices as benchmarks has been criticized as being to general and not
representative for a manager's "habitat" or his style. More elaborate and
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specialized measurements of portfolio performance have been developed. They
will be introduced in the following chapters.
2.3. Traditional Measures of Performance
The foundation of these performance measures is that the return of a portfolio is
adjusted for the risk it bore over the time period under consideration. Traditionally
the adjustment was either based on the security-market-line or on the capital
market line. The security market line based performance measures are Jensen's
Alpha and the Treynor Index. Traditional capital market line based measures of
portfolio performance are the Sharpe Ratio and the RAP (Risk-Adjusted
Performance) Ratio proposed by Modigliani. Morningstar's RAR (Risk-Adjusted
Rating) falls also into this category.
2.3.1. Security-Market-Line based performance measures
In 1965 Jack L. Treynor20 introduced a risk-adjusted measure to rank mutual fund
performance. As a measure of risk he used the beta. Beta reflects the non-
diversifiable portion of a securities total risk and can be calculated from CAPM.
The equation is the following:
)()()()(
pfRpRpTR
β−=
R(p) = Average return of portfolio (p)
R(f) = Average risk free rate (f)
β (p) = Sensitivity of portfolio (p) to market return changes
20 TREYNOR (1965), p. 63 - 75
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The Treynor Ratio gives the slope of the security market line. The higher the TR
the better a portfolio will rank. That can be seen if one introduces indifference
curves of a risk-averse investor. Through a greater TR higher indifference curves
of a risk-averse investor can be reached and the greater will be his utility.
Beta
Return
r2
r1
ß2 ß1
SML1SML2
rf
Beta
Return
r2
r1
ß2 ß1
SML1SML2
rf
Indifference Curves
Figure 4: Relationship between TR and an investors' utility.
The second measure that uses the CAPM as the underlying concept is Jensen's
Alpha.21 Jensen's Alpha measures the positive or negative abnormal return
relatively to the return predicted by the CAPM. With the subsequent formula the
value for Alpha can be calculated.
( )[ ])()()()()()( pfRmRfRpRp βα −+−=
R(p) = Average return of portfolio (p)
R(f) = Average risk free rate (f)
R(m) = Average return of the market (m)
β (p) = Sensitivity of portfolio (p) to market return changes
21 JENSEN (1968), p. 389 - 416
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Alpha represents the return differential between the return of the portfolio and the
return predicted by the CAPM adjusted for the systematic risk of portfolio (p). The
following table shows the popularity of Jensen's Alpha.
1971-75 1976-80 1981-85 1986-90 TotalSharpe 54 63 38 36 191Jensen 51 81 36 52 220Total 105 144 74 88 411Treynor-Mazuy 6 10 8 10 34Friend II 37 31 7 5 80Contradictory Studies* 0 11 11 21 43Grossman-Stiglitz 0 0 78 117 195Source: Institute for Scientific Informaion, Social Science Citation Index , annual.*Studies by McDonald (1974), Mains (1977), Kon and Jen (1979) and Shawky (1982)
Figure 5: Citations for the SR and the Jensen Alpha and some additional studies.
The Treynor Ratio and Jensen's Alpha are related to the systematic risk
component implied by the Sharpe-Lintner Model. There are 2 problems with the
application of these two performance measures:
1) Is the systematic risk the appropriate risk measure for an investor?
2) Does the Sharpe-Lintner Model regard all relevant information in predicting a
securities or portfolios expected return?
The answer to question 1 will depend on whether the investor holds a single
security or a portfolio of securities. In the case that he holds a portfolio of
securities the systematic risk may well be the relevant measure of risk. In the
case of holding a single security the total risk of the specific security will be the
just measure of risk.22 The second question will be addressed in point 2.4.
22 SARPE, ALEXANDER, BAILEY (1998), p. 835
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2.3.2. Capital-market-line based performance measures
When risk-adjusted portfolio performance measures are grounded on the capital-
market-line, the risk adjustment is accomplished by using the total risk of a
portfolio or security. The main difference to security-market-line based
performance measures is, that a capital asset pricing model is not required and
thus alleviating the problem of making assumptions concerning a certain model.
The sole measure of risk is total risk which is equivalent to the statistical measure
of standard deviation or σ. The two traditional measures based thereon are the
Sharpe Ratio and the RAP (Risk-Adjusted Performance) measure.
Another popular measure to rank investment funds in the United States is
Morningstar's RAR (Risk-Adjusted Rating) and as it is also based on a portfolios
total risk adjustment although using a special procedure to adjust for it, it will be
briefly described too.
The Sharpe Ratio23 essentially measures a portfolios average performance over
the risk-free rate per unit of total risk of the portfolio.
)()()()(
pfRpRpSR
σ−=
R(p) = Average return of portfolio (p)
R(f) = Average risk free rate (f)
σ (p) = Ex post standard deviation of portfolio (p)
The Sharpe Ratio's simplicity may be of major appeal to ranking agencies. Even
the Austrian periodical "trendINVEST"24 reports the SR although the funds are
not ranked according to it. Modigliani & Modigliani mention it to be "probably the
23 SHARPE (1966), p. 119 - 13824 trendINVEST (2000), p. 56 - 90
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most popular measure of risk-risk adjusted return"25. Following SR the portfolio
with the highest SR can be considered to be performing best.
Franco Modigliani and Leah Modigliani26 propose a modified version of Sharpe's
measurement approach. They call the ratio they calculate RAP but it is also
referred to as M². In opposite to Sharpe who ranks funds according to the slope
of the capital market line, they lever or un-lever, depending if the sigma of the
portfolio is higher or lower than that of the market, the portfolios risk to equal the
market risk and present the resulting risk-adjusted return as the ranking variable.
This procedure produces the exact same ranking as obtained by applying the
Sharpe Ratio. They justify their approach with the argument that the average
investor who is not familiar with advanced finance techniques can easier
understand RAP. Analytically their approach is the following:
( ) )()()(*)()()( fRfRpR
pmpRAP +−=
σσ
R(p) = Average return of portfolio (p)
R(f) = Average return of the risk-free rate (f)
σ (m) = Ex post standard deviation of market (m)
σ (p) = Ex post standard deviation of portfolio (p)
The relationship between SR and RAP can be shown to be the following:
)()(*)()( fRmpSRpRAP += σ
The benefit of RAP is that it can be readily compared to the market index yield.
The portfolio with the highest value of RAP is corollary the best performing one.
25 MODIGLIANI, MODIGLIANI (1997), p. 5126 MODIGLIANI, MODIGLIANI (1997), p. 45 - 54
20
Morningstar's risk-adjusted rating (RAR) is one of the most popular ratings in the
United States. 27 In 1995 90 % of new money invested in stock funds went into
four-star or five-star ratings awarded by Morningstar. I will not pursue the exact
procedure and its implications on traditional concepts in this project as it is very
complex and lengthy and therefore may be the subject of another work. Rather I
would like to mention the paper28 in which Sharpe analyzed RAR and summarize
his findings.
Sharpe compared the ranking of mutual funds calculated on the basis of RAR to
the ranking obtained through calculating the excess return sharpe ratio. The
excess return sharpe ratio takes the return of a portfolio over the risk free rate
and divides it by the standard deviation differential between the risk-free rate's
standard deviation and the portfolio's standard deviation. Sharpe finds that if
funds have good average historical returns the excess return sharpe ratio ERSR
and RAR are closely related with a correlation coefficient of 0.985.
Figure 6: Correlation between Morningstar's RAR and Excess Return Sharpe Ratio (ERSR).
27 SHARPE (1998), p. 2128 SHARPE (1998), p. 21 - 33
21
He further concludes that RAR should be view as an attempt to determine a best
single fund and assumes that the investor holds only one single fund. The
findings lay out that also in the case of poor overall market performance RAR is
appropriate in determining which fund is best performing assuming an investor
holds only one fund. The weakness Sharpe specifies is that RAR fails to capture
an important property of investors preferences - the desire for portfolios that are
neither the least nor most risky available. He finally concludes that if the only
choice for a measure by which to select funds is between RAR and ERSR, the
evidence favors selecting the ERSR but he acknowledges also that a more
appropriate choice would be to use either a different measure or none at all.
2.4. Weaknesses of Traditional Measures of Performance
The main problem with traditional performance measures is the usage of a
benchmark, especially in estimating the security market line. Whenever the
security market line is incorrectly estimated that means the market index is
inefficient, it can have severe impacts on the outcomes of the Treynor Index and
Jensen's Alpha. The incorrect positioning of the security market line can have
two reasons, neither of which is related to statistical variation:29
1) The true risk free return is different from the risk-free return used in the
model. This problem can be caused by the circumstance that the investor
under consideration can not borrow at the assumed risk-free rate used in the
model. This problem is not only limited to the Treynor Index and Jensen
Alpha as will be explained later on.
29 ROLL (1980), p. 5 - 12
22
2) A non-optimized market index has been employed that means an index
whose expected return differs from the expected return of the optimized index
appropriate for the true risk-free return.
These factors cause the security market line to be positioned incorrectly as
shown below.
Figure 7: Possible performance measurement errors due to mis-specification of the benchmark.
On the basis of these evaluations it can be seen that the Teynor Ratio and
Jensen's Alpha rate funds take on more risk relatively better compared to the
market. Lehmann and Modest30 concluded further that the application of a
specific factor model has major implication on the performance measures yielded
by benchmarks thus fueling the discussion over what is a proper model to
describe return characteristics of securities. At this point it becomes clear that the
relevant problem in determining performance of mutual funds is finding and
providing the correct input measures for the model and assumptions in models
about risk reflection parameters may often not be as clear cut as seeming.
The problem of defining the appropriate risk-free rate has also implications on the
Sharpe Ratio and therefore on RAP. The Sharpe ratio assesses performance in
assuming a linear relationship between total risk and excess return over the risk- 30 LEHMANN, MODEST (1987), p. 233 - 265
23
free rate. If an investor has to pay higher interest rates the higher the presumed
level of risk than that will also lead to a misclassification of funds as his
investment universe compared to the benchmark differs.
24
3. Alternative Measures of Portfolio Performance
Traditional measures have shown several points of concern when applied in
performance evaluation. In this chapter I will introduce alternative approaches to
determine a portfolios required return.
3.1. The Fama and French three & five Factor APT-Model
The Fama and French31 model is built on the Arbitrage Pricing Theory Model
developed by Ross in 1976.32 It states that in an equilibrium market the arbitrage
portfolio must be zero or in other words an arbitrage portfolio can not exist. If this
condition did not hold market participants would sell assets whose expected
return is lower than implied by the detected common risk factors of the market
and buy assets whose expected return is higher than implied by the risk factors.
This process of arbitrage ensures equilibrium market as market participants
engage in it until there is no further possibility in making a riskless profit through
trading one security for another.
On this basis Fama and French tried to define the factors which are relevant in
predicting a securities expected return. The equation to measure a security's
expected return is given below:
kiki FFiR λλλ +++= ...)( 110
R(i) = Return on security (i)
λ (0) = The risk-free rate or zero beta portfolio
λ (ik) = Factor sensitivity of security (i) to factor (k)
F(1-k) = Factors that explain a security's return
31 FAMA, FRENCH (1993), p. 3 - 5632 ROSS (1976), p. 341 - 360
25
Through regression analysis the factors responsible for a security's variation can
be detected. One setback of APT-model is that the model does not specify the
specific risk factors. Fama and French detected three risk factors for stock
portfolios and two risk factors for bond portfolios. The factors for stock portfolios
are
� The excess return of the market over the risk free rate
� The size of the firm
� The book-to-market equity ratio
and for bond portfolios they are
� The time to maturity
� The default risk premium
Fama and French propose their findings as being useful for portfolio performance
evaluation but did not pursue it per se.
Lehmann and Modest33 conducted an extensive study on different benchmarks.
They use the CRSP34 equally weighted and value weighted returns to construct
the different benchmarks. The number of securities they used in the construction
of their benchmarks was 750. The fund returns were taken from 130 mutual
funds over the period of 15 years that is from January 1968 to December 1982.
They compared the Sharpe-Lintner Model's excess return predictions with the
APT-Model's excess return predictions over the above mentioned time period.
They found that the Sharpe-Lintner model produces alphas that are less negative
and less statistically significant than the APT-Models alpha predictions. See table
below.
33 LEHMANN MODEST (1987), p. 233 - 26534 University of Chicago Center for Research in Security Prices (It's files contain complete data on NYSElisted stocks since July 1962)
26
Values in % except for t-value(absolute)
APT-M Alpha S-L-MAlpha
Difference APT - SLM Alpha
VWER EWAR VWERJan. 1968 to Dec. 1972 -4,85 -1,41 -0,15 -3,44(Standard deviation) 3,86 4,37 4,23(t-value) 14,33 3,68 0,40Jan. 1973 to Dec. 1977 -5,45 -0,79 -6,32 -4,66(Standard deviation) 3,6 4,54 4,91(t-value) 17,26 1,98 14,68Jan. 1978 to Dec. 1982 -3,85 1,4 -3,19 -5,25(Standard deviation) 3,3 3,98 3,27(t-value) 13,30 4,01 11,12Figure 8: S-L-M is the Sharpe-Lintner-Model. VWER denotes the excess return when using thevalue weighted CRSP and EWAR denotes the excess return when using the equally weightedCRSP as the benchmark. I calculated the t-value the following: α(i)/(σ(i)/²√130). 130 is thenumber of funds they used in their study.
They found that the Sharpe-Lintner model and APT benchmarks "differ more
than they agree on the Treynor-Black benchmarks over all three periods"35 (they
split the 15 year period in three 5-year periods). On the application of Jensen's
Alpha on the Sharpe-Lintner model benchmark they conclude that this is more
similar to no risk-adjustment at all than it is to the application on the APT
benchmark. The typical rank difference between the APT based Jensen Alpha
and no risk-adjustment was twenty two, nineteen and forty seven positions for
the three 5-year periods. In contrast, the typical rank difference between the
Sharpe-Lintner model based Jensen Alpha and no risk-adjustment are seven,
seven and twelve positions for the three 5-year periods. They conclude that
inferences about mutual fund performance are dramatically affected by the
choice between an APT model benchmark and a Sharpe-Lintner model
benchmark.
Their tests do not say anything about the basic validity of the Sharpe-Lintner
model and the APT mode. The explanation they give for the significant negative
abnormal returns is that their benchmarks, the Sharpe-Lintner model benchmark
35 LEHMANN, MODEST (1987), p. 260
27
and the APT model benchmark, are possibly not mean-variance efficient. They
further acknowledge that the APT model could explain anomalies involving
dividend yield and own variance but could not account for size-effect.
Beyond that they tested different numbers of factors in the APT-Model and found,
that between five, ten and fifteen factors the result-changes were very small. This
can be considered as support for the Fama-French APT approach using five
factors to represent market risk.
Kothari and Warner36 conducted another study that shows the difference in
Jensen Alphas when applying the Sharpe-Lintner model and APT-Model in
defining the benchmark. Kothari and Warner built a 50 stock portfolio through
randomly drawing from the population of the NYSE/AMEX securities. They
repeated this procedure at the beginning of every month over 336 month that is
from January 1964 to December 1991. The portfolio's returns were than tracked
for 36 months. This formed the basis for their benchmark. They found that when
they compared the performance of their randomly selected stock portfolios to a
Sharpe-Lintner model benchmark their random portfolios showed a Jensen Alpha
of over 3 %. The Fama-French APT model performed better as setting a
benchmark by which it only had a Jensen Alpha of -1.2 %. These empirical
results are very similar to the ones found in Lehmann and Modest as their
average performance difference was (3.44% + 4.66% + 5.25%)/3 that is -4.45 %
(APT-M minus SL-M). They conclude that standard mutual fund performance
measures are unreliable and mis-specified.
3.2. The Grinblatt & Titman no Benchmark Model
The encountered problems with benchmarks have led to alternative approaches
to determine a portfolio's performance. Grinblatt and Titman37 pursued one
36 KOTHARI, WARNER (1997), p. 1 - 4437 GRINBLATT, TITMAN (1993), p. 47 - 68
28
where no benchmark is needed and thus alleviating several problems associated
with the use of a benchmark. Their analysis in turn is only applicable if the
evaluator has knowledge about the exact composition of the portfolio under
evaluation. This is in strong contrast to the portfolio performance measures
introduced earlier since they allowed portfolio performance evaluation without
apprehending a portfolio's composition.
The underlying concept of their measure, they call it the "Portfolio Change
Measure"38, is that an informed investor will hold securities that will have a higher
return when they are included in the portfolio than when they are not included.
Further, an informed investor will tilt his portfolio weights towards assets with
expected returns higher than average and away from assets with expected
returns lower than average. This will cause a positive covariance between
portfolio weights and the return of a security for an informed investor whereas it
should not be any covariance between portfolio weights and the return of an
asset for the uninformed investor. The way Grinblatt and Titman propose to
measure this covariance is the following:
( )[ ] TwwRPCMN
j
T
tktjjtjt /
1 1,∑∑
= =−−=
PCM = Portfolio Change Measure
R(jt) = Return of security (j) at time (t)
w(jt) = Weight of security (j) at time (t)
w(j,t-k) = Weight of security (j) at time (t - k)
T = Number of time periods under consideration
38 GRINBLATT, TITMAN (1993), p. 51
29
Under the null hypothesis of no superior information, both current and past
weights are uncorrelated with current returns and thus the PCM measure should
be indistinguishable from zero.
Potential problems with this measure can arise from the violation of the key
assumption to this concept namely that mean returns of assets are constant over
the sample period. Portfolios that specialize in takeover targets or bankrupt
stocks will realize positive performance with this measure because they include
assets whose expected returns are higher than usual. The same holds true for
managers who are exploiting serial correlation in stock returns. One must also
keep in mind that this measure can only be applied if the evaluator knows the
exact composition of the portfolio over time, which may be the cause for its
sparse use.
Despite that the PCM approach overcomes the problems of measuring the SML
as described in 2.4.
Grinblatt and Titman applied the PCM measure on 155 mutual funds over a 10-
year time period from December 31st 1974 to December 31st 1984 on quarterly
holdings. On this basis they formed two portfolios, the first lagged one quarter
and the second lagged 4 quarters. These differenced weights where then
multiplied by CRSP monthly stock returns where the weights were held constant
over 3 months and therefore a time series of monthly portfolio returns was
created for the one quarter and four quarter lagged PCM. For example with the
one quarter lag, the April, May and June returns were multiplied by the difference
between the portfolio weights held on March 31st 1975 and the weights held on
December 31st 1974 and so forth.
They found that for the one quarter lagged PCM measure the value was
statistically indistinguishable from zero which indicates that informed investors
can not realize the benefits of their information in one quarter. The 4 quarters
lagged PCM showed statistically significant abnormal returns indicating that
investors do have superior information and that it is revealed with a one-year lag.
30
The average abnormal returns of the entire sample are about 2% per year. The
table below shows the abnormal returns for different mutual fund categories and
its level of significance.
Performance Measure
Lagged 1 Quarter Lagged 4 Quarters
No. of Mean Wilcoxon Mean Wilcoxon Funds Performance t-statistica Probabilityb Performance t_statistica Probabilityb
Total sample 155.37 1.47 .233 2.04 3.16* .004Aggressive growth funds 45 . 39 .98 .475 3.40 3.55* .004Balanced funds 10 -.48 -1.87 .057 .01 .03 .902Growth funds 44 .66 2.01* .017 2.41 2.94* .009Growth-income funds 37 .14 .61 .095 .83 1.75 .107Income funds 13 .54 1.54 .475 1.33 2.64* .002Special purpose funds 3 -.10 -.16 .233 .21 .19 .711Venture capital/special situation funds 3 1.26 1.07 .812 2.66 1.43 .035
Fl-statistic (Abnormal performance in every category = 0)
F = 3.1438*
Prob > F = .0028c
F2-statistic (Abnormal performance across categories is equal)
F ~ 3.6590*
Prob > F = .0014c
a) The mean over all months divided by the standard error of mean.b) The probability that the absolute value of the Wilcoxon-Mann-Whitney Rank z-statistic is greater than the absolute value of the
observed z-statistic under the null.c) The probability of the F-statistic being greater than the outcome shown, tinder the null hypothesis (Type 1 error).*) Type I error < .05.
Figure 9: Performance estimates for 155 surviving mutual funds grouped by investment objectivecategories (Return in % per year).
Grinblatt and Titman report that the PCM measure results in smaller standard
errors than approaches that use the security market line. They attribute the
increased estimation precision to the higher correlation between their
"benchmark" (the current returns of a funds historical portfolio) and the returns of
the current portfolio than any traditional benchmark portfolio.
31
Their final conclusion was that mutual funds on average achieved positive
abnormal performance during the 10-year period under estimation but that after
considering transaction cost and fund expenses the net abnormal average
performance is close to zero. They further conclude that traditional measures of
performance add noise to true performance and thus bias the measure towards
finding no performance. This is because outside evaluators are not measuring
the true performance of a fund but only the performance of some hypothetical
portfolio that is correlated with the fund instead of evaluating holdings that
correspond to each transaction. Consequently they found that managers who
performed well in one period were likely to do so in a following period thus
inferring manager skill.
3.3. The Sharpe Approach: Asset Allocation and StyleAnalysis
The portfolio evaluation models described in this master's thesis does not require
the knowledge of the exact composition of the portfolio except for the Grinblatt
and Titman model described in 3.2. For an outside evaluator this is of practical
importance as it is the condition for making portfolio evaluation feasible. The
inconvenience is that the resulting portfolio measures are of general nature. In
light of that Sharpe39 developed an "evaluation system" that reflects a higher
degree of specification and regards an investment manager's universe also
labeled his specific "investment style". He argued that it would be more adequate
to measure an investment manager by the asset class returns he invested in
instead of comparing his return to a universal benchmark. This idea of "grouping"
funds was put forth first time by LeClair40. Brinson, Hood and Beebower41 found
in their study in 1986 that the staggering part of the portfolio performance of 91
pension plans came from asset allocation. In 1988 Sharpe introduced a method
39 SHARPE (1992), p. 7 - 1940 LeCLAIR (1974), p. 220 - 22441 BRINSON, HOOD, BEEBOWER (1986), p. 39 - 44
32
to determine a funds "effective asset mix"42 through constrained regression
analysis and thereon he grounded his renowned paper of 1992 titled "Asset
Allocation: Management Style and Performance Measurement".
3.3.1. Determinants of the Model
The main input in Sharpe's asset class factor model is the single asset classes.
Sharpe defined certain standards that an asset class should meet in order to
assure the usefulness of the model. This is not found to be strictly necessary, but
it is desirable for the usefulness of the model. The qualitative exigencies on an
asset class are
1. Mutually exclusive
2. Exhaustive
3. Have returns that differ
The asset class should represent a capitalization-weighted portfolio of securities
in order to mimic return variation created by different weights of asset classes in
the returns of the portfolio under evaluation. Sharpe pointed out further that asset
class returns should either have low correlations with one another or, in cases
where correlations are high, different standard deviations.43 If independent
variables are highly correlated, as two indexes representing different approaches
to investing with the same asset class, the reliability of the estimated coefficients
in meaningfully describing the underlying relationship is very much in doubt.44
The problem of multicollinearity can reduce the explanatory power of a model
and therefore the asset classes should show low correlation, possibly none,
following Sharpe's qualification mentioned above.
The number of asset classes Sharpe uses in his proposed model is twelve. Each
of the twelve indexes is supposed to represent a strategy that could be followed 42 SHARPE (1988), p. 59 - 6943 SHARPE (1992), p. 843 LOBOSCO, DiBARTOLOMEO (1997), p. 80
33
passively and at low cost using an index fund. The possibility of investing in the
index at low cost is of importance as this is the alternative the manager is
measured against. If the benchmark we apply is not a feasible investment
alternative it will be a biased measure. The twelve classes he uses are
1. T-Bills Cash equivalents with less than 3 months to maturity. Index: Salomon
Brothers' 90-day Treasury Bill Index.
2. Intermediate-Term Government Bonds Government bonds with less than
10 years to maturity. Index: Lehman Brothers' Intermediate-Term Government
Bond Index
3. Long-Term Government Bonds Government bonds with more than 10
years to maturity. Index: Lehman Brothers' Long-Term Government Bond
Index
4. Corporate Bonds Corporate bonds with ratings at least Baa by Moody's or
BBB by Standard & Poor's. Index: Lehman Brothers' Corporate Bond Index
5. Mortgage Related Securities Mortgage-backed and related securities.
Index: Lehman Brothers' Mortgage-Backed Securities Index
6. Large-Capitalization Value Stocks Stocks in S&P 500 stock index with high
book-to-price ratios (50% of the stocks in the S&P 500 index). Index:
Sharpe/BARRA Value Stock Index
7. Large-Capitalization Growth Stocks Stocks in the S&P 500 stock index with
low book-to-price ratios (remaining 50% of the stocks in the S&P 500 index).
Index: Sharpe/BARRA Growth Stock Index
8. Medium-Capitalization Stocks Stocks in the top 80% of capitalization in the
US equity universe after the exclusion of stocks in the S&P 500 stock index.
Index: Sharpe/BARRA Medium Capitalization Stock Index
34
9. Small-Capitalization Stocks Stocks in the bottom 20% of capitalization in
the US equity universe after the exclusion of stocks in the S&P 500 stock
index. Index: Sharpe/BARRA Small Capitalization Stock Index
10. Non-US Bonds Bonds outside the US and Canada.
Index: Salomon Brothers' Non-US Government Bond Index
11. European Stocks European and non-Japanese Pacific Basin stocks.
Index: FTA Euro-Pacific Ex Japan Index
12. Japanese Stocks Index: FTA Japan Index
Every six months the equity categories are reclassified. The S&P 500 stocks are
reviewed and if the change in book-to-price ratios implies a change in the
classification, for example a stock that falls from the top 50% (relatively high
book-to-price ratio) into the bottom 50% (relatively low book-to-price ratio) than
the stock is regrouped. Non-S&P stocks, stocks in the medium-cap and small-
cap class, are classified in order that 80% of these stocks are in the medium-cap
class and 20% in the small-cap class. To avoid excessive turnover in the
composition of these indexes of relatively illiquid stocks and an associated high
cost for index tracking, any stock that has "recently crossed over the line"45 a
relatively small distance is allowed to remain in its former index. A relatively small
distance is defined with 20% within the boundary value. The remaining eight
asset classes are self-explanatory all together the twelve asset classes were
constructed to cover the investment universe from which portfolio managers
chose their assets.
The explained variables will be the individual fund returns, which can be
observed in newspapers or bought from specialized research companies.
45 SHARPE (1992), p. 9
35
3.3.2. The Procedure
After asset classes have been defined and the desired history of returns
corresponding to them has been obtained, data analysis is put to work. The
objective is to determine the weights of the previously defined asset classes in
the portfolio of an individual mutual fund through quadratic programming.
Quadratic programming is set to minimize the variance of the residuals under
certain constraints. The usefulness of minimizing the variance and not using
standard regression or constrained regression to determine a portfolios "asset
class exposures" can be seen in the table below.
Regression and Quadratic Programming ResultsTrustees' Commingled Fund-U.S. Portfolio
Januarv 1985 through December 1989Unconstrained
RegressionConstrainedRegression
QuadraticProgramming
Bills 14.69 42.65 0
Intermediate Bonds -69.51 -68.64 0
Long-Term Bonds -2.54 -2.38 0
Corporate Bonds 16.57 15.29 0
Mortgages 5.19 4.58 0
Value Stocks 109.52 110.35 69.81
Growth Stocks -7.86 -8.02 0
Medium Stocks -41.83 -43.62 0
Small Stocks 45.65 47.17 30.04
Foreign Bonds -1.85 -1.38 0
European Stocks 6.15 5.77 0.15
Japanese Stocks -1.46 -1.79 0
Total 72.71 100.00 100.00
R-squared 95.20 95.16 92.22Figure 10: Resulting asset class weights through unconstrained and constrained regression andthrough quadratic programming.
The constraints are that the sum of the weights of the different asset classes in
the portfolio must be 1 and that no short positions are allowed as common
36
mutual funds policies do not allow short positions46. Analytically, the program
looks the following:
[ ] iKikjiji
i
RwRwR
Var
FunctionObjective
ε
ε
+++=
→
⋅
...
)(min
:
10
1
:
1
≤≤
=
⋅
∑=
ij
K
jiK
w
w
tosubject
Var(ε (i)) = Variance of residuals of security (i)
R(i) = Return of portfolio (i)
R(j) = Return of asset class (j)
w(ij) = Weight of asset class (j) in portfolio (i)
ε (i) = Residual return of portfolio (i)
Sharpe defines the residual return of a portfolio as the portfolios "tracking error"
and its variance as the funds "tracking variance"47. In other words, it represents
the value contributed to the total return of a portfolio by a managers stock picking
ability. The other part is explained by the employed asset classes. The style
determined through this method can be regarded as an average of the potentially
changing styles over the examined period. This indicates that the longer the
examination period the more inaccurate this method becomes. To obtain a
46 SHARPE (1992), p. 1147 SHARPE (1992), p. 11
37
clearer picture and especially to see how a style changes over time one needs to
roll a time window over the examination and run the quadratic program for every
time window. The result will be a series of asset class weights that reflect if the
manager changed his "style" or if he kept weights constant over time.
The style weights can now be used to produce a time series of excess returns by
subtracting the "style benchmark" from the actual portfolio return at each single
point in time. The excess returns would be generated by the formula
[ ]tKtpKtjtpjtptp RwRwR ,,,,,, ... ++−=ε
ε (p,t) = Excess return or return due to selection of portfolio (p) at time (t)
R(p,t) = Absolute return of portfolio (p) at time (t)
R(j,t) = Return of asset class (j) at time (t)
w(p,t) = Weight of asset class (j) in portfolio (p) at time (t)
The method of separating a portfolios return in a "style return" and in a "selection
return" makes it possible to distinguish between the performance of the portfolio
manager and the investor. Knowing the styles of investment an investor can
create his individual asset class portfolio. The investment manager on the other
hand will be measured accordingly to his class benchmark and will not have to
bear responsibility for overall unsatisfying returns due to asset allocation.
3.3.3. Criticisms and Improvements
The basic style analysis model averages the styles of an investment manager
over the period under consideration and returns the estimated weights. If an
investment manager changes his investment styles, style analysis will return an
average of his styles and not the accurate composition of his investment portfolio
38
at a time.48 This problem can be overcome through rolling a window over the
period under consideration and thus partly offset the problem of changing styles
of the investment manager.49 Trzcinka50 acknowledges the problem of
inaccuracy but concludes that the strengths of Sharpe's style analysis are its
objectivity, low cost and its practical application.
Lobosco and DiBartolomeo51 have researched the problem of determining the
confidence intervals of Sharpe's style weights. They found that the confidence
interval for a style weight of a particular market index increases with the standard
error of the style analysis, decreases with the number of returns used in the style
analysis and also decreases with the "independence" of the market indexes used
in the analysis. The formula52 they derived to calculate the confidence intervals
is shown below:
1−−×=
knBi
awi σ
σσ
σ (wi) = Standard deviation of the amount of error in the estimate of the style
weight for index (i) through style analysis
σ (a) = Standard deviation of the residuals from determining the style weights
through quadratic programming
σ (Bi) = Standard deviation of the portion of return on index (i) not attributable to
other market indexes
n = Number of data points in the return time series
k = Number of market indexes with nonzero style weights
48 CHRISTOPHERSON (1995), p. 3849 SHARPE (1992), p. 1150 TRZCINKA (1995, p. 4651 LOBOSCO, DiBARTOLOMEO (1997), p. 80 - 8552 LOBOSCO, DiBARTOLOMEO (1997), p 84 -85
39
Through Monte Carlo simulations on a portfolio of market index weights with an
arbitrarily chosen value for the standard error they tried to approximate the true
index weights. The repetition of Monte Carlo simulation generated a probability
distribution for each of the style weights with the mean values showing
approximately the true values chosen in the beginning (before taking an arbitrary
value for the standard error and simulating the different outcomes of style
weights). Lobosco and DiBartolomeo concluded that the predicted values and the
simulated values converge. The confidence interval measure can help to
determine whether different asset classes should be used due to excessive
collinearity and for which situations style analysis through quadratic programming
may not be well suited. Furthermore they suggest using daily returns in order to
reduce the confidence intervals.
The problem of multicollinearity in the chosen asset classes can be coped with
by using less asset classes.53
53 LOBOSCO, DiBARTOLOMEO (1997), p. 82
40
4. Applied Style Analysis
4.1. The Data
4.1.1. Austrian Investment Funds
I will be evaluating six Austrian investment funds using style analysis. The six
funds I choose are all open-end funds. I selected them randomly from a list of
mixed international investment funds published in trendINVEST54. The reason for
choosing international funds was to examine the general validity of style analysis
for Austrian funds and not only for a specific subcategory like stock funds. The
larger number of funds in the international category proved also useful in regard
to obtaining sufficient historical data as such data is often not available. The
funds initially selected were:
Raiffeisen Global Mix Fd (Raiffeisen KAG)
Appollo 4 (Capital Invest KAG - BA)
SparInvest (Erste SparInvest KAG)
Gernerali Mixfond (3-Banken-Generali Invest)
Global Securities Trust (Carl Spängler KAG)
Constantia Privat Invest (Constantia Privatbank KAG)
Volksbanken Inter Invest (Volksbanken KAG)
Aquila International (Gutmann KAG)
The latter two asset management companies, Volksbanken KAG and Gutmann
KAG, did not provide any historical data, resulting in the elimination of these 54 trendINVEST (2000), p. 76 - 77
41
funds in my final "examination group". The return histories gathered from the
remaining asset management companies included reinvested dividends, thus
avoiding possible interruptions in fund return data due to dividend outpayments
made by the fund. The mutual fund data covers the period from 4/95 to 12/99.
Fund data for Austrian investment funds beyond a five-year history often does
not exist and in turn it was the main parameter for choosing an approximate five-
year history. The fact that there are only 56 monthly returns instead of 60, is due
to the data series of Constantia Privat Invest, which had only a history of 56
month by December 31st 1999.
Monthly return data was used to enhance the statistical significance compared to
quarterly data and constrained to the relatively short time period. Although daily
data would be preferable due to the previously mentioned statistical significance
of the estimated style weights55, it was not feasible as daily return histories for
the necessary asset classes could not be obtained. The advantage of this "data
tradeoff" is a reduced computation time especially when "rolling a window" as
described later in 4.2.2.
Returns were computed for all funds and asset classes using the natural
logarithm (ln). Exchange rate problems associated with investments in different
currency areas did not cause any specific difficulty as the individual asset
management companies provided fund return histories in local (Austrian)
currency terms. The returns were not corrected for management fees or any
other administrative cost incurred by the fund.
In the following the descriptive statistics of the investment funds are displayed.
By that a first glance may be gathered concerning the potential composition of a
particular investment fund.
Figure 12 shows the return history of Constantia Privat Invest. I will refer to it as
Con fund. The relatively low mean return of Con fund accompanied by an also
55 LOBOSCO, DiBARTOLOMEO (1997), p. 84
42
low standard deviation is characteristic to this fund indicating a high portion of
fixed income securities.
0
2
4
6
8
10
-0.01 0.00 0.01 0.02
Series: CONSSample 1995:05 1999:12Observations 56
Mean 0.005105Median 0.005561Maximum 0.027207Minimum -0.016894Std. Dev. 0.009842Skewness -0.116157Kurtosis 2.498004
Jarque-Bera 0.713930Probability 0.699797
Figure 12: Descriptive statistics of Constantia Privat Invest fund.
Figure 11 exhibits the descriptive statistics for Appollo 4. The descriptive
statistics reveal a moderate mean monthly return of 0.76% per month with a
relatively small standard deviation of 1.8%, potentially indicating a substantial
share of fixed income securities. The Jarque-Bera test indicates that we cannot
reject the null hypothesis of normal distribution for Appollo 4.
0
2
4
6
8
10
12
14
-0.04 -0.02 0.00 0.02 0.04
Series: A4Sample 1995:05 1999:12Observations 56
Mean 0.007585Median 0.010455Maximum 0.051477Minimum -0.035200Std. Dev. 0.018146Skewness -0.250747Kurtosis 3.153286
Jarque-Bera 0.641651Probability 0.725550
Figure11: Descriptive statistics of Appollo 4 fund
43
Below the descriptive statistics of Generali Mixfund, in short Gen fund, are
displayed. The histogram reflects a relatively larger range of realized returns
indicating greater volatility.
0
2
4
6
8
10
12
-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08
Series: GENSample 1995:05 1999:12Observations 56
Mean 0.011441Median 0.013952Maximum 0.072833Minimum -0.063160Std. Dev. 0.029523Skewness -0.271016Kurtosis 2.795971
Jarque-Bera 0.782664Probability 0.676156
Figure 13: Descriptive statistics of Generali Mixfund.
The histogram in figure 14 reflects the realized returns of Raiffeisen Global Mix
Fund, referred to as Rai Fund. Distinct is the relatively high mean return.
0
2
4
6
8
10
12
-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08
Series: RAISample 1995:05 1999:12Observations 56
Mean 0.013887Median 0.016291Maximum 0.075101Minimum -0.054117Std. Dev. 0.028372Skewness -0.195006Kurtosis 2.604408
Jarque-Bera 0.720073Probability 0.697651
Figure14: Descriptive statistics of Raiffeisen Global Mix fund.
The descriptive statistics of SparInvest from Erste Bank capital asset
management, denominated Ers, reveal a relatively high third and fourth moment
and are therefore at the 100 % level not normally distributed. The Jarque-Bera
44
test for normality follows a Chi-square statistic with two degrees of freedom and
is based on skewness and kurtosis. The high mean and standard deviation
indicate strong exposure to stocks and potentially derivative instruments.
0
2
4
6
8
10
-0.10 -0.05 0.00 0.05
Series: ERSSample 1995:05 1999:12Observations 56
Mean 0.014591Median 0.019496Maximum 0.064690Minimum -0.107150Std. Dev. 0.031988Skewness -1.277064Kurtosis 5.410110
Jarque-Bera 28.77513Probability 0.000001
Figure 15: Descriptive Statistics of SparInvest Fund.
In Figure 16 the descriptive statistics of Global Securities Trust from Carl
Spängler asset management (referred to as Spa) are exhibited. The relatively
low mean and standard deviation could indicate significant exposure to bonds.
0
2
4
6
8
10
-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04
Series: SPASample 1995:05 1999:12Observations 56
Mean 0.010508Median 0.012990Maximum 0.038068Minimum -0.029623Std. Dev. 0.016710Skewness -0.554665Kurtosis 2.706236
Jarque-Bera 3.072791Probability 0.215155
Figure 16: Descriptive statistics of Global Securites Trust fund.
45
4.1.2. Asset Classes
The primary question I faced was how many asset classes to include in the
analysis and what markets to cover. Sharpe specifies that all markets should be
covered and that the resulting asset classes should be mutually exclusive56. It
was not feasible to obtain asset class data for all markets, especially concerning
bond asset classes. In trying to determine the investment universe from which a
typical Austrian investment fund selects its securities, I gathered fund
composition data from some of the funds under evaluation. The fund data
revealed that the funds showed strong exposure to European, US and Japanese
equities and to European bonds. Knowing the approximate exposure, a
restriction to lesser asset classes seemed feasible without endangering the
meaningfulness of the results.
4.1.2.1. Equity Asset Classes
Morgan Stanley Capital International (MSCI) provides historical data on
numerous equity asset classes. MSCI asset classes are constructed to cover at
least 60% of an entire equity market. Extended asset classes offered by MSCI
cover at least 70% of a specific market. The market is consequently split into
value and growth securities. The specification into value stocks or growth stocks
is subject to the Price/Book ratio of the specific security. Those 50% of securities
with the relatively higher Price/Book ratio are grouped into the growth stock class
and the remaining 50% with relatively lower Price/Book are grouped into the
value stock class. MSCI provides semi-annual re-balancing of the growth and
value categories. If Price/Book ratios change in a manner that they would qualify
for the other category, they would have to surpass the 50% separation line by
more than 10% before being reclassified in the other category. This policy
reduces asset turnover and consequently provides a more appropriate
benchmark.
56 SHARPE (1992), p. 8
46
MSCI selects stocks with sufficient liquidity until 60% of the market capitalization
is reached. The indexes are capital-weighted indexes using the Laspeyres price
index formula to calculate the price change.
Furthermore, these equity indexes are also calculated using net and gross
dividends reinvested. The constructed indexes reinvest dividends when they are
paid. The net dividend index seems more apt for investment fund evaluations as
it corrects for certain taxation aspects. MSCI subtracts from gross dividends any
withholding tax retained at the source for foreigners who do not benefit from a
double taxation treaty.57 MSCI indicates further that as withholding taxes may
vary according to the shareholders domicile the most conservative rates are
applied. This master's thesis will use net dividend reinvested indexes as equity
indexes representing the most viable investment alternative available and thus
alleviating some of the problems discussed in chapter 2.
To reflect investment alternatives, asset class indexes had to be converted into
local currency terms. To achieve this, I gathered the necessary exchange rate
data from the homepage of Österreichische Kontrollbank58. Converting the USD
indexes and the JPY index with the appropriate exchange rate series resulted in
the final asset class index of the applied foreign asset classes. Asset class
returns were calculated applying the natural logarithm. The formula for the return
calculation was rt = ln(Pt) - ln(Pt-1).
The equity asset classes employed in this master's thesis are:
Austrian Trading Index Source: Reuters
European Value Stocks Net Dividends Reinvested Source: MSCI Europe
Value Index.
57 INDEX CALCLATION (MSCI 2000), p. 4458 www.oekb.at, June 2000
47
European Growth Stocks Net Dividends Reinvested Source: MSCI Europe
Growth Index.
Japanese Standard Stocks Net Dividends Reinvested Source: MSCI Japan
Standard Index.
North America Standard Stocks Net Dividends Reinvested Source: MSCI
North America Standard Index.
The term "Standard" indicates that the Japanese equity universe was not split
into the subgroups value and growth hence the index represents 60% of the
Japanese equity universe.
Although Austrian stocks are already included in the European value and growth
index it seems useful to employ a separate asset class covering liquid Austrian
stocks. The problem of redundancy is of minor importance since Austrian
equities weight only 2.9 % in the MSCI Standard European Stock Index. In
addition this could only cause stronger collinearity, but could not result in a
biased estimation. The possible loss of information when evaluating Austrian
investment funds justifies this compromise. Besides there is the problem of
neglecting dividends paid when using the ATX as an asset class. The problem
would be especially severe if the funds under evaluation showed significant
exposure to the ATX asset class. It was found later that this was not the case
leading to the use of the ATX as an asset class that provides an interesting
insight in how much of an Austrian portfolio, classified as being internationally
diversified, is invested in Austrian securities. The problem is also partly mitigated
by using net dividends for reinvestment since the dividend-factor does not
influence the return history as severe as when gross dividends are used.
4.1.2.2. Fixed Income Asset Classes
Unfortunately, it was not possible to gather sufficient bond index data to cover
the entire bond universe. Thus resulting in an unsatisfying coverage of the bond
48
universe by only two indexes that could be obtained. Extensive bond indexes
with sufficient histories are generally provided by investment services. Unlike
equity indexes, fixed income indexes are not for free. The two fixed income asset
classes chosen are the following:
Austrian Government Bond Index Interest Reinvested Source:
Österreichische Kontrollbank.
European Government Bond Index Source: Salomon Brothers G7 Government
Bond Index.
The Austrian government bond index is called API59. This index includes all
Austrian government bonds and is also corrected for interest and bond discount
proceeds. These payments are reinvested. This property serves well for this work
as this property makes the index a potential investment alternative and thereby
reduces some of the inaccuracies of a benchmark. The data series was provided
by Österreichische Kontrollbank. The European Government Bond Index was
provided in the tutorial "Kapitalmarktforschung" at the University of Vienna.
Unfortunately, it does not incorporate interest and other proceeds provided by
bonds. The index represents the government bond market of the seven largest
European nations. I was not able to obtain data on European corporate bonds.
After contacting all renowned providers of such data no response was received.
Nevertheless it should not be crucial to this work either.
4.1.2.3. Statistical Properties of the Employed Asset Classes
Asset classes should be mutually exclusive, exhaustive and have returns that
differ60. The aspects of mutual exclusivity and exhaustiveness were discussed in
subchapters 4.1.2.1 and 4.1.2.2. The property of different returns will be
examined in this part. The asset-class return series descriptive statistics are
shown below.
59 Anleihen Performance Index60 SHARPE (1992), p. 8
49
API ATX EGROWTH EVALUE G7GOV JPSTAND NASTAND
Mean 0.0056 0.0040 0.0226 0.0222 0.0120 0.0074 0.0265 Median 0.0072 0.0129 0.0282 0.0277 0.0095 0.0138 0.0334 Maximum 0.0212 0.1241 0.1219 0.1161 0.0725 0.1553 0.1312 Minimum -0.0171 -0.1925 -0.1133 -0.1642 -0.0390 -0.1245 -0.1572 Std. Dev. 0.0082 0.0618 0.0484 0.0496 0.0272 0.0678 0.0545 Skewness -0.6088 -0.6526 -0.7907 -1.2184 0.2862 -0.0164 -0.8685 Kurtosis 2.7356 3.8168 3.7970 5.5491 2.3414 2.1438 4.0103
Jarque-Bera 3.62 5.53 7.32 29.02 1.78 1.71 9.42 Probability 0.1635 0.0629 0.0258 0.0000 0.4114 0.4246 0.0090
Observations 56 56 56 56 56 56 56Figure 17: Descriptive statistics of selected asset classes.
Figure 17 shows that except for the European growth asset class and the
European value asset class where the mean return is similar, the preferred
property of different returns is accomplished. Consequently standard deviations
also differ except for the above mentioned asset classes where they are very
close. For the remaining asset classes the problem of multicollinearity should not
be of major concern. One asset class could probably represent the European
growth and European value asset class but for the sake of possible additional
information, resulting from the constrained regression, they will be applied
individually. Intuitively one would expect correlations between the different asset
classes to be moderate except for the European value and growth classes.
Figure 18 displays the cross-correlations of the individual asset classes.
API ATX EGROWTH EVALUE G7GOV JPSTAND NASTANDAPI 1.00 0.09 0.20 0.18 0.48 0.07 0.28ATX 0.09 1.00 0.71 0.82 0.37 0.46 0.63EGROWTH 0.20 0.71 1.00 0.88 0.51 0.55 0.80EVALUE 0.18 0.82 0.88 1.00 0.48 0.56 0.79G7GOV 0.48 0.37 0.51 0.48 1.00 0.31 0.67JPSTAND 0.07 0.46 0.55 0.56 0.31 1.00 0.56NASTAND 0.28 0.63 0.80 0.79 0.67 0.56 1.00Figure 18: Cross-Correlations of selected asset classes.
50
The cross-correlogram shows the expected higher correlation between the
European growth and value asset classes and interestingly also a relatively high
correlation of these asset classes with the North American asset class. However
Sharpe mentions that in cases where correlations between asset classes are
high, these classes should have different standard deviations61 which is the case
when the European asset classes are compared to the North American asset
class.
4.2. Determining the Funds Style and Selection Return
4.2.1. The Funds Average Composition
The constrained regression procedure described in 3.3.2 was employed on the
return series introduced above. The calculation of style weights was performed in
Excel using Solver. This was necessary since EVIEWS does not allow
constrained regressions involving inequality parameters and does further not
include an explicitly addressable optimizer. Nevertheless it did not cause any
problem concerning the accuracy of the estimates as Solver computes the style
weights to the fourth decimal point.
Setting up a quadratic program in Excel is relatively uncomplicated. The
optimizer Solver in Excel can be used to solve three common optimization
problems: minimizing, maximizing and equalizing a certain parameter. In this
case the minimization feature is used.
61 SHARPE (1992), p. 8
51
Figure 19: Solver set-up to calculate Sharpe style weights.
Figure 19 displays the Solver mask programmed to calculate style weights for
Gen fund. The target is to minimize the variance of the residuals. Thus, one first
needs to calculate the residuals using arbitrary weights plugged into the fields
below the asset class line (C3:I3). Now the variance formula for the residuals can
be inserted into the target cell defined in Solver (in this case K3). The only task
left to calculate the style weights is defining of the constraints, which can be done
easily by clicking on the add button on the Solver surface.
In figure 20 style weights were calculated setting up Solver as described above
and using the selected asset classes.
52
Fund EVALUE EGROWTH NAVALUE NAGROWTH JPSTAND G7GOV API ATX R² adjusted R²Con 0% 1% 0% 0% 0% 3% 89% 7% 0.7899 0.7642A4 7% 16% 3% 1% 2% 13% 58% 0% 0.8221 0.8004Gen 4% 35% 0% 7% 0% 9% 38% 7% 0.8446 0.8256Rai 1% 8% 8% 2% 13% 32% 27% 8% 0.8247 0.8032Ers 0% 0% 0% 0% 0% 0% 88% 12% -0.0366 -0.1635Spa 0% 0% 0% 13% 2% 2% 75% 8% 0.5511 0.4961
Fund EVALUE EGROWTH NASTAND JPSTAND G7GOV API ATX R² adjusted R²Con 0% 1% 0% 0% 3% 89% 7% 0.7899 0.7642A4 8% 15% 4% 2% 13% 58% 0% 0.8218 0.8000Gen 3% 38% 5% 0% 10% 37% 6% 0.8418 0.8224Rai 3% 6% 10% 13% 32% 27% 9% 0.8243 0.8028Ers 0% 0% 0% 0% 0% 88% 12% -0.0366 -0.1635Spa 0% 0% 15% 3% 1% 75% 6% 0.5376 0.4809
Fund EVALUE EGROWTH NAVALUE JPSTAND G7GOV API R² adjusted R²Con 7% 1% 0% 0% 3% 89% 0.7164 0.6817A4 6% 17% 4% 2% 14% 58% 0.8219 0.8001Gen 11% 40% 2% 0% 12% 35% 0.8347 0.8144Rai 8% 9% 11% 13% 31% 27% 0.8150 0.7923Ers 11% 0% 0% 0% 0% 89% -0.0639 -0.1941Spa 0% 5% 13% 4% 3% 74% 0.4927 0.4305
Figure 20: Style weights using different asset classes.
The data above reveals that style weights do not differ significantly when the
North American asset class is split in a value and a growth class. Thus inferring
that a North American composite asset class is sufficient to explain variations in
the returns of the individual funds. Yet when dropping the ATX asset class
weights change notably. It can be regarded as evidence that the investment
funds under evaluation do have exposure to the ATX asset class and that
inclusion in the analysis is essential. The analysis shows that using the seven
asset classes introduced in 4.2.2.1 is reasonable. However, introducing a
European corporate bond asset class may have improved the results. This is
consistent with intuition that Austrian investment funds classified as "International
Mixfunds" should have at least exposure to the asset classes in the middle panel
of figure 20. Figure 15 reveals that Ers fund should be strongly invested in
stocks. The constrained regression analysis over the entire sample period
53
indicates no exposure to stocks at all. The results must be questioned and the
usefulness of a constrained regression for Ers fund is seriously in doubt62.
The R-squared values are reasonable except for Ers fund. The likely reasons are
missing asset classes, changes in style and/or a high asset turnover63. The
analysis of changing styles will be put forth at a later point. Besides, R-squared
values are relatively stable when using different combinations of asset classes.
Notwithstanding a notable difference exists in R-squared between the six asset-
class model and the seven and eight asset-class model thus favoring one of the
latter. It is useful to mention that the objective is not to maximize R-squared but
to infer as much as possible about the fund's exposures to variations in the
returns of asset classes during the period studied64.
According to the evidence the seven asset-class model will be used in the
subsequent analysis.
Residual Series AnalysisThe resulting residual series from the constrained regression performed on the
six investment funds using Solver were exported to EVIEWS and their statistical
properties examined. The accuracy of the following analysis is somewhat limited
due to the constraints employed in the regression analysis and thus may lead to
residual properties different to the ones from standard regression analysis.
However the analysis is performed in order to gather an approximation of the true
results. Plotting the residuals in a histogram and applying a Jarque-Bera test
revealed that most residuals were distributed normal at about the same level as
the underlying fund returns presented in figure 11 to 16. Jarque-Bera test
analysis for Ers fund's return distribution unveiled non-normality at the 100%
level. Residual analysis for normal distribution revealed the same result.
62 LOBOSCO, DiBARTOLOMEO (1997), p. 8463 comp. CHRISTOPERSON (1995), p. 32 - 4364 SHARPE (1992), p. 11
54
A Ljung-Box (LB) test was applied to test for auto-correlation in the residuals.
The LB test follows a Chi-square distribution with k degrees of freedom where
the k degrees of freedom are equal to the number of auto-correlations. The
analysis was performed on the first 10 lags but only results for the first lag are
reported, as probability values concerning auto-correlation at a greater lag were
less significant.
AC PAC Q-Stat ProbCon -0.218 -0.218 2.8098 0.094A4 -0.048 -0.048 0.1339 0.714Gen 0.045 0.045 0.117 0.732Rai -0.244 -0.244 3.5254 0.06Ers -0.167 -0.167 1.6458 0.2Spa 0.006 0.006 0.0024 0.961
Figure 21: Test statistic of first-lag auto-correlation of residuals.
The values in the probability column are all greater than 0.05 revealing no auto
correlation at a α level of 95%. The null hypothesis of the LB statistic is that the
time series is White Noise thus inferring that the regression residuals are White
Noise in regard to auto-correlation and the null can not be rejected at the 95%
significance level.
Finally the mean expected values of the residuals were tested for significance
from zero. This was accomplished with a standard t-test. The t-values are
displayed in figure 22.
E(resid) STDV(resid) t-valueCon -0.00055 0.00452 -0.91294A4 -0.00359 0.00766 -3.51108Gen -0.00276 0.01174 -1.75742Rai 0.00281 0.01190 1.77016Ers 0.00922 0.03257 2.11946Spa 0.00184 0.01136 1.20879
Figure 22: T-values of the estimated residual's mean value for selected Austrian investmentfunds.
55
The critical t-value at the 95% level is (+ -) 1.67 for an approximately normally
distributed variable with k-1 degrees of freedom. Figure 22 shows that residuals
of A4, Gen, Rai and Ers fund are statistically significantly different from zero at
the 95% level. The style analysis for Ers fund returned unlikely results,
consequently the t-value is likely improper. One should keep in mind that the
calculated style weights represent an average over the estimation period leading
to inaccuracies in cases where portfolio management changes its style and
securities are frequently turned over.
4.2.2. Rolling a Window
The problem of changing styles can be overcome by rolling a time window during
the estimation period. In the following, the impact of this operation on styles
estimated in 4.2.1 and the corresponding R-squared values will be disclosed. I
used a 20-month time window hence estimating the style composition of a
portfolio on 36 consequential periods. The resulting style changes are 20-month
average style changes.
Estimate 1: min → ε for the time interval t….t+20
Estimate 2: min → ε for the time interval t+1….t+21
�
Estimate 36: min → ε for the time interval t+35….t+55
Besides, the same procedure as described in 3.3.2 was applied for every single
estimate. In general the R-squared values increased, as one would expect due to
the shorter estimation periods. Con funds style changes from January 1997 to
December 1999 are shown below.
56
Con Style Changes
0%
20%
40%
60%
80%
100%
J-97 A-97 J-97 O-97 J-98 A-98 J-98 O-98 J-99 A-99 J-99 O-99
Months
Styl
e W
eigh
ts
API G7GOV ESTAND ATX
Figure 23: Style changes of Con fund.
When examining the return properties of Con fund in 4.1.1, a low mean return
and a low standard deviation were specific to it. The assumption that Con fund is
mainly invested in fixed income securities was confirmed. Striking 89% of the
funds assets are invested in Austrian government bonds. The style weights from
the 56-month constrained regression were 1% EGROWTH, 3% G7GOV, 89%
API and 7% for the ATX. The corresponding R-squared value was 79%. Thus
inferring that almost 80% of the return generated by Con fund are attributable to
the asset allocation decision of Con fund. The graph in figure 23 reveals no
material changes in the composition of the fund over the estimation period.
In the course of the analysis it turned out that the weights estimates for EVALUE
and EGROWTH were relatively small. Thus the European growth and value
index were combined in the graph for a better perception. The similar mean and
standard deviation of these two asset classes have occasionally caused slightly
erratic style weight allocations between the two asset classes. The problem was
of minor importance though. The combined index is termed ESTAND.
The only slightly higher R-squared of 82.4% when taking the average R-squared
of the 36 estimations compared to the 79% computed initially provides evidence
57
for a stable investment style of Con fund. When performing style change analysis
the R-squared value climbed to about 90 % for the period from April 98 to July 90
and fell to around 70% by December 99. This may either indicate superior
selection skill of the fund management or a shift in asset allocation towards an
asset class not represented in the analysis.
A4 Style Changes
0%
20%
40%
60%
80%
100%
J-97 A-97 J-97 O-97 J-98 A-98 J-98 O-98 J-99 A-99 J-99 O-99
Months
Styl
e W
eigh
ts
API G7GOV ESTAND NASTAND JPSTAND ATX
Figure 24: Style changes of A4 fund.
A4 fund kept a steady 70:30 allocation policy between fixed income assets and
equity assets. Austrian government bonds although were steadily replaced by
European government bonds, reaching about 20% by the end of 1999. The
composition of equity shows diversification to all four equity-classes applied. The
strongest equity exposure is to European stocks with about 15% of total fund
value. Interestingly ATX values were eliminated by the end of 1998 maybe
reflecting the inferior development of ATX-values compared to other asset
classes. Generally when looking at A4 funds descriptive statistics one would
have expected a relatively strong fixed income portion in the portfolio as the
mean return was around 9% and standard deviation was relatively low. These
58
expectations are confirmed here. The composition of the fund can be regarded
as stable and therefore making it especially suitable for the style analysis
approach put fourth by Sharpe.65 R-squared increased when using the 20-month
window technique to an average of about 89% compared to 82% computed
initially. Again, this reveals that rolling a window enhances the explanatory
power of style analysis.
Figure 25 shows the composition change for Gen fund.
Gen Style Changes
0%
20%
40%
60%
80%
100%
J-97 A-97 J-97 O-97 J-98 A-98 J-98 O-98 J-99 A-99 J-99 O-99
Months
Styl
e W
eigh
ts
API G7GOV ESTAND NASTAND ATX
Figure 25: Style changes of Gen fund.
Distinct is the large share of ATX securities in the beginning of the evaluation
period and its reduction to about zero by the end of 1997, in favor of European
stocks. Furthermore the stable split of about 45:55 between bonds and stocks is
striking. In the period form April 1998 to April 1999 the R-squared value was
approximately 95% indicating little asset rotation and/or the application of asset
classes resembling viable passive investment alternatives. The average R-
squared value using the window technique augmented by about 6%, in line with
the findings in the A4 fund analysis. The reduction in Austrian government bonds
65 CHRISTOPHERSON (1995), p. 41
59
in favor of European government bonds somewhat resembles the development
of the fixed income asset classes of A4 fund over the time period studied.
Rai Style Changes
0%
20%
40%
60%
80%
100%
J-97 A-97 J-97 O-97 J-98 A-98 J-98 O-98 J-99 A-99 J-99 O-99
Months
Styl
e W
eigh
ts
API G7GOV ESTAND NASTAND JPSTAND ATX
Figure 26: Style changes of Rai fund.
Remarkable for Rai fund is the abrupt change out of European stocks into
European government bonds and North American stocks in the third quarter of
1997. Rai fund splits its securities at a 55:45 ratio between fixed income
securities and equities. In contrary to Gen fund, Rai fund increased its exposure
to ATX-securities by the end of 1997 but steadily reduced it thereafter. The
cumulative return on the ATX from July 1997 until December 1998 was circa
negative 15%, marking a possible reason for the reduction in exposure to the
ATX. Overall the styles reflected in the graph above are relatively smooth except
for the significant changes in 1997.
The style weights estimated for Ers fund in figure 20 did not seem plausible
because of the almost 90% exposure to the Austrian government bond index. Ers
had an average return of around 17% over the estimation period compared to a
6% average return of the API over the same period thus indicating that the real
style weights are misrepresented. The negative R-squared value is further
60
evidence that the estimated style weights for Ers fund are not representative for
the true style weights. The reason for this severe mis-specification may be the
lack of specific asset classes. This example shows the indispensability of
representing the entire investment universe when selecting asset classes.
Notwithstanding the composition of Ers fund will be displayed below.
Ers Style Changes
0%
20%
40%
60%
80%
100%
J-97 A-97 J-97 O-97 J-98 A-98 J-98 O-98 J-99 A-99 J-99 O-99
Months
Styl
e W
eigh
ts
API G7GOV ESTAND ATX
Figure 27: Style changes of Ers fund.
The consecutive style estimates for Spa fund are exhibited in figure 28. With the
Spa fund the significant increase in R-squared is striking. Compared to the
estimation of style weights over 56 months, R-squared raised by 17% from about
53% to 70% when applying the 20-month time window over the estimation
period. This could reflect a relatively high asset class rotation that could not be
detected with the "static" model. The near disappearance of European bonds and
equities could be doubted and may be the result of a missing asset class.
Unfortunately Spa fund did not provide any composition data in order to confirm
the findings.
61
Spa Style Changes
0%10%20%30%40%50%60%70%80%90%
100%
J-97 A-97 J-97 O-97 J-98 A-98 J-98 O-98 J-99 A-99 J-99 O-99
Months
Styl
e W
eigh
ts
API G7GOV ESTAND NASTAND JPSTAND ATX
Figure 28: Style changes of Spa fund.
4.2.3. Comparison of real and estimated style weights
Raiffeisen capital asset management, Bank Austria capital asset management
and 3-Banken-Gernerali capital asset management provided detailed
composition data for the requested funds. The findings in the preceding analysis
will be compared to the "true" composition of the funds of these 3 asset
management services. The individual securities will be assigned to the
corresponding asset class used in the constrained regression analysis. Problems
arise with European corporate bonds and foreign bonds. European corporate
bonds were assumed in the G7GOV asset class, as a more specific index (asset
class) was unattainable. Therefore the G7GOV is assigned all European bonds -
corporate and government. Non-European bonds were classified according to
their country-assignment by the capital asset management company. Two asset
classes are added in figure 29 to represent the true exposure of the individual
funds to them. Rai fund split European stocks in the supplied composition data
into EUR for the participating Euro countries and the remaining European stocks
according to their country. Hence the explicit portion of ATX securities in the EUR
composite weight provided by Rai fund could not be determined and was set to
62
zero. The data was supplied for December 1999. When comparing the data one
must keep in mind that he will be comparing the "true" composition at the end of
December 1999 with the average composition of each fund over the 20-month
period from May 1998 to December 1999.
TRUE CALC TRUE CALC TRUE CALCESTAND 12% 22% 43% 48% 18% 4%NASTAND 16% 4% 10% 5% 25% 16%ATX 3% 0% 0% 0% 0% 8%JPSTAND 4% 1% 0% 0% 10% 10%API 19% 48% 10% 24% 0% 6%G7GOV 33% 25% 34% 23% 15% 55%NABOND 9% 0% 0% 0% 21% 0%JPBOND 0% 0% 0% 0% 8% 0%CASH 4% 0% 3% 0% 0% 0%
A4 Gen Rai
Figure 29: True and calculated weight comparison of A4, Gen and Rai fund.
Figure 29 unveils a relatively close estimation of Gen funds true weight
exposures. The application of the appropriate asset classes in the constrained
regression analysis may very well be the substantial contributor to this result.
Except for the lack of a non-European bond asset class and a European
corporate bond asset class the passive benchmark asset classes utilized were
exhaustive. The relatively stable style over the period from the end of 1997 to the
end of 1999 may have assisted the close estimation of the true style. These
points refer exactly to what was discussed in 4.1, namely the properties that
asset classes should be exhaustive, mutual exclusive and have low correlations.
The estimated style weights for A4 fund resemble the true allocations to fixed
income assets and to equities reasonably close. An estimated 73% invested in
fixed income assets and the remainder in equity compared to the true 65% in
fixed income assets and 35% invested in equities by December 1999. The style
weight allocations to the single equity asset classes and bond asset classes are
unsatisfactory. The lack of certain bond asset classes may cause significant
63
misallocations in the applied asset classes. For a similar discussion see
LOBOSCO and DiBARTOLOMEO (1997).
With Rai fund the problem is very much the same. Striking 29% of the funds total
asset value was not represented in the constrained regression analysis leading
to the previously discussed misallocations. Interestingly the JPSTAND asset
class was not affected by that problem. Possibly the low correlation of the
JPSTAND asset class with the other asset classes caused the stability of the
estimated weight despite the named obstacles. Now, at the latest, the importance
of carefully and exhaustively specified asset classes becomes evident.
4.2.4. Contribution through Selection
So far the discussion of style analysis revolved around defining an investment
funds style, neglected the skill evaluation of the individual portfolio manager.
Detecting superior and inferior performance will be at the core in this section.
Style analysis qualifies asset allocation performance as variance of portfolio
performance explained. By that means, the R-squared ratio determines the
amount of variance in the returns of a portfolio explained and attributable to asset
allocation. Thus an R-squared value of 80% suggests that 80% of a portfolio's
return is attributable to the asset allocation decision of the investor or the
investment manager. The remaining part (in this case 20%) is what the
investment manager contributed to the overall portfolio return through active
stock selection. In Sharpe's words: "A passive fund manager provides an
investor with an investment style, while an active manager provides both style
and selection."66 Hence, an active manager is only worth the money if his
contribution through stock selection exceeds the higher management fees. In the
following the cumulative selection returns for five of the six funds are displayed.
Ers fund was not considered due to the improper style weight estimates resulting
from the restricted number of asset classes.
66 SHARPE (1992), p. 16
64
Cumulative Excess Return of Con Fund
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
M-95 N-95 M-96 N-96 M-97 N-97 M-98 N-98 M-99 N-99Months
Cum
ulat
ive
Exce
ss
Ret
urn
Figure 30: Cumulative selection return of Con fund.
Asset allocation accounts for 79% of Con funds return. The rest is attributable to
the above displayed selection return, which is negative. The average selection
return for Con fund was negative 0.055% per month with a standard deviation of
0.452% per month. The corresponding t-value is -0.91. The graph reveals that
the investment management may not have been worth its money. Especially in
the first year and a half of the examination period the underperformance is
persistent. From the end of 1997, a more erratic pattern evolves resembling
white noise instead of skill. One drawback is that it was not possible to verify the
estimated style weights by actual composition data since Constantia asset
management did not provide the data. The possible lack of an asset class may
influence the analysis in this paragraph too. However, this has no impact on the
effectiveness of the residual evaluation provided style weights are estimated
correctly.
65
Cumulative Excess Return of A4 Fund
-0.23
-0.18
-0.13
-0.08
-0.03
0.02
M-95 N-95 M-96 N-96 M-97 N-97 M-98 N-98 M-99 N-99Months
Cum
ulat
ive
Exce
ss
Ret
urn
Figure 31: Cumulative excess return of A4 fund.
In opposition to Con fund A4 fund's performance is relatively stable but also
negative. The average under performance is negative 0.36% per month with a
standard deviation of 0.766% per month and a t-value of -3.51. The critical t-
value at the 95% level for 55 degrees of freedom is 1.67, thus indicating that the
under performance is significant. The contribution through security selection to
the overall return of A4 fund was a negative 18%. The return contribution through
asset allocation to the overall return was 82%. A4 funds management provided
composition data for 12/1998 and 12/1999. Subsequently the data will be used
and the weights and results will be referred to as "true".
The process applied was the following: The weights computed for A4 fund when
rolling the window, were roughly steady. It was assumed that the "true" average
portfolio weights would be stable in the same manner. Therefore, the supplied
composition data for December 1998 and December 1999 was averaged and the
resulting weights calculated. Since asset class data for US bonds and cash
holdings were not included in the analysis the average 11% US bonds between
December 1998 and December 1999 were split between the asset classes
G7GOV and API assuming that these two asset classes somewhat resemble the
US-bond market. The 4% average cash were assumed to behave similar to API
and attributed to it. Figure 32 displays the cumulative excess return using the
66
estimation residuals from style analysis termed "cumresestim" and the "true"
cumulative excess return applying the supplied data for A4 fund.
-0.35-0.30-0.25-0.20-0.15-0.10-0.050.000.05
M-95 N-95 M-96 N-96 M-97 N-97 M-98 N-98 M-99 N-99
Months
Cum
ulat
ive
Exce
ss R
etur
n
Cumrestrue Cumresestim
Figure 32: Comparison of performance development of A4 fund using "true" style weights andestimated style weights.
The average performance when utilizing "true" weights was -0.586% per month
with a standard deviation of 1.4% per month. In comparison, the average under
performance using estimated data was -0.36% per month with a standard
deviation of 0.77% per month. The t-values were -3.1 and -3.5 for the "true" and
the estimated weights hence both show statistical significance at the 95% level.
The comparison shows that in this case, although the estimated weights do not
exactly resemble "true" weights, the residual analysis still provides reasonable
approximations for the true performance.
67
Cumulative Excess Return of Gen Fund
-0.23
-0.18
-0.13
-0.08
-0.03
0.02
M-95 N-95 M-96 N-96 M-97 N-97 M-98 N-98 M-99 N-99
Months
Cum
ulat
ive
Exce
ss
Ret
urn
Figure 33: Cumulative excess return of Gen fund.
Gen fund shows an average negative performance of 0.276% per month.
Substantial under performance occurred between May 1995 and December1996.
The fund underperformed by approximately -15% cumulatively over that period.
From there on the under performance was not as severe but still present. The
monthly standard deviation for the overall period was 1.174% and the figure for
the t-value was -1,74. The computed style weights were shown to comply
especially well with what the fund really invested in. This is exhibited in figure 29.
Gen fund did provide composition data but only for December 1999, thus the
"true" composition of Gen fund can not be reasonably defined especially since
style changes occurred as revealed by rolling the 20-month window through the
estimation period. Furthermore figure 28 indicated that the style of Gen fund was
well estimated and thus the residuals plotted in figure 33 should be
representative.
Over the evaluation period 84% of Gen funds return was accounted for by asset
allocation. The remaining 16% were the selection return described above. The
data leads to the conclusion that the security selection ability of the fund
management is not sufficient to justify active asset management fees, even
worse, the contribution through active stock selection rather corresponds to a
negatively sloped trend line. An investor who invested the equivalent portions of
68
the portfolio in the overall index instead of selected securities of the index, would
have outperformed Gen fund's management by approximately 15.5% over the
56-month period.
Cumulative Excess Returns of Rai Fund
-0.05
0.00
0.05
0.10
0.15
0.20
M-95 N-95 M-96 N-96 M-97 N-97 M-98 N-98 M-99 N-99
Months
Cum
ulat
ive
Exce
ss
Ret
urn
Figure 34: Cumulative excess returns for Rai fund.
According to the initial style analysis Rai fund's overall return is to 82%
attributable to Rai fund's the asset allocation decision.
However, residual analysis for Rai fund unveils significant abnormal positive
returns, particularly over the last three years. The t-value for the entire sample
period is 1.75. The average excess return is 0.28% per month with a standard
deviation of 1.19% per month. These findings must be used cautiously. Figure 29
shows that by December 1999 Rai fund was invested in US and Japanese bonds
with 29% of its total fund value. These two asset classes were not represented in
the style analysis computations, hence style weights may be inaccurately
estimated. The relatively high mean return of Rai fund of 1.39% per month or
about 16% p.a. (see Figure 14) may indicate that the portion of fixed income
securities is not 59% but somewhat lower. This would mean that the cumulative
excess return plot is upward biased.
69
To see how sensitive Rai fund's cumulative excess return is to changes in the
style weights two additional residual series are generated using different weight
settings. The impacts are demonstrated in Figure 35.
-0.05
0.00
0.05
0.10
0.15
0.20
M-95 N-95 M-96 N-96 M-97 N-97 M-98 N-98 M-99 N-99
Months
Cum
ulat
ive
Exce
ss R
etur
n
Testres1 Testres2 Raires
Figure 35: Sensitivity of Rai fund's selection return to style weight changes.
Testres1 is the cumulative excess return of a conducted style analysis for Rai
fund reducing the initially calculated fixed income weights of G7GOV and API by
5% each and increasing ESTAND and NASTAND by 5% thus reducing fixed
income asset classes by 10% and increasing equity asset classes by 10%. It is
assumed that JPSTAND is stable due to the distinct mean and standard
deviation characteristics.
In Testres2 the style weight change between the fixed income asset classes and
the equity asset classes was set to 5%. The API asset class was reduced by 3%
and the G7GOV asset class by 2%, while ESTAND was increased by 2% and
NASTAND by 3%.
Even when fixed income asset classes are reduced by 10% in favor of equity
asset classes the valuation reveals above zero cumulative excess returns. The
average excess return for Testres1 is 0.1% per month with a standard error of
1.26%. Testres2 shows an average excess return of 0.17% and a standard
deviation of 1.21%. In both cases the t-statistic is not significant with a t-value
70
below 1.1 and it can not be concluded that Rai's fund management exhibits stock
selection skill.
For reasons stated earlier a more detailed analysis of this point is not possible.
Cumulative Excess Return of Spa Fund
-0.04
0.00
0.04
0.08
0.12
0.16
M-95 N-95 M-96 N-96 M-97 N-97 M-98 N-98 M-99 N-99Months
Cum
ulat
ive
Exce
ss
Ret
urn
Figure 36: Cumulative excess return of Spa fund.
Spa funds asset allocation accounted for only 54% of its overall return, which
may be an indicator for missing asset classes. The computed style weights in
turn are also seriously in doubt as European bonds and equities are estimated to
be zero. Nevertheless the average excess return estimated is 0.18% per month
with a standard error of 1.14% and a t-value of 1.2 thus inferring insignificance
from zero.
For Ers fund no residual analysis will be conducted as the estimated style
weights are unlikely to be good estimates of the "true" style weights.
4.2.5. Summary of Findings
A comparison of the findings obtained through style analysis to common
benchmark measures like the Jensen Alpha, the Sharpe Ratio or the Traynor
Ratio does not seem to be useful due to distinct differences. The previously
mentioned measures are all based on a single benchmark, contrary to style
71
analysis, which splits the benchmark in an asset allocation and a style selection
return. Style analysis is most suited to define a funds composition and an
investment management's stock picking abilities and thereby provides a more
confined and detailed analysis of how the overall return was obtained.
Comparing selection return findings with general return ratios would be
comparing two different sets of information and thus would not be appropriate.
Nevertheless a general overview of the findings is provided below.
56m CumR Avg TR/m SR Avg B/MR/m Avg SelR/m T-V SelR
Con Fund 28.59% 0.51% 0.42 0.57% -0.055% -0.91A 4 Fund 42.47% 0.76% 0.83 1.12% -0.360% -3.51Gen Fund 64.07% 1.14% 1.13 1.42% -0.276% -1.74Rai Fund 77.77% 1.39% 1.35 1.11% 0.280% 1.75Ers Fund 81.71% 1.46% 1.12Spa Fund 58.84% 1.05% 1.31 0.87% 0.180% 1.2
56m CumR: Cumulative total return over the observed 56 months
Avg TR/m: Average total return per month over the observed 56 months
SR: Sharpe Ratio from trendInvest for the last 36 months using the 12-months Euribor
of 3.459% p. a. Avg B/MR/m: Average benchmark return per month over the observed 56 months
Avg SelR/m: Average selection return per month over the observed 56 months
T-V SelR: T-value of the average selection return per month
Figure 37: Summary of findings for Austrian investment funds.
4.3. Some additional insight using US Mutual Funds
The analysis of Austrian investment funds disclosed some critical points, for
example the specification of asset classes or multicollinearity. Thus some
additional evidence is sought using a different investment universe: The US
investment world. Four US mutual funds are analyzed using style analysis. The
funds are displayed below.
72
Acorn FundDreyfuss Growth Opportunity FundFidelity Magellan Fund20th Century Ultra Investors Fund
Figure 38: Selected US mutual funds.
According to the denomination Dreyfuss Growth Opportunity fund should be
mainly invested in equities and Fidelity Magellan fund should have strong
exposure to equities too, if no sever change in style occurred since Sharpe's
analysis. Especially interesting is Fidelity Magellan Fund as its style analysis data
estimated by Sharpe for the period from January 1985 to December 1989 were
displayed in his paper.67 And the time period studied will be from January 1990 to
December 1995 thus revealing possible changes over the entire period.
Asset class data for the period under examination was supplied by the Carlson
School of Management and in particular by Professor Maheswaran. The returns
for the asset classes as well as for the above-mentioned funds were provided
monthly. The returns were computed geometrically. The asset classes are:
US Treasury bills (Cash equivalents with less than 3 month to maturity)
US Intermediate-term government bond. (Government bonds with less than 10
years to maturity)
US Long-term government bonds (Government bonds with more than 10 years
to maturity)
US Corporate bonds
US Value stocks (Russel 1000 value index)
US Growth stocks (Russel 1000 growth index) 67 SHARPE (1992), p. 7 - 19
73
US Small stocks (Russel 2000, which are the 2000 smallest stocks of Russel
3000)
To gather a comprehension of the level of the multicollinearity dimension a
correlation summary is shown.
CORPBDS GRSTCKS IMBDS LTBDS SMSTCKS TBILLS VLSTCKSCORPBDS 1.00 0.56 0.92 0.98 0.43 0.04 0.64GRSTCKS 0.56 1.00 0.41 0.53 0.82 0.05 0.87IMBDS 0.92 0.41 1.00 0.91 0.22 0.14 0.50LTBDS 0.98 0.53 0.91 1.00 0.37 0.01 0.62SMSTCKS 0.43 0.82 0.22 0.37 1.00 -0.15 0.82TBILLS 0.04 0.05 0.14 0.01 -0.15 1.00 -0.12VLSTCKS 0.64 0.87 0.50 0.62 0.82 -0.12 1.00
Figure 39: Correlogram of selected US asset classes.
Interestingly the correlation between US growth and value stocks is about the
same as between European growth and value stocks. A high correlation is also
striking between different bond asset classes. This is what one would anticipate
due to the common sensitivity to interest rate changes and maybe similar
duration characteristics. T-bills in turn almost indicate no correlation at all.
74
CORPBDS GRSTCKS IMBDS LTBDS SMSTCKS TBILLS VLSTCKS
Mean 0.0069 0.0083 0.0061 0.0070 0.0093 0.0039 0.0075 Median 0.0091 0.0073 0.0058 0.0067 0.0197 0.0034 0.0127 Maximum 0.0436 0.1413 0.0270 0.0581 0.1122 0.0069 0.0838 Minimum -0.0383 -0.0959 -0.0258 -0.0450 -0.1338 0.0021 -0.0879 Std. Dev. 0.0176 0.0406 0.0135 0.0230 0.0483 0.0015 0.0347 Skewness -0.3873 0.3758 -0.4347 -0.2582 -0.4443 0.6691 -0.2612 Kurtosis 2.8024 4.3267 2.4504 2.7475 3.1198 2.1130 3.0743
Jarque-Bera 1.60 5.81 2.64 0.83 2.01 6.44 0.70 Probability 0.4499 0.0547 0.2665 0.6616 0.3661 0.0399 0.7061
Observations 60 60 60 60 60 60 60
Figure 40: Descriptive statistics of selected US asset classes.
Asset classes used in the analysis of US mutual funds are shown above. The
mean return of the overall equity universe has sharply increased in the period
form January 1995 to December 1999 compared to the period from January
1990 to December 1994. If one looks at the average return of growths, value and
small stocks - representing the entire US equity universe - than the mean return
is somewhere between 0.75% per month to 0.93% per month for the 1990 to
1995 period. The analysis of the 1995 to 1999 period displayed in figure 17
shows a mean return for North American equities of about 2.65% per month
about three times the average return of the 1990 to 1995 period. Corporate
bonds and long-term bonds are the only two asset classes showing similar mean
returns but standard deviations are different. The correlogram in figure 38
already revealed this information. Striking is also the substantial difference
between median and mean for small and value stocks. For small stocks the
median is more than double the value of the mean marking fewer but larger
negative returns relatively to positive ones.
75
0.0% 20.0% 40.0% 60.0% 80.0% 100.0%
TBILLS
IMBDS
LTBDS
CORPBDS
VLSTCKS
GRSTCKS
SMSTCKS
R-squared
-0.2
-0.1
0
0.1
0.2
0.3
J-90 J-91 J-92 J-93 J-94
Cum
ulat
ive
Exce
ss R
etur
n
Figure 41: Style weights and cumulative selection contribution of Akorn fund.
Akorn fund returned on average 1.13% per month with a standard deviation of
4.41% per month. The asset allocation decision of the fund management
accounted for 88.2% of the overall return. Akorn fund experienced substantial
exposure to small stocks, which returned about 0.93% per month from January
1990 to December 1994 hence possibly revealing stock selection skill. The
volatility of small stocks over that period was 4.83% per month.
The residual analysis shows that Akorn's fund management did contribute value
through stock selection of approximately 0.24% per month and a standard
deviation of 1.51% per month over the studied period. These figures are not
significant at the 95% level as the t-value of 1.25 is below the required 1.67. The
Jarque-Bera test for normality is 0.7 with an associated 70% probability for the
normal distribution assumption under the null hypothesis of Jarque Bera.
76
0.0% 20.0% 40.0% 60.0% 80.0% 100.0%
TBILLS
IMBDS
LTBDS
CORPBDS
VLSTCKS
GRSTCKS
SMSTCKS
R-squared
-0.2
-0.1
0.0
0.1
J-90 J-91 J-92 J-93 J-94
Cum
ulat
ive
Exce
ss R
etur
nFigure 42: Style weights and cumulative selection contribution of Dreyfuss Growth OpportunityFund.
Dreyfuss Growth Opportunity fund's composition data unveils an almost sole
exposure to small and growth stocks. The fund's mean return was 0.53% per
month which somewhat pales when compared to the average returns of the two
asset classes of 0.93% per month for small stocks and 0.83% per month for
growth stocks. The standard deviation of Deryfuss Growth Opportunity fund's
return series is about 4.46% per month, slightly lower than that of small stocks of
4.83% and higher than the 4.1% of the growth stock asset class. About 81% of
Dreyfuss Growth Opportunity fund's return are attributable to asset allocation.
Stock selection accounted for 19% of Dreyfuss Growth Opportunity fund's return
- in this case a negative stock selection return averaging 0.31% per month with a
volatility of 1.94%. The erratic graph of the cumulative excess return suggests no
stock selection skill over the analyzed period. The adjacent t-value is negative
1.25 thus not significant at the 95%. A Jarque-Bera test returned a value of 0.16
with a probability for normal distribution of the residuals of 92%.
77
0.0% 20.0% 40.0% 60.0% 80.0% 100.0%
TBILLS
IMBDS
LTBDS
CORPBDS
VLSTCKS
GRSTCKS
SMSTCKS
R-squared
-0.05
0.00
0.05
0.10
0.15
0.20
J-90 J-91 J-92 J-93 J-94
Cum
ulat
ive
Exce
ss R
etur
nFigure 42: Style weights and cumulative selection contribution of Fidelity Magellan fund.
Fidelity Magellan Fund is substantially invested in value and growth stocks. This
is not surprising as the fund managed about $14 billion at the end of 1989,
making substantial investments in small stocks difficult.68 Sharpe found in his
study that the management of Fidelity Magellan fund reduced the portion of small
stocks successively from 1985 to about mid 1987 and kept it constant thereafter
at a 15 to 18 percent level. Sharpe found an R-squared for Fidelity Magellan fund
of 97.3% for the period from January 1985 to December 1989. The R-squared
value for the period studied here is 93.6% thus asset allocation accounted for
more than 90% of Fidelity Magellan fund's generated return of the last 10 years.
The mean return of Fidelity Magellan fund was 1.03% per month with a standard
deviation of 4.08%.
Remarkable is Magellan fund's stock selection contribution. Although only 6.4%
relative to its asset allocation decision but 0.23% per month in absolute terms
with a volatility of 1.04% per month. Over the period studied by Sharpe the
selection return was 0.57% per month with a standard deviation of 1.05% per
month. Testing the residuals for normality resulted in a 76% probability using the
Jarque Bera test. The t-value for this period was 1.7 indicating significant
68 SHARPE (1992), p. 13
78
outperformance and the t-value found by Sharpe was 3.76.69 This shows Fidelity
Magellan fund's remarkable performance.
0.0% 20.0% 40.0% 60.0% 80.0% 100.0%
TBILLS
IMBDS
LTBDS
CORPBDS
VLSTCKS
GRSTCKS
SMSTCKS
R-squared
-0.2
0
0.2
0.4
0.6
0.8
J-90 J-91 J-92 J-93 J-94C
umul
ativ
e Ex
cess
Ret
urn
Figure 43: Style weights and cumulative excess return of 20th Century Ultra Investors fund.
20th Century Ultra Investors fund generated a mean return of 1.71% per month
during the time period studied. This impressive return was achieved by taking on
substantial risk, namely 6.75% per month. Compared to the previous funds
asset allocation accounts for remarkable smaller part of the overall return. 73.7%
of the overall fund return are attributable to asset allocation.
Style analysis reveals significant stock picking skill of 20th Century Ultra Investors
management. The average excess return was 0.81% per month with a standard
deviation of 3.46. Although the excess return exhibits a considerable standard
deviation the t-value of 1.82 is significant at the 95% level. The fact that the
residuals are only normally distributed with a probability of 43% somewhat
mitigates the aptness of the t-value.
69 SHARPE (1992), p. 18
79
5. Conclusion and Final Remarks
Generally there are two broad ways to approach portfolio performance
evaluation. One is based on knowing the exact composition of the target
portfolio. If access to this information is provided the Grinblatt & Titman model
may be well suited to detect manager skill. Furthermore, the procedure will
produce very accurate results and conventional benchmark problems are
irrelevant, as no benchmark is needed.
The alternative is that no knowledge about an investment fund's composition
exists. This was the subject this master's thesis explored. The main problem
arising from the uncertainty of the investment funds composition is the selection
of an appropriate benchmark. Especially when asset allocation decisions of an
investment management are restricted by investment policies imposed by the
investor. In such cases a "generic" benchmark like the S&P 500 will be
notoriously non-reached by a fixed income fund but likely beaten by a fund taking
on sufficiently high risk like a small stock fund. Traditional measures of
performance do take risk into account but do not regard investment style and
reveal only a relatively small part of the information incorporated in the return
series of funds.
The analysis of six Austrian investment funds using style analysis unveiled
strength and weaknesses of style analysis. Most crucial for obtaining sound
results is the proper choice of asset classes. The fact that asset classes were not
exhaustive as required by Sharpe70 turned out to be very demonstrative in terms
of limitations of the model. When asset classes, the fund is invested in, are not
represented in the constrained regression analysis the estimated weights are
seriously in doubt. Furthermore asset classes with very similar return histories,
similar mean return and standard deviation are also prone to cause erratic style
weight estimates when rolling a time window over the sample period. This
70 SHARPE (1992), p. 8
80
problem was occasionally present with the EVALUE and EGROWTH asset
classes, but was of minor impact. This infers that a correlation of about 0.87 does
not pose a significant problem even when potential asset classes are missing.
Another point of concern is high asset turnover as well as high asset class
turnover because style weights may not be stable and be insufficient estimates of
the "true" style weights. However, rolling a window can mitigate this problem.
When asset classes are specified appropriately the information content revealed
with style analysis is remarkable. Particularly practical is the division of portfolio
returns into an asset allocation contribution and a security selection contribution.
By that, an active investment manager's skill can be isolated from obligations
imposed by an investment policy or independent asset allocation decisions made
by the manager. On this basis the investor has a tool to select a manager with a
style he considers apt for his investment strategy. Furthermore sufficiently large
investment funds can construct a "portfolio manager matrix". Thus hiring
investment managers for the asset allocation decision, who have shown to do
especially well in that field and stock selection managers who were detected to
do well in the stock selection field. An additional advantage of style analysis is its
cost effectiveness and simplicity. The sole input are return series.
One problem the style analysis approach developed by Sharpe does not solve
either is the benchmark problem. Style analysis breaks the benchmark down into
separate benchmarks for each asset class. By that the benchmark problem has
only been broken down into separate "smaller" benchmark inaccuracies, but the
initial problem persists namely that that a mispecified riskless rate and a
mispecified market portfolio (individual asset class portfolios) can lead to the
inaccuracies exhibited in Figure 7.
Overall it seems that style analysis is very useful in cases where composition
data for specific investment fund is not available. Thus, referring to the question
addressed in the beginning about the usefulness of Sharpe's style analysis
model for portfolio performance evaluation, the conclusion was reached that
81
when asset classes are specified carefully, impressive information can be
revealed. Furthermore portfolio evaluation can be very specific resulting in a fair
classification and evaluation of a portfolio management's performance.
I
Figures
Figure 1: Haugen Robert A., "Modern Investment Theory", Third Edition,
Prentice Hall/Upper Saddle River, NJ (1993), p. 207.
Figure 2: Sorenson Eric H., Miller Keith L., Samak Vele, "Allocating between
Active and Passive Management", Financial Analyst Journal, Sept./Oct. (1998),
p. 19.
Figure 3: Treynor Jack L., Mazuy Kay K., "Can Mutual Funds Outguess the
Market?", Harvard Business Review, July/Aug. (1966), p. 133.
Figure 4: "Relationship between the TR and an investors' utility", Created by
Johann Aldrian
Figure 5: Ippolito Richard A., "On Studies of Mutual Fund Performance, 1962 -
1991", Financial Analysts Journal, Jan./Feb. (1993), p. 44.
Figure 6: Sharpe William F., "Morningstar's Risk-Adjusted Ratings", Financial
Analyst Journal, July/Aug., (1998), p. 30.
Figure 7: Roll Richard, "Performance Evaluation and Benchmark Errors (I)", The
Journal of Portfolio Management, Summer (1980), p.6.
Figure 8: Lehmann Bruce N., Modest David M., "Mutual Fund Performance
Evaluation: A Comparison of Benchmarks and Benchmark Comparisons",
Journal of Finance, June, (1987), p. 251.
II
Figure 9: Grinblatt Mark, Titman Sheridan, "Performance Measurement without
Benchmarks: An Examination of Mutual Fund Returns", Journal of Business, vol.
66 no. 1, (1993), p. 56.
Figure 10: Sharpe, William F., "Asset Allocation: Management Style and
Performance Measurement", Journal of Portfolio Management, Winter (1992),
p.10.
Figure 11: Descriptive statistics of Appollo 4 fund.
Figure 12: Descriptive statistics of Constantia Privat-Invest fund.
Figure 13: Descriptive statistics of Generali Mixfund.
Figure 14: Descriptive statistics of Raiffeisen Global Mix fund.
Figure 15: Descriptive statistics of SparInvest fund.
Figure 16: Descriptive statistics of Spängler Securities Trust fund.
Figure 17: Descriptive statistics of selected asset classes.
Figure 18: Cross-Correlogram of selected asset classes.
Figure 19: Solver set-up to calculate Sharpe style weights.
Figure 20: Style weights using different asset classes.
Figure 21: Teststatistic of first-lag auto-correlation of residuals.
III
Figure 22: T-values of the estimated residual's mean value for selected Austrian
investment funds.
Figure 23: Style changes of Con fund.
Figure 24: Style changes of A4 fund.
Figure 25: Style changes of Gen fund.
Figure 26: Style changes of Rai fund.
Figure 27: Style changes of Ers fund.
Figure 28: Style changes of Spa fund.
Figure 29: True and calculated weight comparison of A4, Gen and Rai fund.
Figure 30: Cumulative selection return of Con fund.
Figure 31: Cumulative excess return of A4 fund.
Figure 32: Comparison of performance development of A4 fund using "true"
style weights and estimated style weights.
Figure 33: Cumulative excess return of Gen fund.
Figure 34: Cumulative excess return of Rai fund.
.
Figure 35: Sensitivity of Rai fund's selection return to style weight changes.
Figure 36: Cumulative excess return of Spa fund.
IV
Figure 37: Summary of findings for Austrian investment funds.
Figure 38: Selected US mutual funds.
Figure 39: Correlogram of US asset classes.
Figure 40: Descriptive statistics of US asset classes.
Figure 41: Style weights and cumulative selection contribution of Akorn fund.
Figure 42: Style weights and cumulative selection contribution of Dreyfuss
Growth Opportunity fund.
Figure 42: Style weights and cumulative selection contribution of Fidelity
Magellan fund.
Figure 43: Style weights and cumulative excess return of 20th Century Ultra
Investors fund.
V
References
Brinson Gary P., Hood Randolph L., Beebower Gilbert L., "Determinants of
Portfolio Performance", Financial Analysts Journal, July/August (1986).
Christopherson Jon A., "Equity Style Classifications", The Journal of Portfolio
Management, Spring (1995).
Elton Edwin J., Gruber Martin J., "Modern Portfolio Theory and Investment
Analysis", Fourth Edition, John Wiley & Sons Inc./NY (1992).
Fama Eugene F., French Kenneth R., "Common Risk Factors in the Returns on
Stocks and Bonds", Journal of Financial Economics, February, (1993).
Fama Eugene F., "Efficient Capital Markets: A Review of Theory and Empirical
Work", Journal of Finance, May (1970).
Fischer, Edwin O., "Finanzwirtschaft für Fortgeschrittene", Oldenburg Verlag,
Wien - München, (1996).
Friend Irwin, Blume Marshall E., Crockett Jean, "Mutual Funds and other
Institutional Investors", Mc Graw Hill/NY (1970).
Graham Benjamin, "The Intelligent Investor", HarperBusiness NY, (1949).
Grinblatt Mark, Titman Sheridan, "Performance Measurement without
Benchmarks: An Examination of Mutual Fund Returns", Journal of Business, vol.
66 no. 1, (1993).
VI
Grinblatt Mark, Titman Sheridan, "Mutual Fund Performance: An Analysis of
Quarterly Portfolio Holdings", Journal of Business, 62 (1989).
Hull John C., "Options Futures and other Derivatives", Third Edition, Prentice
Hall/Upper Saddle River, NJ (1997).
Ippolito Richard A., "On Studies of Mutual Fund Performance, 1962 - 1991",
Financial Analysts Journal, Jan./Feb. (1993).
Jensen Michael C., "The Performance of Mutual funds in the Period 1945 -
1964", Journal of Finance, May, (1968).
Kothari S. P., Warner Jerold B., "Evaluating Mutual Fund Performance",
Working Paper SLOAN School of Management (1997); MIT, 50 Memorial Drive,
Cambridge, MA 02142; email:[email protected].
LeClair Robert T., "Discriminant Analysis and the Classification of Mutual
Funds", Journal of Economics and Business, Spring (1974).
Lehmann Bruce N., Modest David M., "Mutual Fund Performance Evaluation: A
Comparison of Benchmarks and Benchmark Comparisons", Journal of Finance,
June, (1987).
Lobosco Angelo, DiBartolomeo Dan, "Approximating the Confidence Intervals
for Sharpe Style Weights", Financial Analysts Journal, July/August (1997).
Modigliani Franco, Modigliani Leah, "Risk-Adjusted Performance", The Journal
of Portfolio Management, Winter (1997).
MSCI, "Index Calculations", www.msci.com/cover/index.html, June 2000.
VII
Österreichische Kontrollbank, "Index Downloads", www.oekb.at, June 2000.
Roll Richard, "Performance Evaluation and Benchmark Errors (I)", The Journal
of Portfolio Management, Summer (1980).
Ross Stephen A., "The Arbitrage Theory of Capital Asset Pricing", Journal of
Economic Theory, December, (1976).
Sharpe William F., "Morningstar's Risk-Adjusted Ratings", Financial Analyst
Journal, July/Aug., (1998).
Sharpe William F., Alexander Gordon J., Bailey Jeffery V., "Investments",
Sixth Edition, Prentice Hall/Upper Saddle River, NJ (1998).
Sharpe, William F., "Asset Allocation: Management Style and Performance
Measurement", Journal of Portfolio Management, Winter (1992).
Sharpe William F., "Determining a Fund's Effective Asset Mix", Investment
Management Review, December (1988).
Sharpe William F., "Mutual Fund Performance", Journal of Business, January
(1966).
Sorenson Eric H., Miller Keith L., Samak Vele, "Allocating between Active and
Passive Management", Financial Analyst Journal, Sept./Oct. (1998).
trendINVEST, "Investment & Fonds", No. 1/Jänner 2000.
Treynor Jack L., Mazuy Kay K., "Can Mutual Funds Outguess the Market?",
Harvard Business Review, July/Aug. (1966).
VIII
Treynor Jack L., "How to Rate Management of Investment Funds", Harvard
Business Review, Jan./Feb., (1965).
Tzcinka Charles A., "Equity Style Classifications: Comment", The Journal of
Portfolio Management, Spring (1995).
IX
Monthly data with dividends reinvested of the analyzed Austrian investment funds in EUR.
Con BA Gen Rai Ers Spa
A-95 39.39 145.31 82.48 344.70 88.77 68.06M-95 40.48 148.51 86.33 353.58 91.03 69.89
J-95 40.48 147.79 85.09 351.13 93.20 72.20
J-95 40.79 150.19 86.55 359.51 93.17 73.14
A-95 41.00 151.71 86.11 370.61 96.02 75.53
S-95 41.14 151.23 84.95 363.28 97.25 75.62
O-95 41.07 150.75 82.77 360.55 98.65 75.84
N-95 41.91 152.66 85.97 373.79 98.49 77.67
D-95 42.44 153.51 87.22 382.20 101.65 76.98
J-96 43.46 156.48 93.25 394.66 104.35 79.55
F-96 43.07 155.97 93.32 392.08 107.55 80.70
M-96 43.31 157.30 93.57 396.83 106.59 82.30
A-96 43.63 160.53 93.99 410.08 108.53 85.23
M-96 43.84 160.89 93.89 412.06 111.17 87.10
J-96 43.70 160.41 92.95 411.52 112.59 86.82
J-96 43.46 157.81 90.48 395.97 113.69 84.62
A-96 43.95 159.91 91.70 400.76 109.06 85.62
S-96 44.54 162.81 93.24 415.57 110.29 88.50
O-96 44.86 163.07 93.37 416.79 115.12 87.67
N-96 45.52 166.59 96.11 439.02 116.96 89.51
D-96 45.94 167.84 97.95 445.54 123.51 90.20
J-97 46.43 173.02 101.76 465.60 123.13 93.21
F-97 47.10 177.11 105.43 485.84 128.59 95.26
M-97 46.68 176.01 105.84 481.12 131.22 95.43
A-97 46.61 178.69 107.47 489.77 130.37 96.68
M-97 47.03 182.78 109.68 512.75 134.76 98.20
J-97 47.52 189.14 114.39 531.22 139.24 99.77
J-97 48.54 199.13 121.18 572.65 147.52 103.64
A-97 47.87 192.24 115.28 558.34 156.04 101.24
S-97 48.54 194.34 119.18 555.36 146.60 102.04
O-97 47.87 187.67 114.13 526.10 152.57 100.01
N-97 48.12 191.34 117.09 530.21 146.39 102.10
D-97 48.68 194.55 120.10 535.73 149.16 102.19
J-98 49.41 198.38 123.19 558.00 154.88 104.53
F-98 49.94 202.42 127.73 575.30 159.03 107.34
M-98 50.57 206.40 133.87 589.90 167.59 107.86
A-98 50.39 204.37 131.65 583.96 175.09 106.53
M-98 51.13 203.89 133.42 580.17 175.36 106.58
J-98 51.38 207.16 137.75 584.84 173.39 108.09
J-98 51.66 207.47 137.24 574.19 175.32 107.62
A-98 51.94 201.41 128.84 547.19 175.87 104.48
S-98 51.83 196.99 123.89 533.62 158.00 105.45
X
Con BA Gen Rai Ers Spa
O-98 51.76 199.71 125.57 545.52 153.47 109.10N-98 52.50 208.91 133.57 579.48 161.16 111.17
D-98 52.88 207.39 135.21 576.47 171.93 113.39
J-99 53.58 211.01 135.14 601.91 174.97 116.31
F-99 53.58 208.64 134.89 613.58 179.13 117.80
M-99 53.34 209.52 135.64 636.46 179.04 119.67
A-99 54.39 213.93 137.93 667.54 187.04 120.90
M-99 53.48 212.35 134.85 655.00 195.58 117.97
J-99 53.34 213.73 138.01 680.58 190.54 119.74
J-99 53.06 210.67 133.48 671.50 196.12 119.14
A-99 52.95 214.08 136.58 685.09 191.35 120.20
S-99 52.39 209.56 133.53 672.57 190.99 119.37
O-99 52.15 210.51 138.45 675.95 186.23 120.09
N-99 52.57 218.79 148.91 724.56 192.34 122.29
D-99 52.43 222.21 156.53 750.20 200.97 122.60
XI
Asset class data with net dividends reinvested in USD applied in Austrian fund's style analysis.
ESTAND EVALUE EGROWTH NASTAND NAVALUE NAGROWTH JPSTAND G7GOV API
A-95 180 105
M-95 0.0203 0.0242 0.0163 0.0380 0.0391 0.0369 -0.0640 190 107
J-95 0.0094 -0.0040 0.0227 0.0235 0.0143 0.0327 -0.0490 186 107
J-95 0.0508 0.0515 0.0501 0.0322 0.0360 0.0282 0.0766 187 108
A-95 -0.0394 -0.0480 -0.0313 -0.0003 0.0074 -0.0082 -0.0410 200 109
S-95 0.0298 0.0279 0.0342 0.0410 0.0313 0.0505 0.0084 196 110
O-95 -0.0047 -0.0067 -0.0028 -0.0012 -0.0137 0.0108 -0.0576 196 111
N-95 0.0071 0.0113 0.0034 0.0423 0.0482 0.0366 0.0575 205 113
D-95 0.0312 0.0357 0.0276 0.0144 0.0212 0.0072 0.0499 205 115
J-96 0.0065 0.0010 0.0116 0.0355 0.0313 0.0395 -0.0132 214 116
F-96 0.0181 0.0183 0.0179 0.0090 0.0062 0.0119 -0.0180 209 115
M-96 0.0119 0.0100 0.0142 0.0102 0.0179 0.0025 0.0347 210 116
A-96 0.0072 0.0101 0.0037 0.0157 0.0150 0.0163 0.0553 218 117
M-96 0.0077 0.0079 0.0071 0.0255 0.0108 0.0402 -0.0528 218 118
J-96 0.0110 0.0083 0.0136 0.0038 -0.0034 0.0109 0.0053 220 117
J-96 -0.0126 -0.0101 -0.0150 -0.0446 -0.0445 -0.0444 -0.0458 213 118
A-96 0.0292 0.0322 0.0263 0.0229 0.0281 0.0174 -0.0457 215 119
S-96 0.0208 0.0145 0.0265 0.0534 0.0402 0.0662 0.0341 226 121
O-96 0.0230 0.0197 0.0254 0.0276 0.0369 0.0171 -0.0694 228 122
N-96 0.0494 0.0534 0.0458 0.0724 0.0732 0.0712 0.0189 236 124
D-96 0.0192 0.0221 0.0170 -0.0203 -0.0189 -0.0216 -0.0716 235 125
J-97 0.0027 0.0082 -0.0018 0.0650 0.0506 0.0792 -0.1152 252 126
F-97 0.0131 0.0210 0.0067 0.0059 0.0057 0.0062 0.0231 260 127
M-97 0.0318 0.0316 0.0320 -0.0464 -0.0364 -0.0567 -0.0335 255 126
A-97 -0.0050 -0.0029 -0.0070 0.0613 0.0394 0.0836 0.0356 268 127
M-97 0.0418 0.0491 0.0346 0.0562 0.0561 0.0562 0.1048 265 127
J-97 0.0488 0.0407 0.0568 0.0429 0.0355 0.0504 0.0721 274 129
J-97 0.0458 0.0465 0.0451 0.0752 0.0818 0.0686 -0.0309 295 130
A-97 -0.0588 -0.0579 -0.0597 -0.0611 -0.0482 -0.0745 -0.0907 288 129
S-97 0.0926 0.0929 0.0922 0.0510 0.0544 0.0475 -0.0152 287 130
O-97 -0.0504 -0.0322 -0.0670 -0.0289 -0.0259 -0.0320 -0.0978 282 130
N-97 0.0152 0.0167 0.0138 0.0433 0.0355 0.0513 -0.0633 291 131
D-97 0.0359 0.0362 0.0358 0.0149 0.0242 0.0054 -0.0589 299 132
J-98 0.0408 0.0391 0.0424 0.0112 -0.0103 0.0328 0.0855 307 134
F-98 0.0753 0.0685 0.0818 0.0682 0.0644 0.0720 0.0052 306 135
M-98 0.0688 0.0949 0.0438 0.0514 0.0493 0.0535 -0.0705 314 135
A-98 0.0192 0.0208 0.0176 0.0111 0.0204 0.0018 -0.0041 306 135
M-98 0.0201 0.0221 0.0181 -0.0201 -0.0248 -0.0152 -0.0565 307 137
J-98 0.0109 -0.0008 0.0222 0.0382 0.0071 0.0684 0.0139 313 138
J-98 0.0196 0.0324 0.0074 -0.0128 -0.0184 -0.0075 -0.0133 310 139
A-98 -0.1344 -0.1607 -0.1099 -0.1538 -0.1746 -0.1350 -0.1211 313 141
S-98 -0.0408 -0.0409 -0.0407 0.0620 0.0551 0.0690 -0.0277 304 143
O-98 0.0770 0.0683 0.0856 0.0751 0.0787 0.0715 0.1550 301 143
XII
ESTAND EVALUE EGROWTH NASTAND NAVALUE NAGROWTH JPSTAND G7GOV API
N-98 0.0518 0.0543 0.0494 0.0644 0.0529 0.0760 0.0447 308 144D-98 0.0428 0.0197 0.0648 0.0555 0.0262 0.0834 0.0380 302 145
J-99 -0.0065 -0.0183 0.0046 0.0433 0.0249 0.0597 0.0072 315 147
F-99 -0.0257 -0.0105 -0.0400 -0.0297 -0.0101 -0.0472 -0.0222 321 146
M-99 0.0109 0.0428 -0.0217 0.0404 0.0252 0.0555 0.1299 330 146
A-99 0.0294 0.0665 -0.0105 0.0375 0.0859 -0.0120 0.0409 339 149
M-99 -0.0492 -0.0589 -0.0385 -0.0240 -0.0201 -0.0282 -0.0580 342 147
J-99 0.0167 0.0175 0.0159 0.0517 0.0273 0.0775 0.0904 341 145
J-99 0.0092 0.0258 -0.0102 -0.0317 -0.0270 -0.0366 0.0952 328 144
A-99 0.0101 0.0053 0.0154 -0.0070 -0.0313 0.0177 -0.0070 332 143
S-99 -0.0077 -0.0110 -0.0042 -0.0282 -0.0463 -0.0109 0.0589 331 143
O-99 0.0361 0.0191 0.0540 0.0632 0.0528 0.0728 0.0420 336 142
N-99 0.0266 -0.0053 0.0594 0.0217 -0.0049 0.0451 0.0420 351 143
D-99 0.0976 0.0776 0.1168 0.0699 0.0202 0.1107 0.0602 352 143
XIII
Monthly exchange rate data from April 1995 to December 1999
ATS-JPY ATS-USD RetJPY RetUSD
A-95 11.549 9.707M-95 11.765 10.094 0.019 0.039
J-95 11.47 9.736 -0.025 -0.036
J-95 11.05 9.719 -0.037 -0.002
A-95 10.62 10.315 -0.040 0.060
S-95 10.127 9.976 -0.048 -0.033
O-95 9.742 9.944 -0.039 -0.003
N-95 9.94 10.065 0.020 0.012
D-95 9.842 10.088 -0.010 0.002
J-96 9.776 10.48 -0.007 0.038
F-96 9.87 10.335 0.010 -0.014
M-96 9.74 10.382 -0.013 0.005
A-96 10.3 10.757 0.056 0.035
M-96 9.964 10.801 -0.033 0.004
J-96 9.838 10.731 -0.013 -0.007
J-96 9.651 10.34 -0.019 -0.037
A-96 9.578 10.418 -0.008 0.008
S-96 9.652 10.747 0.008 0.031
O-96 9.376 10.649 -0.029 -0.009
N-96 9.496 10.801 0.013 0.014
D-96 9.423 10.954 -0.008 0.014
J-97 9.466 11.495 0.005 0.048
F-97 9.872 11.905 0.042 0.035
M-97 9.583 11.819 -0.030 -0.007
A-97 9.61 12.166 0.003 0.029
M-97 10.272 11.962 0.067 -0.017
J-97 10.739 12.283 0.044 0.026
J-97 11.068 12.99 0.030 0.056
A-97 10.615 12.634 -0.042 -0.028
S-97 10.25 12.436 -0.035 -0.016
O-97 10.12 12.148 -0.013 -0.023
N-97 9.748 12.418 -0.037 0.022
D-97 9.767 12.633 0.002 0.017
J-98 10.117 12.837 0.035 0.016
F-98 10.094 12.729 -0.002 -0.008
M-98 9.777 13.001 -0.032 0.021
A-98 9.548 12.627 -0.024 -0.029
M-98 9.02 12.552 -0.057 -0.006
J-98 9.093 12.732 0.008 0.014
J-98 8.692 12.531 -0.045 -0.016
A-98 8.811 12.488 0.014 -0.003
S-98 8.675 11.803 -0.016 -0.056
XIV
ATS-JPY ATS-USD RetJPY RetUSD
O-98 9.97 11.616 0.139 -0.016
N-98 9.758 11.995 -0.021 0.032
D-98 10.290 11.747 0.053 -0.021
J-99 10.417 12.087 0.012 0.029
F-99 10.478 12.489 0.006 0.033
M-99 10.766 12.810 0.027 0.025
A-99 10.843 12.985 0.007 0.014
M-99 10.820 13.160 -0.002 0.013
J-99 11.024 13.323 0.019 0.012
J-99 11.170 12.867 0.013 -0.035
A-99 11.911 13.015 0.064 0.011
S-99 12.213 12.902 0.025 -0.009
O-99 12.556 13.164 0.028 0.020
N-99 13.360 13.628 0.062 0.035
D-99 13.395 13.697 0.003 0.005
XV
Monthly US Mutual fund data from January 1990 to December 1994
20th cent. Ultra Investor Akorn Dreyfuss Fidelity MagellanJ-90 -0.11020 -0.06480 -0.04640 -0.06450
F-90 0.03690 0.01250 0.01240 0.02050
M-90 0.03680 0.03030 0.02150 0.02560
A-90 -0.02450 -0.02890 -0.02910 -0.02530
M-90 0.17090 0.07130 0.07840 0.08890
J-90 0.01610 0.01630 -0.00190 0.00430
J-90 -0.02530 -0.01440 -0.01050 -0.01130
A-90 -0.08880 -0.13460 -0.08910 -0.09800
S-90 -0.05590 -0.10630 -0.05850 -0.06350
O-90 -0.02640 -0.02350 0.00960 -0.01240
N-90 0.13200 0.03970 0.04300 0.07570
D-90 0.06610 0.03310 0.01480 0.03100
J-91 0.15270 0.05960 0.04260 0.06990
F-91 0.08210 0.08170 0.07540 0.08680
M-91 0.12840 0.04050 0.04770 0.03400
A-91 -0.05120 0.03220 -0.02790 0.00370
M-91 0.08370 0.04710 0.04210 0.05700
J-91 -0.08770 -0.04310 -0.04860 -0.05880
J-91 0.12540 0.04620 0.07710 0.06210
A-91 0.06010 0.03790 0.03670 0.03070
S-91 -0.00610 0.00090 0.00180 -0.00380
O-91 0.06660 0.03160 0.04940 0.01340
N-91 -0.04570 -0.02840 -0.01400 -0.04940
D-91 0.17000 0.09670 0.15620 0.11800
J-92 0.01730 0.05360 -0.02040 0.00040
F-92 -0.01640 0.03640 -0.01860 0.02030
M-92 -0.06970 -0.00710 -0.05300 -0.02710
A-92 -0.07370 -0.03030 -0.04800 0.01480
M-92 0.02470 0.00430 0.02610 0.00960
J-92 -0.06270 -0.02440 -0.05000 -0.01760
J-92 0.05710 0.03760 0.05260 0.02810
A-92 -0.05140 -0.01600 -0.02290 -0.02170
S-92 0.02360 0.01770 0.01170 0.01150
O-92 0.04880 0.05280 0.02150 0.00710
N-92 0.07700 0.05800 0.05600 0.02530
D-92 0.05470 0.04140 0.01080 0.01940
J-93 0.02790 0.04300 -0.00080 0.02630
F-93 -0.07590 -0.00280 -0.07010 0.02090
M-93 0.05160 0.05410 0.01060 0.03670
A-93 -0.01770 -0.02510 -0.04380 0.00980
M-93 0.11670 0.08240 0.06860 0.03940
J-93 0.03950 0.01140 -0.00240 0.01390
J-93 -0.00850 -0.00510 -0.00400 0.01100
XVI
20th cent. Ultra Investor Akorn Dreyfuss Fidelity MagellanA-93 0.08120 0.06600 0.06070 0.05880
S-93 0.02850 0.02920 0.03690 0.01080
O-93 -0.02000 0.03810 -0.00870 0.00070
N-93 -0.05090 -0.03780 -0.04660 -0.03290
D-93 0.04290 0.03720 0.02700 0.02990
J-94 0.05940 0.00500 0.02610 0.03950
F-94 -0.01240 -0.01210 -0.01090 -0.00730
M-94 -0.06170 -0.04840 -0.06970 -0.04640
A-94 -0.00570 0.00830 0.00690 0.01000
M-94 -0.02920 -0.01130 0.00980 -0.01150
J-94 -0.04980 -0.03420 -0.02720 -0.04340
J-94 0.01140 0.03230 0.01790 0.03350
A-94 0.05850 0.04610 0.02250 0.04740
S-94 -0.01790 -0.01250 -0.01440 -0.02600
O-94 0.04490 -0.00370 0.01550 0.03380
N-94 -0.04060 -0.03890 -0.05000 -0.05480
D-94 0.01560 -0.00360 0.01320 0.01410
XVII
Monthly US asset class returns from January 1990 to December 1994
Tbills IMBDS LTBDS CORPBDS VLSTCKS GRSTCKS SMSTCKSJ-90 0.0057 -0.0105 -0.0343 -0.0191 -0.0623 -0.0804 -0.0873
F-90 0.0057 0.0007 -0.0025 -0.0012 0.0252 0.0071 0.0310
M-90 0.0064 0.0002 -0.0044 -0.0011 0.0102 0.0398 0.0392
A-90 0.0069 -0.0077 -0.0202 -0.0191 -0.0390 -0.0130 -0.0327
M-90 0.0068 0.0261 0.0415 0.0385 0.0830 0.1039 0.0708
J-90 0.0063 0.0151 0.0230 0.0216 -0.0227 0.0108 0.0026
J-90 0.0068 0.0174 0.0107 0.0102 -0.0087 -0.0091 -0.0438
A-90 0.0066 -0.0092 -0.0419 -0.0292 -0.0879 -0.0959 -0.1338
S-90 0.0060 0.0094 0.0117 0.0091 -0.0484 -0.0539 -0.0889
O-90 0.0068 0.0171 0.0215 0.0132 -0.0137 0.0040 -0.0611
N-90 0.0057 0.0193 0.0402 0.0285 0.0693 0.0674 0.0763
D-90 0.0060 0.0161 0.0187 0.0167 0.0254 0.0354 0.0393
J-91 0.0052 0.0107 0.0130 0.0150 0.0450 0.0512 0.0901
F-91 0.0048 0.0048 0.0030 0.0121 0.0665 0.0794 0.1122
M-91 0.0044 0.0023 0.0038 0.0108 0.0148 0.0394 0.0701
A-91 0.0053 0.0117 0.0140 0.0138 0.0074 -0.0047 -0.0026
M-91 0.0047 0.0059 0.0000 0.0039 0.0373 0.0446 0.0476
J-91 0.0042 -0.0023 -0.0063 -0.0018 -0.0421 -0.0474 -0.0578
J-91 0.0049 0.0129 0.0157 0.0167 0.0419 0.0536 0.0350
A-91 0.0046 0.0247 0.0340 0.0275 0.0182 0.0334 0.0369
S-91 0.0046 0.0216 0.0303 0.0271 -0.0074 -0.0174 0.0078
O-91 0.0042 0.0134 0.0054 0.0043 0.0166 0.0156 0.0264
N-91 0.0039 0.0128 0.0082 0.0106 -0.0513 -0.0255 -0.0463
D-91 0.0038 0.0265 0.0581 0.0436 0.0838 0.1413 0.0800
J-92 0.0034 -0.0195 -0.0324 -0.0173 0.0016 -0.0242 0.0811
F-92 0.0028 0.0022 0.0051 0.0096 0.0245 0.0015 0.0292
M-92 0.0034 -0.0079 -0.0094 -0.0073 -0.0145 -0.0273 -0.0338
A-92 0.0032 0.0098 0.0016 0.0016 0.0431 0.0072 -0.0351
M-92 0.0028 0.0222 0.0243 0.0254 0.0050 0.0074 0.0133
J-92 0.0032 0.0177 0.0200 0.0156 -0.0062 -0.0252 -0.0470
J-92 0.0031 0.0242 0.0398 0.0308 0.0386 0.0448 0.0348
A-92 0.0026 0.0150 0.0067 0.0090 -0.0306 -0.0122 -0.0283
S-92 0.0026 0.0194 0.0185 0.0099 0.0138 0.0116 0.0230
O-92 0.0023 -0.0182 -0.0198 -0.0156 0.0009 0.0150 0.0316
N-92 0.0023 -0.0084 0.0010 0.0069 0.0328 0.0435 0.0766
D-92 0.0028 0.0146 0.0246 0.0228 0.0238 0.0100 0.0348
J-93 0.0023 0.0270 0.0280 0.0250 0.0290 -0.0115 0.0338
F-93 0.0022 0.0243 0.0354 0.0256 0.0352 -0.0158 -0.0231
M-93 0.0025 0.0043 0.0021 0.0025 0.0295 0.0193 0.0324
A-93 0.0024 0.0088 0.0072 0.0052 -0.0128 -0.0400 -0.0275
M-93 0.0022 -0.0009 0.0047 0.0020 0.0201 0.0350 0.0442
J-93 0.0025 0.0201 0.0449 0.0293 0.0221 -0.0092 0.0062
J-93 0.0024 0.0005 0.0192 0.0100 0.0112 -0.0179 0.0138
XVIII
Tbills IMBDS LTBDS CORPBDS VLSTCKS GRSTCKS SMSTCKSA-93 0.0025 0.0223 0.0434 0.0287 0.0361 0.0410 0.0432
S-93 0.0026 0.0056 0.0005 0.0043 0.0016 -0.0075 0.0282
O-93 0.0022 0.0018 0.0096 0.0051 -0.0007 0.0278 0.0258
N-93 0.0025 -0.0093 -0.0259 -0.0188 -0.0206 -0.0066 -0.0326
D-93 0.0023 0.0032 0.0020 0.0067 0.0190 0.0173 0.0342
J-94 0.0025 0.0138 0.0257 0.0202 0.0378 0.0231 0.0313
F-94 0.0021 -0.0258 -0.0450 -0.0286 -0.0342 -0.0182 -0.0036
M-94 0.0027 -0.0257 -0.0395 -0.0383 -0.0372 -0.0483 -0.0527
A-94 0.0027 -0.0105 -0.0150 -0.0097 0.0192 0.0048 0.0059
M-94 0.0032 -0.0002 -0.0082 -0.0062 0.0115 0.0151 -0.0113
J-94 0.0031 -0.0028 -0.0101 -0.0081 -0.0240 -0.0295 -0.0337
J-94 0.0028 0.0169 0.0363 0.0309 0.0311 0.0342 0.0164
A-94 0.0037 0.0026 -0.0086 -0.0031 0.0287 0.0557 0.0557
S-94 0.0037 -0.0158 -0.0331 -0.0265 -0.0332 -0.0137 -0.0034
O-94 0.0038 -0.0023 -0.0025 -0.0050 0.0139 0.0236 -0.0040
N-94 0.0037 -0.0070 0.0066 0.0018 -0.0404 -0.0320 -0.0404
D-94 0.0044 0.0053 0.0161 0.0157 0.0115 0.0168 0.0268