Evaluation of Portfolio Performance What is Required of a Portfolio Manager (PM)? We have two major requirements of a PM: 1. The ability to derive above average returns for a given risk class (large risk-adjusted returns); and 2. the ability to completely diversify the portfolio to eliminate all unsystematic risk. May also desire large real (inflation-adjusted) returns, maximization of current income, high after-tax rate of return, preservation of capital. Requirement #1 can be achieved either through superior timing or superior security selection. A PM can select high beta securities during a time when he thinks the market will perform well and low (or negative) beta stocks at a time when he thinks the market will perform poorly. Conversely, a PM can try to select undervalued stocks or bonds for a given risk class. Requirement #2 argues that one should be able to completely diversify away all unsystematic risk (as you will not be compensated for it). You can measure the level of diversification by computing the correlation between the returns of the portfolio and the market portfolio. A completely diversified portfolio correlated perfectly with the completely diversified market portfolio because both include only systematic risk. Some portfolio evaluation techniques measure for one requirement (high risk-adjusted returns) and not the other; some measure for complete diversification and not the other; some measure for both, but don't distinguish between the two requirements. Composite Equity Portfolio Performance Measures
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Evaluation of Portfolio Performance
What is Required of a Portfolio Manager (PM)?
We have two major requirements of a PM:
1. The ability to derive above average returns for a given risk class (large risk-adjusted returns); and
2. the ability to completely diversify the portfolio to eliminate all unsystematic risk.
May also desire large real (inflation-adjusted) returns, maximization of current income, high after-tax rate of return, preservation of capital.
Requirement #1 can be achieved either through superior timing or superior security selection. A PM can select high beta securities during a time when he thinks the market will perform well and low (or negative) beta stocks at a time when he thinks the market will perform poorly.
Conversely, a PM can try to select undervalued stocks or bonds for a given risk class.
Requirement #2 argues that one should be able to completely diversify away all unsystematic risk (as you will not be compensated for it). You can measure the level of diversification by computing the correlation between the returns of the portfolio and the market portfolio. A completely diversified portfolio correlated perfectly with the completely diversified market portfolio because both include only systematic risk.
Some portfolio evaluation techniques measure for one requirement (high risk-adjusted returns) and not the other; some measure for complete diversification and not the other; some measure for both, but don't distinguish between the two requirements.
Composite Equity Portfolio Performance Measures
As late as the mid 1960s investors evaluated PM performance based solely on the rate of return. They were aware of risk, but didn't know how to measure it or adjust for it. Some investigators divided portfolios into similar risk classes (based upon a measure of risk such as the variance of return) and then compared the returns for alternative portfolios within the same risk class.
We shall look at some measures of composite performance that combine risk and return levels into a single value.
Treynor Portfolio Performance Measure (aka: reward to volatility ratio)
This measure was developed by Jack Treynor in 1965. Treynor (helped developed CAPM) argues that, using the characteristic line, one can determine the relationship between a security and the market. Deviations from the characteristic line (unique returns) should cancel out if you have a fully diversified portfolio.
Treynor's Composite Performance Measure: He was interested in a performance measure that would apply to ALL investors regardless of their risk preferences. He argued that investors would prefer a CML with a higher slope (as it would place them on a higher utility curve). The slope of this portfolio possibility line is:
A larger Ti value indicates a larger slope and a better portfolio for ALL INVESTORS REGARDLESS OF THEIR RISK PREFERENCES. The numerator represents the risk premium and the denominator represents the risk of the portfolio; thus the value, T, represents the portfolio's return per unit of systematic risk. All risk averse investors would want to maximize this value.
The Treynor measure only measures systematic risk--it automatically assumes an adequately diversified portfolio.
You can compare the T measures for different portfolios. The higher the T value, the better the portfolio performance. For instance, the T value for the market is:
In this expression, b m = 1.
Demonstration of Comparative Treynor Measures: Assume that you are an administrator of a large pension fund (i.e. Terry Teague of Boeing) and you are trying to decide whether to renew your contracts with your three money managers. You must measure how they have performed. Assume you have the following results for each individual's performance:
Investment
Manager
Average Annual Rate of Return
Beta
Z 0.12 0.90
B 0.16 1.05
Y 0.18 1.2
You can calculate the T values for each investment manager:
Tm (0.14-0.08)/1.00=0.06
TZ (0.12-0.08)/0.90=0.044
TB (0.16-0.08)/1.05=0.076
TY (0.18-0.08)/1.20=0.083
These results show that Z did not even "beat-the-market." Y had the best performance, and both B and Y beat the market. [To find required return, the line is: .08 + .06(Beta).
One can achieve a negative T value if you achieve very poor performance or very good performance with low risk. For instance, if you had a positive beta portfolio but your return was less than that of the risk-free rate (which implies you weren't adequately diversified or that the market performed poorly) then you would have a (-) T value. If you have a negative beta portfolio and you earn a return higher than the risk-free rate, then you would have a high T-value. Negative T values can be confusing, thus you may be better off plotting the values on the SML or using the CAPM (in this case, .08+.06(Beta)) to calculate the required return and compare it with the actual return.
Sharpe Portfolio Performance Measure (aka: reward to variability ratio)
This measure was developed in 1966. It is as follows:
It is VERY similar to Treynor's measure, except it uses the total risk of the portfolio rather than just the systematic risk. The Sharpe measure calculates the risk premium earned per unit of total risk. In theory, the S measure compares portfolios on the CML, whereas the T measure compares portfolios on the SML.
Demonstration of Comparative Sharpe Measures: Sample returns and SDs for four portfolios (and the calculated Sharpe Index) are given below:
Portfolio Avg. Annual RofR SD of return Sharpe measure
B 0.13 0.18 0.278
O 0.17 0.22 0.409
P 0.16 0.23 0.348
Market 0.14 0.20 0.30
Thus, portfolio O did the best, and B failed to beat the market. We could draw the CML given this information: CML=.08 + (0.30)SD
Treynor Measure vs. Sharpe Measure. The Sharpe measure evaluates the portfolio manager on the basis of both rate of return and diversification (as it considers total portfolio risk in the denominator). If we had a fully diversified portfolio, then both the Sharpe and Treynor measures should given us the same ranking. A poorly diversified portfolio could have a higher ranking under the Treynor measure than for the Sharpe measure.
This measure (as are all the previous measures) is based on the CAPM:
We can express the expectations formula (the above formula) in terms of realized rates of return by adding an error term to reflect the difference between E(Rj) vs actual Rj:
By subtracting the risk free rate from both sides, we get:
Using this format, one would not expect an intercept in the regression. However, if we had superior portfolio managers who were actively seeking out undervalued securities, they could earn a higher risk-adjusted return than those implied in the model. So, if we examined returns of superior portfolios, they would have a significant positive intercept. An inferior manager would have a significant negative intercept. A manager that was not clearly superior or inferior would have a statistically insignificant intercept. We would test the constant, or intercept, in the following regression:
This constant term would tell us how much of the return is attributable to the manager's ability to derive above-average returns adjusted for risk.
Applying the Jenson Measure. This requires that you use a different risk-free rate for each time interval during the sample period. You must subtract the risk-free rate from the returns during each observation period rather than calculating the average return and average risk-free rate as in the Sharpe and Treynor measures. Also, the Jensen measure does not evaluate the ability of the portfolio manager to diversify, as it calculates risk premiums in terms of systematic risk (beta). For evaluating diversified portfolios (such a most mutual funds) this is probably adequate. Jensen finds that mutual fund returns are typically correlated with the market at rates above .90.
Application of Portfolio Performance Measures
Calculated Sharpe, Treynor and Jenson measures for 20 mutual funds. Using the Jenson measure, only 3 managers had superior performance (Fidelity Magellan, Templeton Growth Funds, and Value Line Special Situations Fund) while 2 managers had inferior performance (Oppenheimer Fund and T. Rowe Price Growth Stock Fund).
Relationship among Portfolio Performance Measures
For all three methods, if we are examining a well-diversified portfolio, the rankings should be similar. A rank correlation measure finds that there is about a 90% correlation among all three measures. Reilly recommends that all three measures. [In my opinion the Jensen measure is the most stringent. It is testing for statistical significance, whereas the other methods are not. The other methods are also examining average returns, whereas the Jensen measure uses actual returns during each observation period.]
Factors that Affect Use of Performance Measures
You need to judge a portfolio manager over a period of time, not just over one quarter or even one year. You need to examine the manager's performance during both rising and falling markets. There are also other problems associated with these measures:
w Measurement Problems: All of these measures are based on the CAPM. Thus, we need a real world proxy for the theoretical market portfolio. Analysts typically use the S&P500 Index as the proxy; however, it does not constitute a true market portfolio. It only includes common stocks trading on the NYSE. Roll, in his 1980/1981 papers, calls this benchmark error.
We use the market portfolio to calculate the betas for the portfolios. Roll argues that if the proxy used for the market portfolio is inefficient, the betas calculated will be inappropriate. The true SML may actually have a higher (or lower) slope. Thus, if we plot a security that lies above the SML it could actually plot below the "true" SML.
w Global Investing: Incorporating global investments (with their lower coefficients of correlation) will surely move the efficient frontier to the left, thus providing diversification benefits. It may also shift the efficient frontier upward (increasing returns). [However, we have no proxy to measure global markets.]
Salomon Smith Barney small-cap /growth 50 131 196 246
Salomon Smith Barney small-cap /value 50 113 183 239
S&P large-cap /growth 162 27 38 45
S&P large-cap/value 338 27 34 44
S&P small-cap /growth 234 136 187 226
S&P small-cap /value 366 132 189 234
Note: 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. 708
12
TREYNOR , MAZUY (1966), p. 131 - 13611
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. 4612
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. Jensen
15
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 - 138
15
JENSEN, (1968), p. 389 - 416
16
GRINBLATT, TITMAN (1989), p. 393 - 416
17
SORENSON, MILLER, SAMAK (1998), p. 18
18
ELTON, GRUBER (1992), p. 70513
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. 4414
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 and15
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. Treynor
20
introduced a risk-adjusted measure to rank mutual fund
performance. As a measure of risk he used the beta. Beta reflects the nondiversifiable portion of a securities total risk and can be calculated from CAPM.
The equation is the following:
( )
( ) ( )
( )
p
R p R f
TR p
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 - 7516
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
SML1
SML2
rf
Beta
Return
r2
r1
ß2 ß1
SML1
SML2
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
β (p) = Sensitivity of portfolio (p) to market return changes
21
JENSEN (1968), p. 389 - 41617
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 Total
Sharpe 54 63 38 36 191
Jensen 51 81 36 52 220
Total 105 144 74 88 411
Treynor-Mazuy 6 10 8 10 34
Friend II 37 31 7 5 80
Contradictory Studies* 0 11 11 21 43
Grossman-Stiglitz 0 0 78 117 195
Source: 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. 83518
2.3.2. Capital-market-line based performance measures
When risk-adjusted portfolio performance measures are grounded on the capitalmarket-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 Ratio
23
essentially measures a portfolios average performance over
the risk-free rate per unit of total risk of the portfolio.
( )
( ) ( )
( )
p
R p R f
SR p
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 - 138
24
trendINVEST (2000), p. 56 - 9019
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 Modigliani
26
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:
( ) ( ) ( ) ( ) *
( )
( )
( ) R p R f R f
p
m
RAP p = − +
σ
σ
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:
RAP( p) = SR( p) *σ (m) + R( f )
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. 51
26
MODIGLIANI, MODIGLIANI (1997), p. 45 - 5420
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 paper
28
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. 21
28
SHARPE (1998), p. 21 - 3321
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 - 1222
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 Modest
30
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 - 26523
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 French
31
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:
i ik k
R(i) = λ0
+ λ 1
F1
+ ... + λ F
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 - 56
32
ROSS (1976), p. 341 - 36025
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 Modest
33
conducted an extensive study on different benchmarks.
They use the CRSP
34
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 - 265
34
University of Chicago Center for Research in Security Prices (It's files contain complete data on NYSE
listed stocks since July 1962)26
Values in % except for t-value
(absolute)
APT-M Alpha S-L-M
Alpha
Difference APT - SLM Alpha
VWER EWAR VWER
Jan. 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 0,40
Jan. 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 14,68
Jan. 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 11,12
Figure 8: S-L-M is the Sharpe-Lintner-Model. VWER denotes the excess return when using the
value weighted CRSP and EWAR denotes the excess return when using the equally weighted
CRSP as the benchmark. I calculated the t-value the following: α(i)/(σ(i)/²√130). 130 is the
number 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. 26027
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 Warner
36
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 Titman
37
pursued one
36
KOTHARI, WARNER (1997), p. 1 - 44
37
GRINBLATT, TITMAN (1993), p. 47 - 6828
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:
PCM [R w( ) w ] T
N
j
T
t
jt jt j t k
/
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. 5129
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 31
st
1974 to December 31
st
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 31
st
1975 and the weights held on
December 31
st
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
Special purpose funds 3 -.10 -.16 .233 .21 .19 .711
Venture 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 = .0028
c
F2-statistic (Abnormal performance across categories is equal)
F ~ 3.6590*
Prob > F = .0014
c
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 objective
categories (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 Style
Analysis
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 Sharpe
39
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 LeClair
40
Brinson, Hood and Beebower .
41
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 - 19
40
LeCLAIR (1974), p. 220 - 224
41
BRINSON, HOOD, BEEBOWER (1986), p. 39 - 4432
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 - 69
43
SHARPE (1992), p. 8
43
LOBOSCO, DiBARTOLOMEO (1997), p. 8033
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 smallcap 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. 935
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 Results
Trustees' Commingled Fund-U.S. Portfolio
Januarv 1985 through December 1989
Unconstrained
Regression
Constrained
Regression
Quadratic
Programming
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 92.22
Figure 10: Resulting asset class weights through unconstrained and constrained regression and
through 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 common36
mutual funds policies do not allow short positions
46
Analytically, the program .
looks the following:
i ij j
[ ] ik K i
i
R w R w R
Var
Objective Function
ε
ε
+ + + =
→
⋅
...
min ( )
:
0 1
1
:
1
≥ ≥
=
⋅
∑
=
ij
K
j
iK
w
w
subject to
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. 11
47
SHARPE (1992), p. 1137
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
p t p t pj t j t
[ ] pK t K t ,
R ,
w ,
R ,
w ,
R ,
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 portfolio38
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
Trzcinka
50
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 DiBartolomeo
51
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 formula
52
they derived to calculate the confidence intervals
is shown below:
× − −1
=
Bi
n k
a
wi
σ
σ
σ
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. 38
49
SHARPE (1992), p. 11
50
TRZCINKA (1995, p. 46
51
LOBOSCO, DiBARTOLOMEO (1997), p. 80 - 85
52
LOBOSCO, DiBARTOLOMEO (1997), p 84 -8539
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. 8240
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 trendINVEST
54
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
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 - 7741
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 fiveyear 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 31
st
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 weights
55
, 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. 8442
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: CONS
Sample 1995:05 1999:12
Observations 56
Mean 0.005105
Median 0.005561
Maximum 0.027207
Minimum -0.016894
Std. Dev. 0.009842
Skewness -0.116157
Kurtosis 2.498004
Jarque-Bera 0.713930
Probability 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: A4
Sample 1995:05 1999:12
Observations 56
Mean 0.007585
Median 0.010455
Maximum 0.051477
Minimum -0.035200
Std. Dev. 0.018146
Skewness -0.250747
Kurtosis 3.153286
Jarque-Bera 0.641651
Probability 0.725550
Figure11: Descriptive statistics of Appollo 4 fund43
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: GEN
Sample 1995:05 1999:12
Observations 56
Mean 0.011441
Median 0.013952
Maximum 0.072833
Minimum -0.063160
Std. Dev. 0.029523
Skewness -0.271016
Kurtosis 2.795971
Jarque-Bera 0.782664
Probability 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: RAI
Sample 1995:05 1999:12
Observations 56
Mean 0.013887
Median 0.016291
Maximum 0.075101
Minimum -0.054117
Std. Dev. 0.028372
Skewness -0.195006
Kurtosis 2.604408
Jarque-Bera 0.720073
Probability 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-Bera44
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: ERS
Sample 1995:05 1999:12
Observations 56
Mean 0.014591
Median 0.019496
Maximum 0.064690
Minimum -0.107150
Std. Dev. 0.031988
Skewness -1.277064
Kurtosis 5.410110
Jarque-Bera 28.77513
Probability 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: SPA
Sample 1995:05 1999:12
Observations 56
Mean 0.010508
Median 0.012990
Maximum 0.038068
Minimum -0.029623
Std. Dev. 0.016710
Skewness -0.554665
Kurtosis 2.706236
Jarque-Bera 3.072791
Probability 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 exclusive
56
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. 846
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 Kontrollbank
58
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. 44
58
www.oekb.at, June 200047
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 bond48
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 API
59
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
differ
60
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 Index
60
SHARPE (1992), p. 849
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
Probability 0.1635 0.0629 0.0258 0.0000 0.4114 0.4246 0.0090
Observations 56 56 56 56 56 56 56
Figure 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 NASTAND
API 1.00 0.09 0.20 0.18 0.48 0.07 0.28
ATX 0.09 1.00 0.71 0.82 0.37 0.46 0.63
EGROWTH 0.20 0.71 1.00 0.88 0.51 0.55 0.80
EVALUE 0.18 0.82 0.88 1.00 0.48 0.56 0.79
G7GOV 0.48 0.37 0.51 0.48 1.00 0.31 0.67
JPSTAND 0.07 0.46 0.55 0.56 0.31 1.00 0.56
NASTAND 0.28 0.63 0.80 0.79 0.67 0.56 1.00
Figure 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 deviations
61
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. 851
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 R²
Con 0% 1% 0% 0% 0% 3% 89% 7% 0.7899 0.7642
A4 7% 16% 3% 1% 2% 13% 58% 0% 0.8221 0.8004
Gen 4% 35% 0% 7% 0% 9% 38% 7% 0.8446 0.8256
Rai 1% 8% 8% 2% 13% 32% 27% 8% 0.8247 0.8032
Ers 0% 0% 0% 0% 0% 0% 88% 12% -0.0366 -0.1635
Spa 0% 0% 0% 13% 2% 2% 75% 8% 0.5511 0.4961
Fund EVALUE EGROWTH NASTAND JPSTAND G7GOV API ATX R² adjusted R²
Con 0% 1% 0% 0% 3% 89% 7% 0.7899 0.7642
A4 8% 15% 4% 2% 13% 58% 0% 0.8218 0.8000
Gen 3% 38% 5% 0% 10% 37% 6% 0.8418 0.8224
Rai 3% 6% 10% 13% 32% 27% 9% 0.8243 0.8028
Ers 0% 0% 0% 0% 0% 88% 12% -0.0366 -0.1635
Spa 0% 0% 15% 3% 1% 75% 6% 0.5376 0.4809
Fund EVALUE EGROWTH NAVALUE JPSTAND G7GOV API R² adjusted R²
Con 7% 1% 0% 0% 3% 89% 0.7164 0.6817
A4 6% 17% 4% 2% 14% 58% 0.8219 0.8001
Gen 11% 40% 2% 0% 12% 35% 0.8347 0.8144
Rai 8% 9% 11% 13% 31% 27% 0.8150 0.7923
Ers 11% 0% 0% 0% 0% 89% -0.0639 -0.1941
Spa 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 period53
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 doubt
62
.
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 turnover
63
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 assetclass 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 studied
64
.
According to the evidence the seven asset-class model will be used in the
subsequent analysis.
Residual Series Analysis
The 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. 84
63
comp. CHRISTOPERSON (1995), p. 32 - 43
64
SHARPE (1992), p. 1154
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 Prob
Con -0.218 -0.218 2.8098 0.094
A4 -0.048 -0.048 0.1339 0.714
Gen 0.045 0.045 0.117 0.732
Rai -0.244 -0.244 3.5254 0.06
Ers -0.167 -0.167 1.6458 0.2
Spa 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-value
Con -0.00055 0.00452 -0.91294
A4 -0.00359 0.00766 -3.51108
Gen -0.00276 0.01174 -1.75742
Rai 0.00281 0.01190 1.77016
Ers 0.00922 0.03257 2.11946
Spa 0.00184 0.01136 1.20879
Figure 22: T-values of the estimated residual's mean value for selected Austrian investment
funds.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
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 to62
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 CALC
ESTAND 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 significant63
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. 1664
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-99
Months
Cumulative Excess
Return
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-99
Months
Cumulative Excess
Return
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 tvalue 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 the66
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
-0.05
0.00
0.05
M-95 N-95 M-96 N-96 M-97 N-97 M-98 N-98 M-99 N-99
Months
n Cumulative Excess Retur
Cumrestrue Cumresestim
Figure 32: Comparison of performance development of A4 fund using "true" style weights and
estimated 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
Cumulative Excess
Return
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 of68
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
Cumulative Excess
Return
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
Cumulative Excess Return
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-value70
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-99
Months
Cumulative Excess
Return
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 style71
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.