1.
INTRODUCTION
The existence of risk
means that the investor cannot associate a simple figure, or return, with an
investment in any asset. Analysis must be made through a series of returns and
associated with its probability of occurrence, called frequency distribution or
distribution of returns.
In an investment analysis
the two most used attributes of a distribution are a measure of central
tendency, called expected return, and a risk measure; the standard deviation is
the most used. Investors should not and do not have a simple asset as an
investment, they have groups or portfolios of assets. One important aspect in
risk analysis is that the risk of a portfolio is more complex than a simple
mean of the risk of individual assets; it is influenced by the level of
dependence among the assets.
This need to think in
terms of risk-return, that is, a bi-dimensional analysis, brings about a
certain difficulty. In order to make analysis easy, performance indexes were
created, which generally are ratios between a measure of return and a measure
of risk. The risk measures, in turn, can be very different according to the
perception of “what is risk?”
The most used performance
measure is the Sharpe Index (SHARPE, 1966), which uses the standard deviation
as a measure of risk. Another index that appeared in that same time was
proposed by Treynor (1965), who uses the Beta of the CAPM (SHARPE,1964) model
as a measure of risk.
Jensen (1968) suggests a
change to the CAPM model by not fixing the intercept as zero. This intercept
becomes known as Jensen’s Alfa and is used as a performance index.
As from the separation of
the concept of risk from that of uncertainty, Sortino (SORTINO & VAN DER
MEER, 1991) defines risk as the possibility of occurrence of an event, that is,
he separates the bad uncertainty and proposes the use of the negative variance
(FISHBURN, 1977) as a measure of risk.
Entropy is used as a
measure of risk by Philippatos (PHILIPPATOS & WILSON, 1972), and by using
this concept, a performance index is proposed by Caldeira (2005).
The objective of this
paper is to introduce the main performance indexes and detail their ideas and
formulations. Since no performance index is absolute because it depends mainly
on the definition of risk it uses, performance indexes of financial assets
analysis will be compared in order to observe their similarities and
differences.
2.
MAIN
PERFORMANCE INDEXES
2.1.
The Sharpe Index
This Sharpe (1966) Index
stems from the theory of portfolio analysis through the ratio between return
and risk, as measured by the mean and standard deviation respectively. This
performance index was made formal through the work of William Sharpe (1966).
Empirical studies show
that the market responds very quickly to new information by affecting the value
of the stocks; by this, the market tends to be perfectly informed and each
player uses the information in the same way.
The performance forecast
of a portfolio is described by two measures: the expected Rate of Return (RP)
and the forecast of variability, or risk, expressed by the standard deviation
of the return (sP). A hypothesis is that all investors are able to invest funds in a
risk-free common asset and borrow funds at the same rate. At some point, all
investors share the same forecasts regarding the future performance of the
assets (hence, of the portfolios). On these conditions, all efficient
portfolios will converge to a linear function given by:
(1)
Where RF is the return of
the risk-free asset and b is the risk premium.
If we highlight b, we
have:
(2)
This relation became known
as the Sharpe Index (ISharpe)
The best portfolio will be
the one with the highest ISharpe. If more than one portfolio is efficient, all
of them must be on the same straight line, that is, have equal indexes.
This index deals with
future performance forecasts. At some points, the best projection of return is
the mean of the historical returns. The same happens with the forecast for the
standard deviation, for which we use the historical standard deviation.
Therefore, it is usually used as:
(3)
In the event of a negative
Sharpe Index, it is not applicable, since it represents a risk premium, and
investors are contrary to risk, the premium must be positive; that is why it
loses interpretability.
Considering that the
Sharpe Index represents a risk premium, the following restriction exists:
, since a negative premium means loss.
2.2.
The Treynor Index
Treynor (1965) proposes to separate risk according to its nature. A part of
the risk is produced by the general floating of the market (volatility of the
stock market), and the other part results from the floatation of a certain
asset of the fund. The consequences of practical importance for each one of
these mentioned risks are:
I. The
effect of management of the rate of return of an investment made in a period of
time is normally amortized by the floatation of the market. Market oscillations
imply in better (or worse) profitability for the case of more unsteady funds
that will have better (or worse) profitability than those les unsteady funds.
For any example with a reasonable-sized interval, the mean of return is still
dominated by market trend.
II.Measures
of medium return do not provide support for those investors that are contrary
to risk. There is some importance in the floatation in certain stocks, for
portfolios with few stocks. In order to avoid this risk, the investors
diversify their portfolios. The importance tied to the risk varies from
investor to investor; for that reason, no absolute measure of fund performance
is viable.
The first step to obtain a
satisfactory measure of performance is to relate the expected return of the
portfolio (RP) with the return of a mean market index (RM), that is, take out
the influence of the market from the asset return. This relation is called
characteristic line and is defined by the following linear function:
(4)
The parameters of the
characteristic line can be estimated through statistical methods (such as that
of Square Minimums). Interesting is that this trend is stationary with time,
not considering the floatation of the short-term return.
It contains information
both on the rate of return and risk; its inclination measures volatility. A
steep inclination means that the rate of return of the fund involved is market
floatation-sensitive; a smooth inclination indicates that the fund involved is
relatively market floatation-insensitive.
This line might not be
very clear or significant (for instance, low R2). It means that not all of the
risk of the fund involved is explained through market floatation.
The performance of a fund
can be measured by the tangent of the angle of inclination. The demonstration of
the formula stems directly from geometry
(5)
where is the angle of
inclination of the characteristic line, and
is the symbol of
volatility (measured by the inclination of the characteristic line).
2.3.
Jensen´s Alpha
Jensen’s Alpha stems
directly from the application of the theoretical results of CAPM, independently
derived by Sharpe (1964), Lintner (1965) and Treynor. (1965) Consider that the return of portfolio P can be
written as:
(6)
Where:
Rate of return of
portfolio P,
Risk-free rate
Measurement of risk
Expected rate of
return from the market
Normal error id
Should one wish to
estimate the systemic risk of a non-managed asset or portfolio, estimate
through regression is an efficient one. Nevertheless, care must be taken when
applying this equation to managed portfolios. If a manager has better forecasts
(due to privileged information, for instance), he will systematically tend to
select portfolios that provide . Therefore, his portfolio will earn more than the “ordinary”
risk premium for its level of risk. We should be aware of such possibility on
estimating the systemic risk of a managed portfolio.
The relaxation of the
equation for such possibility can be done simply by breaking off the need of
regression to go through origin. Therefore, existence of a constant in equation
(1) by using (2) as an estimated equation is allowed.
(7)
The new error term will now have
.
So, if the portfolio
manager is able to foresee the prices of the assets, the intercept, , in equation (7), will be positive. Definitely, this
represents an average rate of increment in the portfolio for a time that is
solely due to the ability to foresee the future price of the assets. It is
interesting to notice that an ingenuous random selection of assets is expected
to have an intercept that equals zero. Therefore, if a manager does not make
good choices, Alpha tends to be negative.
2.4.
The Sortino Index
A number of scholars and
analysts complain that the standard deviation and beta are not relevant
measures in many investment situations because they do not distinguish good
risk from bad risk.
Sortino (1991) maintains
that the knowledge developed to make investment decisions under conditions of
uncertainty, and for which Harry Markowitz, William Sharpe, and Merton Miller
received the Nobel Prize, requires a clear distinction between risk and
uncertainty.
Risk and return are
inseparable components of the concept of uncertainty. Uncertainty in the
financial market is described in terms of the variance of the possible returns
and their chance of occurrence. The distribution of probability describes the
form of uncertainty.
In many investment
decisions there is a minimum expected return to achieve a target like, for
example, the portfolio of an insurance company needs a minimum return to meet
its obligations with its insurants. We can describe it like the minimum return
that must be achieved in order to avoid bad consequences. Those returns higher
than the minimum acceptable return (MAR) ensure meeting the target; therefore,
there is no risk. The more the return surpasses the MAR, the lower the risk of
a bad result.
We can say that the
standard deviation captures the risk associated to reach the mean, but it can
be completely dissociated from bad results, for only those returns below the
MAR incur in risk, and the lower below the MAR, the greater is the risk.
Therefore, an asset A that
has a minimum return above the MAR has no risk. If compared to an asset B that
has a lower standard deviation and a minimum return value below the MAR, this
asset B has a higher risk.
Hence, in situations where
there is an identifiable MAR, which is economically significant, we must tell
the good volatility from the bad one.
Measuring the bad
volatility can be done through negative variance (FISHBURN, 1977); therefore,
risk is shown as:
(8)
Consequently, the
risk-return index proposed by Sortino is given by:
(9)
2.5.
The Caldeira Index
This index was proposed
due to the comparison of uncertainty through variance not be valid for
distributions of different families. As for the indexes proposed by Sharpe
(1966) and Treynor (1965), there is a hypothesis that the data are Gaussian.
By observing the
limitation in the use of variance as a measure of risk, Philippatos and Wilson
(1972) propose replacement of variance by entropy. This measure of risk, in
turn, is more general and allows for the utilization of assets with any
distribution.
Still on this line,
Philippatos and Gressis (1975) conduct a comparative study between the
mean–variance model and the mean–entropy model, where he observes that the
models are equivalent for normal and uniform distributions. Concerning the
log–normal distribution, the results are not equivalent, since entropy becomes
a function not only of variance, but also of mean. This result was achieved due
to the asymmetry of this distribution. Nevertheless, this model, despite being
innovative, did not move forward due to the difficulty in implementing the
theory.
The basic principle of the
index proposed by Caldeira (2005) is to allow for the comparison of data
generated by any distribution without the need of the initial knowledge of its
generating function, thus turning this model into a generalization of the
Sharpe model. The generating function is estimated through the Parzen Windows (PARZEN,
1962). Therefore, this index can be written as:
(10)
Where h(P) is the
differential entropy of the portfólio.
3. EMPIRICAL
STUDY
This essay proposes to
compare the performance indexes presented in the previous chapter to financial
assets (it can be used with portfolios as well). Ten financial assets traded at
BOVESPA were selected and their properties studied. The data concerning the
daily returns of these assets were collected from the page of “Easy Invest”
brokers (www.easyinvest.com.br). The time span of the data ranges from August
10th 2000 to September 27th 2004.
The descriptive statistics
(see Table 1) show the big differences among these assets. Assets ARCZ3, CRUZ3
and VALE5 had average daily profitability above 0.15%, whereas assets BRAP4 and
TNLP4, they had averages lower than 0.07%. With relation to the first measure
of risk, the standard deviation, asset ELET6 is the one with the highest risk
followed by BRAP4 and EMBR4; and those assets with the lowest risk are VALE5,
ARCZ3 and PETR4, respectively.
As for the second measure
of risk, the semi standard deviation (square root of the semi variance), asset
ELET6 is the most risky, followed closely by EMBR4, while asset ARCZ3 has the
lowest level of risk, followed by VALE5. As for the risk measured by the semi
standard deviation, asset ARCZ3 is less risky than VALE5, which differs from
the perception of risk measured by the standard deviation. This difference is
explained by the high asymmetry coefficient of asset ARCZ3, which, under the
optics of loss, becomes more attracting. This same difference can be perceived
when comparing assets AMBV4 and CRUZ3.
Asset ITAU4 is the one
with the highest risk under the optics of entropy followed by ELET6 and TNLP4.
Asset ARCZ3, again, is the one with the lowest risk; the distribution function
estimated through the Parzen Windows for this asset, see Figure 3, shows a great
concentration of return around the mean, and for that reason its kurtosis is
very high.
Table 1:
Descriptive statistics of returns
Stocks PETR4, TNLP4 and
ELET6 (Figures 1, 2 and 3, respectively) are slightly asymmetrical and their
tails are a little heavier than the normal curve. Stocks EMBR4 and VALE5 (Figures
4 and 5, respectively) are also asymmetrical with the negative values of the
latter being well concentrated until -4%. Assets CRUZ3, AMBV4 and ITAU4 (Figures
6, 7 and 10, respectively) show a greater concentration in the middle and a
slight asymmetry with a greater excess kurtosis.
Stocks ARCZ3 and BRAP4 (Figures
8 and 9, respectively) have very different characteristics from the rest. The
former has its values very concentrated around its mean, displacing itself a
lot from the Normal curve. As for BRAP4, it is concentrated toward the center,
much lighter than ARCZ3, but it is a tri-modal distribution and its asymmetry
is perceived mainly due to a ‘gap’ between 2.5 and 5, which does not exist in
the negative part of the distribution.
It is possible to see that
all of the curves are a little out of normal; because of that, the Jarque-Bera
(Table 2) test rejected this hypothesis. In addition to this rejection, the
difference in asset distributions is clear, each distribution with a more
remarkable feature. Therefore, the estimates given by Parzen Windows maintain
important characteristics of distribution such as asymmetry and high kurtosis.
Figure 1: Estimate of the fsd of asset Petr4 via
Parzen Windows (H1=5).
Figure 2: Estimate of the fsd of asset TNLP4 via Parzen Windows (H1=5).
Figure 3:
Estimate of the fsd of asset ELET6
via Parzen Windows (H1=5).
Figure 4: Estimate of the fsd of asset EMBR4 via Parzen Windows (H1=5).
Figure 5: Estimate of the fsd of asset VALE5 via Parzen Windows (H1=5).
Figure 6: Estimate of the fsd of asset CRUZ3 via Parzen Windows (H1=5).
Figure 7: Estimate of the fsd of asset AMBV4 via Parzen Windows (H1=5).
Figure 8: Estimate of the fsd of asset ARCZ3 via
Parzen Windows (H1=5).
Figure 9: Estimate of the fsd of asset BRAP4 via
Parzen Windows (H1=5).
Figure 10: Estimate of the fsd of asset ITAU4 via
Parzen Windows (H1=5).
Table 2 :
Jarque-Bera Normality Test
Table 3 shows that, with
relation to the Beta risk, asset ELET6 is the riskiest, followed by TNLP4.
Under this risk viewpoint, the least risky is asset ARCZ3. The problem with
this measure of risk is that not all assets have a great adherence (adjusted
R2) to a market index (IBOVESPA). Therefore, the estimate for the Beta may not
represent the risk involved well.
Table 3: Beta
in relation to the IBOVESPA
From the measures of risk
and return, the performance indexes were calculated (Table 4). Due to the differences in terms of risk measurement, the absolute
results become somehow incomparable. In order to compare the behavior of these
indexes, it is necessary to use common references and thus create comparable
metrics.
One such way to compare
these performance indexes is the classification of investments through each
index (Table 4) and so compare the results.
Table 4:
Performance Indexes
One such way to compare these
performance indexes is the classification of investments through each index (Table 5) and so compare the results.
Table 5:
Classification of assets according to the performance indexes
As to the classification
of that asset with the poorest performance result, all the indexes selected
asset TNLP4, and likewise for the second poorest result, asset BRAP4. The
eighth place (except classification by the Caldeira Index) is for asset ELET6. As
for the other positions, the concordance differences are greater.
Asset AMBV4 is in seventh
place according to the Sharpe Index; however; some distribution characteristics
of its returns make this asset to go up at least one position regarding the
other indexes. By analyzing this asset from the risk and loss viewpoint
(Sortino Index), or referring to entropy as a measure of risk, it gains one
position to the sixth place.
Regarding asset ITAU4,
while the Sortino Index ranks this asset in fifth place, the Sharpe Index ranks
it sixth, the Traynor and the Jensen Indexes rank it seventh, and the Caldeira
Index rank it eighth. All this difference is due to differences in risk
measuring; the returns of this asset have positive asymmetry (Table 1), which
reduces the semi standard deviation (measure of risk of the Sortino Index).
Regarding the Caldeira Index, the entropy of this distribution is the highest
of all, making it the riskiest. This measure of risk makes this investment less
attractive from this point of view.
Asset EMBR4 (ranked as the
fifth best asset by the Sharpe Index) loses two positions when the Sortino
Index is used; this is due to the asymmetry of this distribution. But if the
Caldeira Index is used, this is the third best asset, since its entropy has a
positive influence.
Both the Sharpe and the
Sortino Indexes rank asset Petr4 the fourth best asset. This similarity can be
explained through the low coefficient of asymmetry. As to the classification by
the Caldeira Index, this asset is prejudiced by entropy, thus ranking fifth
place.
Asset ARCZ3 is ranked the
third best asset by the Sharpe Index and occupies the same position by the
Sortino Index, despite its high asymmetry. Nevertheless, the Caldeira Index
considers this asset the best investment due to its low risk (entropy), mainly.
According to the Sharpe
metrics, the best and the second best assets are respectively CRUZ3 and VALE5,
a result that repeats for the index. For the Caldeira metrics, these assets are
classified as second and third respectively. This difference occurs mainly due
to the low entropy of asset ARCZ3, which makes it the best asset according to
this index.
The results for the
Traynor Index are not significant, since the CAPM model does not show great
adherence for most of the assets. The Jensen’s Alpha is significant only in
assets VALE5, CRUZ3 and ARCZ3. And this is the respective classification order.
Table 6 shows the result
of the correlations among the indexes. Generally speaking, these correlations
are high, at a maximum of 99.69% and minimum of 70.52%.
These results show an
almost perfect correlation (99.69%) between the Sharpe and Sortino Indexes.
This result is due to the fact that these assets are only slightly
asymmetrical. Regarding the Sharpe Index and the Jensen’s Alpha, this result
also shows a high correlation (93.77%). As for the correlation between the
Traynor and Caldeira Indexes, it falls to 73.83% and 70.52%, respectively. This
deviation in relation with the Caldeira Index can be explained by the fact that
the data do not have a Gaussian distribution.
The Traynor Index has a
high correlation with the Caldeira Index (97.8%), which is reduced when it is
compared to the Jansen’s Alpha and the Sortino metrics (80.00% and 77.83%,
respectively). The correlation between the Jensen’s Alpha and the Sortino Index
is pretty high (93.28%), and with relation with the Caldeira metrics, this
correlation is even lower (75.75%). And the correlation between the Sortino and
Caldeira Indexes is 74.50%.
By and large, the Caldeira
Index is the one that has the most differences regarding the other ones. This
is so because it does not have to work with Gaussian data, which is a
hypothesis for the other indexes.
Table 6:
Correlation among the indexes
4.
CONCLUSIONS
This paper shows that
there are different metrics to assess investments. The ones shown herein differ
regarding the understanding of risk. Risk is a subjective concept that we try
to decode by adopting objective metrics. However, the understanding as to what
risk metrics é the best adopted one shall depend mainly on who is involved in
the matter.
The standard deviation
used as a measure of risk by the Sharpe Index represents well the dispersion
around the mean; however, Sortino presents a feature that was not predicted by
Sharpe, asymmetry. This asymmetry creates the necessity to separate risks; as a
risk of loss, the semi standard deviation is adopted by the index proposed by
Sortino.
The index developed by
Caldeira aims to measure all the uncertainty involved in estimating the mean,
thus bringing entropy as a measure of risk and allowing for the comparison of
any distribution. This index has a similar problem to that of the Sharpe Index:
it does not separate good uncertainty from bad uncertainty (the problem that
originated the Sortino Index). This can be an alteration to be used in this
index.
The indexes using the CAPM
to be estimated have a major problem. If the model is not well adjusted, this
metrics will be not. In this essay only three Jensen’s Alphas were significant,
whereas only one model had R2 above 70% and only three above 50%.
By and large, the indexes
had pretty close results in terms of investment classification, and this can be
seen in the matrix of correlation of the indexes, which has the lowest
correlation around 70%. This means that, apparently, all of the risk metrics
used tend to be an approximation to the subjective term risk.
The Sortino Index can be a
more robust index if the objective is measuring of the index of loss; however,
despite the Caldeira Index does not separate risk of loss from risk of gain,
its measure of risk allows for the comparison of any distribution.
Joining the ideas from
both the Sortino and the Caldeira Indexes can bring about an even more robust
index that allows for measuring entropy for those values below an established
minimum (risk of loss).
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