Potential Predictors of Small Cap Stocks: Sharpe Ratio, Sortino Ratio, Alpha
Bethanie Royality-Lindman; Felicia Lin; Laura Meadows
Abstract:
One major debate surrounding the stock market is whether to follow active or passive trading.
Passive investors believe that, according to the Law of One Price and the Efficient Market
Hypothesis, there is no opportunity for making arbitrage profit and the best course of action is to
create a portfolio that reflects the overall market to balance gains and losses and to have a chance
at every successful stock. Active trading involves speculating stocks to see which will have the
best performance as well as finding differences in prices of the same stock to make an arbitrage
profit. Active investors have for many years been trying to find risk adjusted predictors of the
stock market in order to gain the most profit. We choose Sharpe Ratio, Sortino Ratio, and Alpha
after researching the most frequently used risk-adjusted measurements on various hedge-fund
websites. Via linear regression, those three measurements are tested for their significance in
determining and predicting the outcome of future full market cycles as risk-adjusted
measurements.
Introduction:
In 1959 Harry Markowitz created a model to explore if mean and variance of return are good
measures of an investor’s utility. Markowitz’s Mean-Variance Model suggests investors look to
minimize variance of return and maximize return. Given this assumption, many mathematicians
and money managers have tried to either predict the stock market as a whole (active managing)
or explain why this is not possible (passive investing).
William F. Sharpe argues for passive investing. Sharpe suggests that at any one point in time, the
market return is the weighted average of returns on all the possible investments within the
market. A passive investor by definition will have the same return as the market return, since a
passive portfolio reflects the same securities in the certain market with the same proportions of
investments that the whole market contains. Sharpe explains that an active investor must also
have the same return as the market return, given that the entire market’s return is a weighted
average of all securities, including those actively managed. Although individual active investors
may do better than or worse than the market, on average active investors do as well as the
market. Passive investing merely involves mimicking the market, but to invest actively one must
put in time and effort to speculate on the market and choose the best investments, or pay
someone to do this work. Since the return on active and passive investments must be exactly the
same, and the cost of active management is higher than that of passive, an actively managed
dollar is worth less than a passively managed dollar. With this, Sharpe concludes that passive
investing provides greater profit than active investing.
Sharpe created a ratio in order to try to predict the future of a stock’s success. This ratio is widely
used and involves subtracting the risk-free rate of return over a certain period of time from the
return on the investment being investigated, divided by the standard deviation of the stock’s
return over time. The standard deviation is an attempt at measuring the volatility of the stock.
upside volatility, or deviating from the mean return in a positive direction, is desired but is
accounted for in Sharpe’s ratio. In an attempt to improve upon the Sharpe ratio, Dr. Frank
Sortino created a ratio of his own, accounting only for the downside, and unwanted, volatility.
Sortino works closely with retirement funds in most of his work. He argues that not every person
has the same needs but that every investor obviously wants positive returns. Since it is not
necessary to beat the market to gain, passive investing is a viable option for earning a profit
while minimizing risk. Sortino believes in the looking carefully at downside capture, as shown in
his ratio, when predicting the outcome of stocks. Sortino notes that CAPM and passive investing
are limited but also explains that with current knowledge, they are the best options for
minimizing risk and maximizing return.
Two professors, mathematicians, and money managers that build off Sharpe’s work and argue
for CAPM are Eugene F. Fama and Kenneth R. French. In their paper “The Capital ASset
Pricing Model: Theory and Evidence” Fama and French explain the shortcomings of the version
of CAPM that was created by Lintner and Sharpe. The two think that there is more that must be
explored than just return, risk free return, and volatility to predict an investment’s outcome.
Fama and French mention the need to look at groups of stock that are similar and size and type,
or if the investment is a growth investment. Fama and French believe in the Law of One Price
and in passive investing, but struggle, as do many others, to find the perfect predictor of an
investment’s return.
In our investigation we looked at 296 small cap funds over a full market cycle (about 7.5 years)
to identify possible predictors of small cap investments’ future outcomes. Specifically, our
research looked into the functionality of the Sharpe Ratio, Sortino Ratio, and alpha to predict the
market.
Sharpe Ratio Analysis:
Sharpe Ratio is developed to measure risk-adjusted performance in any asset or portfolio. It is
calculated by determining the return of a risk-free asset normally using United States Treasury
Bills and subtracting that from the expected return of the asset or portfolio. That difference is
then divided by the standard deviations of the asset’s or portfolio’s return. The Sharpe Ratio
describes how much in “excess” return an investor is receiving for the extra risk/volatility that an
investor beholds when investing in riskier assets or portfolio. Investors look to be compensated
for investing in additional risk assets rather than for holding risk-free asset or portfolio like US
Treasury Bills or Bonds.
The Formula for Sharpe Ratio:
Sharpe Ratio =
𝑅𝑅 −𝑅𝑅𝑅
𝑅𝑅
𝑅𝑅 = Expected asset return
𝑅𝑅𝑅 = Risk-free rate of return
𝜎𝑅 = Asset standard deviation
The returns measured by this ratio have no defined time length as long as the assets or portfolio
are normally distributed. Any abnormalities like skewed data or multi-modal on the distribution
curve can be a problem. Asset standard deviation would potentially represent a weakness
depending on the distribution.
The risk-free rate of return is usually based on United States Treasury Bill. Investing with
additional risk is usually compensated, so a risk-free rate of return is used to see if the investor is
properly rewarded for taking that risk. While the risk-free asset and the investor’s target
asset/portfolio should be equal in duration, it doesn’t necessarily have to be as long as both of the
assets or portfolio can be annualized.
Since the full market cycle is 7.5 years, a long-dated IPS (inflation protected security) would
provide the best comparison for the investment, however that would result in different values for
the ratio due to the length of the duration of the investment; IPS would have a higher return than
United States Treasury Bills.
Since the ratio is calculated with the intention of finding the “excess” return or rather
compensation based on risk, the higher the ratio the better the investment for the investor is from
a risk to reward perspective.
The Sharpe Ratio is a risk adjusted measure of return that is used to evaluate the performance of
an asset or of a portfolio. The ratio can be used to compare one portfolio with another portfolio
by making a change to incorporate the risk difference. In the future, we will refer to First Market
Cycle as FMC and Second Market Cycle as SMC.
Descriptive Statistics: Standard Deviation FMC
Variable
Standard Deviation FMC
N
295
N*
0
Variable
Standard Deviation FMC
Median
27.460
Mean
27.708
Maximum
55.910
SE Mean
0.180
StDev
3.095
Variance
9.579
Minimum
18.515
Regression Analysis: Cycle Growth% versus Full Market Cycle Sharpe
The regression equation is
Cycle Growth% = 19.2 - 21.4 Full Market Cycle Sharpe
246 cases used, 49 cases contain missing values
Predictor
Constant
Full Market Cycle Alpha
S = 41.9102
R-Sq = 0.6%
Coef
19.15
-21.39
SE Coef
14.51
17.06
T
1.32
-1.25
P
0.188
0.211
VIF
1.000
R-Sq(adj) = 0.2%
𝑅0 :𝛽1 = 0
𝑅1 :𝛽1 ≠0
To analyze the relationship, we regress the Cycle Growth % between the first full market cycle
and the second full market cycle against the calculated Sharpe Ratio of a full market cycle. We
first check the standard deviation of the assets to ensure that the Sharpe Ratio is applicable; the
histogram of the standard deviation shows normal curve. We determine a fitted line equation of
Cycle Growth% = 19.2 - 21.4 Full Market Cycle Sharpe. From the R-Sq value of 0.6%, we can
see that practically none of the variability in Cycle Growth% was associated with Full Market
Cycle Sharpe. Also, given the resulting p-value of 0.211, we fail to reject the null hypothesis that
the slope is equal to zero at the 0.05 level of significance. We conclude that there is insufficient
evidence to conclude that there is a significant relationship between Full Market Cycle Sharpe
and Cycle Growth%, and thus the full market Sharpe Ratio is not a significant variable in
determining or predicting the outcome of full market cycles.
Sortino Ratio Analysis:
Investors’ usual approach to building a portfolio is by determining the amount of risk they are
willing to take on then selecting the appropriate investment that is in their range of risk tolerance;
in other words, the allocation of investment is driven by the amount of risk or “return” of
investment they desire.
The Sortino Ratio is the exact opposite of such an approach. This ratio focuses on the investors’
need for the amount of return on their investment not the amount of risk they wish to take on.
Sortino Ratio unlike the Sharpe Ratio measures only the downside volatility. The calculation is
similar to Sharpe except it uses the negative asset standard deviation for the denominator instead
of the standard deviation of the desired assets or portfolio. The risk-free return of the asset is
determined by the investor, for our situation it will be United States Treasury Bill to achieve
conformity throughout the rest of the calculations.
The Formula for Sortino Ratio:
Sortino Ratio =
𝑅𝑅 −𝑅𝑅𝑅
𝑅𝑅
𝑅𝑅 = Expected asset return
𝑅𝑅𝑅 = Risk-free rate of return
𝜎𝑅 = Negative asset standard deviation
This focus on the negative asset standard deviation makes the Sortino Ratio very relevant to
investors because it looks at potential losses instead of simply the risk or volatility of an
investment. The 𝑅𝑅𝑅 is also attractive due to the lack of limits on the parameter of what the
benchmark can be. The benchmark usually is used to compare volatility against risk-free
however it may not be every investor’s main objective. In regards to the Sortino Ratio, the
investor has the freedom of choice in determining which benchmark aligns with the focus of
making a bigger return on investment; thus the investor may choose a higher risk benchmark
versus the regular United States Treasury Bill.
Descriptive Statistics: Downside Standard Deviation FMC
Variable
Downside Standard Deviat
N
295
N*
0
Variable
Downside Standard Deviat
Median
11.760
Mean
11.895
Maximum
24.770
SE Mean
0.0823
StDev
1.414
Variance
2.000
Minimum
8.370
Regression Analysis: Cycle Growth% versus Full Market Cycle Sortino
The regression equation is
Cycle Growth% = 22.7 - 14.4 Full Market Cycle Sortino
246 cases used, 49 cases contain missing values
Predictor
Constant
Full Market Cycle Alpha
S = 41.8564
R-Sq = 0.9%
Coef
22.67
314.45
SE Coef
14.66
9.733
T
1.55
-1.48
P
0.123
0.139
VIF
1.000
R-Sq(adj) = 0.5%
𝑅0 :𝛽1 = 0
𝑅1 :𝛽1 ≠0
To analyze the relationship, we regress the Cycle Growth % between the first full market cycle
and the second full market cycle against the calculated Sortino Ratio of a full market cycle. We
first check the standard deviation of the negative assets to ensure that the Sortino Ratio is
applicable; the histogram of the standard deviation shows normal curve. We determine a fitted
line equation of Cycle Growth% = 22.7 - 14.4 Full Market Cycle Sortino. From the R-Sq value
of 0.9%, we can see that practically none of the variability in Cycle Growth% was associated
with Full Market Cycle Sortino. Also, given the resulting p-value of 0.139, we fail to reject the
null hypothesis that the slope is equal to zero at the 0.05 level of significance. We conclude that
there is insufficient evidence to conclude that there is a significant relationship between Full
Market Cycle Sortino and Cycle Growth%, and thus the full market Sortino Ratio is not a
significant variable in determining or predicting the outcome of full market cycles.
Alpha Analysis:
Alpha or the Jensen Index shares some resemblance to the CAPM (Capital Asset Pricing Model).
While CAPM equation is used to calculate the required ROI (return of investment), CAPM is
also used to evaluate the performance of various assets and portfolio. Alpha is used to figure out
how much the realized return of the assets and portfolio from the expected market return as
referenced in CAPM.
The formula for Alpha is as follows:
𝛼 = 𝑅𝑅 − [(𝑅𝑅 − 𝑅𝑅𝑅 )𝑅]
𝑅𝑅 = Expected asset return
𝑅𝑅 = Expected market return
𝑅𝑅𝑅 = Risk-free rate of return
Alpha is measures the premiums of risk in terms of β (beta); for our data we average out the 5
year β and 10 year β for the approximate of a full market cycle. It is assumed that the asset or
portfolio being measured is varied. Alpha also requires different risk-free rate of return for each
time interval to be measured. In this case the risk-free rate of return will be using T-Bill (United
States Treasury Bill); since our full market cycle is calculated to be 7.5 years, we will be using
an equivalent time duration for the T-Bill. We will examine the asset’s or portfolio’s returns
minus the risk-free asset’s return and relate that to the return of the market portfolio minus the
same risk-free asset’s return.
Regression Analysis: Cycle Growth% versus Full Market Cycle Alpha
The regression equation is
Cycle Growth% = 0.76 - 44.6 Full Market Cycle Alpha
246 cases used, 49 cases contain missing values
Predictor
Constant
Full Market Cycle Alpha
S = 41.5219
R-Sq = 2.5%
Coef
0.761
-44.62
SE Coef
2.655
17.94
T
0.29
-2.49
P
0.775
0.074
VIF
1.000
R-Sq(adj) = 2.1%
𝑅0 :𝛽1 = 0
𝑅1 :𝛽1 ≠0
To analyze the relationship, we regress the Cycle Growth % between the first full market cycle
and the second full market cycle against the calculated Alpha of a full market cycle. We
determine a fitted line equation of Cycle Growth% = 0.76 - 44.6 Full Market Cycle Alpha. From
the R-Sq value of 2.5%, we can see that practically none of the variability in Cycle Growth% was
associated with Full Market Cycle Alpha. Also, given the resulting p-value of 0.074, we fail to
reject the null hypothesis that the slope is equal to zero at the 0.05 level of significance. We
conclude that there is insufficient evidence to conclude that there is a significant relationship
between Full Market Cycle Alpha and Cycle Growth%, and thus the full market Alpha is not a
significant variable in determining or predicting the outcome of full market cycles.
While alpha is a good measure of performance that uses realized return and risk-free return to
calculate the risk on the investor, however calculation without further diversifying the asset and
portfolio weakens the correlation that can be provided by alpha.
Comparison of the Three Methods:
The classic model for determining risk of a portfolio is the Sharpe ratio. Since the Sharpe ratio is
often considered a base for comparison, I will begin by describing the Sharpe ratio, and then
compare the Sortino and Alpha to the Sharpe ratio. The Sharpe ratio essentially tells us the
origins of an investment's returns--either from excess risk, or carefully thought-out investment
decisions. As mentioned earlier, the Sharpe ratio is calculated by subtracting the risk free rate
from the average rate of return of the portfolio, and dividing it by the standard deviations of the
return of the investment. Essentially, the Sharpe ratio is a measurement of how much return an
investor is receiving, in exchange for the amount of risk that they are taking for a particular
investment. A limitation of the Sharpe ratio is that it is expressed as a raw number. This means
that having one fund’s Sharpe ratio is not meaningful, unless compared to another fund’s ratio.
The Sharpe ratio will give an idea of the risk adjusted return in comparison to other funds, but
will not be helpful on its own.
The Sortino ratio can be described as a modification of the Sharpe ratio. The difference between
the Sharpe and Sortino ratio is that Sortino takes into account “bad volatility” by separating the
positive and negative returns. Negative returns on an investment is described as bad volatility,
while good volatility is caused by positive returns and are desirable by investors.
Mathematically, the Sortino ratio takes the portfolio’s return and subtracts the risk-free rate, and
divides it by the downside deviation. The downside deviation is similar to standard deviation,
except that instead of using the mean, it only considers returns that fell below a minimum
acceptable return value. Therefore the difference between the Sharpe and Sortino ratios is that
the Sortino ratio does not discriminate based on all volatility, only the “bad”, or what falls below
a certain threshold. This threshold, or the minimum acceptable return value, is set by the investor
based on his or her comfort level. Therefore the Sortino ratio gives a more personalized rating of
an investment, based upon each individual investor. The Sortino ratio is most commonly used for
those investors who are risk-averse. It allows investors to set their minimum acceptable return
value and assess negative risk, as opposed to taking into account all volatility. Depending on
whether the investor wants to concentrate on downside deviation or standard deviation, it can be
difficult to say which ratio is more useful. Sortino ratio is often better for analyzing highly
volatile portfolios because it takes into account downside deviation, and therefore does not need
as many data points as standard deviation offers. On the other hand, the Sharpe ratio is best for
analyzing portfolios with low volatility because it requires more data points, that standard
deviation will give, while the downside deviation will not offer enough.
The alpha value takes into account a portfolio’s beta. It is the the measurement of the actual
return and expected return, taking its beta into consideration. In other words, it is the excess risk,
taking into account the risk that the investor took. Beta measures volatility that cannot be
eliminated through diversification. If an investment yields a higher return than what was
expected given its beta, the value will be positive. One of the drawbacks of alpha in comparison
to the Sortino and Sharpe ratio is that the value of beta must be meaningful. For instance, since
the alpha ratio depends on beta, if the beta of a particular investment has an R-squared value that
is too low, then alpha will not be meaningful. The Sharpe ratio does not have this limiting factor,
and therefore will always be meaningful. The Sharpe ratio, which uses standard deviation, can be
used to compare funds of varying types, since standard deviation is calculated using the same
methods. However, since alpha uses beta, and can be calculated based on different benchmarks
for stocks or bonds, it is not always useful when comparing the two. Sharpe does not have this
problem and therefore can be used when comparing both stock and bond investments.
Why is Sharpe Ratio the Standard?:
The Sharpe ratio is one of the most commonly used ratios to measure risk of a portfolio. One
reason is that it can be used to measure both stock and bond funds because it uses standard
deviation. Since standard deviation is calculated using the same methods, it allows the
measurements to be comparable. The Sharpe ratio is a simplified version of the Sortino ratio and
does not distinguish between “good” and “bad” volatility. Whereas this creates obvious
limitations, it keeps the ratio uniform and simple. It does not depend on the investor’s level of
comfort or a particular minimum acceptable return value. The Sharpe ratio is intended to be used
to measure diversified, liquid investments, and will give the most accurate measurement for
investments that fall into this category. However hedge funds are not normally distributed, and is
an example of why other ratios are more useful for determining risk.
Although the Sharpe ratio has its drawbacks, perhaps the reason why it is the most widely used is
due to its simplicity. It was developed in 1966 and won a nobel prize, which enhanced investor’s
beliefs in its ability to assess risk. Many other ratios are based upon the Sharpe ratio, or are
variations of the Sharpe ratio. This makes the Sharpe ratio an easy, common measurement from
which to compare and base other determinants of risk. Although it is understood that it may not
be the most useful in method of determining risk universally, it will always remain the simplest.
Conclusion:
Sharpe, Sortino, Fama, French and many other credible professionals have attempted to find the
perfect formula for predicting return on a stock. Sharpe insisted that it is best to minimize cost
when investing, which is most easily done through passive investing. Sortino brings to the table
the idea that it is only necessary to look at downside volatility when looking at an investment’s
future return, because upside capture is desirable so should not be considered when looking at a
stock’s volatility. Fama and French explore the many ratios and possible predictors of securities.
In our research, we explored small cap stocks and their performances in a complete market cycle
of about seven years. In an attempt to find the most accurate predictor of return, we looked at the
Sharpe Ratio, the Sortino Ratio, and alpha and compared their predictions to the actual return on
these investments.
Below is the two-sample T-Test for the Sharpe ratio of the FMC against the SMC and the
Sortino ratio of the FMC against the SMC. The resulting p-value of 0 for both tests shows that
we must reject the null hypothesis that states that the difference between the average of the
Sharpe ratio of the FMC and the SMC is 0. Therefore, the difference is significant. The FMC is
different from the SMC in both cases, indicating that both the Sharpe ratio and Sortino ratio are
incapable of predicting such large differences and therefore would not be good predictors of any
market cycle.
Two-Sample T-Test and CI: Sharpe FMC, Sharpe SMC
Two-sample T for Sharpe FMC vs Sharpe SMC
N
Mean StDev SE Mean
Sharpe FMC 295 1.564 0.966
0.056
Sharpe SMC 295 0.931 0.240
0.014
Difference = mu (Sharpe FMC) - mu (Sharpe SMC)
Estimate for difference: 0.6333
95% CI for difference: (0.5194, 0.7471)
T-Test of difference = 0 (vs not =): T-Value = 10.93
Both use Pooled StDev = 0.7039
P-Value = 0.000
DF = 588
Two-Sample T-Test and CI: Sortino FMC, Sortino SMC
Two-sample T for Sortino FMC vs Sortino SMC
N
Mean
StDev SE Mean
Sortino FMC 295
1.061
0.616
0.036
Sortino SMC 295 0.3503 0.0935
0.0054
Difference = mu (Sortino FMC) - mu (Sortino SMC)
Estimate for difference: 0.7104
95% CI for difference: (0.6391, 0.7816)
T-Test of difference = 0 (vs not =): T-Value = 19.59
Both use Pooled StDev = 0.4404
P-Value = 0.000
DF = 588
Regression Analysis: Sharpe FMC versus Sharpe SMC
Analysis of Variance
Source
DF
Regression
1
Sharpe SMC 1
Error
293
Total
294
Adj SS Adj MS
6.666 6.6655
6.666 6.6655
267.698 0.9136
274.363
F-Value P-Value
7.30
0.007
7.30
0.007
Model Summary
S
0.955848
R-sq
2.43%
R-sq(adj)
2.10%
R-sq(pred)
0.51%
Coefficients
Term
Constant
Sharpe SMC
Coef
0.981
0.626
Regression Equation
SE Coef T-Value P-Value
VIF
0.223 4.40
0.000
0.232 2.70
0.007 1.00
Sharpe FMC = 0.981 + 0.626 Sharpe SMC
The graph above shows the regression of the Sortino FMC versus the Sortino SMC. After
running the regression of the Sortino FMC to the SMC, we found a non-existent, or weak
correlation. The r^2 was only .0216, meaning that only 2.16% of the variation in the
Sortino FMC can be explained by the Sortino SMC. Therefore, the correlation is not
significant to conclude a linear relationship between the two.
Regression Analysis: Sortino FMC versus Sortino SMC
Analysis of Variance
Source
DF
Adj SS Adj MS F-Value P-Value
Regression
1
2.407 2.4074 6.47
0.012
Sortino SMC 1
2.407 2.4074
6.47
0.012
Error
293 109.091 0.3723
Total
294 111.499
Model Summary
S
0.610185
R-sq
2.16%
R-sq(adj)
1.83%
R-sq(pred)
0.00%
Coefficients
Term
Constant
Sortino SMC
Coef
0.722
0.968
SE Coef T-Value P-Value
0.138 5.23
0.000
0.381 2.54
0.012 1.00
VIF
Regression Equation
Sortino FMC = 0.722 + 0.968 Sortino SMC
Looking at the scatterplots of “Cycle Growth% Versus Full Market Cycle Sharpe” and “Cycle
Growth% Versus Full Market Cycle Sortino”, we investigated both ratios’ effectiveness on
predicting the full market cycle. We calculated the growth % between full market cycles and
regressed it as a response to the factor of Sharpe Ratio and Sortino Ratio. The resulting
coefficient of determination, R-sq, is 0.6% and 0.9% respectively. A mere 0.6% and 0.9% of the
variability in stock returns can be associated with a change in Sharpe Ratio and Sortino Ratio.
The resulting p-value is 0.211 and 0.139 which leads us to fail to reject the null hypothesis of
𝑅0 :𝛽1 = 0. This suggests that Sharpe Ratio and Sortino Ratio plays no part in predicting the return
of the full portfolio over the duration of a full market cycle. Therefore, if given any Sharpe or
Sortino ratio for a small cap stock, it would not be possible to predict the future value after a full
market cycle of the asset or portfolio.
Additionally, the graph below shows the regression line of the Sharpe ratio versus the Sortino
ratio. We regressed the Sharpe ratios against the Sortino ratios and found a strong, positive,
linear relationship between the two with an r^2 of 96.3. This means that 96.3% of the variability
in Sharpe can be described by the Sortino ratio. Therefore, both ratios are equally poor predictors
of the performance of small cap stocks.
Lastly, we investigated alpha’s effectiveness on predicting the full market cycle. We were able to
plot individual alphas for each stock we investigated against the return of each investment over a
duration of 7.5 years, the length of a full market cycle. The resulting coefficient of
determination, R^2, is 2.5%. A mere 2.5% of the variability in stock returns can be associated
with a change in alpha. The resulting p-value is 0.074 which leads us to fail to reject the null
hypothesis of 𝑅0 :𝛽1 = 0. This suggests that alpha plays no part in predicting the return of the full
portfolio over the duration of a full market cycle.
In the future, our investigation could be improved upon by adding a variable that includes a
measure of value and/or growth of each stock. Past research has shown that it may be important
to divide the stocks into sections, grouped by similarities of the style of the stocks, in terms of
their ability to add value or growth to a portfolio. Based on the definitions of growth and value
stocks, these types of investments perform differently. Therefore, separating these types of
stocks and then testing our methods on the new groupings may provide evidence that alpha, the
Sharpe ratio, or the Sortino ratio are capable of predicting future performance of certain types of
securities.
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