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Diversification Benefits of Risk Models in Managing Portfolios:
A Case of Taiwan’s Stocks
ABSTRACT
How to effectively fashion investment strategies is a core issue for modern finance. In this paper,
we investigate the over-time benefits of mean-variance (MV), mean-absolute deviation (MAD),
downside risk (DSR), value-at-risk (VaR), and conditional value-at-risk (CVaR) models in
portfolio management by using the daily data of Taiwan’s stocks. We further consider investment
constraints in portfolios to ensure the feasibility of their applications. Using four performance
criteria, we find the risk models, particularly the CVaR, yield higher ex ante and ex post
performance than a naïve buy-and-hold portfolio. The two-stage regressions show that high return
benefits are associated with a bearish market while high reduction in risk is positively related to
high volatility. Though VaR is regarded as a standard model applied in the real world, our findings
suggest that CVaR can serve as a good alternative.
JEL Classifications: C61, G11
Keywords: Finance; Diversification Benefits; Risk Modeling, VaR, CVaR.
* Corresponding author
1
1. Introduction
How to fashion optimal investment strategies is a core issue for modern finance. Though the
analytical framework of Markowitz [1] has provided the foundations for portfolio theory, several issues
challenge its application. Since portfolio theories are frequently questioned by their usefulness in the
real world, what are the economic values of exercising these strategies? How does the conclusion
regarding portfolio performance vary as we try to achieve different investment goals? How do the
benefits of risk-return portfolio models vary under different stages in business cycle? Given the
complexity of computing a huge number of quadratic programming, is the optimal portfolio that uses
simplified estimation less effective than that uses full estimation of variance-covariance matrix? Since
the process of portfolio optimization is sensitive to the risks, how do the methods of modeling risk
affect portfolio performance? Furthermore, given value-at-risk (VaR) is widely regarded as a standard
model in risk management and conditional value-at-risk (CVaR) has been increasingly popular, how
does modeling investors’ risk tolerance in portfolio selection affect the portfolio’s performance? Does
CVaR show empirical superiority over VaR model? The above issues are critical both in academia and
on Wall Street to determine how and when these models should be used. In this paper, we evaluate the
over-time performance of various portfolio models.
This paper differs from previous studies in three aspects. First, this paper examines the performance
of the optimal portfolio models that are advanced by the traditional mean-variance (MV) framework.
We use the naïve buy-and-hold (BH) market portfolio as the benchmark and compare its performance
with those generated by the risk portfolio models. They include Markowitz [1] mean-variance (MV),
mean-absolute deviation (MAD), downside risk (DSR), VaR, and CVaR models. The later four
methods represent the modifications for the various issues that complicate and bias the implementation
of the authentic MV model. For instance, VaR has been heavily implemented by financial institutions
while CVaR is regarded as an advance for risk management. In addition, the MV optimization has been
proved to depart from its theoretic optimal return.1 By using various performance criteria, our paper
provides investors the insight in selecting portfolio models according to their objective.
Second, the results of the various measurements allow us to evaluate the performance of the portfolio
strategies from different angles. The majority of previous literature has evaluated the benefits of
1 For instance, Bai, Liu, and Wong [2] develop bootstrap-corrected estimations and prove that their method provides
proportionally consistent estimates comparing to theoretic estimates.
2
diversifying portfolios from an ex ante view, such as the expected return, but seldom stress the
realization value of these strategies. In an identical manner to how these models can be applied in the
finance industry, we form portfolio strategies by using data in previous period to generate the
parameters for modeling asset weights and to rebalance the portfolios in the next period over time.2 Our
findings of ex post portfolio performance, the realization profit of strategy, provide a comprehensive
perspective on how to select models for investors with different investment objectives, whose points of
view also represent the various motivations for diversifying investments.
Third, in order to ensure the feasibility of our strategies and the usefulness of our findings, our tests
incorporate certain conditions that may hinder the use of the portfolio models in the finance industry. It
has been documented that the fat-tail in return distribution and corner solutions may cause difficulties
in exercising the optimal portfolios in practice. We investigate the benefits of more flexible portfolios
with/without allowing short-sales. In addition we apply VaR and CVaR models that consider return
distribution in constructing portfolio. We also apply multiple objective programming by Yu and Lee
[6] on the portfolio selection. The strategies with consideration of these different scenarios help ensure
the comparability of the results.
Modeling risks for portfolios can be challenging. As the number of assets grows, the increasing
variance-covariance matrix yields the computational complexity. Konno and Yamazaki [7] propose a
mean-absolute-deviation (MAD) in which risk is defined as the mean of the absolute value of the
difference in return. To enhance the efficiency of calculation, Simaan [8] suggests that the MAD model
can be linearized without the need to calculate the covariance matrix. Recent studies, such as Leung,
Ng, and Wong [9] and Hirschberger, Qi, and Steuer [10], also focus on how to enhance the
computational efficiency in portfolio optimization, particular a large scale of assets. Harlow [11] and
Gotoh and Takeda [12], on the other hand, suggest that risk is the value below a benchmark of return.
Therefore, in the downside risk (DSR) model, only lower partial moments (LPMs) serve as the
portfolio risk. DeMiguel et al. [13] suggest that there is no portfolio model consistently yields
significantly higher out-of-sample performance than the naïve diversification strategy (1/N portfolio).
Our study evaluates the performance of the models that are developed from the Markowitz’s mean-
variance model [1] while includes the conditions that should be considered in the real world.
2 See Briec and Kerstens [3], Li, Sarkar, and Wang [4], and Woodside-Oriakhi, Lucas, and Beasley [5] for detailed
discussion.
3
The above models deal with asset allocation but do not soundly respond to the call of portfolio
selection in risk management. The linear relationship between the portfolio return and the returns of its
composing securities challenges ex post performance measurement. The MV, MAD, and DSR models
consider the historical volatility but fail to model the future potential downside risk. To improve this
issue, the value-at-risk (VaR) model is applied to estimate the loss that the investor can tolerate given a
certain confidence level [14]. However, the VaR measurement bears some limitations in managing
portfolio volatility. First, Rockafellar [15] and Rockafellar and Uryasev [16] point out that the results
of the VaR may be biased when there are discontinuities in distributions. Second, Mausser and Rosen
[17] suggest that the VaR can generate multiple local minima and may generate a misleading global
optimum solution. Third, the VaR analysis only shows a sub-additivity property if it is under an
assumption of elliptical distribution, e.g., a Gaussian or multivariate normal distribution, of loss.
Therefore, the effectiveness of VaR relies primarily on the accuracy of modeling of the joint
distribution of asset returns.
The conditional value-at-risk (CVaR) model is an improvement to the VaR in the above issues. The
CVaR is less sensitive to the shape of the loss distribution while, more importantly, respects the
properties of coherent risk measure with convex nature.3 Rockafellar and Uryasev [20] document that
the CVaR can quantify downside risk more precisely than traditional MV models since the CVaR
models asymmetry, or fat tail, in asset return distribution. Since CVaR can be linearized and can reach
the global optimum, it demonstrates better theoretical properties in risk management than VaR.
To ensure the feasibility of the results, we also consider the impact of short-sale constraints in
portfolio construction. Green and Hollifield [21], Jagannathan and Ma [22], and Kwan [23] have
documented that adding weight constraints yields the potential to improve a portfolio’s risk-return
trade-off. Previous studies have taken into account the limitation of short selling, particularly in the
research related to the economies of developing countries (De Roon, Nijman, and Werker [24]; Li,
Sarkar, and Wang [4]).
We use the naive diversification, the equally-weighted (1/N) portfolio in DeMiguel et al. [13], as the
benchmark to evaluate the performance of various portfolio models. Our empirical results confirm the
benefits of risk model portfolios, particularly CVaR model. We also find that CVaR model outperform 3 Coherent risk measure means a risk measure have four desired properties: monotonicity, sub-additivity, homogeneity, and
translational invariance. See Artzner, Delbaen, Eber, and Heath [18] and Rockafellar, Uryasev, and Zabarankin [19] for
detailed discussion.
4
VaR model in generating higher performance. Though VaR is a standard model applied in the real
world, our findings suggest that CVaR can serve as a good alternative. The two-stage regressions show
that return benefits are negatively associated with the market return while reduction in risk is positively
associated with market volatility. This confirms the usefulness of these portfolio models during market
downturn. We find that CVaR model with an between 50% and 90% demonstrates higher
performance. The non-linear relationship between risk tolerance and diversification effectiveness
indicate that the goals of portfolio management determine the optimal in VaR and CVaR models.
The structure of the rest of the paper is as follows. Section 2 presents the models and their empirical
applications in this study. Section 3 describes how to evaluate the effectiveness of the diversifying
models. Section 4 presents the data. Section 5 reports the major empirical results. The analyses of the
market condition’s impact on the effectiveness of diversification and the impact of on VaR and
CVaR models are presented in Section 6 and Section 7, respectively. Section 8 concludes.
2. Risk Portfolio Models and their Empirical Applications
We evaluate the economic value of the mean-variance (MV) model and its deviations—the mean-
absolute deviation (MAD) model, the downside risk (DSR) model, the value-at-risk (VaR) model and
the conditional value-at-risk (CVaR) model—in managing portfolios. The portfolios are rebalanced
with considering situations in which short-sales are allowed and they are prohibited. Since Markowitz’s
mean-variance model [1] is one of risk portfolio models in this paper, we use historical mean and
variance to estimate the parameters.
2.1. No short-sale allowed
2.1.1. Mean-variance (MV) model
Suppose a representative local investor wants to minimize the volatility of her portfolio and gain a
given return by allocating funds among various assets. The expected returns and the variance-
covariance of n asset returns, R, can be expressed as a vector E(R) = [E(R1), E(R2), …, E(Rn)] and a
positive definite matrix V, respectively. The variance of asset j and the covariance of asset i and j are
2
j and ij , respectively. Let S be the set of all real vectors of weighting wT in
n such that
w 1T w1+w2+……wn=1, where 1 is an n-vector of ones. The problem of Markowitz optimal portfolio
selection is expressed as follows [1]:
5
min
n
i
n
jij
jiijj
n
j
jP www1 1
2
1
22
(1)
s.t. ,)()(1
n
j
Pjj REwRE
(2)
,11
n
j
jw
(3)
,0jw , ..., ,1 nj (4)
where 2
P is the portfolio variance; 2
j is the variance of the return on securities j;ij is the
covariance between the returns of securities i and j;jR is the return on securities j;
jw is the weight in
security j; and )( PRE is the portfolio required return. In this setting, short-sales are prohibited.
Following Ballestero and Pla-Santamaria [25], Briec and Kerstens [3], and Woodside-Oriakhi, Lucas,
and Beasley [5], we further develop our portfolio rebalancing model by adding additional constraints:
,0,
jjjj llww
, ..., ,1 nj (5)
,2.005.0 jjj uwu
, ..., ,1 nj (6)
0, 1 ,ju , ..., ,1 nj (7)
where 0,jw represents the initial weight of the jth asset;
jl denotes the weight of the long position of
the jth asset after rebalancing;
jl denotes the short position of the jth weight after rebalancing; and ju
is a binary variable indicating whether the jth asset should be invested or not.
2.1.2. Mean-absolute deviation (MAD) model
Konno and Yamazaki [7] propose the MAD model in which variances are replaced by measures of
mean-absolute deviation. Due to the need to solve a large-scale of quadratic programming, the
conventional MV model will lack efficiency as the number of assets increases. The objective function
is:
min 1 1
1( ( ))
T n
jt j j
t j
r E R wT
(8)
6
We split the absolute deviation term,
ttt
n
j
jjjt dddwREr1
))((
and ,0
tt dandd then
n
j
jjjt wREr1
))(( tt dd , Chang [26] suggests that the objective function in the MAD model can
be linearized as:
min
T
t
tdT 1
1
(9)
s.t.
,0))((1
n
j
jjjtt wRErd , ..., ,1 Tt (10)
,0))((1
n
j
jjjtt wRErd , ..., ,1 Tt (11)
constraints (2) – (7),
where T is the ending period; jtr is the return on security j in period t; and td is the deviation between
the return and the average return4.
2.1.3. Downside risk (DSR) model
Grootveld and Hallerbach [27] suggest the risk that investors should care about is possible loss in
portfolio value. The objective function of the DSR model is:
min 1 1 1
1( ( ( )) ( ( )) )
2
T n n
jt j j jt j j
t j j
r E R w r E R wT
(12)
s.t. constraints (2) – (7).
The objective of DSR model can be linearized as:
t
tttt
n
j
jjjt
n
j
jjjt
ddddd
wRErwREr
22
))(())((11
(13)
Therefore, the DSR model is reformulated as following linear model:
min
T
t
tdT 1
1 (14)
4 We acknowledge a reviewer’s comment on the issue of linearization.
7
s.t.
td ,0))((1
n
j
jjjt wREr , ..., ,1 Tt (for 0
td ), (15)
0
td , , ..., ,1 Tt (16)
constraints (2) – (7).
2.1.4. Value-at-risk model (VaR)
The Value-at-Risk (VaR) model has been widely applied by financial institutions. Among various
VaR portfolio models, Benati and Rizzi [28] use VaR to replace variance in MV model and structure as
a mixed integer linear programming. We apply the “Problem 4” (P4) model by Lin [29], which
improves Benati and Rizzi model [28], to construct and rebalance VaR portfolios.
max rVaR
(17)
s.t. ,1
n
j
jtjt rwx , ..., ,1 Tt
(18)
,)( t
MinVaRMin
t yrrrx , ..., ,1 Tt
(19)
,)1(1
1
VaRT
t
tyT
(20)
1
1( ),
T
t p
t
x E RT
(21)
ty {0, 1}, (22)
constraints (2) – (7).
The observed portfolio return in time t is
n
j
jtjt rwx1
; (0 1), ; Minr be the minimum return that
can be observed in the market; jw is the percentage of wealth that is allocated to asset j, variables tx is
the portfolio observed return in time t. The decision maker will not accept any investment whose VaR
is less than rVaR
.
2.1.5. Conditional value-at-risk model (CVaR)
8
Artzner, Delbaen, Eber, and Heath [18] and Artzner, Delbaen, Eber, and Heath [30] suggest that the
VaR bears some undesirable mathematical characteristics.5 To improve the above issues, Pflug [31] and
Rockafellar, Uryasev, and Zabarankin [19] suggest CVaR model. The relation between the VaR and the
CVaR is shown in Figure 1. Ogryczak and Ruszczynski [32] suggest that VaR criterion is equivalent to
the first-order stochastic dominance while Ma and Wong [33] show that CVaR is equivalent to the
second-order stochastic dominance. Therefore, investor’s utility function satisfies the following
properties, ′0 and ′′0.6 )(CVaR w is the conditional expectation of the loss greater than a critical
loss value ( )w (Rockafellar and Uryasev, [16]). Specifically, when the probability that f(w, y)
(w) is equal to 1-,
1
( , )
( ) (1 ) ( , ) ( )f w y
CVaR w f w y p y dy
(23)
where the random vector y stands for the risk and ),( ywf is the loss associated with the decision
vector w. The underlying probability distribution of y int , the uncertainties in domain y in the future,
is )(yp . The cumulative probability function that ),( ywf does not exceed a threshold is defined as
),( w .
[INSERT Figure 1 ABOUT HERE]
To simplify the complexity in calculating the CVaR value, , Rockafellar and Uryasev [16] extend the
function and linearize it as follows:
1( , ) (1 ) ( ( , ) ) ( )ty
F w f w y p y dy
(24)
For Equation (24), Rockafellar and Uryasev [16] propose using multiple vector sets {y1, y2, …, yt} to
replace the original possibility function p(y) and yield the approximate function:
T
t
tywfT
wF1
]),([)1(
1),(
~
(25)
5 For instance, a lack of sub-additivity and convexity will drive the volatility of the sum of a portfolio to be larger than the
sum of the variances of the individual assets. This attribute may discourage diversification as it presents an increase in
portfolio risk.
6 We appreciate reviewer’s suggestion.
9
The above function can be transformed to a linear function by changing ),( tywf to auxiliary
variables t. The optimization question can be rewritten as follows:
min
T
t
tT
wF1)1(
1),(
~
(26)
s.t. ,0t (27)
,),( tt ywf . ..., ,1 Tt (28)
After linearizing, yt denotes the return on scenario t, and f(w, y) = -[w1r1t + … + wnrNT ] denotes the loss
function. The CVaR is designed to solve the optimization of the following equation system:
min
T
t
tT 1)1(
1
(29)
s.t. ,0t (30)
,
1
n
j
jtjt rw
, ..., ,1 Tt (31)
constraints (2) – (7).
The calculation of the VaR and the CVaR is usually in monetary values. In our study, we will present
the rate of return and asset values according to the corresponding performance measures. In addition,
previous studies do not propose how to determine risk aversion coefficient in practice. We start our
study by setting moderate risk-averse=50% and then compare the values of various levels of
expected shortfall.
2.2. Short-sale allowed
We further consider portfolio rebalancing scenarios in which short-sales are allowed. When the
expected returns of all available assets are negative, short-sale prohibited models may not generate a
solution if the expected return of the portfolio is set to be non-negative. Allowing short-sales not only
expands the efficient frontier but also helps to fashion more flexible asset allocation strategies,
particularly during periods of market downturn (e.g., Kwan [34]). White [35] and Angel, Christophe
and Ferri [36] also suggest that short-selling grants speculation opportunity and may decrease portfolio
10
volatility if the short-sale proportion is not excessive. Therefore, the proportion of short selling should
be considered in optimization to manage portfolio risk. Given there can be more than one objective, we
apply a simple weighted method for multiple objective programming.
2.2.1. Mean-variance with short-sales (MV_S) model
When the optimal portfolio is constructed by the MV_S model, the objective function needs to
incorporate the short selling portion of the portfolio in the minimization of portfolio variance. The
weight of asset j is decomposed into weights of long position (+) and short position (–), wj = jj ww .
The optimization will solve the following equations:
min
n
j
j
n
i
n
j
jjiiij wpwwwwp11 1
)()))(()(1(
(32)
s.t. ,)())((1
n
j
Pjjj REwwRE
(33)
,1)(1
n
j
jj kww
(34)
,0,
jjjj llww
, ..., ,1 nj (35)
,0,
jjjj ssww
, ..., ,1 nj (36)
,2.005.0 jjj uwu , ..., ,1 nj (37)
,2.005.0 jjj vwv
, ..., ,1 nj (38)
,1 jj vu (39)
ju , jv {0, 1}, , ..., ,1 nj (40)
where p is the proportion of weight for the two objectives, portfolio variance and short-selling weight;
k is the initial margin requirement for short selling;
0,jw is the long position weight of security j prior to
portfolio rebalancing; and
0,jw is the short position weight of security j prior to portfolio rebalancing.
With each rebalancing,
jl is the buying weight of security j;
jl is the short-selling weight of security j;
js is the short weight of security j; and
js is the repurchasing weight of security j. Other than to use
the binary variable uj and vj to indicate long and short position, we also consider the upper bounds of
the weight in Equations (37) and (38) to ensure the feasibility of the portfolios.
11
We apply a simple weighted method to combine the two objectives in the MV_S model as the
objective function. Regarding the constraints in the model, Equation (34) specifies the budget allocated
to buying and short selling; (35) shows the long position after rebalancing; (36) represents the short
selling position after rebalancing; and (37) and (38) define the upper and lower bounds of the total
weights of each security of long position and that of short selling position, respectively. The definition
of the binary variables in Equation (39) ensures that long and short positions do not happen
simultaneously.
2.2.2. Mean-absolute deviation with short-sales (MAD_S) model
We modify Konno and Yamazaki [7] as a MAD_S model. As the weights of the long and short
position are separate, the objective is as follows:
min ))()()((1
n
j
jjjjtjj REwwrww (41)
The objective can be transformed into a linear combination of the optimal portfolio variance and the
short-sale weights:
min )()1
)(1(11
n
j
j
T
t
t wpdT
p
(42)
s.t. 0)()()(1
n
j
jjjjtjjt REwwrwwd , (43)
0)()()(
1
n
j
jjjjtjjt REwwrwwd ,
, ..., ,1 Tt (44)
constraints (33) – (40).
2.2.3. Downside risk with short-sales (DSR_S) model
The setting of the DSR_S model is similar to the structure of the MV_S and the MAD_S in its
combination of risk and short-sale weights. Specifically,
min
T
t
n
j
jt wpdT
p1 1
)(1
)1( (45)
s.t.
td ,0)))(((1
n
j
jjjjt wwREr , ..., ,1 Tt (for 0
td ),
,0
td , ..., ,1 Tt
12
constraints (33) – (40).
2.2.4. Value-at-risk with short-sales (VaR_S) model
A mixed integer linear rebalancing VaR Model with short selling is as following:
max
T
t
n
j
jt wpxT
p1 1
)()1
)(1( (46)
s.t. ,)(1
n
j
jtjjt rwwx , ..., ,1 Tt (47)
,)( t
MinVaRMin
t yrrrx , ..., ,1 Tt (48)
,)1(1
1
T
t
tyT
(49)
ty {0, 1}, , ..., ,1 Tt (50)
constraints (33) – (40).
2.2.5. Conditional value-at-risk with short-sales (CVaR_S) model
We modify Rockafellar and Uryasev [16] as a CVaR_S model by transforming the objectives, the
portfolio variance and short-selling weights, into a linear function. Specifically,
min )())1(
1)(1(
11
n
j
j
T
t
t wpT
p
(51)
s.t. ,)(1
n
j
jjjtt wwr , ..., ,1 Tt (52)
,0 t , ..., ,1 Tt (53)
constraints (33) – (40).
3. Measuring Diversification Performance
To evaluate the effectiveness of the diversifying strategies that are formed by the above risk-return
portfolio models, we use the naïve buy-and-hold strategy as the benchmark. Specifically, we measure
the diversification benefits by using the differences in expected return, the volatility of return, Sharpe
ratio, and the realized return between the diversified portfolios and the overall market portfolio. For a
naïve investor without the knowledge of optimization techniques, asset allocation that follows the
13
market potentially can be a feasible diversification strategy. We also consider scenarios in which short-
sales are and are not allowed. The first measure of diversification benefit is the increase in expected
return brought by following risk-return models,
E( *r ) = E( *r ) - E( BHr ), (54)
where E( *r ) is the expected return generated by a risk-return model (e.g., MV, MAD, DSR, VaR or
CVaR models), and E( BHr ) is the expected return yielded by the buy-and-hold market portfolio.
The second measure of the benefits of diversification is the reduction in volatility. Elton, Gruber,
Brown, and Goetzmann [37] suggest that investors may seek to reduce the variance of a portfolio due
to the lack of predictability of its expected returns. The decrease in volatility caused by diversifying
investments is:
* = BH - * , (55)
where * is the standard deviation of return generated by a risk-return model and BH is the volatility
of the buy-and-hold (BH) market portfolio strategy.
The third measure is the difference in Sharpe ratio (SR) between the risk model portfolio (*) and the
naïve portfolio. We follow Bai, Hui, Wong, and Zitikis [38] and Chiou ([39] and [40]), the measure is:
*SR = *SR - .BHSR (56)
We further evaluate the performance to realize the diversification strategies. Previous studies, such
as DeMiguel, Garlappi, and Uppal [13], indicate that a poor estimation of asset returns challenges the
application of portfolio models. Thus, evaluating the out-of-sample performance, or the realization
return, of the optimal portfolio benefits the selection of the optimal portfolio. Our fourth performance
measure is the returns by realizing diversified portfolios. We rebalance the portfolios according to the
weights of the strategies and compute their market values. The increase in the realized return brought
by the risk-return models is:
R( *r ) = R( *r ) - R( BHr ), (57)
where R( *r ) is the realized return generated by a risk-return model and R( BHr ) is the realized return
yielded by the buy-and-hold market portfolio strategy in the period.
14
We also compare the excess portfolio values of the risk-return models to the buy-and-hold market
portfolio. In every period, the portfolios are marked to market when they are rebalanced according to
the optimization results. The excess portfolio value (EPV) of a risk-return model portfolio is:
*EPV = *PV - BHPV , (58)
where *PV is the portfolio market value of a risk-return model and BHPV is the buy-and-hold stock
market portfolio value in the period.
4. Data
We collect the daily return of the stocks in the Taiwan 50 Index from 11/02/2006 to 1/30/2012 from
the database of the Taiwan Economic Journal (TEJ). According to World Federation of Exchanges in
December 2012, Taiwan’s stock market is among top 20 largest in the world. We try to maximize the
contribution to literature by examining the performance of the portfolio models during the market
downturn. We exclude the stocks with missing data and use 37 stocks in this study. Table 1 shows the
summary statistics of the sample stocks. The major industries in our sample are electronic parts and
components, computer manufacturers, and financials. This distribution is similar to the distribution of
stocks in Taiwan’s exchange market. The total market value of these selected stocks represents more
than 50% of the overall capitalization in Taiwan’s stock market during the sample period. This ensures
that the portfolio strategies presented in this research are feasible, as the stocks included are of high
liquidity.
[INSERT TABLE 1 ABOUT HERE]
5. Empirical Results
We form and rebalance the optimal portfolios by using the models presented in Section 2. We
consider conditions of short-sales and use four measures presented in Section 3 to compare the
performance of the various strategies. We rebalance each of the portfolios every 20 trading days by
using the data of the previous 60 trading days. The initial investments on 1/25/2007 are assumed to be
$1 million NT dollars.
5.1. Short-sales prohibited
15
Table 2 reports the benefits of diversification when short-sales are prohibited. We compare the
performance between the naïve buy-and-hold (BH) portfolio and the portfolios that are constructed by
the MV, the MAD, the DSR, VaR and the CVaR models. VaR model generates highest E(r) though it
is not significantly higher than that yielded from CVaR. For SR and R(r), the CVaR model
outperforms the other four models and demonstrates a higher yield than the passive buy-and-hold
market portfolio. For the decrease in volatility (), the MV and MAD models provide greater benefits
than the other portfolio strategies. Since the objectives for the DSR and CVaR models do not
emphasize minimizing the weighted average of the distance between the observation and the mean
return, they provide less reduction in volatility. In general, the realized portfolio return demonstrates a
higher uncertainty than the in-the-sample ex-ante return and Sharpe ratio over time. Among them,
CVaR model demonstrates highest realized profit.
[INSERT TABLE 2 ABOUT HERE]
Figure 2 demonstrates the time-variation of the diversification benefits. Graph A shows that though
the excess expected return for the various strategies changes significantly over time, the VaR and
CVaR models outperform the other three in most periods. On the other hand, Graph B shows that the
MV model seems to yield a superior reduction in portfolio volatility. Shown in Graph C, the CVaR
consistently yield higher Sharpe ratio than other risk portfolio models though it does always dominate
buy-and-hold strategy. Because the portfolio models are more suited to explaining historical trends than
modeling future asset returns, none of these models consistently show higher realized returns than the
other three, as is shown in Graph D.
[INSERT FIGURE 2 ABOUT HERE]
Figure 2 also shows that the time-series patterns of high diversification benefits vary across the
measures. In general, the benefits of the expected return and Sharpe ratio are low during late 2008 and
late 2009, but the volatility reductions are high. Though the realized return benefits shown in Graph D
are more volatile over time, most values during late 2008 and late 2009 are negative too. The size of
diversification benefits seems to be impacted by the market condition. In Graph B, the MV and MAD
models consistently provide a greater reduction in volatility than the other strategies over the sample
period.
5.2. Short-sales allowed
16
The values of diversification in situations in which short-sales are allowed are reported in Table 3.
The CVaR_S model yields the highest expected return, highest Sharpe ratio, and the highest realized
return when comparing the other risk-return models. The ex post returns and Sharpe ratio generated by
diversification strategies increase when short selling assets is allowed in all models except the MAD_S.
The portfolio feasibility also slightly enhances the proportion of periods of positive excess return, both
measured by the ex ante expected return and the realized return. Though the MV model still provides a
higher decrease in risk than the other models, allowing asset short-sales decreases the reduction in
portfolio volatility. The results also show the superiority of CVaR to VaR in managing portfolio.
[INSERT TABLE 3 ABOUT HERE]
The time-variation of diversification benefits with asset short-sales is presented in Figure 3. In
general, the orders of effectiveness for the five risk portfolio models are similar to the findings of short-
sales prohibited. However, the time-variation of diversification benefits of short-selling portfolios is
larger than their corresponding portfolios that prohibit short-sales.
[INSERT FIGURE 3 ABOUT HERE]
5.3. Excess portfolio values
The realization value of risk portfolio strategies is one of the most concerned issues to finance
professionals. We further compute the cumulative excess portfolio values (EPVs) of the risk models
over the sample period. The time trends for the portfolios in which short-sales are prohibited and their
summary statistics are presented in Figure 4. The hypothetical initial investing value of each portfolio
is $1 million NT dollars. The fact that all risk-model portfolios yield positive terminal EPVs suggests
that diversifying strategies are preferable to the naïve buy-and-hold market portfolio over the long
term. Among them, the CVaR model yields the highest EPV and, interestingly, always outperforms the
market portfolio. The MAD model demonstrates high sensitivity to market conditions over time.
[INSERT FIGURE 4 ABOUT HERE]
Figure 5 shows the EPVs when short-sales are allowed. The flexibility of the portfolio brought by
allowing short-sales does not necessarily enhance their effectiveness but, more or less, shirks the
volatility of the realized values. Though the CVaR_S model still yields the highest EPV as compared to
the other strategies, the rank of the EVPs of the other strategies with short-sales is different from the
results of those with no short-sales shown in Figure 4. The lower performance suggests that the VaR
model may not be as feasible as CVaR is in managing portfolio, particularly during bear market.
17
[INSERT FIGURE 5 ABOUT HERE]
The empirical results regarding portfolio values confirm the economic value of risk-model
portfolios, particularly the CVaR. Though the diversification benefits are time-varying, the
effectiveness of the strategies appears to be affected by the market condition.
6. The impact of market conditions on the effectiveness of diversification
The previous empirical results have shown the superiority of the risk models, particularly the CVaR
model, in determining asset allocation as compared to the passive 1/ N buy-and-hold portfolio strategy.
However, there are still questions that need to be clarified: What drives the variation of the
effectiveness of the diversified portfolios over time? Do the benefits of diversification increase or
decrease during the sample period? In this section, we evaluate the extent to which the market
condition impacts the potential gains of risk-modeling investments. Specifically, we are interested in
ascertaining how the return and the volatility of the market affect the values of diversification. Such
knowledge is useful for portfolio managers in determining the timing that one should apply the
investing models.
We examine the relationship between diversification benefits and the market condition and study
whether the effectiveness of diversification demonstrate a long-term trend. We adapt Carrieri, Chaieb,
and Errunza [41] to investigate the benefits of diversification in a domestic market. Since the errors in
these dependent variables very likely are correlated with the market return and volatility, we perform a
two-stage least squared (2SLS) regression to analyze the time-series of the diversification benefits.
Specifically:
E( *r ) = 321 rta , (59)
* = 321 rta , (60)
SR = 321 rta , (61)
R( *r ) = 321 rta , (62)
where a is a constant, r is the market return in the current period, is the volatility in the current
period, and t is the time period.7
7 We rebalance each of the portfolios every 20 trading days by using the data in the previous 60 trading days.
18
Table 4 reports the regression results of the benefits that are generated from the risk model portfolios
in which short-sales are prohibited. For the four measures, a high return in the market is associated with
low benefits generated from the diversification strategies. It makes sense that investors can easily profit
during a bullish market period without prior knowledge of portfolio modeling. The effectiveness of the
portfolios however does not show a significant time trend over the sample period. This finding suggests
the usefulness of risk portfolio models during the bearish market period.
[INSERT Table 4 ABOUT HERE]
The impact of volatility on the benefits of diversification varies according to different measures and
risk models. Though the reduction in volatility increases when the market becomes riskier, the two
return benefit measures show an opposing but statistically-insignificant relationship. Since high
volatility is generally associated with bearish market, the increases of the expected return and the
realized return generated by the portfolio models become relatively small during that period.
The results of the benefits that are generated from the portfolios with short-sales allowed are
presented in Table 5. Compared to the results of portfolios in which short-sales are prohibited, the signs
of the decrease in market volatility and the sign of the increase in expected return differ. When the
portfolio construction becomes more flexible, investors have a chance to increase the ex ante portfolio
performance by short selling stocks during a bearish market period. However, similar to Carrieri et al.
[41], the excess realized return is still negatively associated with market volatility due to the difficulty
of forecasting the expected return and the risk. The regression results of VaR are similar to CVaR.
[INSERT Table 5 ABOUT HERE]
Table 6 reports the results of the two-stage least squared (2SLS) regressions of the excess portfolio
values (EPVs) for the risk model portfolios as compared to the naïve buy-and-hold (BH) portfolio.
Similar to the above regressions (59) - (62), the test of the EPV for any model * is:
*EPV = 321 rta , (63)
where a is a constant, and t is the time. The time trends indicate that these portfolio strategies yield a
higher return than the passive market portfolio over the long term. The p-values also confirm the
robustness of the long-term outperformance of these portfolio models. Similar to the previous findings,
the benefits are large when the market is bearish. The market volatility is positively correlated with the
EPVs of all portfolios except CVaR and VaR models.
[INSERT Table 6 ABOUT HERE]
19
Our empirical results confirm the usefulness of risk portfolio models, particularly in a bear market.
Similar to DeMiguel, Garlappi, and Uppal [13], we use naïve diversified portfolio as the benchmark in
the study but our results show their value, particularly CVaR model.
7. Impact of on the effectiveness of the VaR and CVaR models
Previous studies such as Rockafellar and Uryasev [16] suggest the superiority of CVaR in theory but
do not propose how to determine risk aversion coefficient in practice. In this section, we investigate
how the risk tolerance coefficient, , affects the diversification effectiveness of the VaR and CVaR
models. Though our empirical results have documented that CVaR portfolios demonstrate the highest
performance among the models tested, a study on various ’s helps portfolio managers determine the
level on risk aversion.
Our empirical results in Section 4 have shown that the CVaR model with an of 50% demonstrates
the highest performance comparing with other models. An increase in means that the investor
becomes more risk averse, as shown in Figure 1. How to select an that fits an investors’ goal to
diversify his investments, however, is an empirical issue. Using the criteria described above, we
compare the effectiveness of both cases in which short-sales are allowed and cases in which they are
prohibited.
Table 7 reports the benefits of diversification of CVaRs with different ’s when short-sales are
prohibited. Similarly, we compare the performance of the naïve market portfolio and the various CVaR
portfolios. Though E(r) and SR are the highest when equals 50%, the other two performance
measures ( and R(r)) suggest that the of 70% may generate better risk reduction and higher
realized return. In addition, ex ante conservativeness in investments does not lead to higher reduction in
risk and greater improvements in performance. For the decrease in volatility (), the of the 70%
model not only provides greater benefits than the other portfolio strategies but also most frequently
yields reductions in risk. For the realized return, the proportions of periods of positive excess return
are higher when the ’s are 70% and 90%.
[INSERT TABLE 7 ABOUT HERE]
Table 8 presents the effectiveness of risk portfolios in which short-sales are allowed. The three return
measures, E(r), SR, and R(r), suggest that an of 50% is the optimal risk tolerance though a CVaR
20
model with an of 70% yields the highest average . The patterns in Tables 7 and 8 also show that
the optimal may vary due to the measure of the benefits but, in general, is between 50% and 90% for
the sample.
[INSERT Table 8 ABOUT HERE]
The economic value of executing the CVaR portfolios with various ’s when short-sales are
prohibited is presented in Figure 6. Among them, the portfolios with ’s of 50% and 70% yield
positive EPVs over the entire sample period. In addition, the maximum loss for the other three
portfolios increases as the portfolio becomes more risk averse. The fact that the patterns for the EPVs
differ across portfolios suggests that the relation between the benefits and the risk aversion factors is
not linear.
[INSERT FIGURE 6 ABOUT HERE]
Figure 7 shows the EPVs when short-sales are allowed. Like the results of the other risk models, the
flexibility of portfolio short-sales does not increase the portfolio value but, more or less, decreases the
volatility of the realized values. Though the 70% portfolio still yields the highest EPV in comparison to
the other strategies, the rank of the EPVs of the other high portfolios with short-sales differs from the
results in Figure 6.
[INSERT FIGURE 7 ABOUT HERE]
Table 9 presents the effectiveness of VaR models in which short-sales are prohibited. When an
=70%, the ex ante return and vitality are higher than those of other risk aversions. However the ex post
performance is the highest when =90%. This is consistent with the finding in Figure 8 that the market
value of executing VaR models with various ’s. When =90% and 95%, the EPVs yielded by no-
short-sale VaR models outperform the models of other ’s. Their market values are also lower than
those generated by corresponding CVaR models.
[INSERT Table 9 ABOUT HERE]
[INSERT FIGURE 8 ABOUT HERE]
Table 10 presents the effectiveness of VaR models in which short-sales are allowed. All measures
suggest that an of 90% may be the optimal risk tolerance. One may also find nonlinear patterns
between the benefits generated from the CVaR portfolios and their risk tolerances. The fact that CVaR
models outperform VaR models in all aspects confirms the superiority of CVaR in managing portfolio
in practice.
21
[INSERT Table 10 ABOUT HERE]
Figure 8 shows that the market value of executing VaR models with various ’s when short-sales are
allowed. The patterns for the EPVs confirm nonlinear relation between the benefits and the risk
aversion factors. In addition, EPVs yielded by VaR models are constantly lower than those generated
by CVaR models. Unlike CVaR, VaR models not always yield positive EPVs but suffer low
performance during 2009 and 2010 when market recovers from the financial crisis.
[INSERT FIGURE 9 ABOUT HERE]
In sum, we investigate the impact of the risk tolerance coefficient in the CVaR and VaR models on
the effectiveness of portfolio diversification. Our empirical results show CVaR models outperform
VaR models in all the performance measures and confirm the superiority of CVaR in managing
portfolio. In general, an between 50% and 90% may be optimal for diversifying strategies, depending
on performance criteria. A high risk tolerance coefficient, e.g., higher than 90%, may decrease the
effectiveness of CVaR portfolios.
8. Conclusion
In this paper, we study the over-time benefits of various widely-applied models—namely the mean-
variance (MV), the mean-absolute deviation (MAD), the downside risk (DSR), value-at-risk models
(VaR), and the conditional value-at-risk models (CVaR)—in managing investment portfolios. The
naïve buy-and-hold market portfolio is used as the benchmark. To ensure the feasibility of the
applications, we also consider the impact of various investment constraints. Using the price data of the
most liquid stocks in Taiwan from 2007 to 2012, our empirical results of excess expected returns,
reduction in volatility, increase in Sharpe ratio, realized return, and excess portfolio value confirm the
benefits of risk model portfolios. Among them, the CVaR portfolio demonstrates the highest
performance. Though VaR is a standard model applied in the real world for risk management, CVaR
can be considered as a good alternative.
We further examine how market conditions influence the benefits of diversification. The two-stage
regressions show that return benefits are negatively associated with the overall market return while
reduction in risk is positively related to market volatility. We also find that a VaR or a CVaR model
with an between 50% and 90% demonstrates better performance. The non-linear relations between
22
the risk tolerance and diversification effectiveness indicate that the goals of portfolio management
determine the optimal in the VaR and CVaR models. Our results also empirically show the
superiority of CVaR model to VaR model in managing portfolio.
We add to the current literature by evaluating the effectiveness of widely-applied diversification
models from both ex ante and ex post views. This paper synthesizes the major concepts and modi
operandi of the previous research and maximizes the practicality of managing investing portfolios. The
results support the performance superiority of risk model portfolios in comparison to the naïve buy-
and-hold market portfolio strategy. The findings regarding the impact of market conditions on the
diversification benefits confirm the usefulness of portfolio models, in particular during a bearish market
period. We also provide evidence to apply CVaR model to replace VaR model in managing portfolio.
Our empirical study regarding the risk tolerance coefficient in the VaR and CVaR models is helpful to
finance industry. Future research may evaluate the performance of other risk portfolios and risk
management models in different scenarios.
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25
Table 1. Summary Statistics of Sample Stocks
The summary statistics and the industry of the stocks that are included in this study during the period from 11/02/2006 to 1/30/2012 are
reported. The mean and standard deviation of each stock’s return are annualized.
Code Stock Industry Mean St Dev Skewness Kurtosis 1101 Taiwan Cement Corp. Cement 0.0658 0.4229 -0.02 70.70 1102 Asia Cement Corporation Cement 0.0529 0.4129 -0.17 1.82 1216 Uni-President Enterprises Corp. Food 0.0787 0.3737 -0.26 2.69 1301 Formosa Plastics Corporation Plastics 0.1096 0.3082 -0.31 3.32 1303 Nan Ya Plastics Corporation Plastics 0.0515 0.3089 -0.16 2.33 1326 Formosa Chemicals & Fibre Corporation Textiles 0.0996 0.3019 -0.12 3.16
1402 Far Eastern New Century Corporation Textiles 0.0726 0.4411 -0.05 0.49 1722 Taiwan Fertilizer Co., Ltd. Textiles 0.0677 0.4596 -0.03 0.39 2002 China Steel Corporation Steel 0.0000 0.2818 -0.20 3.80 2105 Cheng Shin Rubber Ind., Co., Ltd. Rubber 0.1467 0.4356 -0.85 7.58 2308 Delta Electronics, Inc. Electronic Parts/Components -0.0418 0.3743 -0.46 2.27 2311 Advanced Semiconductor Engineering, Inc. Electronic Parts/Components 0.0041 0.4202 -0.18 1.47
2317 Hon Hai Precision Ind. Co., Ltd. Electronic Parts/Components -0.1589 0.4083 -0.63 4.18 2324 Compal Electronics, Inc. Electronic Parts/Components 0.0331 0.3815 -0.24 1.53 2325 Siliconware Precision Industries Co., Ltd. Electronic Parts/Components -0.0387 0.4221 -0.09 1.21 2330 Taiwan Semiconductor Manufacturing Co., Ltd. Electronic Parts/Components 0.0510 0.3040 0.02 2.17 2347 Synnex Technology International Corp. Electronic Parts/Components 0.1669 0.4394 -0.11 1.21 2353 Acer Incorporated Computer -0.0712 0.4009 -0.09 0.92
2354 Foxconn Technology Co., Ltd. Computer -0.1954 0.4903 -0.30 1.86 2382 Quanta Computer Inc. Computer 0.0438 0.3498 -0.28 3.21 2409 AU Optronics Corp. Electronic Parts/Components -0.2010 0.4112 -0.11 0.85 2454 MediaTek Inc. Electronic Parts/Components -0.0278 0.4313 -0.08 0.46 2498 HTC Corporation Electronic Parts/Components -0.1050 0.5154 -1.53 13.20 2801 Chang Hwa Commercial Bank, Ltd. Financial -0.0393 0.3538 0.02 1.54
2880 Hua Nan Financial Holdings Co., Ltd. Financial -0.0474 0.3482 -0.07 2.12 2881 Fubon Financial Holding Co., Ltd. Financial 0.0407 0.3781 -0.07 1.56 2882 Cathay Financial Holding Co., Ltd. Financial -0.1229 0.3908 0.00 1.23 2883 China Development Financial Holding Corp. Financial -0.0744 0.3575 -0.06 1.81 2885 Yuanta Financial Holdings Financial 0.0325 0.4506 -0.04 0.39 2886 Mega Financial Holding Co., Ltd. Financial -0.0248 0.3619 -0.09 1.56
2888 Shin Kong Financial Holding Co., Ltd. Financial -0.2239 0.4362 0.00 0.75 2890 Sinopac Financial Holdings Company Limited Financial -0.1152 0.3972 -0.05 1.31 2891 Chinatrust Financial Holding Company Ltd. Financial -0.0450 0.4198 -0.17 1.50 2892 First Financial Holding Co. Ltd. Financial -0.0413 0.3685 0.04 1.74 2912 President Chain Store Corporation Retailing 0.1495 0.3420 0.09 3.18 3231 Wistron Corporation Computer 0.0279 0.4421 -0.34 1.52
6505 Formosa Petrochemical Corp Chemicals 0.0673 0.2942 -0.09 2.27
26
Table 2. Benefits of Diversified Portfolios: Short-sales Prohibited
The table reports the benefits of the diversification of portfolios that are constructed by the Markowitz (1952) mean-variance (MV), the
mean-absolute deviation (MAD), the downside risk (DSR), the value-at-risk (VaR) with =50%, and the conditional value-at-risk
(CVaR) with =50% models in which short-sales are prohibited. To measure the benefits, we compare the increase in expected return
(E(r)), the decrease in volatility (), the increase in Sharpe ratio (SR), and the increase in realized return (R(r)) of the
aforementioned diversified portfolios with the corresponding values generated by the passive buy-and-hold market portfolio. We also
report the distribution of the benefits over the sample period.
MV-BH MAD-BH DSR-BH CVaR-BH VaR-BH
Increase in Expected Return (E(r))
Mean 0.0039 0.0043 0.0000 0.0208 0.0154 Max 0.0584 0.0591 0.0623 0.0651 0.0907 Min -0.0331 -0.0282 -0.0431 -0.0104 -0.0224 SD 0.0045 0.0045 0.0053 0.0046 0.0061 Skewness 0.90 1.00 0.90 0.74 1.25 Kurtosis 1.93 1.89 2.30 0.70 2.13 Distribution (%)
E(r) 0 45.16 46.77 48.39 6.45 19.35
0<E( r) 0.01 25.81 24.19 30.65 16.13 29.03
0.01<E( r) 0.02 16.13 19.35 12.90 35.48 24.19
0.02E( r) 12.90 9.68 8.06 41.94 27.42
Decrease in Volatility ()
Mean 0.0111 0.0097 0.0066 0.0072 0.0039 Max 0.0265 0.0248 0.0218 0.0212 0.0134 Min 0.0022 0.0011 -0.0076 -0.0030 -0.0068 SD 0.0049 0.0049 0.0058 0.0054 0.0049 Skewness 0.54 0.60 0.18 0.32 -0.26 Kurtosis 0.58 0.34 0.28 -0.58 -0.73 Distribution (%)
0 0.00 0.00 11.29 6.45 24.19
0< 0.01 1.61 6.45 12.90 16.13 14.52
0.01< 0.02 8.06 16.13 19.35 22.58 19.35
0.02 90.32 77.42 56.45 54.84 41.94
Increase in Sharpe Ratio (SR)
Mean 0.0260 0.0268 -0.0144 0.1433 0.0910 Max 0.3035 0.2304 0.2032 0.5097 0.3345 Min -0.1586 -0.1525 -0.2247 -0.0512 -0.0904 SD 0.0829 0.0821 0.1028 0.0984 0.0958 Skewness 0.46 0.20 -0.20 1.06 0.26 Kurtosis 0.93 -0.07 -0.42 2.89 -0.33 Distribution (%)
SR 0 37.10 35.48 46.77 6.45 12.90 0< SR 0.1 40.32 40.32 41.94 24.19 46.77 0.1<SR 0.2 19.35 19.35 8.06 48.39 25.81 0.2SR 3.23 4.84 3.23 20.97 14.52
Increase in Realized Return (R(r))
Mean 0.0697 0.0748 0.0589 0.0782 0.0407 Max 1.5202 1.1350 1.6979 1.1222 2.7829 Min -1.2552 -1.6296 -1.5533 -1.7933 -2.5471 SD 0.1316 0.1333 0.1629 0.1342 0.3264 Skewness -0.20 -0.58 -0.01 -0.83 -0.05 Kurtosis 1.76 1.84 1.51 3.03 -0.24 Distribution (%)
R( r) 0 45.16 46.77 41.94 45.16 48.39
0<R( r) 0.01 3.23 3.23 3.23 1.61 0.00 0.01<R( r) 0.02 0.00 0.00 3.23 1.61 1.61 0.02R( r) 51.61 50.00 51.61 51.61 50.00
27
Table 3. Benefits of Diversified Portfolios: Short-sales Allowed
The table reports the benefits of the diversification of portfolios that are constructed by the Markowitz (1952) mean-variance (MV), the
mean-absolute deviation (MAD),the downside risk (DSR),the value-at-risk (VaR) with =50%, and the conditional value-at-risk (CVaR)
with =50% models in which short-sales are allowed. To measure the benefits, we compare the increase in expected return (E(r)), the
decrease in volatility (), the increase in Sharpe ratio (SR), and the increase in realized return (R(r)) of the aforementioned diversified
portfolios with the corresponding values generated by the passive buy-and-hold market portfolio. We also report the distribution of the
benefits over the sample period.
MV-BH MAD-BH DSR-BH CVaR-BH VaR-BH Increase in Expected Return (E(r))
Mean 0.0100 0.0104 0.0076 0.0234 -0.0001
Max 0.0907 0.0907 0.0907 0.0907 0.0000 Min -0.0219 -0.0214 -0.0290 -0.0104 -0.0063 SD 0.0064 0.0064 0.0068 0.0058 0.0002 Skewness 1.58 1.61 1.46 1.05 -7.87
Kurtosis 2.56 2.63 2.19 1.25 62.00
Distribution (%)
E(r) 0 37.10 38.71 38.71 6.45 1.61
0<E( r) 0.01 30.65 29.03 32.26 16.13 98.39
0.01<E( r) 0.02 11.29 11.29 11.29 35.48 0.00
0.02E( r) 20.97 20.97 17.74 41.94 0.00
Decrease in Volatility ()
Mean 0.0100 0.0089 0.0063 0.0068 0.0046 Max 0.0330 0.0313 0.0275 0.0313 0.0546 Min -0.0229 -0.0257 -0.0286 -0.0280 -0.0340
SD 0.0099 0.0099 0.0098 0.0099 0.0117
Skewness -0.37 -0.42 -0.51 -0.35 0.80
Kurtosis 2.17 2.47 2.08 2.16 6.01 Distribution (%)
0 9.68 11.29 20.97 19.35 32.26
0< 0.01 6.45 9.68 12.90 14.52 8.06
0.01< 0.02 14.52 9.68 9.68 14.52 17.74
0.02 69.35 69.35 56.45 51.61 41.94
Increase in Sharpe Ratio (SR)
Mean 0.072 0.071 0.037 0.072 0.071 Max 0.418 0.418 0.418 0.418 0.418 Min -0.151 -0.113 -0.245 -0.151 -0.113
SD 0.114 0.111 0.133 0.114 0.111
Skewness 0.894 0.954 0.522 0.894 0.954
Kurtosis 0.766 0.618 0.522 0.766 0.618 Distribution (%)
SR 0 22.58 29.03 33.87 22.58 29.03
0< SR 0.1 46.77 33.87 43.55 46.77 33.87
0.1<SR 0.2 12.90 19.35 6.45 12.90 19.35
0.2SR 17.74 17.74 16.13 17.74 17.74
Increase in Realized Return (R(r)) Mean 0.0748 0.0669 0.0851 0.0976 0.0709
Max 1.3225 1.3225 1.9122 1.3225 1.7645 Min -1.7036 -1.5946 -1.7220 -1.5930 -1.7031 SD 0.1312 0.1355 0.1539 0.1291 0.1531
Skewness -0.74 -0.39 0.36 -0.65 0.06
Kurtosis 3.02 1.60 2.87 2.49 2.66
Distribution (%)
R( r) 0 46.77 48.39 38.71 40.32 46.77
0<R( r) 0.01 1.61 0.00 1.61 1.61 0.00
0.01<R( r) 0.02 1.61 0.00 6.45 3.23 0.00
0.02R( r) 50.00 51.61 53.23 54.84 53.23
28
Table 4. Regression Analysis of the Benefits of Diversified Portfolios: Short-sales Prohibited
The table reports the results of the two-stage least squared (2SLS) regressions that analyze the time-series of diversification benefits for
each risk model portfolio as compared to the naïve buy-and-hold (BH) market portfolio when short-sales are prohibited. The testing
models for each diversification benefit—increase in expected return (E(r)), decrease in volatility (), the increase in Sharpe ratio
(SR), and increase in realized return (R(r))—are:
321
* )( rtarE ,
321
* )( rtar ,
321
* )( rtarR ,
where the a are constant, and t is the time. The adjusted R-square of each model is shown.
Increase in Expected Return (E(r))
MV-BH MAD-BH DSR-BH VaR-BH =50%) CVaR-BH =50%) Coeff p-value Coeff p-value Coeff p-value Coeff p-value Coeff p-value
C 0.001 0.241 0.000 0.339 0.000 0.855 0.002 0.000 0.002 0.002 Time 0.000 0.762 0.000 0.789 0.000 0.857 0.000 0.235 0.000 0.272 r -0.405 0.000 -0.390 0.000 -0.424 0.000 -0.439 0.000 -0.331 0.000
-0.011 0.643 -0.003 0.892 0.010 0.745 -0.004 0.476 0.018 0.526
Adj R-Sq 0.517 0.488 0.451 0.269 0.381
Decrease in Volatility ()
MV-BH MAD-BH DSR-BH VaR-BH =50%) CVaR-BH =50%) Coeff p-value Coeff p-value Coeff p-value Coeff p-value Coeff p-value
C 0.001 0.213 0.000 0.471 -0.001 0.095 0.001 0.677 0.000 0.485 Time 0.000 0.124 0.000 0.110 0.000 0.285 0.001 0.241 0.000 0.150
r -0.122 0.050 -0.123 0.065 -0.076 0.315 -0.115 0.797 -0.172 0.021
0.150 0.000 0.138 0.000 0.187 0.000 0.234 0.001 0.126 0.000 Adj R-Sq 0.445 0.377 0.397 0.243 0.359
Increase in Sharpe Ratio (SR)
MV-BH MAD-BH DSR-BH VaR-BH =50%) CVaR-BH =50%) Coeff p-value Coeff p-value Coeff p-value Coeff p-value Coeff p-value C 0.056 0.146 0.027 0.472 -0.058 0.207 0.218 0.000 0.226 0.000
Time 0.000 0.722 0.001 0.303 0.000 0.457 0.001 0.782 0.000 0.624 r -19.197 0.000 -19.420 0.000 -21.890 0.000 -38.84 0.000 -18.725 0.001
-2.647 0.191 -1.292 0.505 1.911 0.430 -7.053 0.001 -6.675 0.008
Adj R-Sq 0.225 0.269 0.273 0.259 0.179
Increase in Realized Return (R(r))
MV-BH MAD-BH DSR-BH VaR-BH =50%) CVaR-BH =50%) Coeff p-value Coeff p-value Coeff p-value Coeff p-value Coeff p-value
C 0.027 0.129 0.024 0.178 0.027 0.238 0.015 0.263 0.039 0.039 Time 0.000 0.829 0.000 0.844 0.000 0.895 0.001 0.902 0.000 0.485
r -7.016 0.001 -6.688 0.003 -6.591 0.016 -1.256 0.378 -5.538 0.014
-1.398 0.142 -1.202 0.216 -1.482 0.225 -1.579 0.256 -1.903 0.059
Adj R-Sq 0.123 0.103 0.050 0.040 0.068
29
Table 5. Regression Analysis of the Benefits of Diversified Portfolios: Short-sales Allowed
The table reports the results of the two-stage least squared (2SLS) regressions that analyze the time-series of diversification benefits for
each risk model portfolio as compared to the naïve buy-and-hold (BH) market portfolio when short-sales are allowed. The testing models
for each diversification benefit—increase in expected return (E(r)), decrease in volatility (, the increase in Sharpe ratio (SR), and
increase in realized return (R(r))—are:
321
* )( rtarE ,
321
* )( rtar ,
321
* )( rtarR ,
where the a are constant, and t is the time. The adjusted R-square of each model is shown.
Increase in Expected Return (E(r))
MV-BH MAD-BH DSR-BH VaR-BH =50%) CVaR-BH =50%) Coeff p-value Coeff p-value Coeff p-value Coeff p-value Coeff p-value
C 0.000 0.522 0.000 0.540 0.000 0.263 0.002 0.000 0.002 0.001 Time 0.000 0.618 0.000 0.502 0.000 0.695 0.001 0.072 0.000 0.160 r -0.654 0.000 -0.638 0.000 -0.682 0.000 -0.732 0.001 -0.500 0.000
0.045 0.031 0.050 0.022 0.074 0.000 0.001 0.978 0.038 0.162
Adj R-Sq 0.828 0.815 0.875 0.801 0.625
Decrease in Volatility ()
MV-BH MAD-BH DSR-BH VaR-BH =50%) CVaR-BH =50%) Coeff p-value Coeff p-value Coeff p-value Coeff p-value Coeff p-value
C -0.001 0.719 -0.001 0.602 -0.002 0.161 0.001 0.580 -0.002 0.249 Time 0.000 0.143 0.000 0.134 0.000 0.172 0.001 0.144 0.000 0.063
r 0.323 0.067 0.278 0.115 0.271 0.113 -0.128 0.217 0.203 0.245
0.157 0.051 0.150 0.061 0.196 0.013 0.145 0.002 0.155 0.053
Adj R-Sq 0.052 0.041 0.070 0.258 0.050
Increase in Sharpe Ratio (SR)
MV-BH MAD-BH DSR-BH VaR-BH =50%) CVaR-BH =50%) Coeff p-value Coeff p-value Coeff p-value Coeff p-value Coeff p-value
C 0.082 0.027 0.074 0.039 -0.040 0.290 0.218 0.000 0.273 0.000 Time 0.000 0.508 0.000 0.449 0.001 0.216 0.001 0.798 0.000 0.720
r -38.332 0.000 -37.824 0.000 -42.176 0.000 -38.84 0.000 -33.28 0.000
-1.657 0.398 -1.205 0.522 3.804 0.063 -7.053 0.001 -8.179 0.001 Adj R-Sq 0.591 0.608 0.681 0.559 0.388
Increase in Realized Return (R(r))
MV-BH MAD-BH DSR-BH VaR-BH =50%) CVaR-BH =50%) Coeff p-value Coeff p-value Coeff p-value Coeff p-value Coeff p-value
C 0.033 0.075 0.027 0.160 0.033 0.127 0.027 0.321 0.038 0.037 Time 0.000 0.842 0.000 0.741 0.000 0.786 0.001 0.902 0.000 0.553
r -5.175 0.019 -4.895 0.033 -5.882 0.023 -2.548 0.254 -4.183 0.054
-1.790 0.070 -1.337 0.193 -1.691 0.146 -1.969 0.109 -1.838 0.062
Adj R-Sq 0.058 0.031 0.043 0.003 0.036
30
Table 6. Regression Analysis of EPVs
The table reports the results of the two-stage least squared (2SLS) regressions that analyze the time-series of excess portfolio values
(EPVs) for each risk model portfolio as compared to the naïve buy-and-hold (BH) market portfolio. The testing model for the EPV is:
321
* rtaEPV ,
where the a are constant, and t is the time. The unit of EPVs in the regression is $1 million NT dollars. The adjusted R-square of each
model is shown.
Short-sales Prohibited
MV-BH MAD-BH DSR-BH CVaR-BH =50%) VaR-BH =50%) Coeff p-value Coeff p-value Coeff p-value Coeff p-value Coeff p-value
C -0.054 0.106 -0.058 0.116 -0.046 0.287 0.024 0.292 -0.026 0.472
Time 0.005 0.000 0.006 0.000 0.003 0.000 0.006 0.000 0.002 0.001
r -12.073 0.003 -12.309 0.006 -13.767 0.009 -16.988 0.001 -11.87 0.001
3.523 0.053 4.499 0.026 5.247 0.026 -0.416 0.851 -0.568 0.678
Adj R-Sq 0.699 0.676 0.411 0.681 0.302
Short-sales Allowed
MV-BH MAD-BH DSR-BH CVaR-BH =50%) VaR-BH =50%) Coeff p-value Coeff p-value Coeff p-value Coeff p-value Coeff p-value
C -0.003 0.951 -0.020 0.602 -0.025 0.610 0.059 0.193 0.019 0.562
Time 0.001 0.037 0.002 0.008 0.001 0.046 0.003 0.000 0.002 0.004
r -16.261 0.002 -14.022 0.004 -15.262 0.010 -17.659 0.002 -14.27 0.002
2.180 0.344 3.085 0.145 5.786 0.031 -0.553 0.821 -0.135 0.545
Adj R-Sq 0.229 0.265 0.250 0.393 0.252
31
Table 7. Benefits of CVaR Portfolios: Short-sales Prohibited
The table reports the benefits of CVaR model portfolios with different risk tolerance coefficients () in which short-sales are prohibited.
We compare the increase in expected return (E(r)), the decrease in volatility (), the increase in Sharpe ratio (SR), and the increase in
realized return (R(r)) of the various CVaR model portfolios with the corresponding values generated by the passive buy-and-hold market
portfolio to measure the benefits. We also report the distribution of the benefits over the sample period.
CVaR = 50% 70% 90% 95% 99%
Increase in Expected Return (E(r))
Mean 0.0208 0.0170 0.0118 0.0102 0.0100
Max 0.0651 0.0642 0.0515 0.0477 0.0410
Min -0.0104 -0.0244 -0.0154 -0.0265 -0.0266
SD 0.0046 0.0048 0.0043 0.0041 0.0043 Skewness 0.74 0.58 0.66 0.11 0.07 Kurtosis 0.70 0.92 0.51 0.23 -0.15 Distribution (%)
E(r) 0 6.45 12.90 17.74 22.58 25.81
0<E( r) 0.01 16.13 22.58 29.03 25.81 25.81
0.01<E( r) 0.02 35.48 33.87 32.26 27.42 25.81
0.02E( r) 41.94 30.65 20.97 24.19 22.58
Decrease in Volatility ()
Mean 0.0072 0.0078 0.0071 0.0058 0.0040 Max 0.0212 0.0249 0.0158 0.0168 0.0134
Min -0.0030 -0.0046 -0.0053 -0.0068 -0.0068
SD 0.0054 0.0056 0.0049 0.0046 0.0049
Skewness 0.30 0.40 -0.39 -0.34 -0.28 Kurtosis -0.61 0.43 -0.22 0.69 -0.70 Distribution (%)
0 6.45 6.45 8.06 8.06 22.58
0< 0.01 16.13 9.68 9.68 14.52 16.13
0.01< 0.02 22.58 27.42 19.35 20.97 17.74
0.02 54.84 56.45 62.90 56.45 43.55
Increase in Sharpe Ratio (SR)
Mean 0.153 0.133 0.095 0.084 0.078
Max 0.618 0.565 0.468 0.408 0.384
Min -0.041 -0.094 -0.060 -0.105 -0.101
SD 0.108 0.107 0.098 0.097 0.095
Skewness 1.361 0.946 0.854 0.476 0.476
Kurtosis 4.857 3.278 1.944 0.794 0.545
Distribution (%)
SR 0 6.45 9.68 17.74 20.97 24.19
0< SR 0.1 19.35 25.81 33.87 33.87 32.26
0.1<SR 0.2 45.16 38.71 35.48 32.26 33.87
0.2SR 29.03 25.81 12.90 12.90 9.68
100.00 100.00 100.00 100.00 100.00
Increase in Realized Return (R(r)) Mean 0.0782 0.0985 0.0714 0.0500 0.0769 Max 1.1222 1.1383 1.0106 1.2194 2.3771 Min -1.7933 -1.5484 -1.2904 -1.3033 -1.4222 SD 0.1342 0.1362 0.1209 0.1234 0.1512 Skewness -0.83 -0.72 -0.48 -0.42 1.21
Kurtosis 3.03 1.69 1.34 1.41 6.17
Distribution (%)
R( r) 0 45.16 40.32 38.71 43.55 45.16
0<R( r) 0.01 1.61 0.00 0.00 3.23 0.00
0.01<R( r) 0.02 1.61 0.00 1.61 1.61 0.00
0.02R( r) 51.61 59.68 59.68 51.61 54.84
32
Table 8. Benefits of CVaR Portfolios: Short-sales Allowed
The table reports the benefits of CVaR model portfolios of different risk tolerance coefficients () in which short-sales are allowed. We
compare the increase in expected return (E(r)), the decrease in volatility (), the increase in Sharpe ratio (SR), and the increase in
realized return (R(r)) of the various CVaR model portfolios with the corresponding values generated by the passive buy-and-hold market
portfolio to measure the benefits. We also report the distribution of the benefits over the sample period.
CVaR = 50% 70% 90% 95% 99%
Increase in Expected Return (E(r))
Mean 0.0234 0.0206 0.0162 0.0156 0.0152
Max 0.0907 0.0907 0.0907 0.0907 0.0907
Min -0.0104 -0.0214 -0.0154 -0.0214 -0.0224
SD 0.0058 0.0059 0.0060 0.0060 0.0061 Skewness 1.05 1.07 1.42 1.35 1.24 Kurtosis 1.25 1.55 2.26 2.37 2.07 Distribution (%)
E(r) 0 6.45 9.68 12.90 17.74 20.97
0<E( r) 0.01 16.13 25.81 32.26 29.03 25.81
0.01<E( r) 0.02 35.48 29.03 30.65 24.19 25.81
0.02E( r) 41.94 35.48 24.19 29.03 27.42
Decrease in Volatility ()
Mean 0.0070 0.0074 0.0070 0.0059 0.0042 Max 0.0279 0.0279 0.0279 0.0279 0.0279
Min -0.0084 -0.0084 -0.0084 -0.0069 -0.0068
SD 0.0061 0.0062 0.0058 0.0058 0.0060
Skewness 0.63 0.49 0.28 0.55 0.86 Kurtosis 1.54 1.52 2.27 2.80 2.62 Distribution (%)
0 8.06 8.06 9.68 11.29 24.19
0< 0.01 16.13 9.68 9.68 14.52 17.74
0.01< 0.02 19.35 25.81 17.74 16.13 14.52
0.02 56.45 56.45 62.90 58.06 43.55
Increase in Sharpe Ratio (SR)
Mean 0.166 0.150 0.115 0.107 0.100
Max 0.618 0.565 0.468 0.418 0.418
Min -0.041 -0.077 -0.060 -0.105 -0.084
SD 0.114 0.113 0.110 0.110 0.110
Skewness 1.089 0.850 0.794 0.605 0.633
Kurtosis 3.031 2.018 0.880 0.333 0.250
Distribution (%)
SR 0 4.84 6.45 12.90 16.13 19.35
0< SR 0.1 20.97 24.19 33.87 32.26 33.87
0.1<SR 0.2 37.10 35.48 29.03 27.42 25.81
0.2SR 37.10 33.87 24.19 24.19 20.97
Increase in Realized Return (R(r))
Mean 0.0976 0.0815 0.0449 0.0406 0.0223 Max 1.3225 1.3219 1.3219 1.3219 1.3219 Min -1.5930 -1.5554 -1.1929 -1.3438 -1.3669 SD 0.1291 0.1294 0.1231 0.1287 0.1240
Skewness -0.65 -0.53 -0.14 -0.23 -0.21
Kurtosis 2.49 2.06 1.72 1.37 1.65
Distribution (%)
R( r) 0 40.32 43.55 40.32 45.16 46.77
0<R( r) 0.01 1.61 0.00 0.00 3.23 0.00
0.01<R( r) 0.02 3.23 0.00 3.23 1.61 0.00
0.02R( r) 54.84 56.45 56.45 50.00 53.23
33
Table 9. Benefits of VaR Portfolios: Short-sales Prohibited
The table reports the benefits of VaR model portfolios of different risk tolerance coefficients () in which short-sales are prohibited. We
compare the increase in expected return (E(r)), the decrease in volatility (), the increase in Sharpe ratio (SR), and the increase in
realized return (R(r)) of the various VaR model portfolios with the corresponding values generated by the passive buy-and-hold market
portfolio to measure the benefits. We also report the distribution of the benefits over the sample period.
CVaR = 50% 70% 90% 95% 99%
Increase in Expected Return (E(r))
Mean 0.0152 0.0293 0.0212 0.0185 0.0152 Max 0.0907 0.0946 0.0907 0.0907 0.0907 Min -0.0224 -0.0063 -0.0177 -0.0225 -0.0224 SD 0.0217 0.0215 0.0218 0.0208 0.0217 Skewness 1.24 1.07 1.08 1.43 1.25 Kurtosis 2.06 1.38 1.34 2.35 2.08 Distribution (%)
E(r) 0 20.97 4.84 11.29 9.68 20.97
0<E( r) 0.01 0.00 0.00 0.00 0.00 0.00
0.01<E( r) 0.02 3.23 1.61 1.61 0.00 3.23
0.02E( r) 75.81 93.55 87.10 90.32 75.81
100.00 100.00 100.00 100.00 100.00
Decrease in Volatility ()
Mean 0.0039 -0.0048 0.0027 0.0049 0.0042 Max 0.0134 0.0287 0.0279 0.0279 0.0279 Min -0.0068 -0.0405 -0.0157 -0.0084 -0.0068 SD 0.0049 0.0102 0.0075 0.0063 0.0060 Skewness -0.27 0.49 0.64 0.49 0.88 Kurtosis -0.69 4.37 1.93 2.02 2.63 Distribution (%)
0 22.58 72.58 32.26 17.74 24.19
0< 0.01 8.06 6.45 3.23 4.84 8.06
0.01< 0.02 1.61 3.23 11.29 3.23 3.23
0.02 67.74 17.74 53.23 74.19 64.52
100.00 100.00 100.00 100.00 100.00
Increase in Sharpe Ratio (SR)
Mean 0.088 0.139 0.119 0.111 0.088 Max 0.334 0.523 0.555 0.478 0.334 Min -0.090 -0.028 -0.072 -0.083 -0.090 SD 0.096 0.099 0.108 0.097 0.096 Skewness 0.31 0.96 1.17 1.21 0.31 Kurtosis -0.35 2.37 3.06 2.69 -0.35 Distribution (%)
SR 0 14.52 6.45 8.06 6.45 14.52
0< SR 0.1 48.39 25.81 48.39 46.77 48.39 0.1<SR 0.2 25.81 43.55 24.19 33.87 25.81
0.2SR 11.29 24.19 19.35 12.90 11.29
100.00 100.00 100.00 100.00 100.00
Increase in Realized Return (R(r))
Mean 0.0028 0.0006 0.0052 0.0020 0.0007 Max 0.1932 0.1207 0.1316 0.1207 0.1207 Min -0.1138 -0.1451 -0.1254 -0.1254 -0.1338 SD 0.0465 0.0431 0.0407 0.0400 0.0385 Skewness 0.85 -0.63 0.03 -0.55 -0.38 Kurtosis 4.90 1.95 2.63 2.06 2.67 Distribution (%)
R( r) 0 45.16 45.16 41.94 45.16 46.77 0<R( r) 0.01 3.23 4.84 3.23 1.61 1.61
0.01<R( r) 0.02 0.00 0.00 0.00 1.61 0.00 0.02<E( r) 51.61 50.00 54.84 51.61 51.61
100.00 100.00 100.00 100.00 100.00
34
Table 10. Benefits of VaR Portfolios: Short-sales Allowed
The table reports the benefits of VaR model portfolios of different risk tolerance coefficients () in which short-sales are allowed. We
compare the increase in expected return (E(r)), the decrease in volatility (), the increase in Sharpe ratio (SR), and the increase in
realized return (R(r)) of the various VaR model portfolios with the corresponding values generated by the passive buy-and-hold market
portfolio to measure the benefits. We also report the distribution of the benefits over the sample period.
VaR = 50% 70% 90% 95% 99%
Increase in Expected Return (E(r))
Mean 0.0155 0.0154 0.0155 0.0154 0.0154
Max 0.0907 0.0907 0.0907 0.0907 0.0907 Min -0.0224 -0.0224 -0.0224 -0.0224 -0.0224 SD 0.0062 0.0061 0.0061 0.0061 0.0061
Skewness 1.29 1.25 1.24 1.25 1.25
Kurtosis 2.26 2.13 2.02 2.13 2.14
Distribution (%)
E(r) 0 19.35 19.35 19.35 19.35 19.35
0<E( r) 0.01 27.42 27.42 27.42 27.42 27.42
0.01<E( r) 0.02 25.81 25.81 25.81 25.81 25.81
0.02E( r) 27.42 27.42 27.42 27.42 27.42
Decrease in Volatility ()
Mean 0.0042 0.0042 0.0043 0.0042 0.0042 Max 0.0279 0.0279 0.0279 0.0279 0.0279 Min -0.0068 -0.0068 -0.0068 -0.0068 -0.0068 SD 0.0060 0.0060 0.0060 0.0060 0.0060 Skewness 0.86 0.87 0.85 0.87 0.88
Kurtosis 2.63 2.60 2.60 2.65 2.62
Distribution (%)
0 25.81 25.81 24.19 24.19 25.81
0< 0.01 16.13 16.13 17.74 17.74 16.13
0.01< 0.02 14.52 14.52 12.90 14.52 14.52
0.02 43.55 43.55 45.16 43.55 43.55
Increase in Sharpe Ratio (SR)
Mean 0.090 0.090 0.091 0.090 0.090
Max 0.334 0.334 0.334 0.334 0.334
Min -0.090 -0.090 -0.090 -0.090 -0.090
SD 0.096 0.096 0.096 0.096 0.096
Skewness 0.276 0.278 0.270 0.273 0.277
Kurtosis -0.357 -0.356 -0.374 -0.342 -0.337
Distribution (%)
SR 0 14.52 14.52 14.52 14.52 14.52
0< SR 0.1 46.77 46.77 46.77 46.77 46.77
0.1<SR 0.2 25.81 25.81 24.19 25.81 25.81
0.2SR 12.90 12.90 14.52 12.90 12.90
Increase in Realized Return (R(r))
Mean 0.0076 0.0115 0.0198 0.0085 0.0130 Max 1.4291 1.4291 1.4291 1.4291 1.4291 Min -1.7850 -1.7513 -1.7513 -1.7513 -1.7513 SD 0.1395 0.1392 0.1393 0.1390 0.1393
Skewness -0.53 -0.51 -0.55 -0.49 -0.52
Kurtosis 2.74 2.58 2.60 2.60 2.57 Distribution (%)
R( r) 0 46.77 46.77 45.16 46.77 46.77
0<R( r) 0.01 0.00 0.00 0.00 0.00 0.00
0.01<R( r) 0.02 0.00 0.00 0.00 0.00 0.00
0.02R( r) 51.61 51.61 53.23 51.61 51.61
35
Figure 1. VaR and CVaR
A: Increase in Expected Return
B: Decrease in Volatility
C: Increase in Sharpe Ratio
D: Increase in Realized Return
Figure 2. Effectiveness of Diversified Portfolios over Time: Short-sales Prohibited
The figure shows the over-time differences of portfolio value between the passive buy-and-hold (BH) strategy and the strategies using the
mean-variance (MV), the mean-absolute deviation (MAD), the downside risk (DSR), the value-at-risk (VaR) with =50%, and the
conditional value-at-risk (CVaR) with =50% models in which short-sales are prohibited. To measure the benefits, we compare the
increase in expected return (E(r)), the decrease in volatility (), the increase in Sharpe ratio (SR), and the increase in realized return
(R(r)) of the aforementioned diversified portfolios with the corresponding values generated by the passive buy-and-hold market
portfolio. We rebalance each of the portfolios every 20 trading days by using the data of the previous 60 trading days.
VaR
Loss
CVaR
Maximum Loss 1
Probability
36
A: Increase in Expected Return
B: Decrease in Volatility
C: Increase in Sharpe Ratio
D: Increase in Realized Return
Figure 3. Effectiveness of Diversified Portfolios over Time: Short-sales Allowed
The figure shows the over-time differences of portfolio value between the passive buy-and-hold (BH) strategy and the strategies using the
mean-variance (MV), the mean-absolute deviation (MAD), the downside risk (DSR), the value-at-risk (VaR) with =50%, and the
conditional VaR (CVaR) with =50% models in which short-sales are allowed. To measure the benefits, we compare the increase in
expected return (E(r)), the decrease in volatility (), the increase in Sharpe ratio (SR), and the increase in realized return (R(r))
of the aforementioned diversified portfolios with the corresponding values generated by the passive buy-and-hold market portfolio. We
rebalance each of the portfolios every 20 trading days by using the data of the previous 60 trading days.
MV-BH MAD-Bh DSR-BH VaR-BH CVaR-BH
Max 403,218 409,860 381,252 277,592 608,265 Min -62,684 -75,786 -36,428 -63,328 0
Figure 4. Excess Portfolio Value over Time: Short-sale Prohibited The figure shows the over-time differences of portfolio value between the passive buy-and-hold (BH) strategy and the strategies using the
mean-variance (MV), the mean-absolute deviation (MAD), the downside risk (DSR), the value-at-risk (VaR) with =50%, and the
conditional value-at-risk (CVaR) with =50% models in which short-sales are prohibited. We rebalance each of the portfolios every 20
trading days by using the data of the previous 60 trading days. The initial investments on 1/25/2007 are assumed to be $1 million NT
dollars. The values shown in the figure are the value of the MV model minus the value of the BH, the value of the MAD model minus the
value of the BH, the value of the DSR model minus the value of BH, the value of the VaR(=50%) model minus the value of the BH, and
the value of the CVaR(=50%) model minus the value of the BH.
37
MV-BH MAD-BH DSR-BH VaR-BH CVaR-BH Max 256,853 271,463 350,244 209,242 449,896 Min -96,437 -85,689 -48,101 -103,756 15,283
Figure 5. Excess Portfolio Value over Time: Short-sale Allowed
The figure shows the over-time differences of portfolio value between the passive buy-and-hold (BH) strategy and the strategies using the
mean-variance (MV), the mean-absolute deviation (MAD), the downside risk (DSR), value-at-risk (VaR) with =50%, and the
conditional value-at-risk (CVaR) with =50% models in which short-sales are allowed. We rebalance each of the portfolios every 20
trading days by using the data of the previous 60 trading days. The initial investments on 1/25/2007 are assumed to be $1 million NT
dollars. The values shown in the figure are the value of the MV model minus the value of the BH, the value of the MAD model minus the
value of the BH, the value of the DSR model minus the value of the BH, the value of the VaR(=50%) model minus the value of the BH,
and the value of the CVaR(=50%) model minus the value of the BH.
50% 70% 90% 95% 99% Max 593,343 720,772 387,635 281,152 423,699 Min 13,946 18,220 -10,850 -48,535 -57,355
Figure 6. Excess Portfolio Value of CVaR Models over Time: Short-sale Prohibited The figure shows the over-time differences of portfolio value between the passive buy-and-hold (BH) strategy and the CVaR models with
various risk tolerance coefficients () in which short-sales are prohibited. We rebalance each of the portfolios every 20 trading days by
using the data of the previous 60 trading days. The initial investments on 1/25/2007 are assumed to be $1 million NT dollars. The values
shown in the figure are the values of the CVaR models minus the value of the BH.
38
50% 70% 90% 95% 99% Max 449,896 607,345 259,845 341,512 218,077 Min 15,283 18,220 -59,813 -71,548 -78,856
Figure 7. Excess Portfolio Value of CVaR Models over Time: Short-sale Allowed
The figure shows the over-time differences of portfolio value between the passive buy-and-hold (BH) strategy and the CVaR models with
various risk tolerance coefficients () in which short-sales are allowed. We rebalance each of the portfolios every 20 trading days by
using the data of the previous 60 trading days. The initial investments on 1/25/2007 are assumed to be $1 million NT dollars. The values
shown in the figure are the values of the CVaR models minus the value of the BH.
50% 70% 90% 95% 99%
Max 277,592 143,134 524,974 310,583 201,404 Min -63,328 -158,532 -44,713 -41,899 -106,421
Figure 8. Excess Portfolio Value of VaR Models over Time: Short-sale Prohibited The figure shows the over-time differences of portfolio value between the passive buy-and-hold (BH) strategy and the VaR models with
various probabilities () in which short-sales are prohibited. We rebalance each of the portfolios every 20 trading days by using the data
of the previous 60 trading days. The initial investments on 1/25/2007 are assumed to be $1 million NT dollars. The values shown in the
figure are the values of the VaR models minus the value of the BH.
50% 70% 90% 95% 99%
Max 209,242 233,111 236,371 212,012 218,077 Min -103,756 -85,876 -44,353 -99,548 -95,618
Figure 9. Excess Portfolio Value of VaR Models over Time: Short-sale Allowed
The figure shows the over-time differences of portfolio value between the passive buy-and-hold (BH) strategy and the VaR models with
various probabilities () in which short-sales are allowed. We rebalance each of the portfolios every 20 trading days by using the data of
the previous 60 trading days. The initial investments on 1/25/2007 are assumed to be $1 million NT dollars. The values shown in the
figure are the values of the VaR models minus the value of the BH.