Diversification Benefits of Risk Models in Managing Portfolios: A …€¦ · Diversification Benefits of Risk Models in Managing Portfolios: A Case of Taiwan’s Stocks ABSTRACT

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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)


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)


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)


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


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)


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)


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.


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.


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.


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.


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


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.


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.


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%.


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.


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.


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.


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.



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


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.


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


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


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


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.


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



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


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



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.



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



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



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



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%


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


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


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


CVaR = 50% 70% 90% 95% 99%


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


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


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


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


realized return (R(r)) of the various VaR model portfolios with the corresponding values generated by the passive buy-and-hold market


CVaR = 50% 70% 90% 95% 99%


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


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


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


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


realized return (R(r)) of the various VaR model portfolios with the corresponding values generated by the passive buy-and-hold market


VaR = 50% 70% 90% 95% 99%


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


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


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


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


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



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


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


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.

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