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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.19 No.3 pp.498-509
DOI : https://doi.org/10.7232/iems.2020.19.3.498

Mean-Var Investment Portfolio Optimization Under Capital Asset Pricing Model (CAPM) with Nerlove Transformation: An Empirical Study Using Time Series Approach

Sukono, Kankan Parmikanti, Lisnawati, Sonny Hersona Gw, Jumadil Saputra*
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Jatinangor 45363, Bandung-Sumedang, Indonesia
Faculty of Economics and Business, Universitas Singaperbangsa, Karawang, Tanggerang, Indonesia
Faculty of Business, Economics, and Social Development, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Kuala Terengganu, Malaysia
May 6, 2020 June 22, 2020 June 30, 2020

ABSTRACT

This paper discusses the problems of Mean-VaR investment portfolio optimization under the Capital Asset Pricing Model with Nerlove transformation by using a time series approach. It assumed that the market return follows the pattern of the time series, whereas stocks return analyzed following the pattern of capital asset pricing models with Nerlove transformation. So that the mean of the market return can be estimated by using the model of Autoregressive Moving Average (ARMA), while volatility can be estimated using a model of Generalized Autoregressive Conditional Heteroscedasticity (GARCH). As a risk-free asset here used the interest rate of Indonesia Bank (BI rate), together with the market return is used to estimate the CAPM regression equation for each stock analyzed. Based on the estimator of the CAPM regression equation is used to estimate the mean and variance of the return value of each stock. Estimator mean and the variance value of each stock, as well as the estimator value of the covariance between the stocks, will be used for investment portfolio optimization process. Investment portfolio optimization process is based on a model of Mean-VaR by risk tolerance factor. The goal is to determine the weight composition of the optimum of the investment portfolio. Based on the optimization process was obtained the composition weight of investment portfolio optimum. This composition weight can be taken into consideration in decision making for investors to invest in particular on the assets that were analyzed here.

1. INTRODUCTION

In the investment world, almost all investments contain elements of uncertainty or so-called risk. Investors do not know with certainty the results obtained from the investment. As stated by Cerović et al. (2015), in such circumstances the investors face risks in its investments. Youth who can do is estimate how the expected profit from the investment, and how far the possibility that actual results will be deviated from the expected results, as described by Džaja and Aljinović (2013). The first issue concerned with the calculation of the expected value and the second relates the measurement of the spread of values. Because investors face a risky investment opportunity, investment choices cannot rely solely on the expected profit rate alone; this is explained by Campbell et al. (2001). If the investor expects to earn a high-profit level, then he should be willing to bear the risk that high anyway. According to Ramos et al. (2020), measurement of the level of risk of the most commonly used is the variance and standard deviation. Both measures the actual return differs from the average return. Variance measures the average squared difference between the actual return and the average return.

The higher the value of the variance, the actual return is much different from the average returns. Based on Sukono (2011), Azevedo and Reis (2019), however, please note, that the variance is a measure of average risk. Therefore, the variance is a measure that cannot accommodate all risk events occurring. Appeared an idea, that to measure the level of risk by using quantile, from now on known as Value-at-Risk (VaR). Refer to Sarjono et al. (2020), Value-at-Risk is one measure of the risk of investment losses. Value-at-Risk, in fact, is a measure of how much change (volatile) an asset. Based on Hekmatpour et al. (2017), it needed to be known and understood to estimate the change in policy and future risk.

One characteristic of investments in securities is ease in forming an investment portfolio. That is, investors can easily spread (diversify) investment in a variety of investment opportunities. In practice investors in securities often diversify their investments. They combine a variety of securities in the investment; this is explained by Campbell et al. (2001) and Yazdanshenas et al. (2015). In other words, they form the investment portfolio. So the portfolio is nothing but a collection of investment opportunities. Why do they diversify? There is a saying that wise investors do not put all Reviews their eggs into just one basket. As explained by Panjer et al. (1998) and Oke (2013), they do diversification to reduce risk. Therefore, the need to understand how to calculate the level of expected return and risk of the combination of several investments, or from the formation of the investment portfolio. According to the explanation of Campbell et al. (2001) and Sukono (2011), calculation of expected profit rate of a portfolio is relatively easy, because nothing else is a weighted average (weighted) of the expected profit rate of each of the securities in the portfolio formation. Instead, calculation of the relative risk in the portfolio is rather complicated because it contains the elements of a correlation or covariance between the returns of the securities that make up the portfolio. As said by Barberis et al. (2015), Golestani and Fallah (2019) and Çelik (2012), as a rational investor, would choose an efficient investment opportunity. Therefore, investors need to assess the magnitude of the risk and profitability of all existing alternative investment. If the investment appraisal process is used the same guidelines, then it will be able to formulate a general equilibrium model. According to Khan et al. (2012), one of the general equilibrium models is Capita Asset Pricing Model (CAPM). The model will explain the relationship between risk and profitability of all existing assets on the market.

Džaja and Aljinović (2013) stated that the general equilibrium model Capital Asset Pricing Model (CAPM) is useful for explaining the relationship between risk and profitability, as well as determine the size of the relevant risks for each asset, are also useful in the process of determining asset returns. Based on Choudhary and Choudhary (2010) and Muhamma et al. (2014), Capital Asset Pricing Model (CAPM) the price of the asset is determined by regression the excess return of assets against the risk-free asset return (called the risk premium of assets), while the difference between the market return to the return of risk-free assets (so-called market risk premium). The principles of assets return assessment using standard CAPM, the risk premium when the asset is affected by the risk premium on the market today.

However, the risk premium assets at this time allow also influenced by the market risk premium on moments before. Problems like so is a form of distributed lag issues. One of the methods of analysis for the distributed lag issues this can be done using a Nerlove model approach. As described by (Gosalamang et al., 2012;Baykalova et al., 2018;Kirsten and Verrecchia 2019), the Nerlove models also referred to as a model of stock adjustment or partial adjustment. Furthermore, the market return often fluctuates in line with the change of time. Thus it can be said that the characteristic pattern of changes in the market return is following the model of the time series. If a change of market returns follow a time series model, then the mean the market return, in general, can be estimated using a model autoregressive moving average (ARMA), this is explained by (Hossain et al., 2015;Sheykh and Emadian, 2019) and Tsay (2005). While the volatility can generally be estimated using models of generalized autoregressive conditional heteroscedastic (GARCH). Based on the description above, the problem in this research is how to estimate the weight of the weighing allocation of funds in each asset returns in the formation of the portfolio, to obtain optimal portfolio with a risk tolerance factor of each investor. Because the risk is measured by using a model of Value-at-Risk (VaR), then the model portfolio optimization to be performed is shaped Mean-VaR. Where the return of assets following the shape of the capital asset pricing model (CAPM), which transformed Nerlove, by using a time series model approach. Thus the purpose of this study is the optimization of the optimum weights of Mean-VaR portfolio under the CAPM with Nerlove transformation using a time series model approach. For completing the achievement of these objectives, this research formulated CAPM regression with Nerlove transformation, which is used to estimate the value of return on assets in a portfolio of securities. As a numerical illustration, we analyzed some of the securities that are traded on stock markets in Indonesia.

2. MATERIAL AND METHOD

Steps being taken in the methodology here sequentially include discussions of the calculation of assets return. The estimated value of the mean and variance of the market return by using the time series model approach, the formulation of CAPM regression with the Nerlove transformation to estimate the average value and variance of a return of the asset, the estimated value of covariance between the assets return, and portfolio optimization model to determine optimum weights.

2.1 Calculation Model of Asset Returns

There are several models for the calculation of the return of an asset, but in this paper, the calculation of the return of assets completed by using a model of log returns. As described by Tsay (2005), if we let Pt the asset price at the time t (t = 1, 2, ..., T with T the number of data observations), the return of an asset rt can be calculated using the equation:

$r t = ln ( P t P t − 1 )$
(1)

Similarly, to the calculation of the market return, if we let Mt the stock price index at a time t (t = 1, 2, ..., T with T the number of data observations), then the market return xt can be calculated using the equation:

$x t = ln ( M t M t − 1 )$
(2)

with $ln A = log e ( A )$ , and it is assumed that $P 0 = M 0 = 1$. According to Tsay (2005), selection of this log return model, because that model of log returns have advantages, one of which is that if A small, then the value of ln(1+ A)≈ A. For example, note the value of return approaching the value of log returns 0.02, ln(1+0.02)= 0.0086 ≈ 0.02.

2.2 Time Series Models of Market Return

In this study, it is assumed that the pattern of changes in market returns follow a time series. So that the mean estimates, in general, can be done using the models of autoregressive moving average (ARMA). Meanwhile, volatility estimation, in general, can be done using the models of generalized autoregressive conditional heteroscedastic (GARCH). Starting the first estimate of the mean models following:

Mean Model of Market Return. The mean model estimation is done by using the model of autoregressive moving average (ARMA). Refer to Hossain et al. (2015), suppose xt the market return at a time t (t =1, 2, ..., T with T the number of data observations) that stationary, the general model of ARMA (p, q), can be expressed in the equation following:

$x t = ϕ 0 + ∑ i = 1 p ϕ i x t − i + ε t − ∑ j = 1 q θ j ε t − j$
(3)

where φ0 the constant term, φi (i = 1, ..., p) and θj (j = 1, ..., q) constant coefficients, and {εt] residual sequence of the ARMA models, that assumed normally distributed white noise with mean zero and variance $σ ε 2$. Non-negative integers p and q is the order of the ARMA model. The AR and MA are a special case the model of ARMA(p,q).

Mean modeling stages. As explained by Tsay (2005), the stages of modelling the mean in broad outline is follow: (i) model identification, determining the value of the order, using plot of autocorrelation function (ACF) and partial autocorrelation function (PACF); (ii) the parameters estimation can be made with the maximum likelihood method; (iii) diagnostic test, with white test noise and lack of serial correlation of the residuals εt; and (iv) prediction, if the model is suitable, it can be used for prediction performed recursively.

Volatility Model of Market Return. Based on Belhaj and Abaoub (2015), the volatility model estimation is done by using the models of autoregressive conditional heteroscedastic (GARCH). Refer to Raza and Arshad (2015); Bollerslev introduces the GARCH models in 1986, which are a common form or generalization of ARCH models. Generally, GARCH(m,s) models can be expressed as follows:

$ε t = σ t u t σ t 2 = ω 0 + ∑ i = 1 m ω i ε t − i 2 + ∑ j s ψ j σ t − j 2 + u t$
(4)

where ω0 constant term, ωi (i=1, ..., m) and ψj (j = 1, ..., s) constant coefficients, and {ut} residual sequence of GARCH models, that assumed normally distributed white noise with mean zero and variance of one, and {εt} residual sequence of ARMA models.

Volatility Modelling Stages. As explained by Tsay (2005), the volatility modeling stages are as follows: (i) estimate the mean models with time series models approach (i.e., ARMA models); (ii) use of residual from mean model to test the effects of ARCH; (iii) if no ARCH effects, identify and estimate volatility models, as well as form a joint estimation of the mean and volatility models; (iv) perform a diagnostic test, to test the suitability of the model; and (v) if the model is suitable, used for prediction performed recursively.

2.3 Formulation of CAPM Regression with Nerlov e Transformation

In this section performed regression formulation of the capital asset pricing model (CAPM) with Nerlove transformation. However, as the underlying stages, first need to be addressed on a standard CAPM, and then needs to be discussed about the CAPM distributed lag as following.

Regression Equations of CAPM. Standard CAPM is a basic equation in mind that the balance of the capital market will be indicated by the line asset markets, where the line connecting the investment portfolio of a risk-free opportunity, with the opportunity of the risky investment portfolio. This relationship applies to all assets, both efficient and inefficient. To determine the position of this market portfolio, need to be combined between risk assets. According to Panjer et al. (1998) and Oke (2013) if we let rft risk-free asset return at the time t, then the expectation of the risk-free asset is $μ f = E ( r f t )$, and the variance of the risk-free asset is $σ f 2 = V a r ( r f t ) = 0$. All investors are assumed to be invested in the same portfolio, namely at the market portfolio. This assumption is valid because the assumptions at CAPM, that all investors use the same analysis, which uses Markowitz’s method. In a state of equilibrium, all risky assets should be on the market portfolio, because all investors would hold that portfolio.

Refer to Barberis et al. (2015) if the portfolio is composed of all the assets in the market, and supposes that xt the market return at the time t, then the market return expectation is $μ m = E ( x t )$ and the variance of the market return is $σ m 2 = V a r ( x t )$. The difference between the market return expectations, with the expectations of the risk-free assets, return $[ E ( x t ) − μ f ]$ called the market risk premium, and the ratio between the market risk premium and market risk σm, that is $[ E ( x t ) − μ f ] / σ m$ the line slope of capital market equation. If we let rpt the portfolio return of capital markets at the time t, then the portfolio return expectations of the capital market are $μ p = E ( r p t )$, and the variance of the portfolio return of the capital markets is $σ p 2 = V a r ( r p t )$. The capital market portfolios can be expressed as follow:

$E ( r p t ) = E ( r f ) + [ E ( x t ) − E ( r f ) ] σ m σ p$
(5)

Slope $[ E ( r m ) − μ f ] / σ m$ the market price of efficient risk portfolio. The market price shows the additional return required by the market.

Furthermore, based on Choudhary and Choudhary (2010), suppose rt asset returns at the time t, with the expectations of assets return are $μ t = E ( r t )$ and variance $σ t 2 = V a r ( r t )$. Based on the concept of capital market portfolio line mentioned above, the relationship between E(rt), E(xt) and E(rft), can be expressed as

$E ( r t ) − E ( r f t ) = β { E ( x t ) − E ( r f t ) }$
(6)

with β is a slope. The difference between the expected return of assets to the expected return of risk-free assets $[ E ( r t ) − E ( r f t ) ]$ called the risk premium of assets.

According to Sukono (2011) and Panjer et al. (1998), equation (6) cannot be empirically tested statistically since the equation (6) is an equation of expectations, is a value that has not been observed. Therefore, in order to CAPM regression equation can be empirically tested to be amended as follows:

$r t − r f t = β 0 + β 1 ( x t − r f t ) + e t$
(7)

Therefore the risk-free asset return has flats constant, it can be written as $μ f = E ( r f t )$. Because a risk-free asset, then the variance $σ f 2 = V a r ( r f t ) = 0$. So that equation (7) can be expressed as

$r t − μ f = ω 0 + β 0 ( x t − μ f ) + e t$
(8)

where β0 is the constant term, β1 is the slope, and et is the residual. Residual sequence {et} assumed to be white noise that is a normal distribution with mean zero and variance $σ e 2$. To estimate equation (8) can be done by the least squares method.

The CAPM Regression Equation Lag Distributed. The assumptions in the preparation of standard CAPM is still used in the preparation of CAPM lag distributed. A critical distinction that the CAPM distributed to accommodate a possible lagged effect of the risk premium on some past period, while the standard CAPM cannot do it.

According to Sukono (2011), suppose rt the assets return at the time t, and xt market index return at the time t. It is known that the risk-free asset return at the time t, rft has the mean $μ f = E ( r f t )$ is constant, and $σ f 2 = V a r ( r f t ) = 0$. The CAPM regression equation is shaped as a lagged distribution:

$r t − μ f = ω 0 + β 0 ( x t − μ f ) + β 1 ( x t − 1 − μ f ) + ... + β l ( x t − l − μ f ) + e t$
(9)

where ω0 constant term, market risk premium coefficient, and {et} residuals sequence from a regression equation of CAPM lag distributed, which is assumed to be normally distributed white noise with mean zero and variance $σ e 2$. In the CAPM regression equation lag distribution of this, the difficulty is how practical way to determine the length of lag.

Modeling of CAPM Regression with Nerlove Transformation. Nerlove transformation is done with the aim to further simplify the equation (9), and to facilitate the estimation of the length of the lag because the result of Nerlove transformation model simply involves the first lag only. The CAPM equation of the Nerlove transformation obtained by transforming the equation (9) based on the method of Nerlove transformation as follows. To simplify assumed that all forms of assets are a linear function of the market return, to obtain the following equation. The first lag of the distribution equation (8) is:

$r t − 1 − μ f = ω 0 + β 0 ( x t − 1 − μ f ) + e t − 1$
(10)

Also based (7) forms the equation:

$r t ∗ − μ f = ω 0 + β 0 ( x t ∗ − μ f ) + e t ∗$
(11)

If equation (7) subtracted by (9), it can be obtained by the equation following:

$( r t − μ f ) − ( r t − 1 − μ f ) = β 0 [ ( x t − μ f ) − ( x t − 1 − μ f ) ] + ( e t − e t − 1 )$
(12)

If equation (10) subtracted by (9), it can be obtained by the equation following:

$( r t ∗ − μ f ) − ( r t − 1 − μ f ) = β 0 [ ( x t ∗ − μ f ) − ( x t − 1 − μ f ) ] + ( e t ∗ − e t − 1 )$
(13)

Furthermore, if the equation (12) multiplied by δ, it can be obtained by the equation following:

$δ [ ( r t ∗ − μ f ) − ( r t − 1 − μ f ) ] = δ β 0 [ ( x t ∗ − μ f ) − ( x t − 1 − μ f ) ] + δ ( e t ∗ − e t − 1 )$
(14)

According to Gosalamang et al. (2012), Nerlove in 1966 assume an equation as follows:

$( r t − μ f ) − ( r t − 1 − μ f ) = δ β 0 [ ( r t ∗ − μ f ) − ( r t − 1 − μ f ) ]$
(15)

If equation (13) is substituted into (14), it can be obtained by the following equation:

$( r t − μ f ) − ( r t − 1 − μ f ) = δ β 0 [ ( x t ∗ − μ f ) − ( x t − 1 − μ f ) ] + δ ( e t ∗ − e t − 1 ) ] = δ β 0 ( x t ∗ − μ f ) − δ β 0 ( x t − 1 − μ f ) + δ e t ∗ − δ e t − 1$
(16)

From equation (7) is obtained that the $β 0 ( x t − 1 − μ f ) = ω 0 + ( r t − 1 − μ f ) − e t − 1$. Furthermore, if the equation is substituted into the equation (16), it can be obtained by the following equation:

$( r t − μ f ) − ( r t − 1 − μ f ) = δ β 0 ( x t ∗ − μ f ) − δ [ ω 0 − ( r t − 1 − μ f ) − e t − 1 ] + δ e t ∗ − δ e t − 1 = δ β 0 ( x t ∗ − μ f ) − δ ω 0 − δ ( r t − 1 − μ f ) + δ e t − 1 + δ e t ∗ − δ e t − 1$
(17)

So simplification of equation (16), can be expressed as the following equation:

$r t = ( μ f − δ ω 0 ) + δ β 0 ( x t ∗ − μ f ) + ( 1 − δ ) ( r t − 1 − μ f ) + δ e t ∗$
(18)

Equation (17) is called the regression of CAPM lag distributed with Nerlove transformation models. To estimate the regression model equation (17) can be accomplished by using the Least Square or Maximum Likelihood Estimator. Based on the equation (17), is then used to estimate the values of mean and variance of asset returns.

Estimate the Values of Mean and Variance Assets Return. Based on the equation (18) the mean return of the asset can be estimated as follows:

$μ t = E [ r t ] = E [ ( μ f − δ ω 0 ) + δ β 0 ( x t ∗ − μ f ) + ( 1 − δ ) ( r t − 1 − μ f ) + δ e t ∗ ] , = ( μ f − δ ω 0 ) + δ β 0 { E [ x t ∗ ] − μ f } + ( 1 − δ ) { E [ r t − 1 ] − μ f } + δ E [ e t ∗ ] = ( μ f − δ ω 0 ) + δ β 0 ( μ m ∗ − μ f ) + ( 1 − δ ) ( μ t − 1 − μ f )$
(19)

Also based on the equation (17) the variance of asset returns can be estimated as follows:

$σ t 2 = E [ ( r t − μ t ) 2 ] = E [ δ 2 β 0 2 ( x t ∗ − μ m ) 2 + ( 1 − δ ) 2 ( r t − 1 − μ t − 1 ) 2 + δ 2 ( e t ∗ ) 2 + 2 δ β 0 ( 1 − δ ) ( x t ∗ − μ m ) ( r t − 1 − μ t − 1 ) + 2 δ 2 β 0 e t ∗ ( x t ∗ − μ m ) + 2 δ ( 1 − δ ) e t ∗ ( r t − 1 − μ t − 1 ) ] = δ 2 β 0 2 E [ ( x t ∗ − μ m ) 2 ] + ( 1 − δ ) 2 E [ ( r t − 1 − μ t − 1 ) 2 ] + δ 2 E [ ( e t ∗ ) 2 ] + 2 δ β 0 ( 1 − δ ) E [ ( x t ∗ − μ m ) ( r t − 1 − μ t − 1 ) ] + 2 δ 2 β 0 E [ e t ∗ ( x t ∗ − μ m ) ] + 2 δ ( 1 − δ ) E [ e t ∗ ( r t − 1 − μ t − 1 ) ] σ t 2 = δ 2 β 0 2 ( σ m ∗ ) 2 + ( 1 − δ ) σ t − 1 2 + δ 2 ( σ t ∗ ) 2 + 2 cov ( x t ∗ , r t − 1 )$
(20)

2.4 Mean-VaR Portfolio Optimization Model

In this section analyzed the Mean-VaR portfolio optimization model by risk tolerance factor τ. Suppose rt asset return of stock i at the time t, where i=1,...,N with N the number of shares of assets which were analyzed, and t = 1, ..., T with T the number of asset price data observed. Also suppose $w ′ = ( w 1 , ⋯ , w N )$ transpose of the weight vector, $r ′ = ( r 1 t , ⋯ , r N t )$ transpose of asset returns vector, and e'=(1,...,1) transpose of a unit vector. Portfolio return can be expressed as $r p = w ′ r$ with $w ′ e = 1$; this is explained by Maringer (2007) and Panjer et al. (1998). Suppose $μ ′ = ( μ 1 t , … , μ N t )$, the mean return of the portfolio μp can be expressed as:

$μ p = E [ r p ] = w ' μ$
(21)

Suppose the covariance matrix $Σ = ( σ i j ) i , j = 1 , ... , N$, where $σ i j = C o v ( r i t , r j t )$, then the variance of portfolio returns can be expressed as:

$σ p 2 = w ' Σ w$
(22)

Based on Khindanova and Rachev (2005), the risk of portfolio Return as measured by Value-at-Risk (VaR), can be calculated using the equation:

$V a R p = − W 0 { μ p + z α σ p } = − W 0 { w ' μ + z α ( w ' Σ w ) 1 / 2 }$
(23)

where W0 initial capital invested in the formation of the portfolio, and zα percentile of the standard normal distribution with a significance level (1−α)%.

Furthermore, Mean-VaR portfolio optimization model by a risk tolerance factor τ formulated concerning the following definitions.

Definition.Panjer et al., (1998). A portfolio p*called (Mean-VaR) efficient if there is no portfolio p with$μ p ≥ μ p *$and$V a R p ≤ V a R p *$.

According to Liu and Xu (2010), and Björk et al. (2011), Mean-VaR Portfolio optimization model by a risk tolerance factor τ, typically uses the objective function as follows:

$2 τ μ p − V a R and τ ≥ 0$
(28)

where τ risk tolerance factor. That is, an investor with a risk tolerance factor τ, the initial capital invested W0 =1 unit, must complete an optimization problem as follows:

(24)

with the constraint $w ' e = 1$

Completing the system of equations (23), for all values of τ ∈[0,∞) the set of efficient portfolios will be obtained. The set of points in the diagram- $( μ p , V a R p )$ establish an efficient portfolio, which is often referred to as surface efficient (efficient frontier), this is explained by Björk et al. (2011), and Sukono (2011).

Refer to Panjer et al. (1998), equation (23) is a convex quadratic optimization problem. Thus, the Lagrangian multiplier function of the portfolio optimization problem is given by:

$L ( w , λ ) = ( 2 τ + 1 ) w ' μ + z α ( w ' Σ w ) 1 / 2 + λ ( w ' e - 1 )$
(25)

Based on the Kuhn-Tucker theorem, the optimality condition of equation (24) is $∂ L ∂ w = 0$ and $∂ L ∂ λ = 0$. Resolving these two equations optimality condition, obtained the optimal portfolio weight vector equation as follows:

$w * = ( 2 τ + 1 ) Σ − 1 μ + λ Σ − 1 e ( 2 τ + 1 ) e ' Σ − 1 μ + λ e ' Σ − 1 e ; τ ≥ 0$
(26)

with

$λ = − b ± b 2 − 4 a c 2 a ; λ > 0$
(27)

where $a = e ' Σ − 1 e$; $b = ( 2 τ + 1 ) e ' Σ − 1 μ$; $c = ( 2 τ + 1 ) 2 μ ' Σ − 1 μ − ( z α ) 2$, with $Σ − 1$ the inverse of a matrix Σ.

Furthermore, based on Björk et al. (2011), by substituting w* into the equation (20) and (22) respectively, obtained the estimator values of mean and Value-at-Risk of the investment portfolio. As a numerical illustration, discussed some of the assets of shares traded at the stock market in Indonesia, below.

3. RESULTS AND DISCUSSION

The final goal of this discussion is the result and obtaining a combination of weighting allocations to each asset in the formation of the investment portfolio. Steps to achieve these objectives, this study successively discussed: asset data that analyzed, the calculation of assets return, the estimation models of mean and volatility of market return, the estimation of CAPM with Nerlove transformation model for securities assets, and investment portfolio. Starting with a discussion of asset data is analyzed as follows.

3.1 The Analysis of Assets Data

The data used in the analysis of numerical illustrations include closing prices of securities assets, the market price index data, and risk-free asset data. Securities asset data analyzed consisted of five securities include ASII, BBRI, HMSP, TLKM, and BBCA. While the stock market index data used is the composite stock price index (CSPI), and the risk-free asset data used is the interest rate of Indonesia Bank (BI rate). Data of securities assets and the stock price index are accessible via the website: http://finance.yahoo.com, during the period January 2, 2013, until July 29, 2018. While the interest rate data of Indonesia Bank (BI) accessible via website: https: / /www.bi.go.id, in the same period.

Furthermore, five securities price data is determined to return each using equation (1). While the data composite stock price index (CSPI) determined by using the market return (See equation 2). Meanwhile, the interest rate data of Indonesia Bank (BI rate) during the period of observation, the value is relatively constant with a mean of μf = 0.083582, and the variance is assumed to be $σ f 2 = 0$. Further, the asset return analyzed by using Mean and Volatility of Market Return as follows:

3.2 Model Estimation of Mean and Volatility of M arket Return

In this section, the model intends to estimate the average and volatility of the market return. Estimate the average market return is generally done using the ARMA model refers to equation (3). While the estimated volatility of the market return is generally done using GARCH refer to equation (4). Starting the first estimate of the average in the time series models as follows.

Model Estimation of Mean Market Return. Estimation of the mean models done with the help of software Eviews 6. First, test the data stationary market return xt using the unit root test, or Augmented Dickey- Fuller test. The test results show that at variable market return xt there is no unit root. It can be shown from the value of the probability variable market return xt less than the significance level of 0.05. It can be concluded that market return data xt has been stationary. Second, the tentative model identification by using a market return correlogram, and the results showed that the plot autocorrelation function (ACF) cuts off significantly at lag 1, and plot Partial Autocorrelation Function (PACF) also cuts off at lag 1. This indicates that the market return model follows AR(1). Third, based on the model identification, parameter estimation performed on tentative AR(1) models. Estimation process performed by explorative, and the results show that the market return data following the AR(1) model by the equation:

$x t = 0.088662 x t − 1 + ε t$
(29)

Fourth, based on the verification test of the coefficient of xt-1 conducted by using statistical tests-t, the coefficient xt-1 has a probability of 0.0003, which is smaller than the 0.05 significance level. It suggests that the coefficient xt-1 is significant. Similarly, based on the diagnostic test performed by using statistics - F, and the assumption of white noise to the random variable of residual εt, shows that the model AR(1) mentioned above is significant.

Volatility Model Estimation of Market Return. Estimates volatility model done with the help of software Eviews 6. Estimated volatility model is done using data residual εt of the average models. First, test the existence of an element of autoregressive conditional heteroscedasticity (ARCH) against residual εt, using statistical tests of ARCH-LM. Tests suggest that the probability of statistical F(1.1341) is equal to 0.0000 so that the probability of statistical Chi-Square (1) is equal to 0.0000. If the is determined significance level of 0.05, it indicates that the data residual εt there are elements of ARCH. Second, the tentative model identification is made by using the plot correlogram of squared residuals $ε t 2$. Correlogram plot shows that ACF cuts off at lag 1, and PACF also cuts off at lag 1. It indicates that the residual data squares $ε t 2$ following the GARCH(1, 1) model. Third, the model parameter estimation performed on tentative of GARCH(1,1) model, and generates the model parameter estimation equation:

$σ t 2 = 00000922 + 0.140806 ε t − 1 2 + 0.82585 σ t − 1 2 + u t$
(30)

Fourth, based on the verification test: constant parameters, the coefficient of $ε t − 1 2$and the coefficient of $σ t − 1 2$, conducted by using statistical tests-t. The test result of each parameter has a probability of 0.0000, which is smaller than the 0.05 significance level. It suggests that each parameter is significant. Similarly, based on a diagnostic test performed by using statistics-F, and the assumption of white noise to the random variable residual ut, shows that the GARCH(1,1) mentioned above is significant. Fifth, test the presence of the ARCH element against the random variable residual ut, done using statistical ARCH-LM. Tests suggest that the probability of statistical F(1.1341) is equal to 0.6538 so that the probability of statistical Chi-Square (1) is equal to 0.6535. If the is determined significance level of 0.05, it indicates that the data residual ut there is no element of ARCH.

Forecasting. After the estimation model of the average market return and volatility done, and significantly generating mean and volatility models to follow AR (1)-GARCH (1.1) with the estimator equation:

$x ^ t = 0.088662 x t − 1 σ ^ t 2 = 00000922 + 0.140806 ε t − 1 2 + 0.82585 σ t − 1 2$
(31)

Furthermore, the mean and volatility estimator models are used to forecast one period to the next, the value of the average market return $μ m = x ^ T + 1 ( 1 )$= 0.000322, and variance value $σ m 2 = σ ^ T + 1 2 ( 1 )$= 0.000107.

3.3 Estimation of CAPM Regression with Nerlove Transformation

In this section, the regression equation estimate of return of each stock based Capital Asset Pricing Model (CAPM) lag distributed with the Nerlove transformation. The regression equation estimation process is as follows. First, the data estimator market return and the value of the mean risk-free assets return (BI rate), is used to determine the amount of market risk premium return $x t − μ f$. Meanwhile, the risk premium returns the magnitude of each of the securities is determined using $r i t − μ f$ and $r i t − 1 − μ f$ (i=1,...,N with N the number of securities to be analyzed). Secondly, in turn, the risk premium return of each security it $r i t − μ f$(i=1,...,N with N the number of securities to be examined) regressed against variables of $x t ∗ − μ f$ and $r i t − 1 − μ f$.

Estimation of regression equation using the method of least squares, with SPSS Statistics 17.0, and the results as given in Table 1. The results include the estimated the CAPM regression equation with Nerlove transformation, a probability value (PV) from statistic-t for each parameter estimator (written below), the statistic of determination coefficient R2, the statistic of distribution F, and probability value (PV) of the statistic of distribution F. Third, each estimator of the CAPM regression equation with Nerlove transformation, tests of verification and validation. Verification tests carried out on the constant parameters, the coefficient on the variable parameters of the market risk premium $x t ∗ − μ f$, and the coefficient of market risk premium variable $x t − 1 − μ f$, for each of the regression equation. When the specified level of significance α = 0.05, it seems clear that PV each parameter in each regression equation is smaller than the significance level α = 0.05. It suggests that each parameter in each regression equation is significant.

While the validation test performed using the statistical distribution of F, and the probability value (PV) of statistical distribution F. Similarly, in this test, the results show that PV of the statistic of distribution F value for each regression equation is also smaller than the level of significance α = 0.05. It indicates that each regression equation in Table 1 is significant. So that each regression equation can be used to estimate the values of mean and variance of each securities returns which are analyzed.

Furthermore, based on the regression equation in Table 1, are used to estimate the values of mean and variance for each of the securities were analyzed. Estimate the mean value μi performed recursively by equation (18), and the estimated value of the variance $σ i 2$ performed using equation (19). The estimation results mean and variance values for each return are given in Table 2.

Here, by using the mean and variance estimator values of each of the securities, also estimated Value-at- Risk (VaR) securities are concerned, and the results are also given in Table 2. Estimator values of the mean and variance of each return these securities will be used for investment portfolio optimization process.

3.4 Mean-VaR Portfolio Optimization Process

In this section, we analysis of Mean-VaR investment portfolio optimization process, to estimate the weight of an optimal portfolio. This portfolio optimization process is done through the following steps. First, the mean value estimator μi (i=1,...,5) are given in Table 2, is used to establish the mean vector, i.e., μ' = (0.000155 0.003313 0.000263 0.000953 0.003742). Since the number of securities to be analyzed consists of five pieces, the vector unit formed as e' =(1 1 1 1 1). Secondly, using the estimator of variance values $σ i 2$ (i=1,...,5 ) in Table 2, and the value of the covariance estimator of inter securities return, formed covariance matrix Σ , then is determined inverse matrix Σ−1, as follows:

$Σ = 10 − 3 ( 0.5420 0.0360 0.0710 0.0130 0.0210 0.0360 0.8560 0.0450 0.0290 0.0170 0.0710 0.0450 0.6950 0.0370 0.0620 0.0130 0.0290 0.0370 0.3860 0.0540 0.0210 0.0170 0.0620 0.0540 0.8930 )$
(32)

and

(33)

Third, the process of portfolio weight optimization is done referring to equation (9) and (10), in which the values of risk tolerance factor τ determined the simulation begins τ = 0.00 with an increase of 0.01. But in Table 3 were shown to increase 0.2, and only partially to the increase in 0.01. The process of portfolio weight optimization was done performed by using software Mat lab 7.0. In the process of this portfolio optimization in addition to determining the optimum weight composition, also estimated the mean value of the portfolio μp using equation (20), and the estimated Value-at-Risk of portfolio VaRp using equation (22), and the ratio μp/VaRp as illustrated in Table 3.

Paying attention to the portfolio optimization process results in Table 3, it appears that any change in the value of tolerance risk factor for investors will result in a change in the composition of investment portfolios optimum weight. If it is assumed that short sales are not allowed, then the simulation value is the highest risk tolerance factor τ = 4.07. Because for τ > 4.07, in the composition of portfolio weights their negative value. It is contrary to the assumption that short sales are not allowed. Each change the value of risk tolerance factor which lies in the interval 0.00≤τ≤4.07 produces estimator of the portfolio mean μp and the risk of the portfolio as VaRp also different. The plot of a pair of points (μp, VaRp) forming a curve of the efficient portfolio (efficient surface), as given in Figure 1. While Figure 2, is a curve of the ratio of mean with Value-at-Risk portfolio return.

In Figure 1. it appears that efficient surface curve of the portfolio return has increased in the interval 0.00≤τ≤4.07E, and does not degrade over start value of risk tolerance factors τ = 0.00 until τ = 4.07. This indicates that the minimum portfolio is achieved when the value of the risk tolerance factor τ = 0.00, with composition of investment portfolio weight as w' = (0.2186 0.1616 0.1415 0.3282 0.1500). The weight composition produces the expected return of the minimum portfolio is μp = 0.00150, and portfolio risk as Value-at- Risk VaRp = 0.01880. Meanwhile, the maximum portfolio achieved when the value of risk tolerance factor τ = 4.07, with a weight composition of the portfolio as w' = (0.0319 0.03662 0.0008 0.2115 0.3896). Namely to produce expected return of the maximum portfolio is μp = 0.00290, and the risk of maximum portfolio return VaRp = 0.02500. Furthermore, if the maximum portfolio is also the optimum global portfolio, it can be discussed as follows.

Logically, the optimum global portfolio occurs when the mean of portfolio return μp achieve a great value, with the value of the degree of risk VaRp relatively small, so that the ratio of portfolio return means μp and the level of risk VaRp the highest value. Therefore, to determine the optimum global portfolio is necessary to determine the values of the ratio μp / VaRp , and the results can be seen in Table 3. Paying attention to the values of the ratio μp / VaRp are given in Table 3, it appears that the greatest value of the ratio occurs when the value of risk tolerance factor τ = 4.07, with weight composition of the portfolio as w' = (0.0319 0.03662 0.0008 0.2115 0.3896), and produce expected return of the portfolio μp = 0.00290, as well as portfolio risk VaRp = 0.02500. These results also occur at the maximum portfolio, thus, in this case, the maximum portfolio is also the optimum global portfolio.

Therefore, for investors who form the investment portfolio with a combination of securities that are analyzed here, to acquire an optimum portfolio, then the composition of the weight allocation for each security is 3.19% on ASII, 36.62% BBRI, 0.08% HMSP, 21.15% TLKM, and 38.96% BBCA. The composition of the allocation of these funds is expected to produce the mean portfolio return for μp = 0.00290, the level of risk as measured by Value-at-Risk for VaRp = 0.02500.

4. CONCLUSION

This paper has been discussing the Mean-VaR investment portfolio optimization problems under CAPM with Nerlove transformation by using time series approach. Based on the discussion, it can be concluded that: first, the market index return significantly following the model AR (1)-GARCH (1.1); second, the return of five securities are analyzed each significantly following the CAPM regression with Nerlove transformation. CAPM regression estimator with Nerlove transformation such, are useful for determining the values of mean and variance estimator of each security, which further along estimator values of covariance between securities return, used for investment portfolio optimization process. From the results of the investment portfolio optimization process, obtained that the maximum portfolio is also an optimum global portfolio is achieved when the value of risk tolerance factor τ = 4.07, with a weight composition of the portfolio as w' = (0.0319 0.03662 0.0008 0.2115 0.3896). In the weight, composition produces the expected return of the portfolio μp = 0.00290, and portfolio risk VaRp = 0.02500. Efficient portfolio curve can be formed of pair of points ( μp , VaRp ), and this curve can be used as guidelines by investors in investing, especially in a portfolio consisting of five securities mentioned above were analyzed.

ACKNOWLEDGMENTS

Acknowledgments conveyed to the Rector, Director of Directorate of Research, Community Involvement and Innovation, and the Dean of Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, with whom the Internal Grant Program of Universitas Padjadjaran was made possible to fund this research. The grant is a means of enhancing research and publication activities for researchers at Universitas Padjadjaran, and we thank to Universiti Malaysia Terengganu that has suppoted this publication

Figure

Efficient frontier of portfolio return.

The ratio of mean and var portfolio return.

Table

Estimators of CAPM regression equation with Nerlove transformation

Estimators of μi, σi2, VaRi and Securities Return

Mean-VaR portfolio optimization process

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