A Nonlinear Interval Portfolio Selection Model and Its Ap

J Syst Sci Complex

A Nonlinear Interval Portfolio Selection Model and Its Application in Banks∗ YAN Dawen · HU Yaxing · LAI Kinkeung

DOI: 10.1007/s11424-017-6070-3 Received: 6 April 2016 / Revised: 3 January 2017 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2017 Abstract In classical Markowitz’s Mean-Variance model, parameters such as the mean and covariance of the underlying assets’ future return are assumed to be known exactly. However, this is not always the case. The parameters often correspond to quantities that fall within a range, or can be known ambiguously at the time when investment decision must be made. In such situations, investors determine returns on investment and risks etc. and make portfolio decisions based on experience and economic wisdom. This paper tries to use the concept of interval numbers in the fuzzy set theory to extend the classical mean-variance portfolio selection model to a mean-downside semi-variance model with consideration of liquidity requirements of a bank. The semi-variance constraint is employed to control the downside risk, filling in the existing interval portfolio optimization model based on the linear semi-absolute deviation to depict the downside risk. Simulation results show that the model behaves robustly for risky assets with highest or lowest mean historical rate of return and the optimal investment proportions have good stability. This suggests that for these kinds of assets the model can reduce the risk of high deviation caused by the deviation in the decision maker’s experience and economic wisdom. Keywords

Downside-risk management, interval return, portfolio selection, semi-variance, simulation.

YAN Dawen Faculty of Management and Economics, Dalian University of Technology, Dalian 116024, China; School of Mathematical Science, Dalian University of Technology, Dalian 116024, China; Bank of Dalian, Dalian 116001, China. Email: [email protected]. HU Yaxing School of Mathematical Science, Dalian University of Technology, Dalian 116024, China. Lai Kinkeung (Corresponding author) International Business School, Shaanxi Normal University, Xi’an 710062, China; College of Management, Xidian University, Xi’an 710126, China. Email: [email protected]. ∗ This research was supported by the National Natural Science Foundation of China under Grant Nos. 71301017,71731003, 71671023, 11301050 and 51375067, the National Social Science Foundation of China under Grant No. 16BTJ017, China Postdoctoral Science Foundation Funded Project under Grant No. 2016M600207 and the Doctoral Fund of Liaoning Province under Grant No. 20131017. This paper was recommended for publication by Editor WANG Shouyang.

YAN DAWEN · HU YAXING · LAI KIN KEUNG

2

1

Introduction

A new world of modern asset portfolio optimization research has unfolded since the publication of Markowitz’s landmark paper ’portfolio selection’[1] . Over the past 60 years, this area has received growing attention from both researchers and investors[2] . Asset portfolio optimization is a process in which portfolio yields are maximized in the context of established risks, or the risks are minimized to earn certain yields by asset allocation, or the expected utility of asset allocation weights is maximized, i.e., the linear combination of portfolio yield and negative variance are maximized[2] . Irrespective of which of the above model is used, without doubt, in the process of model constructing and solving, it is inevitable to deal with the parameter problem, namely, depicting the expected value of the future yield and risk of the portfolio. Therefore, according to the different presentations of these uncertain parameters, the portfolio selection models are roughly classified into the following three categories. The first category is the static portfolio optimization with deterministic parameters, represented by the Markowitz mean-variance model. Its extensions include the mean-absolute deviation model[3, 4] , semi-absolute deviation model[5, 6] , minimax model[7, 8] and multi-factor portfolio optimization model[9] . The second is the dynamic portfolio choice problem in discrete or continuous time with stochastic income[10–12] . The third is fuzzy portfolio selection problem, taking the vagueness besides the stochastic factors of the returns into consideration[13–15] . Huang, et al.[16–18] proposed a definition of the fuzzy random variable of assets’ future returns and broadened the assets allocation method by considering the double uncertainties of randomness and fuzziness. When using the aforementioned portfolio selection models, we usually assume that the joint distributional function or the membership function of assets returns is exactly known. However, in many real-world problems, it is difficult to specify reliably the complete distribution of uncertain parameters[2] . It may also be the case that quality of the uncertain information is not good enough to be expressed as probability distributions or fuzzy membership[19] . Thus, the above portfolio selection approaches are always accompanied with another kind of modeling risk due to the uncertainty in determining the distribution of asset returns[20] . And solution of many investment decision-making problems can be misleading. It may be why some researchers point out that the ancient 1/N rule outperforms Markowitz principle sometimes, even if many sophisticated optimization models can be applied to the portfolio selection problem[21, 22] . Consequently, another formulation for portfolio selection, called interval portfolio selection model, is put forward to cope with the parameter uncertainty. This approach, closely in accordance with people’s intuition, is derived from the fuzzy set theory. It only needs decision maker to provide a proper range of the uncertainty parameter such as a single asset’s future return based on experience and economic wisdom, not necessarily the complete distribution or membership function that are difficult to obtain. Interval programming can not only facilitate the communication of experience and the wisdom accumulated by decision maker in the long term of practice into the optimization process but also compensate for shortage of distributional infor-

INTERVAL PORTFOLIO SELECTION MODEL AND APPLICATION

3

mation. Therefore, this kind of models has been widely applied to transportation problem[23] , investment decision making[24] , resources allocation[19, 25] and so forth. For interval portfolio selection problem, based on the distinct risk measure, the existing relevant researches can be roughly classified into two streams. The first stream uses the risk measure of the portfolio variance to formulate the nonlinear interval portfolio selection model. Zhang, et al.[26–28] employed interval numbers to depict the assets’ future rates of return and extended the classical expected utility model to an interval portfolio optimization. Ida[29] proposed a multi-objective interval portfolio selection model in the Markowitz mean-variance framework. Bhattacharyya, et al.[30] added the factor of skewness and presented a mean-variance-skewness portfolio selection problem with interval coefficients. The second stream believes that the downside risk can reflect the real loss of the portfolio and uses semi-absolute deviation as a reliable risk measure to formulate the linear interval portfolio optimization models. Lai, et al.[31, 32] incorporated the interval returns in the semi-absolute deviation risk measure and proposed an interval investment strategy. Downside risk is defined as the difference between risky asset’s expected rate of return and its possible returns that are less than the expected value. Compared with variance, it only focuses on the rate of return to the extent it is lower than expectation, which is more in line with practical considerations of investors. Therefore, the downside risk research has drawn an increasing interest from scholars[20, 33–35] . Unfortunately, existing portfolio optimization models under downside risk measure are either formulated with complete distribution and deterministic coefficients or are mainly dependent on the linear risk measure of mean semi-absolute deviation. To the best of our knowledge, none has considered the interval portfolio selection model with the downside semi-variance. And none has applied such kind of model to a bank’s assets allocation problem and analyzed the performance of the model. In this paper, we will address these problems step by step. Firstly, we build and solve a mean-downside-semi-variance portfolio optimization model with interval coefficients, called IMDSV (interval mean downside semi-variance) optimization model. We use interval numbers, which not only reflect the decision maker’s experience and economic wisdom but also the real changes in the future rate of return, to derive future earnings and risk characteristics of the loans portfolio. Moreover, from theoretical analysis and numerical simulation we present the advantages of the use of portfolio selection method proposed by this study in comparison with existing relevant methods including classical mean-variance portfolio model, downside semi-variance portfolio model with pre-specified parameters and an interval linear portfolio model under semi-absolute deviation risk measure. Furthermore, we apply IMDSV model to a hypothetical commercial bank’s assets allocation problem and test the sensitivity of the model’s parameters, including the rate of return interval of the single asset, investor’s optimism factor and risk tolerance level, which provide management insights for a bank’s manager under different economic conditions. The results show that decision maker’s attitude and expectations of future economic conditions and industrial development influence their investment decisions. When investors pre-judge that the economy may go into a recession, the portfolio risk corresponding to one unit yield


4

generated by our model is generally lower than the existing numerical model. This means assets allocation based on our model is safer for the bank, which is vital for a bank’s stable operations, especially during economic downturns. Compared to the classical model, our model is able to obtain a higher loan portfolio yield and guarantees profitability of the bank during economic upturns, if the bank appropriately increases the tolerance risk level. Conversely, the existing numerical model is insensitive to change in real economic conditions, which may expose the bank to a greater investment risk during economic downturns and lead to failure to capture available opportunities for higher earnings during economic upturns. On the other hand, under different economic conditions, the response to change in the expected interval of rate of return is different for different kinds of loans. For risky assets with highest or lowest mean historical rate of return, the optimal investment proportion has good stability. On the contrary, for the remainder whose historical rates of return are in the middle of all loans, the optimal results appear highly sensitive to change of return interval. These results suggest that for loans whose historic yields are at the highest or lowest level of all loans, the model can reduce the risk of high deviation caused by variations in decision maker’s experience and economic wisdom. However, for those in the middle, managers are required to judge each future return more carefully so as to avoid high deviation of decision, thus providing more evidence for the asset allocation problem. The remainder of the paper is organized as follows. Section 2 presents the basic arithmetic and order of relationships for interval numbers. In Section 3, the mean semi-variance portfolio selection model with interval coefficients is constructed, for controlling the downside risk. Section 4 transforms IMDSV optimal model into an indefinite optimization problem with quadratic constraints with deterministic coefficients and applies the modified Augmented Lagrangian algorithm to obtain an ε-global optimal solution for the model. Section 5 presents an application of the model in the assets allocation problem of a commercial bank and provides numerical simulation, focusing on influence of changes of single asset’s anticipated rate of return interval on the optimal investment proportions of the IMSVO model. Section 6 concludes.

2

Interval Arithmetic

Suppose interval number A = [a− , a+ ], B = [b− , b+ ]. Both the lower and the upper limit of these two intervals are real numbers and a− ≤ a+ , b− ≤ b+ should be satisfied. k and c are constants, then we have[36, 37] : A(+)B = [a− + b− , a+ + b+ ], −

+

+

−

(1)

A(−)B = [a − b , a − b ],

(2)

A ± c = [a− ± c, a+ ± c],

(3)

−

+

+

−

kA = [ka , ka ] (k > 0),

(4)

kA = [ka , ka ] (k ≤ 0),

(5)

A(×)B = [min{a− b− , a+ b− , a− b+ , a+ b+ }, max{a− b− , a+ b− , a− b+ , a+ b+ }],

(6)


5

| A |= [max{0, a− } − min{0, a+ }, max{a+ , −a− }].

(7)

Also, the interval numbers can be described as A = m(A), w(A) or A = m(A), a+ [36] , where m(A) = (a− + a+ )/2;

w(A) = (a+ − a− )/2.

(8)

The order relation of the interval number has been discussed extensively[36, 37] . In this article, we introduce the order relation as[36] : A ≤ B ⇔ a+ ≤ b + ,

m(A) ≤ m(B).

(9)

In formulation of the optimization model, definition of order relation in (9) is applied to construction of the downside risk constraint.

3

Portfolio Selection Models

In this section, we review the two classical portfolio selection models, identified by Model (11) and Model (16), and deduce the interval mean semi-variance portfolio optimization model, named Model (18). 3.1

Involving Notations

Considering a financial market with both risky and risk-free assets, some necessary notations are listed as follows. N

The number of risky assets;

H

The number of risk-less assets;

ξ

The (N +H )-dimensional vector of assets’ rate of return, which includes two parts of N risky assets’ rate of return, random variable, H risk-less assets’ rate of return, and the deterministic constant;

xi

The proportion of total investment in i-th asset which is a decision variable, i = 1, 2, · · · , N + H;

rit

The historic rate of return of i-th asset at year t, t = 1, 2, · · · , T ;

ri

The mean historical rate of return for i-th asset;

σij

The consistent estimation of the covariance between ξi and ξj , i.e., cov(ξi , ξi ); i, j = 1, 2, · · · , N and (σij )N ×N is semi-positive;

− σij

The sample estimation of the downside semi-covariance between ξi and ξj , − i, j = 1, 2, · · · , N and (σij )N ×N is semi-positive;

rAi

The estimated interval of the rate of return for the i-th risky asset,a preset interval given by bank managers and characterized by rAi = [ril , riu ];


6 Iσij

The estimated interval of the downside semi-covariance between ξi and ξj , − − i, j = 1, 2, · · · , N , characterized by Iσij = [σijl , σiju ].

3.2

Mean-Variance Portfolio Model

One form of the classical Markowitz model is presented through maximizing the expected rate of return of portfolio under the tolerance risk level VG that is measured by the variance of the portfolio. It can be described as follows: max

N +H

xi Eξi

i=1

s.t. E

N +H

xi ξi − E

i=1 N +H

N +H

2 xi ξi

i=1

xi = 1,

N

=

xi xj E(ξi − Eξi )(ξj − Eξj ) ≤ VG ,

(10)

i,j=1

xi ≥ 0, i = 1, 2, · · · , N + H.

i=1

Since the rate of return of riskless assets is constant, H items of the riskless assets drop out of the variance constraint. Basically, the precise distribution of risky assets’ future return is not known at the time when investors must make the investment decision. Data of historical return of the assets are readily available and can be used to estimate the expected rate of return and the variance of risky assets. Thus, Model (10) is transformed into the following programming problem: max

N +H

xi ri

i=1

s.t.

N

xi xj σij ≤ VG ,

(11)

i,j=1 N +H

xi = 1,

xi ≥ 0, i = 1, 2, · · · , N + H.

i=1

where σij =

T 1 (rit − ri )(rjt − rj ), T t=1

(12)

and (σij )N ×N is the sample covariance matrix, and a semi-positive matrix. Thus, Model (11) is a convex problem and has a global optimal solution. 3.3

Mean Downside Semi-Variance Portfolio Model

Replace the portfolio’s variance in the classical Markowitz mean-variance model (10) with its semi-variance and then we have the mean semi-variance portfolio selection model with downside


7

risk control: max

N +H

xi Eξi

i=1

s.t.

E

N +H

xi ξi − E

N +H

i=1 N +H

− 2 xi ξi ) ≤ VG ,

(13)

i=1

xi = 1,

xi ≥ 0,

i = 1, 2, · · · , N + H,

i=1

where (·)− is a convex function with the following form: ⎧ ⎨ 0, a ≥ 0, (a)− = max{−a, 0} = ⎩ −a, a < 0.

(14)

According to the general properties of a convex function, the left hand side of the downside semi-variance inequality constraint in Model (13) satisfies: E

N +H i=1

xi ξi − E

N +H

− 2 xi ξi

=E

i=1

N

− 2 xi (ξi − Eξi )

i=1

≤E

N

−

2

xi (ξi − Eξi )

i=1

=

N

xi xj E(ξi − Eξi )− (ξj − Eξj )− ,

(15)

i,j=1

where E(ξi − Eξi )− (ξj − Eξj )− is called downside semi-covariance between the i-th and j-th risky assets. As we said before, it is quite difficult to determine distributions of future return of both single risky asset and a portfolio of assets in advance. However, even though the accurate and complete distribution of the future’s assets’ returns is known, solving Model (13) is a quite difficult problem[38] . It is not easy either to solve the model (13) with sampling or historical simulation methods, because of the difficulty to determine algebraic equation of semi-variance for the portfolio of assets. More specifically, it is difficult to directly identify coefficients of multiplication of any two unknown decision variables, xi xj (i, j = 1, 2, · · · , N + H) in the semivariance inequality constraint of Model (13) due to the characteristic of the function (·)− (see definition of the function in (14)), although semi-variance for the portfolio of assets should be the quadratic function of decision variables. References [35, 39] also remind the difficulty in determining the algebraic equation of semi-variance for a portfolio of stock. References [35, 39] computed the coefficient of xi xj as the difference between the covariance of i-th asset and j-th asset returns and the multiplication of Sharpe’s betas for assets i, j and the semi-variance (above the mean) for market portfolio. Finally, algebraic equation of semi-variance for the portfolio of stocks can be obtained by this way.


8

However, the method using beta coefficients is not suitable for the case of making investment decision for a bank. The opportunity set of assets may not be all stocks but include probably all kinds of assets, such as the loans from non-listed companies. This kind of loan’s return from non-listed companies would totally not be correlated with stock market index. Consequently, these loans’ beta coefficients should be equal to zero. If i-th asset’s Sharpe’s beta equals to zero, the coefficient of xi xj in the semi-variance for the portfolio of assets becomes the covariance of the i-th asset’s and the j-th asset’s returns for any j = 1, 2, · · · , N + H, no longer being the semi-covariance according to the method proposed by References [35, 39]. The deviance of the coefficients of decision variables in the portfolio selection model can not only lead to the erroneous downside risk measure for a portfolio of assets, but also the erroneous portfolio decision[2, 40] . For addressing the difficult tractability of Model (13), we can replace the left hand side of the quadratic inequality constraint in Model (13) with the last expression in (15). Furthermore, taking the sample estimator of E(ξi −Eξi )− (ξj −Eξj )− , downside semi-variance portfolio model can be changed into the form below: max

N +H

xi ri

i=1

s.t.

N i,j=1 N +H

− xi xj σij ≤ VG ,

xi = 1,

xi ≥ 0,

(16) i = 1, 2, · · · , N + H,

i=1

where − σij =

T T 1 1 (rit − ri )− × (rjt − rj )− = (max{ri − rit , 0} × max{rj − rjt , 0}), T t=1 T t=1

(17)

− and it is the consistent estimator of E(ξi −Eξi )− (ξj −Eξj )− with a nonnegative value. (σij )N ×N is a semi-positive matrix, thus, Model (16) is still a convex programming problem. In sum, we obtain an algebraic equation of semi-variance for the portfolio of assets, as shown by the last expression in (15), based only on the knowledge of convex analysis. The problem (13) is transformed into an analytical model (16) like Markowitz mean-variance model, which mitigates computational complexity of solution to Problem (13). Existing studies that use beta coefficient method to compute semi-variance matrix of assets’ return and construct an analytical model are more applicable to the portfolio selection problem of stock, while ours is applicable to the more general portfolio selection problem without any limitation on portfolio composition. At the same time, it is guaranteed that the solution of Model (16) is also a solution of original problem (13). This means the optimal solution of Model (16) can be regarded as reliable investment decision policy under downside risk measure. Therefore, all of these can enhance the extension and applicability of portfolio selection model based on downside risk measure of the portfolio semi-variance.


3.4

9

Mean Downside Semi-Variance Model with Interval Coefficients

Based on the mean downside semi-variance model (16), we propose a mean semi-variance portfolio selection model with interval coefficients. It is presented as follows: max

N +H

xi [ril , riu ]

i=1

s.t.

N i,j=1 N +H

− − xi xj [σijl , σiju ] ≤ [VGl , VGu ],

xi = 1,

xi ≥ 0,

(18)

i = 1, 2, · · · , N + H,

i=1

where, the rates of return of risk-less assets are deterministic constants, i.e., ril = riu = ri = rit , i = N, N + 1, · · · , N + H, t = 1, 2, · · · , T . [VGl , VGu ] is a risk tolerance interval. The detailed explanations about [VGl , VGu ] and downside semi-variance interval constraint in Model (18) are listed in the end of this subsection and in the end of (ii) of Subsection 3.6 respectively. Besides, we add some reasonable suggestions for setting interval [VGl , VGu ] in the investment decision making practice and the relevant references on risk tolerance measure in the end of the subsection, too. − − The calculation of Iσij = [σijl , σiju ] in Problem (18) is the key point of the construction of this model and can be implemented through the following procedures. Let’s go back to (17). We replace ri , the sample estimation of i-th asset’s expected return in (17), with estimated interval rAi = [ril , riu ] given by investors based on their experience and economic wisdom, and then we have − Iσij =

T 1 (rit − [ril , riu ])− (×)(rjt − [rjl , rju ])− T t=1

T 1 = [rit − riu , rit − ril ]− (×)[rjt − rju , rjt − rjl ]− T t=1

=

T 1 [(rit − ril )− , (rit − riu )− ](×)[(rjt − rjl )− , (rjt − rju )− ] T t=1

T 1 [(rit − ril )− × (rjt − rjl )− , (rit − riu )− × (rjt − rju )− ] T t=1 T T 1 − − 1 − − = (rit − ril ) × (rjt − rjl ) , (rit − riu ) × (rjt − rju ) . T t=1 T t=1

=

(19)

This can be deduced by the interval arithmetic as well as the monotonically decreasing property of the function of (14). − − Adopt the symbol of σijl and σiju to denote the two limits of the semi-covariance interval,


10

then, the lower and the upper limits are: − σijl

T T 1 1 − − = (rit − ril ) × (rjt − rjl ) = max{ril − rit , 0} × max{rjl − rjt , 0}, T t=1 T t=1

− σiju =

(20)

T T 1 1 (rit − riu )− × (rjt − rju )− = max{riu − rit , 0} × max{rju − rjt , 0}. (21) T t=1 T t=1

− − (σiju ) corresponds As (20)–(21) show, the lower (upper) limit of portfolio downside risk σijl to the lower (upper) limit of anticipated rate of return interval of single asset ril (riu ). There is the same correspondence between the portfolio return and risk interval, which can be easily deduced from the expression of portfolio’s rate of return interval shown by the objective function in Model (18). The intervals of portfolio return and risk in Model (18) depict an appealing characteristic that portfolio return is positively proportional to risk, which follows the general investment rule. According to the previous analysis in this subsection, the portfolio downside semi-variance in the constraint of Model (18) reaches the lower (upper) limit, when every single asset’s expected return reaches its lower (upper) limit. Therefore, VGl represents the tolerated risk level when the expected returns are predicted most pessimistically, while VGu represents the tolerated risk level when the expected returns are predicted most optimistically. If an investor tends to invest in risky assets with higher return, he/she has to tolerate the higher risk. The explanation about [VGl , VGu ] in this paper is analogous to the explanation about the non-negative tolerated risk level interval in [31] that is used as the limit of semi-absolute deviation interval of the portfolio. To some extent, [VGl , VGu ] can be obtained by calculating the square of the tolerated risk level interval in [31]. The definition of multiplication operation on interval numbers can be found in (6) of our paper. The risk tolerance level can be set according to willingness of bank’s shareholders to accept risk and bank’s ability to bear risk in the investment decision making practice. A bank manager or financial planner may determine a percentage interval corresponding to the range of acceptable revenue loss of portfolio comprising of candidate assets based on the bank’s financial condition and economic condition. The acceptable maximum (minimum) ratio of revenue loss to the bank’s total investment (or total asset) is regarded as the interval’s upper (lower) limit. Furthermore, let VGu , VGl be the square of the maximum and the minimum ratio correspondingly. Risk tolerance level is an important factor because it influences the investment decision making, which can be reflected by numerical results of this study. Here we just provide an idea of construction of risk tolerance level for decision-maker in practice. Other researchers have supplied numerous references about risk tolerance measure. For existing studies related to the construction of risk tolerance level, we refer the readers to [41–43].

3.5

Comparative Analysis Between the Exiting Downside Semi-Variance Portfolio Model and Interval Downside Semi-Variance Portfolio Model

In fact, existing Model (16) is only a special case of Model (18) proposed in this paper when the expected rate of return interval of risky asset reduces to a single value, i.e., ril =


11

riu = ri . Once this occurs, the portfolio’s rates of return and downside semi-variance intervals in Model (18) turn into the forms of their respective counterparts in Model (16). In addition, a more detailed description of difference and correlation between portfolio return and risk in Model (16) and Model (18) respectively is illustrated, under the following three conditions: 1) All rate of return intervals of risky assets are located on the left of the assets’ mean historical rates of return chosen as reference points, i.e., riu < ri , for all i. This condition is described by part (a) of Figure 1 and corresponds to economic downturns, in which expected returns of all kinds of assets tend to decrease. (a) ril

(b) riu

Figure 1

ri

ril

(c)

riu

ri

ri

ril

riu

Location relationship between the mean historical rate of return and expectation interval given by the investors

For the first case of anticipation of economic downturn, since riu < ri , rju < rj for all i, j and the function of max{x − rit , 0} increases with x, we have max{riu − rit , 0} × max{rju − rjt , 0} ≤ max{ri − rit , 0} × max{rj − rjt , 0}, − σiju =

1 T

T

(22)

max{riu − rit , 0} × max{rju − rjt , 0}

t=1

T 1 − ≤ max{ri − rit , 0} × max{rj − rjt , 0} = σij . T t=1

(23)

Further, since investment proportion allocated to any single asset xi (i = 1, 2, · · · , N ) is nonnegative, we have N +H i=1 N i,j=1

xi riu ≤

N +H

xi ri ,

(24)

i=1 − xi xj σiju ≤

N i,j=1

− xi xj σij ,

(25)

where the left hand sides of (24) and (25) are upper limits of the overall portfolio rate of return and its downside semi-variance estimated intervals in Model (18), respectively; the right hand side of (24) and (25) are the ones in Model (16). These results tell us that when investors anticipate an economic downturn, a more conservative anticipated return on the assets can be derived. While the problem (18) can provide the relatively smaller estimator of the portfolio downside risk, on the contrary, in this case, Model (16) overestimates the overall portfolio risk which results in deviation of decision and the loss of profit supposed to be obtained by the investor.

12


2) All rate of return intervals of risky assets contain the reference points, assets’ mean historical rates of return, i.e., ri ∈ rAi = [ril , riu ] for all i. This condition is described by Part (b) of Figure 1 and corresponds to a smoothly running economy in which expected return intervals of all kinds of assets contain their mean historical return. For the second case of expectation of a smooth economy, since ril < ri < riu , rjl < rj < rju for all i, j and the function of max{x − rit , 0} × max{y − rjt , 0} increases with x, y, we have max{ril − rit , 0} × max{rjl − rjt , 0} ≤ max{ri − rit , 0} × max{rj − rjt , 0} ≤ max{riu − rit , 0} × max{rju − rjt , 0},

(26)

and − σijl =

≤

T 1 max{ril − rit , 0} × max{rjl − rjt , 0} T t=1 T 1 max{ri − rit , 0} × max{rj − rjt , 0} T t=1

− = σij

≤

T 1 − max{riu − rit , 0} × max{rju − rjt , 0} = σiju . T t=1

(27)

Further, for the same reason as Case 1), the proportion invested in any single asset xi (i = 1, 2, · · · , N ) is non-negative, and we have N +H i=1 N i,j=1

xi ril ≤

N +H

xi ri ≤

i=1 − xi xj σijl ≤

N i,j=1

N +H

xi riu ,

(28)

i=1 − xi xj σij ≤

N i,j=1

− xi xj σiju ,

(29)

where the left (right) hand side of (28) and (29) are lower (upper) limits of the overall portfolio rate of return and downside semi-variance estimated interval in Model (18), respectively; the middle of (28) and (29) are the ones in Model (16). In this case, with any single asset’s expected rate of return increasing, the portfolio’s rate of return and risk interval in Model (18) increase. The location relationship between the overall portfolio rate of return, downside semi-variance estimated interval in Model (18) and the portfolio mean historical rate of return, downside semi-variance in Model (16) continues to have characteristics of Part (b) in Figure 2. In fact, the following third case has similar results, which can be easily seen. This reflects how the single asset’s return anticipated by the investor influences portfolio return and risk interval, in problem (18). 3) All rate of return intervals of risky assets are located on the right of the reference points, i.e., ril > ri , for all i. This condition is described by Part (c) of Figure 1 and corresponds to economic upturns when expect returns for all kinds of assets tend to increase.


13

For the last case of economic upturn anticipation, since ril > ri , rjl > rj for all i, j and the function of max{x − rit , 0} increases with x and then we have: max{ril − rit , 0} × max{rjl − rjt , 0} ≥ max{ri − rit , 0} × max{rj − rjt , 0}, − σijl =

≥

1 T

T

(30)

max{ril − rit , 0} × max{rjl − rjt , 0}

t=1

T 1 − max{ri − rit , 0} × max{rj − rjt , 0} = σij . T t=1

(31)

Further, similar to Case 1) and Case 2) above, we have N +H

xi ril ≥

N +H

i=1 N i,j=1

xi ri ,

(32)

i=1 − xi xj σijl

≥

N i,j=1

− xi xj σij .

(33)

For this situation, the analysis results suggest that when investors expect economic upturn that could result in a more radical anticipation of return, thus Model (18) can provide the relatively larger estimate of the portfolio downside risk; in contrast, Model (16) underestimates the overall portfolio risk which results in the inappropriate decision and puts the investor at greater risk. In short, Model (16) may not be able to represent the real change of the overall portfolio return and risk caused by variation in expectations of future economic conditions and thus may lead to inappropriate investment decisions. Conversely, in addition to application of information of assets’ historical rate of returns, Model (18) indirectly reflects the impacts of different economic conditions on investment decisions by modifying the anticipations about assets’ future return. Therefore, Model (18) can reflect the significance of the experience and wisdom of investors and has more flexibility than Model (16). 3.6

Transformation of the Model (i) Transformation of objective function in Model (18) The objective function in Model (18) can be described as follows: max

Z=

N i=1

xi ril +

N +H i=N +1

xi ri ,

N i=1

xi riu +

N +H

xi ri .

(34)

i=N +1

The objective function consists of two parts, the risky assets portfolio’s expected return interval and the risk-less assets portfolio’s yield. Its maximum reflects the bank’s shareholders’ need of maximizing the future return. Let μ denote an investor’s optimism factor on portfolio return and satisfy 0 ≤ μ ≤ 1. μ reflects the investors’ attitude about the economic conditions. The closer it is to 1, the more optimistic investors are about the future financial market[44] . Let Z = [Z − , Z + ] be the


14

portfolio rate of return interval, through linear average of the lower and upper boundary of portfolio’s return interval in (35) weighted by 1 − μ and μ, the interval-based objective function is transformed into the tractable linear objective function below[44] . max

Z = (1 − μ)Z − + μZ + = μ

N

xi riu + (1 − μ)

i=1

N

xi ril +

i=1

N +H

xi ri .

(35)

i=N +1

In fact, the introduction of this parameter of μ is not only for the purpose of transformation of the interval programming. According to [10], we know that the movements of a financial market on the whole can be viewed as a composition of a primary movement and secondary movement. And in the primary movement, market trends go up or down in a long period. In the secondary market movement, the market has an obviously opposite trend to the primary movement in short period. In this paper, the determination of the expected rate of return intervals [ril , riu ] represents the primary movement in the market trend or economic trend. The corresponding relation has been presented in the previous text. When the intervals of return are set, the secondary movement can be represented through adjusting the value of μ. Thus, with Reference [10] and these detailed explanations, the practical meanings of the parameter in this paper can be clearer for readers. (ii) Transformation of interval constraint in Model (18) Let VG = [VGl , VGu ] be the risk tolerance level interval, given by the decision makers based on investor’s risk tolerance. According to the definition of order relation in (9), the constraint of the portfolio’s downside risk in Model (18) can be transformed into the following (36)–(37). N i,j=1

− xi xj σiju ≤ VGu ,

(36)

N 1 1 − − xi xj (σijl + σiju ) ≤ (VGl + VGu ). 2 i,j=1 2

(37)

Using the inequality constraints (36) and (37), we translate the quadratic interval constraint in Model (18) into solvable forms. At the same time, by (36)–(37), the function of the downside semi-variance interval constraint in Model (18) can clearly been reflected. The downside semivariance interval constraint in Model (18) expresses that the average downside risk and the risk in the worst case scenario are less than or equal to the average value and the maximal possible value of the risk tolerance level, respectively. (iii) Constraints on liquidity risk control Some laws and regulations applicable to commercial banks should be considered, besides portfolio’s downside risk constraint in assets allocation. These laws and regulations are generally implemented to control a bank’s liquidity risk and guarantee its operations[45, 46] . Thus, a group of deterministic linear constraints can be set as follows: N +H i=1

asi xi ≤ (or =, ≥) cs ,

for

s = 1, 2, · · · , U ,

(38)


15

where asi and cs are the coefficients of i-th asset and the constant in the s-th constraint, respectively. According to the requirements of regulatory authorities such as the Basel Committee, all coefficients, including the number of constraints, U , can be determined in advance.

4

Modified Augmented-Lagrangian Solution Method

Based on the order relation between intervals (9) and the application of optimism factor, Model (18) is eventually transformed into an indefinite optimization problem with quadratic constraints shown as (36)–(37). The difficulty in this kind of non-convex problem is intractability of the global optimization solution. The existing solvers such as Matlab toolbox for optimization are generally based on the Interior-Point Method[47–49] , Trust-Region Method[50–52] and Line Search Filter Method[53, 54] , et al., which can guarantee only a local optimal solution for non-convex problems, i.e., they do not offer global optimal solution. Thus, if we use the existing solvers to solve the problem directly, the algorithm that has been implemented in the solvers may end up quickly in a local optimum, which results in lower maximum return. This damages the shareholders’ interests to some extent. In this paper, we employ a novel global optimization method proposed recently by Birgin, et al. in [55] to solve the non-linear and non-convex optimization problem with objective function (35) and constraints (36)–(38). This algorithm that combines Augmented Lagrangian method with αBB method[56] can guarantee that the difference between the objective function value corresponding to approximate global optimizer and real optimal value is less ε (a sufficiently small positive number). In this sense, with the algorithm we could obtain a reasonable approximate of the global optimizer that is called ε-global minimizer. Furthermore, a sufficient condition that guarantees that the ε-global minimizer of the Augmented Lagrangian algorithm can always be found is the compactness of the feasible set. In portfolio optimal selection problems, box constraints, e.g., 0 ≤ xi ≤ x0 , are usually added to the feasible set of the problem for diversification purposes. Here, x0 is an upper limit to be imposed on every portfolio weight. Clearly, after intersection with a box, the feasible set becomes compact. Thus, the method is very applicable to the nonlinear interval portfolio selection problem of this paper that is finally transformed into a non-convex minimization problem with quadratic constraints. By this method, this study can always provide a decision maker with an approximate global optimal investment policy. Moreover, professor Birgin in [55] listed the code related to Augmented Lagrangian in his homepage, which guarantees implementation of the algorithm in practical problems. The relevant notations and the specific steps of the algorithm are as follows. Let h(x) and g(x) denote the equality and inequality nonlinear but continuous constraints, respectively, consisting of hi (x) and gi (x). Besides these constraints, decision variable x satisfies x ∈ Ω, where Ω ⊂ Rn , a closed set and consists of linear and box constraints. λ and η are the vectors of Lagrange multipliers satisfying η > 0. ρ is the penalty parameter, a positive real number. k is the number of iterations, a positive integer. γ and τ are updating factors, satisfying γ > 0, 0 < τ < 1.


16

Then the Augmented Lagrangian methodology integrated with the deterministic global optimization method of αBB is briefly summarized as follows. Algorithm Let λmin < λmax , ηmax > 0, {εk } be a sequence of non-negative numbers subject to limk→∞ εk = ε ≥ 0. Let m and p denote the number of the nonlinear equality and inequality constraints, respectively, λ1i ∈ [λmin , λmax ], i = 1, 2, · · · , m, ηi1 ∈ [0, ηmax ], i = 1, 2, · · · , p, and ρ1 > 0. Initialize k ← 1. Step 1 Let Pk ⊂ Rn be a closed set such that a global minimizer z (the same for all k) belongs to Pk . Use the deterministic global optimization approach of αBB to find an εk -global minimizer xk of the problem Min Lρk (x, λk , η k ) subject to x ∈ Ω ∩ Pk . That means xk ∈ Ω ∩ Pk is such that: Lρk (xk , λk , η k ) ≤ Lρk (x, λk , η k ) + εk , for all x ∈ Ω ∩ Pk ,

(39)

where, the Lρk (x, λ, η) is the Augmented Lagrangian function. Step 2 Update the penalty parameter ρ. If k = 1 or satisfies the following inequality max{ h(xk ) ∞ , V k ∞ } ≤ τ max{ h(xk−1 ) ∞ , V k−1 ∞ } define ρk+1 = ρk . Otherwise, define ρk+1 = γρk , where,

ηik k k , i = 1, 2, · · · , p. Vi = max gi (x ), − ρk

(40)

(41)

Step 3 Update the Lagrangian multipliers by the following rules: λk+1 = min{max{λmin , λki + ρk hi (xk )}, λmax }, i ηik+1 = min{max{η k + ρk gi (xk )}, ηmax }.

(42)

Set k ← k + 1 and go to Step 1. Every limit of the sequence {xk } obtained by the above algorithm is an ε-global optimization solution of the primary problem[55] .

5

An Application to a Bank’s Asset Allocation

In this section, we apply the model to the assets allocation problem of a hypothetical commercial bank called ABC operating in China and show the process of the construction and solution of the model with quadratic interval constraint and analyze the sensitivity of the related parameters. 5.1 5.1.1

Data, Variables and General Description of Parameters Sensitivity Testing The Basic Information of the Bank’s Assets

Tables 1 and 2 list the information of new assets probably invested in by the bank ABC. The bank’s risk-free assets’ expected return is set to be a constant and equal to the corresponding

INTERVAL PORTFOLIO SELECTION MODEL AND APPLICATION Table 1 Information of the risk-free assets of the bank Risk-free assets A∗i

Interest rate ri (%)

1 Cash A∗1

0.00

2 Required reserve 3 Excess reserve

A∗2

1.89

A∗3

1.62

4 Due from banks in the system

A∗4

2.95

Table 2 Information of the new loans of the bank Risky assets Ai

Type of loan

1 AAA (8 years)

Project loan

2 AA (6 months)

Security dealer financing

3 A (5 years)

Term loans to support the purchase of equipment and structures

4 BBB (3 months)

Self liquidating loan stock

5 BB (1 year)

Retailer financing

6 BBB (2 years)

Real estate development loan

7 BB (6 months)

Loans to support acquisitions of other business firms

Table 3 Historical rate of return of loans rit (%) Year

A1

A2

A3

A4

A5

A6

A7

2013

9

10.5

8.2

8

7.4

5.4

4.8

2012

10.7

9.1

8.9

8.5

7.6

6.1

5.4

2011

8.8

8.2

7.6

7.3

7.2

7

-3.4

2010

8.4

8.6

7.4

7.4

6.7

6

-3.9

2009

8.6

9.3

7.8

6.7

7

6.2

4.1

2008

8.9

9.5

7.9

7

7.9

5.5

7.7

2007

8.3

9.1

8.1

7.1

6.9

4.8

6

2006

8.1

9

8.2

7.6

7.4

0.4

4.5

2005

7.5

8.9

7.4

6.8

6.8

-0.7

-4.7

2004

7.8

8.8

7.6

7.2

7

4.4

-3.5

2003

10.4

8.8

7.9

7.5

7.3

5.4

7

2002

8.1

9.2

8.1

6.9

6.8

5.8

-1.4

2001

8.5

8.7

8.2

7.3

7.1

6.6

4

2000

8.3

8.3

7

7.1

7.5

6.1

5

1999

8.4

8.1

7.7

7.4

7.2

5.1

4.7

17


18

interest rate ri (i = 1, 2, 3, 4) in Table 1, as there is nearly no volatility in its expected return. The bank’s risky assets are composed of different types of loans and their original credit ratings and maturities are listed in Table 2. The historical return data of each loan is available and the annual time series rit for the rate of return are given in Table 3 for loan i = 1, 2, · · · , 7, between 1999 and 2013, where t = 1, 2, · · · , T , and t = 1 corresponds to 2013, t = T = 15 to 1999. 5.1.2

Parameters Sensitivity Testing

As one of the main works in the following context, we use the simulation approach to reflect the impact of changes of loans’ expected return intervals on the optimal asset allocation. Differentiated by the relative locations of rAi and ri , sensitivity of parameter rAi in Model (18) is tested in the three cases. (i) rAi contains ri for all i, i.e., ril < ri < riu ; (ii)rAi is on the left of ri , i.e., riu < ri for all i; rAi is on the right of ri , i.e., ril > ri for all i. Each of the above cases is simulated 50, 100 and 200 times stochastically. During the testing process, the effects of other parameters in the asset selection model based on Model (18), including the investor’s optimism factor on portfolio return μ and the target interval of portfolio downside risk VG , are examined also. Therefore, calculation of most of variables below, construction and the solution of the model are implemented based on the three kinds of conditions stated above. 5.1.3

Deterministic Semi-Covariance Matrix of the Loans’ Rate of Return

According to the historical rate of return of each loan during the past 15 years (Table 3), the arithmetic mean and standard deviation of each loan’s rate of return ri (i = 1, 2, · · · , 7) are calculated and listed in the second column of Table 4. Taking the i-th (i = 1, 2, · · · , 7) loan’s historical rate of return data rit (t = 1, 2, · · · , 15) and the mean rate of returns ri in the second − column of Table 4 into (17), the semi-covariance between the i-th and j-th loan, σij , can be computed (Table 5). Table 4 Historic mean and expected intervals of each risky asset’s rate of return under three situations No.

1 Initial credit grade and maturity

2 Mean and standard deviation of historical rate of return, ri (%)

3 Expected return interval [ril , riu ] containing ri for all i(%)

4 Expected return interval [ril , riu ] on the left of ri for all i(%)

5 Expected return interval [ril , riu ] on the right of ri for all i(%)

1

AAA (8 years)

8.6533 (0.87)

[8.60, 8.69]

[6.93, 8.37]

[9.65, 9.75]

2

AA (6 months)

8.9400 (0.59)

[8.46, 9.78]

[7.57, 8.93]

[9.72, 10.05]

3

A (5 years)

7.8667 (0.45)

[7.54, 8.18]

[7.56, 7.71]

[8.87, 8.92]

4

BBB (3months)

7.3200 (0.46)

[6.28, 7.38]

[5.24, 6.70]

[8.32, 8.50]

5

BB (1 year)

7.1867 (0.34)

[6.19, 7.29]

[5.17, 6.50]

[9.01, 9.20]

6

BBB (2 years)

4.9400 (2.18)

[3.73, 6.07]

[2.74, 3.37]

[5.21, 5.30]

7

BB (6 months)

2.4200 (4.41)

[1.64, 2.48]

[0.19, 1.48]

[4.42, 5.31]

Notes: All expected rate of return intervals rAi are randomly obtained and the length is less than 2%.

19


Data in Table 5 are applied to construct the asset selection model based on Model (16) as the comparison of the model based on Model (18). − Table 5 Deterministic semi-covariance matrix of loans’ historic rate of return, σij (%)

A1

A2

A3

A4

A5

A6

A7

A1

0.00205

0.000485

0.000824

0.000751

0.000711

0.006351

0.011318

A2

0.000485

0.001251

0.000738

0.000132

0.000152

0.000201

0.005046

A3

0.000824

0.000738

0.000908

0.000341

0.000313

0.001851

0.006269

A4

0.000751

0.000132

0.000341

0.000697

0.000378

0.002019

0.004089

A5

0.000711

0.000152

0.000313

0.000378

0.000464

0.001548

0.005608

A6

0.006351

0.000201

0.001851

0.002019

0.001548

0.035155

0.028902

A7

0.011318

0.005046

0.006269

0.004089

0.005608

0.028902

0.116099

Table 6 Interval semi-variance matrix of the loans’ expected return in the situation of rAi containing ri for all i(%) A1

A2

A3

A4

A5

A6

A7

A1

[0.0017, 0.0023]

[0.0001, 0.0031]

[0.0002, 0.0016]

[0, 0.0010]

[0, 0.0010]

[0.0044, 0.0092]

[0.0091, 0.0120]

A2

[0.0001, 0.0031]

[0.0001, 0.0100]

[0.0001, 0.0042]

[0, 0.0015]

[0, 0.0017]

[0, 0.0104]

[0.0009, 0.0208]

A3

[0.0002, 0.0016]

[0.0001, 0.0042]

[0.0002, 0.0026]

[0, 0.0009]

[0, 0.0009]

[0.0004, 0.0048]

[0.0011, 0.0118]

A4

[0, 0.0010]

[0, 0.0015]

[0, 0.0009]

[0, 0.0009]

[0, 0.0006]

[0, 0.0033]

[0, 0.0050]

A5

[0, 0.0010]

[0, 0.0017]

[0, 0.0009]

[0, 0.0006]

[0, 0.0008]

[0, 0.0030]

[0, 0.0076]

A6

[0.0044, 0.0092]

[0, 0.0104]

[0.0004, 0.0048]

[0, 0.0033]

[0, 0.0030]

[0.0205, 0.0564]

[0.0187, 0.0410]

A7

[0.0091, 0.0120]

[0.0009, 0.0208]

[0.0011, 0.0118]

[0, 0.0050]

[0, 0.0076]

[0.0187, 0.0410]

[0.0880, 0.1184]

5.1.4

Semi-Covariance Matrix of the Loans’ Expected Rates of Return with Interval Element

Taking ri (i = 1, 2, · · · , 7) in Table 4 as the corresponding reference points, loans’ interval semi-covariance matrix can be calculated in three cases as mentioned in Subsection 5.1.2 above. Let’s take the case of rAi containing ri for all i, i = 1, 2, · · · , 7, as an example. Suppose the expected rates of return interval for each loan provided by the decision makers are as in Column 3 of Table 4. Applying historical return data rit (t = 1, 2, · · · , 15) in Table 3 and expected return interval rAi in Column 3 of Table 4 to (20)–(21), the interval semi-covariance


20

matrix under the situation of rAi containing ri for all i, i = 1, 2, · · · , 7, can be computed and shown as in Table 6. The interval semi-covariance matrixes under other the two situations can be obtained by the process similar to the expected rates of return interval in Columns 4 and 5 of Table 4. Results are listed in Tables 7 and 8. Table 7 Interval semi-variance matrix of the loans’ expected return in the situation of rAi falling on the left of ri for all i(%) A1

A2

A3

A4

A5

A6

A7

A1

[0, 0.0008]

[0, 0.0001]

[0, 0.0003]

[0, 0]

[0, 0]

[0, 0.0029]

[0, 0.0060]

A2

[0, 0.0001]

[0, 0.0012]

[0, 0.0004]

[0, 0]

[0, 0]

[0, 0.0001]

[0, 0.0041]

A3

[0, 0.0003]

[0, 0.0004]

[0.0002, 0.0005] [0, 0]

[0, 0]

[0.0004, 0.0008] [0.0010, 0.0031]

A4

[0, 0]

[0, 0]

[0, 0]

[0, 0]

[0, 0]

[0, 0]

[0, 0]

A5

[0, 0]

[0, 0]

[0, 0]

[0, 0]

[0, 0]

[0, 0]

[0, 0]

A6

[0, 0.0029]

[0, 0.0010]

[0, 0004, 0.0008] [0, 0]

[0, 0]

[0.0115, 0.0169] [0.0112, 0.0168]

A7

[0, 0.0060]

[0, 0.0041]

[0.0010, 0.0031] [0, 0]

[0, 0]

[0.0112, 0.0168] [0.0464, 0.0827]

Table 8 Interval semi-variance matrix of the loans’ expected return in the situation of rAi falling on the right of ri for all i(%) A1

A2

A3

A4

A5

A6

A7

A1

[0.0158, 0.0182]

[0.0098, 0.0143]

[0.0125, 0.0140]

[0.0128, 0.0159]

[0.0214, 0.0253]

[0.0149, 0.0162]

[0.0408, 0.0517]

A2

[0.0098, 0.0143]

[0.0089, 0.0154]

[0.0094, 0.0131]

[0.0086, 0.0138]

[0.0155, 0.0233]

[0.0063, 0.0094]

[0.0264, 0.0420]

A3

[0.0125, 0.0140]

[0.0094, 0.0131]

[0.0120, 0.0130]

[0.0113, 0.0137]

[0.0187, 0.0216]

[0.0089, 0.0097]

[0.0338, 0.0416]

A4

[0.0128, 0.0159]

[0.0086, 0.0138]

[0.0113, 0.0137]

[0.0120, 0.0159]

[0.0190, 0.0244]

[0.0093, 0.0111]

[0.0317, 0.0438]

A5

[0.0214, 0.0253]

[0.0155, 0.0233]

[0.0187, 0.0216]

[0.0190, 0.0244]

[0.0343, 0.0416]

[0.0157, 0.0178]

[0.0558, 0.0730]

A6

[0.0149, 0.0162]

[0.0063, 0.0094]

[0.0089, 0.0097]

[0.0093, 0.0111]

[0.0157, 0.0178]

[0.0393, 0.0407]

[0.0402, 0.0481]

A7

[0.0408, 0.0517]

[0.0264, 0.0420]

[0.0338, 0.0416]

[0.0317, 0.0438]

[0.0558, 0.0730]

[0.0402, 0.0481]

[0.0207, 0.2587]

5.2

The Model Construction

The establishment of the bank’s asset selection model in this section considers only the situation where all expected return intervals rAi contain the mean of historical return ri as an example. Other cases discussed above share the same procedure.


21

First, let us take the lower and upper boundaries of each loan’s future expected return interval in the third column of Table 4, the yield of each risk-less asset in the second column of Table 1 and the investment manager’s optimism factor μ in (35), forming the objective function. − − Second, take the boundary values of interval semi-covariance between two loans, σijl and σiju in Table 6, as well as the extreme limits of the portfolio’s risk tolerance interval which is set in advance by the manager, VGl and VGu , in (36)–(37), then downside risk constraint of the portfolio is built. Eventually, according to the laws and management requirements[44, 45] , bring in the capital budget constraint of x1 + x2 + · · · + x11 = 1, the diversification constraint of 0 ≤ xi ≤ 0.2, i = 1, 2, · · · , 7, and 0 ≤ xi ≤ 1, i = 8, 9, 10, 11, the restriction for the ratio of total loans of x1 + x2 + · · · + x7 ≤ 75%, constraint of excessive reserve ratio of x8 + x10 ≥ 5%(x1 + x2 + · · · + x7 ), reserve requirement ratio of x9 ≥ 6% and the constraint of cash holding ratio respectively based on liquidity risk and the profitability of 0.06% ≤ x8 ≤ 1.5%, then the IMDSV portfolio selection model is constructed. 5.3 5.3.1

Solution and Sensitivity Analysis of Parameters of the Model Optimal Allocations

Suppose the optimism factor on the portfolio yield μ = 0, the risk tolerance interval VG = [0.00152, 0.00192]. Optimal allocations of the indefinite optimization problem are listed in Column 1 and Row 3 in Table 9. Reset μ = 0.5 and μ = 1 and other parameters keep the same. Again achieve the optimal proportions of assets. The corresponding results are listed at the second and third columns of the third row of Table 9. Take the optimal allocations obtained under different values of the optimism factor μ into the objective function and then the corresponding maximum expected yield z can be obtained; see the first three columns of the fourth row of Table 9. Reset VG = [0.012 , 0.0132] and recalculate the model under the same three cases of the optimism factor μ. The corresponding results of asset allocations are listed in Columns 4 and 6 and Row 3 in Table 9 and the maximum expected returns z are in Columns 4 and 6 and Row 4 in Table 9. Similar discussion results for the other two cases, i.e., the expected return interval rAi on the left of the historical mean ri , rAi on the right of ri , are listed in Rows 3 and 4 in Tables 10 and 11. Portfolio’s Downside Semi-Variance Risk Interval V . Take the number of loans N = 7, assets allocation vector X and the lower and upper limits of semi-covariance interval − − between any two loans, σijl and σiju , in Table 6 into the left hand side of downside semi-variance constraint in Model (18), then the portfolio’s downside risk interval V under this situation can be obtained. Results are as listed in Row 5 in Table 9. For other cases, corresponding results can be seen in Rows 5 of Tables 10 and 11. √ Risk Interval Per Unit Yield V /z. Extract the square roots of the two boundaries of √ downside risk interval V and divide it by the expected yield z, then per unit yield risk V /z under all the situations can be obtained: See the last row of Table 9 and Tables 10 and 11. Sensitivity Analysis of μ and VG . In Row 4 of Table 9, as VG increases from [0.00152,


22

0.00192] to [0.012 , 0.0132], portfolio’s rate of return goes up and the downside risk interval also has a right move, as shown in the fifth row. Compared with Rows 4 and 5 in Tables 10 and 11, the same change trends are presented, irrespective of the tolerance interval VG . Therefore, the model follows the general rule of risk being positively proportional to the yield in portfolio’s optimization, and the downside risk of portfolio can be controlled by adjusting the risk tolerance interval. A reduction in VG leads to the decrease in V . Table 9

Influence of μ on the optimization results in the situation of rAi containing ri for all i

Numbers

1

2

3

4

5

6

1 Optimism factor on return μ

0

0.5

1

0

0.5

1

2 Target inter- [0.00152 , val of portfolio 0.00192 ] downside risk VG

[0.00152 , 0.00192 ]

[0.00152 , 0.00192 ]

[0.012 , 0.0132 ]

[0.012 , 0.0132 ]

[0.012 , 0.0132 ]

3 Optimal asset X=(0.1921, allocation pro- 0, 0.0060, portions X, W 0.2, 0.2, 0, 0) W =(0.0006, 0.06, 0.0293, 0.3120)

X=(0.1620, 0, 0.0423, 0.2, 0.2, 0, 0) W =(0.0006, 0.06, 0.0296, 0.3055)

X=(0.1386, 0, 0.0687, 0.2, 0.2, 0, 0) W =(0.0006, 0.06, 0.0298, 0.3024)

X=(0.2, 0.2, 0.2, 0.0794, 0.0706, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)

X=(0.2, 0.2, 0.2, 0.0791, 0.0709, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)

X=(0.2, 0.2, 0.2, 0.15, 0, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)

4 Objective function z

5.27%

5.51%

5.57%

6.48%

6.77%

7.06%

5 Portfolio downside semi-variance interval V

[0.00082 , 0.00192 ]

[0.00072 , 0.00192 ]

[0.00062 , 0.00192 ]

[0.00112 , 0.00392 ]

[0.00112 , 0.00392 ]

[0.00112 , 0.00392 ]

[1.28, 3.45]

[1.08, 3.30]

[1.65, 6.03]

[1.58, 5.77]

[1.51, 5.53]

6 Risk interval [1.53, 3.60] per unit yield √ V /z

The risk level per unit yield also has the same changing trend as the risk tolerance interval VG ; see Row 6 in Table 9 and Tables 10 and 11. In other words, the smaller VG is, the smaller √ V /z will be. In summary, if the risk tolerance interval decreases, downside risk and the expected return of assets portfolio will diminish. What’s more, risk level per unit yield will be decreased, according to the principle that a smaller risk level per unit yield is better. Therefore, decision makers can set a relatively small target interval in order to bear less risk. As to the effect of the optimism factor on return μ, still take Table 9 as an example. With

23


the rise of the return optimism factor μ, under a fixed target risk interval VG , assets allocations vary, and the expected yield z gradually increases (Row 4), which reflects greater optimism for the future, i.e., a higher yield is to be expected. 5.3.2

Comparison of the Models and Sensitivity Analysis of Single Loan’s Rate of Return Interval

1) A comparison with the existing mean-variance portfolio selection model Table 10

Influence of μ on the optimization results in the situation of rAi on the left of ri for all i

Numbers

1

2

3

4

5

6


0

0.5

1

0

0.5

1


[0.00152 , 0.00192 ]

[0.00152 , 0.00192 ]

[0.012 , 0.0132 ]

[0.012 , 0.0132 ]

[0.012 , 0.0132 ]

3 Optimal asset X=(0.2, allocation pro- 0.2, 0.2, portions X, W 0.0781, 0.0719, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)

X=(0.2, 0.2, 0.2, 0.081, 0.069, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)

X=(0.2, 0.2, 0.2, 0.15, 0, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)

X=(0.2, 0.2, 0.2, 0.0784, 0.0716, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)

X=(0.2, 0.2, 0.2, 0.15, 0, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)

X=(0.2, 0.2, 0.2, 0.15, 0, 0, 0) W = (0.0006, 0.06, 0.0369, 0.1525)


6.22%

6.63%

5.82%

6.23%

6.63%

5 Portfolio [0.00032 , downside semi- 0.00132 ] variance interval V

[0.00032 , 0.00132 ]

[0.00032 , 0.00132 ]

[0.00032 , 0.00132 ]

[0.00032 , 0.00132 ]

[0.00032 , 0.00132 ]


[0.50, 2.06]

[0.47, 1.93]

[0.54, 2.20]

[0.50, 2.06]

[0.47, 1.93]

5.82%


24 Table 11

Influence of μ on the optimization results in the situation of rAi on the right of ri for all i

Numbers

1

2

3

4

5

6


0

0.5

1

0

0.5

1


[0.00152 , 0.00192 ]

[0.00152 , 0.00192 ]

[0.012 , 0.0132 ]

[0.012 , 0.0132 ]

[0.012 , 0.0132 ]

3 Optimal asset X=(0.0288, allocation pro- 0.1240, portions X, W 0.0016, 0, 0, 0, 0) W =(0.0006, 0.06, 0.0071, 0.7779)

X=(0.0227, 0.1313, 0, 0, 0, 0, 0) W =(0.0006, 0.06, 0.0071, 0.7882)

X=(0.0136, 0.1401, 0, 0, 0, 0, 0) W =(0.0006, 0.06, 0.0071, 0.7786)

X=(0.2, 0.2, 0.15, 0, 0.2, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)

X=(0.2, 0.2, 0.15, 0, 0.2, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)

X=(0.2, 0.2, 0.15, 0, 0.2, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)


3.92%

3.94%

3.96%

7.63%

7.70%

7.76%

5 Portfolio downside semi-variance interval V

[0.00152 , 0.00192 ]

[0.00152 , 0.00192 ]

[0.00152 , 0.00192 ]

[0.00942 , 0.01062 ]

[0.00942 , 0.01062 ]

[0.00942 , 0.01062 ]

[3.77, 4.82]

[3.70, 4.80]

[12.30, 12.91]

[12.19, 13.79]

[12.09, 13.67]


Mean-variance portfolio selection model represented by Model (11) should be the most classical portfolio selection approach. Many portfolio selection techniques such as multistage portfolio optimization[10] have developed on the basis of Markowitz’s mean-variance formulation. Therefore, let us first compare the numerical results in this paper to the model (11). Take all loans’ expected rate of return represented in Column 2 of Table 4 and the risk-free assets’ rates of return represented in Table 1 into the objective function of Model (11). And then construct the quadratic constraint on portfolio rate of return variance with the loans’ historical rates of return data represented in Table 3 and risk tolerance level of VGl + VGu . In this paper VGl (VGu ) denotes the minimum (maximum) downside risk tolerance, thus VGl + VGu = (2VGl + 2VGu )/2 could be used to represent the mean tolerated level of variance of the return on whole portfolio. Integrate the objective function and the variance constraint with liquidity risk constraints proposed in Subsection 5.2 to build the bank’s asset selection model. The optimal allocation and related results are listed in Table 12.

INTERVAL PORTFOLIO SELECTION MODEL AND APPLICATION Table 12

25

The assets allocation of the existing numerical mean-variance optimization model

Numbers

1

2

1 The risk tolerance level VG

(0.00152 + 0.00192 )/2

(0.012 + 0.0132 )/2

2 Optimal asset allocation proportions X, W

X=(0, 0.2, 0.1731, 0.1470, 0.2, 0, 0) W =(0.0006, 0.06, 0.0354, 0.1839)

X=(0.2, 0.2, 0.2, 0.15, 0, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)


6.38%

6.81%

4 Loans portfolio downside semivariance V √ 5 Risk level per unit yield V /z

2

0.0024

0.00342

3.76

4.99

The optimal allocation results in Tables 9–11 and in Table 8 present the obvious difference between Model (11) and Model (18) that with Model (11) investors do not invest in the first asset (a high yield and high asset), while with Model (18) investors always puts the money more or less in the asset even though the anticipated economic condition is not good enough. The difference is derived from either the difference of use of risk measure for the two models or the difference of the way to deal with parameter. Existing mean-variance portfolio models including both singlestage and multistage portfolio selection model[1, 10] assume that the probability distribution of asset returns in the future is exactly known in advance and thus the parameters such as expected return for assets and variance and their estimations should be certain. However, in practice, some of uncertain information obtained by decision-maker is too vague to be expressed in probability distribution. Therefore, these estimated parameters should be uncertain, but not deterministic. On the other hand, in the existing mean-variance models the risk tolerance is also presumed as a deterministic parameter. Whereas, the real tolerated risk level in the future could deviate from the predetermined value, as the economic condition and market environment change. Consequently, the use of models with certain parameters leads to missing of available earnings compared with the model with interval parameters. For example, let us compare the results listed in the first column of Table 8 to the ones listed in the first three columns of Table 7.2. In this case, managers believe that the economy is going down and set a relevantly lower risk tolerance. The problem (11) with certain objective function and constraints provides decision makers with an only investment strategy. In contrast, Model (18) with uncertain objective and constraints provide a group of alternative investment strategies by easily adjusting the optimism factor of μ. Moreover, with μ increasing, Model (18) can offer the asset allocation strategy with a relevantly higher return. 2) A comparison with the existing mean semi-variance portfolio selection model First, let’s take the mean of a single loan’s historical rate of return ri (i = 1, 2, · · · , 7) in Column 2 of Table 4 and the risk-free assets’ rates of return represented in Table 1 into the objective function of Model (16). Second, we take the sample estimation of the downside semi− covariance between i-th loan and j-th loan, σij , in Table 5, into downside variance constraint of Model (16) and replace risk tolerance level of VG in Model (16) with the value or (VGl + VGu )/2.


26

Combine the objective function and downside risk constraint with liquidity risk constraints proposed in Subsection 5.2 to construct the bank’s asset selection model based on the existing mean downside semi-variance portfolio selection model with deterministic parameters. The optimal allocation and related results are listed in Table 13. Comparison of optimal asset allocations shown in Tables 9–11 and in Table 13 suggests that the proposed model is sensitive to pre-judging of future return on assets. The more accurate the judgment of bank managers on economic trends and industry’s development is, the more precise is the forecasting of future loan’s yield, which, in turn, contributes to a more conducive investment decision for the bank. For example, during economic downturns, risk level corresponding to one unit yield generated by our model is generally lower than the existing numerical models (Table 10 and Table 13). It means assets allocation based on our model is safer for the bank, which is vital for a bank’s stable operations, especially during economic downturns. Compared to the classical model, our model is able to obtain a higher loan portfolio yield and guarantee the profitability of the bank during economic upturns, if the bank appropriately increases the target of loan portfolio risk. Table 13

The assets allocation of the existing numerical MDSV optimization model

Numbers

1

2

1 The risk tolerance level VG

(0.00152 + 0.00192 )/2

(0.012 + 0.0132 )/2

2 Optimal asset allocation proportions X, W

X=(0.0873, 0.2, 0.1169, 0.1624, 0.1834, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)

X=(0.2, 0.2, 0.2, 0.15, 0, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)


6.59%

6.81%

4 Loans portfolio downside semivariance V √ 5 Risk level per unit yield V /z

2

0.0017

0.00212

2.60

3.01

In short, our model can more realistically reflect the change of return on assets as economic cycles change. Therefore, our model stands out for getting investment decision adapting to economic trends and industry’s development while existing numerical models are insensitive to change of real economic conditions, which may expose the bank to a greater investment risk during economic downturns and lead to missing of available earnings during an economic upturn. 3) A comparison with the interval linear portfolio selection model Besides the portfolio selection models with deterministic coefficients based on risk measure of portfolio variance and downside semi-variance, a linear interval programming model discussed by Lai, et al.[31] is selected and applied to the same portfolio problem for the purpose of comparison. The fundamental difference is that the model proposed in [31] is based on portfolio semi-absolute deviation in the interval framework while ours is based on semi-variance. We defer the specific analyses and the numerical results to the appendix because this analytical process


27

is a little bit long which would cause the inconvenience for reading the whole Subsection 5.3.2. We summarize roughly the results of the comparison and list below. Firstly, the investment policy based on the existing model provided in [31] is only applicable to a special case in this paper where the expected return interval of asset j is on the right of mean historical return of asset j for all j. Secondly, the investment policy based on the existing model provided in [31] is too conservative which results in the reduction of the bank’s revenue at the good economic condition from the perspectives of both theoretical analysis and numerical simulation. In sum, the portfolio selection model based on semi-variance risk measure would be more efficient and effective than the one based on the semi-absolute deviation in the downside framework and interval framework. However, we do not see any evidence for the opposite conclusion from the perspective of our analysis. 4) Sensitivity analysis in the case of rAi containing ri Since our model shows great sensitivity to pre-judging of the range in which future expected rate of return of assets falls, the focus is to test it through simulation: What is the effect that deviation of anticipated return interval has on optimal allocations under different economic conditions; whether the response to change of the expected interval of rate of return is consistent for different kinds of loans. Fix VG = [0.012 , 0.0132], μ = 0.5. Derive the expected return interval of each loan by taking ri (i = 1, 2, · · · , 7) as the reference point, on the left side of the range of 1%, namely [ri −1%, ri ], generating the lower limit ril randomly and on the right side of 1% range, namely [ri , ri + 1%], − − generating the other extreme riu randomly. Take ril , riu into (20)–(21), getting σijl and σiju , bring them into the objective function (35) as well as the downside risk constraints (36)–(37), with liquidity constraints to construct and solve the model. Repeat the same simulation 50 times. Allocations for the four risk-free assets are always 0.0006, 0.06, 0.0369 and 0.1525. Proportions for the first three loans, having a relatively higher historical mean return ri , reach to the highest value 0.2 limited by the diversification constraint in Subsection 5.2, and for the last two risky loans, having a relatively lower historical return mean ri , reach to 0. Although the expected return intervals of the loans change, proportions of these five with relatively extreme historical returns remain stable. However, results for the fourth and the fifth loans, having middle mean values among the seven risky assets, vary obviously with the change of the expected return interval. This paper employs the form of heptagons to show the simulation allocations along with changes of the expected return interval, see Figure 2(a). Seven blue radii in unit circle represent seven different axes, where ‘0’ is at the seven intersections of blue radius and circumference, ‘1’ is in the center of the circle. Allocations for seven loans xi (i = 1, 2, · · · , 7) are presented at each axis anticlockwise. Linking each adjacent point, we get a heptagon which depicts a group of allocation proportions. By the same way, 50 heptagons are performed. Furthermore, we add the allocations obtained by the classical MDSV model, Model (16) under the same risk tolerance interval of [0.012 , 0.0132] on the corresponding red dashed axes for further comparison. The purpose of the use of the heptagons in Figures 2 to 4 is to save space immensely and reflect the management insights intuitively. The 7 vertexes of a heptagon represent the


28

proportions invested in the bank’s 7 risky assets with a group of the expected return interval from the bank manager. If the vertexes of heptagons cross at one point, it indicates that the assets allocations do not change with varying the expected return interval. Conversely, if the vertexes of heptagons are dispersed, it indicates that the assets allocations fluctuate as the expected return interval changes. Figures 2 to 4 are the cases of the numbers of the simulation equal to 50, 100 and 200 simulations, respectively. As shown in Figure 2(a), vertexes of 50 heptagons on axis of x1 , x2 , x3 and x6 , x7 cross at one point, while the vertexes on axis of x4 , x5 have some changes. Meanwhile, most heptagons coincide with the red dotted heptagon. Figure 2(a) reflects the three characteristics of the IMDSV model in the case of expected return interval rAj containing the historical mean of return ri : (i) For loans with historical mean of rate of return either higher or lower, the investment proportions keep stable; (ii) For the remainder whose historical rates of return are in the middle of all loans, the optimal investment proportions appear to be greatly sensitive to the change of the expected return interval; and (iii) most of the investment proportions converge to allocation results of the existing model (16). 5) Sensitivity analysis in the case of rAi on the left of ri Fix VG = [0.00152, 0.00192], μ = 0.5. Construct the random interval of rate of return by taking ri (i = 1, 2, · · · , 7) as the upper extreme riu , and on the left side of 2% range, namely [ri − 2%, ri ], generating the lower extreme ril . Repeat the simulation 50 times and the results show a similar feature as 4): Besides risk-less assets, investment proportions of these loans with relatively higher (the first two loans) or relatively lower historical rates of return (the last two loans) perform quite well and stably. However, results of the third to the fifth loans whose historic yields are in the middle of seven loans have an obvious volatility with the change of the anticipated return interval. In Figure 2(b), these corresponding vertexes of 50 heptagons on x1 , x2 and x6 , x7 cross at one point, while on x3 , x4 , x5 it begins to disperse. 6) Sensitivity analysis in the case of rAi on the right of ri Fix VG = [0.012 , 0.0132 ], μ = 0.5. Construct the random yield interval rAj = [ril , riu ] by taking ri (i = 1, 2, · · · , 7) as the lower extreme ril , and on the right side of 2% range, namely [ri , ri + 2%], generating the upper extreme riu . Same as the above process, 50 allocation results can be obtained. Investment proportions of the seven loans are shown in Figure 2(c). From the numerical simulation results in the case of rAi on the right side of ri , we can see features similar to the above two cases. 0

0 x3

0 x4

0

(a)

x2

x3

0

1

x2

1

x4

x1

x5

x5

0

x6

0

x7

0 0

0

1

x4

x5

x7 x6

0 0

x2

x3

0

(c)

0

0 x1

0

0

(b)

0

x6

x1

0

x7

0

Figure 2 Heptagons of seven loans’ allocations and number of simulations equals 50

0

29


Increase the number of simulations from 50 to 100 and 200 and repeat the same process as 4) to 6). The Results are correspondingly presented in Figure 3 as well as Figure 4. 0

0 x3

0

x2

x4

x1

1

x2

1

0

x7

x6

0

x1

0

x5

0

0

x2

1

0

x5

x7

x6

x4

x1

(c)

0 x3

0

0

x5

0

(b)

0 x3

0

x4

0

0

(a)

x7

x6

0

0

0

0

Figure 3 Heptagons of seven loans’ allocations and number of simulations equals 100 0

0 x3

0

x2 x1

1

x5 x6

x4

x2

1

0

0

x1

x5 x6

0

0

x4

0

x7

x2

1

0

x1

0

x5 x6

0

(c)

0 x3

0

0

x7

0

(b)

0 x3

0

x4

0

0

(a)

x7

0

0

Figure 4 Heptagons of seven loans’ allocations and number of simulations equals 200

7) Comparative analysis Through the comparison of the figures above, the robustness of the investment policy proposed by this study can be naturally reflected. In all cases where the anticipated interval rAi contains the mean of historical rate of return ri , or being on the left of ri , or on the right of ri for all i, the proportions invested in the loans whose historic yields are the highest or the lowest of all loans remain stable with the change of expected return intervals, which says that using IMDSV model can reduce the risk of great deviation caused by decision maker’s deviation of experience and economic wisdom. However, proportions invested in the loans with the middle historical rate of return vary obviously with the change of the expected return interval, which requiring managers to judge the future return of this kind of assets more carefully so as to avoid great deviation. These results may provide more evidence for bank asset allocation under the different economic conditions.

6

Conclusions

The main purpose of this article is to have a preliminary discussion on two unresolved problems in the portfolio selection model with interval parameters. Firstly, we deduce a class of mean downside semi-variance optimization model with interval parameters, which may fill in the blanks left by the existing interval portfolio studies which are based only on the linear semi-absolute deviation to depict the downside risk. The model brings

30


the investor’s experiential wisdom of judgments and anticipation of the economic conditions, development of specific industry sectors, even a specific borrower accumulated by the long practice into the classical mean semi-variance portfolio optimal framework. Its use can well reflect change of the return and downside risk of both a single asset and the portfolio with economic and industrial conditions varying, compared with the existing semi-variance portfolio models with deterministic parameters that depend more on historical information of an asset’s return. Therefore, we think that this model offers more flexibility in the investment decision. Secondly, we have applied the model to a bank’s assets allocation and tested how the lender’s anticipation of a loan’s rate of return influences the lending decision. The numerical simulation results show the optimal result of the model behaves robustly, i.e., for risky assets with highest or lowest mean historical rate of return, the optimal investment proportion has good stability; on the contrary, for the remainder whose historical rates of return are in the middle of all loans, the optimal results appear to be greatly sensitive to the change of return interval under the following three conditions. (i) All rate of return intervals of risky assets contain the loans’ mean historical rates of return which are chosen as reference points. This condition corresponds to smooth economic running in which all industrial loans’ expected return intervals contain their mean historic return. (ii) All rates of return intervals of risky assets are located on the right of the reference points. The condition corresponds to economic upturns, in which all kinds of loans’ expected returns tend to increase. (iii) All rates of return intervals of risky assets are located on the left of the reference point. This condition corresponds to economic downturns, in which all kinds of loans’ expected returns tend to decrease. The overall simulation results suggest that for loans whose historic yields are the highest or the lowest of all loans, the model can reduce the risk of high deviation caused by variations in decision makers’ deviation of experience and economic wisdom, however, for those in the middle, managers are required to judge each future return more carefully so as to avoid great deviation. These research findings may provide more evidence for stakeholders in an uncertain investment environment. Although an interval nonlinear portfolio selection model under risk measure of downside semi-variance has been presented by this paper, the behavior of this kind of interval nonlinear optimization model is still to be understood and more analysis is obviously required. To our knowledge, a multistage interval linear optimization approach has been put forward and applied to some practical problem[19] , while a multistage interval nonlinear optimization models and their application in portfolio selection field have not yet been considered. This could be a future research direction that is worthy of consideration.

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[25]

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Appendix A comparison with the linear interval portfolio selection model from the perspectives of theoretical analysis and numerical simulation. We apply a linear interval programming model discussed by Lai, et al.[31] to the same portfolio problem for the purpose of comparison. Lai, et al. incorporated the interval returns with the downside risk and proposed a linear interval programming model for portfolio selection that is very similar to our topic. The fundamental difference is that the model proposed in [31] is based on portfolio semi-absolute deviation while ours is based on semi-variance. We show the difference and the connection between the two models through theoretical analysis and numerical simulation as below. (a) Comparison of the two models from the perspective of theoretical analysis We present a model for optimal portfolio selection with the interval coefficients subject to semi-absolute deviation constraint considered by Lai, et al. in [31]. We have marked the model

34


as Model (18). max

N +1

j xj R

j=1

s.t.

T 1 w t ≤2 [ω, ω], T t=1 N +1

xj = 1,

(43)

xj ≥ 0, j = 1, 2, · · · , N + 1.

j=1

t is a down side In Model (43), xj is the proportion invested in j-th asset, decision variable; w semi-absolute deviation interval of the return of the portfolio at past period t, t = 1, 2, · · · , n and can be computed as w t = max

N

(Rj − rjt )xj , 0 ,

(44)

j=1

j is the expected rate of return interval of asset j corresponding to notation rAj in the where R objective function of our model (18); rjt is the historic rate of return of asset j at the past t-th year, t = 1, 2, · · · , T . As for “ ≤2 ”, it represents an interval order relation and has the same definition as (9) of this paper. [ω, ω] is the tolerated risk level. In Subsection 3.4 of our paper, we also define a risk tolerance level interval [VGl , VGu ] that has an analogous description to [ω, ω] and can be computed by its square. In [31], the authors assume that the investor wants to allocate wealth among N risky assets and one riskless asset. Naturally, Model (43) can be extended and applied to more general portfolio selection problem where H (H ≥ 1) riskless assets exist in a set of candidate assets discussed by this paper. Now, we analyze the relationship between the semi-absolute deviation constraint and the portfolio semi-variance constraint under interval’s order relation defined as (9). j are accurately estimated, w When the expected returns R t should be a deterministic value. According to the Jensen’s inequality and convexity of quadratic function and function max{·, 0}, we have 2

N 2 T T T +H 1 1 1 2 max w t ≤ (w t ) = (Rj − rjt )xj , 0 T t=1 T t=1 T t=1 j=1 2 T N +H 1 ≤ xj max{Rj − rjt , 0} T t=1 j=1 =

=

T N +H 1 j − rjt , 0} × max{R i − rit , 0} xj xi max{R T t=1 i,j=1 N +H i,j=1

xj xi

T 1 max{Rj − rjt , 0} × max{Ri − rit , 0} , (45) T t=1


35

where, the left hand side of the first “≤” of these inequalities and equalities above is the square of the portfolio downside semi-absolute deviation; the right hand side of the last “=” is the portfolio semi-variance presented in Model (16). j = Rj , where Rj is the maximum expectation for return of asset j, by (45) and When R the definition of w t above, we have

2

2 T T N 1 1 w t = max (Rj − rjt )xj , 0 T t=1 T t=1 j=1 ≤

N +H

xj xi

i,j=1

T 1 max{Rj − rjt , 0} × max{Ri − rit , 0} . T t=1

(46)

j = R , where R is the minimum expectation for return of asset j, we Similarly, when R j j have 2

2 T T N 1 1 w = max (Rj − rjt )xj , 0 T t=1 t T t=1 j=1 ≤

N +H

xj xi

i,j=1

T 1 max{Rj − rjt , 0} × max{Rj − rit , 0} . T t=1

(47)

j becomes an uncertain return interval but a deterministic value, left hand Obviously, when R sides of the two inequalities above become the square of upper and lower limits of portfolio semiabsolute deviation interval respectively while the right hand sides of the two inequalities above are the upper and lower limits of portfolio semi-variance interval corresponding to (20)–(21) of this paper. Furthermore, with the definition of the intervals’ order relation, we conclude that a solution of Model (18) proposed by this paper also satisfies the following inequalities:

2 T N 1 max (Rj − rjt )xj , 0 ≤ VGu T t=1 j=1

T N 1 ⇔ max (Rj − rjt )xj , 0 ≤ VGu = ω T t=1 j=1

(48)

and ( T1

T

t=1

max{

N

j=1 (Rj

ω2 + ω2 VGl + VGu = . ≤ 2 2

− rjt )xj , 0})2 + ( T1

T

t=1

max{

N

j=1 (Rj

− rjt )xj , 0})2

2 (49)

This reflects that the solution of Model (18) can meet the portfolio semi-absolute deviation constraint of Model (43) in the square sense presented above. If some additional constraints are added into Models (18) and (43), the optimal solution of Model (18) may also be the feasible solution of Model (43). For instance, we can add the constraint that the lower limit of the downside risk interval does not exceed the lower limit of the risk tolerance interval. The use


36

of the constraint reflects the limitation on the downside risk when the expected returns are predicted most pessimistically and are reasonable. In a word, the previous theoretical analysis shows that the portfolio selection model based on semi-variance risk measure would be more efficient and effective than the one based on the semi-absolute deviation in the downside framework and interval framework, if we add some reasonable constraints to both models. However, from this perspective we do not see any evidence for the opposite conclusion. (b) Comparison of the simulation results Now, for the purpose of further comparison we present the asset allocation results based on Model (50) used in [31]. In order to transform Model (43) into a linear programming problem with the algebraic equation of portfolio semi-absolute deviation, [31] took two technical means. First, replace one constraint of the portfolio semi-absolute deviation over the past T periods in Model (43) with T constraints of portfolio semi-absolute deviation in each period in the past. However, the price of the replacement is that the investment policy provided in [31] would be too conservative. Second, [31] assumed that all risky assets have a tendency toward rising prices and the lower limits of the expected return intervals of all risky assets are always the mean historical rate of return. By this assumption, [31] provided a proposition and transforms Model (43) into a tractable linear programming problem represented as Model (50) below. However, the price of using the assumption is that the investment policy provided in [31] is only applicable to a special case in this paper where the expected return interval of asset j is on the right of mean historical return of asset j for all j. max

N +1

j xj R

j=1

s.t.

N

j − rjt ) ≤2 [ω, ω], xj (R

t = 1, 2, · · · , T,

(50)

j=1 N +1

xj = 1,

xj ≥ 0, j = 1, 2, · · · , N + 1.

j=1

According to definitions of the intervals’ order relation, Model (50) is further transformed into the following linear optimal model (51) that is applied to solve the same asset allocation problem as in this study. For more details of the transformation, please refer to [31]. max

N

((Rj + Rj )xj − λ(Rj − Rj )xj )/2 +

j=1

s.t.

N

N +H

Ri xi

i=N +1

xj (Rj − rjt ) ≤ ω,

(51)

j=1 N j=1

((Rj + Rj )/2 − rjt )xj ≤ (ω + ω)/2,

t = 1, 2, · · · , T,

INTERVAL PORTFOLIO SELECTION MODEL AND APPLICATION N +H

xj = 1,

37

xj ≥ 0, j = 1, 2, · · · , N + 1.

j=1

Given the assumption that all risky assets have the tendency to rising prices in [31], we make comparison for the case of all expected returns being on the right of the mean of the historical returns. Table 14 Results of assets allocation based on mean semi-absolute deviation portfolio model used in [31] in the situation of rAi falling on the right side of ri for all i Numbers

1

2

3

4


0

0.5

0

0.5

2 The tolerance risk level VG

(0.00152 + 0.00192 )/2

(0.012 + 0.0132 )/2

(0.00152 + 0.00192 )/2

(0.012 + 0.0132 )/2

3 Asset allocation proportions X, W

X=(0.0483, 0.0580, 0.0097, 0, 0, 0, 0) W =(0.0006, 0.0600, 0.0052, 0.8183)

X=(0.0483, 0.0580, 0.0097, 0, 0, 0, 0) W =(0.0006, 0.0600, 0.0052, 0.8183)

X=(0.2, 0.2, 0.2, 0.0629, 0.0871, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)

X=(0.2, 0.2, 0.2, 0.15, 0, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)


3.61%

3.55%

7.25%

6.81%

5 Return interval

[0.0355, 0.0368]

[0.0355, 0.0368]

[0.068, 0.0770]

[0.0681, 0.0764]

6 Loans portfolio semi-absolute deviation interval

[0.0002, 0.0013]

[0.0002, 0.0013]

[0.0012, 0.009]

[0.0012, 0.0083]

7 Risk level per √ unit yield V /z

[0.68, 3.53]

[0.69, 3.59]

[1.68, 12.43]

[1.82, 12.17]

Accordingly, in Model (51), N (the number of the loans)=7, H (the number of riskless assets)=4, and parameter λ ∈ [0, 1]. First, let us take the mean of historical rate of return in Column 2 of Table 4, the upper limits of expected return intervals in Column 5 of Table 4, and the interest rates of the risk-free assets in Table 1 into the objective function of Model (51). Second, we take the historical rate of return of loans (Table 3) and the other variables into the semi-absolute deviation constraints in Model (51) and replace the tolerated risk level with the square roots of VGl and VGu respectively. Combine the objective function and the semi-absolute deviation constraints with diversification and liquidity risk constraints proposed in Subsection 5.2 to construct the bank’s asset allocation model based on the portfolio semi-absolute deviation. The solution and more relevant results are listed in Table 14.


38

Table 15 Results of assets allocation based on mean downside semi-variance portfolio model in the situation of rAi falling on the right side of ri for all i (ril being replaced with ri and other parameters unchanged) Numbers

1

2

3

4


0

0.5

0

0.5

2 The tolerance risk level VG

(0.00152 + 0.00192 )/2

(0.012 + 0.0132 )/2

(0.00152 + 0.00192 )/2

(0.012 + 0.0132 )/2

3 Asset allocation proportions X, W

X=(0.0119, 0.1417, 0, 0, 0, 0, 0) W =(0.0006, 0.06, 0.0071, 0.7786)

X=(0.0105, 0.1431, 0, 0, 0, 0, 0) W =(0.0006, 0.06, 0.0071, 0.7787)

X=(0.2, 0.2, 0.2, 0.15, 0, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)

X=(0.2, 0.2, 0.2, 0, 0.15, 0, 0) W =(0.0006, 0.06, 0.0369, 0.1525)


3.79%

3.88%

6.81%

7.27%

5 Return interval

[0.0355, 0.0368]

[0.0355, 0.0368]

[0.068, 0.0770]

[0.0681, 0.0764]

6 Loans portfolio semi-absolute deviation interval

[0.0379, 0.0396]

[0.0379, 0.0396]

[0.0681, 0.0764]

[0.0679, 0.0775]

7 Risk level per √ unit yield V /z

[0.00052, 0.00192]

[0.00052, 0.00192]

[0.00212, 0.0092]

[0.0022, 0.01022]

It is worth to note that the results with λ = 0 listed in Table 15 should be compared to the ones with μ = 0.5 listed in Table 14, while the results with λ = 1 listed in Table 14 should be compared to the ones with μ = 0 listed in Table 15. The results of the comparison show that the bank can obtain higher return from the portfolio during an economic upturn, by the asset allocation policy based on Model (18) proposed in this paper. With Model (51), the bank manager tends to invest more funds in risk-less assets, which results in reduction of the bank’s revenue during good economic conditions. The conservation of investment policy based on Model (51) is consistent with the previous theoretical analyses expect. In sum, compared with the existing relevant studies, we can provide an appealing asset allocation policy which can help the bank gain more under the same risk tolerance.

A Nonlinear Interval Portfolio Selection Model and Its Ap

A Nonlinear Interval Portfolio Selection Model and Its Ap

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