We apply Markowitz portfolio theory to Mongolian economy in order to define optimal budget structure. We assume that the government revenue is a portfolio consisting of seven major taxes and non-tax revenues. We minimize the variance of the portfolio under fixed return of the government revenue. This optimization problem has been solved by the conditional gradient method on MATLAB. Computational results based on Mongolian economic data are provided.

Markowitz Model
1. Introduction

Financial portfolio optimization is widely used in mathematics, statistics, economics and engineering. Fundamental breakthrough in the problem of asset allocation and portfolio optimization is dated to Markowitz’s Modern Portfolio Theory  . It considers rational investors and models with the problem of minimizing the mean-variance of the portfolio with a fixed value for the expected return on the entire portfolio. The model also assumes a market without any taxes or transaction costs, and where short selling is disallowed but assets are infinitely divisible and can be traded with any non-negative fractions.

There are many works devoted to optimization methods and algorithms for solving the portfolio variance minimization problem. This problem belongs to the convex optimization problem so any stationary point found by an optimization method provides a global solution to the problem. Also, the Markowitz model has been extended in various ways in the literature  -  . Tobin James’s work  considers the inclusion of risk-free assets in Markowitz model by the development of the Separation theorem which states that in the presence of a risk-free asset, the optimal risky portfolio can be obtained without any knowledge of the investor’s preferences.

Sharpe’s Capital Asset Pricing Model (CAPM)  takes into account the asset’s sensitivity to non-diversifiable risk while it is being added to an already existing well-diversified portfolio. It considers the importance of the covariance structure of the returns, the variance of the portfolio and the market premium. The model assumes that the investors are rational and risk-averse, are broadly diversified across a range of investments, and that they cannot influence the prices of the assets. Assumptions regarding trade or transaction costs, short-selling and trades with non-negative fractions do apply from the traditional Markowitz’s framework.

Considering the equity markets in perspective, Fernholzs Stochastic Portfolio Theory  discusses a descriptive theory that provides a framework for analyzing portfolio behavior and equity market structure that has both theoretical and practical applications.

Portfolio optimization problems have been studied in     and  . Formulation of Markowitz’s portfolio optimization problem is viewed as a quadratic optimization problem.  and  provides comprehensive literature to convex and numerical optimization methods to solve such a formulation.

 explores a global optimization approach to scenario generation and portfolio optimization looking at them as individual problems.  proposes a stochastic programming approach for multi-period portfolio optimization.  presents a multi-period scenario generation approach to support portfolio optimization and  discusses scenario generation, mathematical models and algorithms for the portfolio optimization problem.  explores portfolio selection using hierarchical Bayesian analysis and Markov Chain Monte Carlo (MCMC) methods.  discusses the portfolio optimization with an envelope-based multi-objective evolutionary algorithm with a variety of non-convex constraints.

 solves the portfolio optimization problem using genetic algorithm.  applies genetic algorithms in a multi-stage portfolio optimization system.  solves the problem with the same method taking into account transaction costs and minimum transaction lot constraints.

 examines constrained Markowitz portfolio selection using ant colony optimization.  considers multi-objective particle swarm optimization approach to the portfolio optimization problem. In this paper, for solving the variance minimization problem, we use the conditional gradient method  which uses a series of linear programming problems. The paper is organized as follows. In Methodology Section, we introduce briefly Markowitz portfolio theory and show how to apply the theory to Mongolian government budget. In Data Description Section, we use Mongolian economic data and construct matrix tables for the proposed model. In the last section, we implement Markowitz model for Mongolian government budget.

2. Methodology

Assume that a government revenue consists of n revenues

A = ∑ i = 1 n A i ,

where A is a total government revenue, and A i is i-th type of revenue, i = 1 , 2 , ⋯ , n .

We can consider A as a portfolio of n assets with weights x i which means A i = x i A ,   i = 1 , 2 , ⋯ , n .

Clearly,

∑ i = 1 n x i = 1 , x i ≥ 0 , i = 1 , 2 , ⋯ , n .

Let r 1 , r 2 , ⋯ , r n be rates of the tax revenues returns.

These have expected values

E ( r 1 ) = r ¯ 1 , E ( r 2 ) = r ¯ 2 , ⋯ , E ( r n ) = r ¯ n .

Then the rate of return of the portfolio is

r = ∑ i = 1 n x i r i .

We denote the variance of the return of i-th tax revenue by σ i 2 , the variance of the return of the portfolio by σ 2 , and the covariance of the return of i-th revenue with j-th revenue by σ i j . It is well known that  

σ 2 = ∑ i = 1 n ∑ j = 1 n x i x j σ i j .

To find a minimum-variance portfolio, we fix the mean value at same arbitrary value r ¯ . Then we find the optimal portfolio by solving the following minimization problem   :

min 1 2 ∑ i = 1 n ∑ j = 1 n x i x j σ i j (1)

subject to

∑ i = 1 n x i r ¯ i = r ¯ (2)

∑ i = 1 n x i = 1 (3)

x i ≥ 0 ,     i = 1 , 2 , ⋯ , n (4)

Note that problem (1)-(4) is convex from a view point of optimization theory. It can be checked that the matrix of covariance C n × n = ( σ i j ) is positive defined. In order to find a solution to problem (1)-(4), we need to write the Lagrangian as

L = 1 2 ∑ i = 1 n ∑ j = 1 n x i x j σ i j + λ 1 ( ∑ i = 1 n x i r ¯ i − r ¯ ) + λ 2 ( ∑ i = 1 n x i − 1 ) + ∑ i = 1 n μ i x i

taking into account condition (4).

Then if we apply Karush-Kuhn-Tucker optimality condition to problem (1)-(4), we have

{ ∂ L ∂ x i = ∑ i = 1 n σ i j x j + λ 1 r ¯ i + λ 2 + μ i = 0 ,     i = 1 , 2 , ⋯ , n μ i x i = 0 ,     i = 1 , 2 , ⋯ , n λ 1 2 + λ 2 2 + ∑ i = 1 n μ i 2 > 0 ,     μ i ≥ 0 ,     i = 1 , 2 , ⋯ , n (5)

To find an optimal solution, we combine system (5) with (2)-(4). It means that

{ ∑ i = 1 n σ i j x j + λ 1 r ¯ i + λ 2 + μ i = 0 ,     i = 1 , 2 , ⋯ , n ∑ i = 1 n x i r ¯ i = r ¯ ∑ i = 1 n x i = 1 μ i x i = 0 ,     i = 1 , 2 , ⋯ , n μ i ≥ 0 ,     i = 1 , 2 , ⋯ , n (6)

This nonlinear system has ( 3 n + 2 ) linear and nonlinear equations with ( 2 n + 2 ) unknowns. So it is better to solve problem (1)-(4) by convex optimization methods and algorithm. For instance, it is convenient to solve problem (1)-(4) by conditional gradient method  since at each iteration of the algorithm we solve just a linear programming problem.

3. Data Description

For numerical analysis we use the following Mongolian economic data for period 1991-2018 which shows structure of government revenue consisted of tax and nontax revenues (Tables 1-3).

Weight of government revenue
X1X2X3X4X5X6X7
YearIncome taxSocial security contributionsProperty taxesTaxes on domestic goods & servicesTaxes on foreign tradeOther taxesNon-tax revenue
19910.3580.0990.0010.3010.0410.0130.187
19920.4270.0710.0000.2430.1130.0150.131
19930.4930.0490.0000.2450.1140.0110.087
19940.3720.0730.0000.2270.0880.0240.217
19950.3360.1090.0000.1940.0660.0240.270
19960.2800.1130.0000.2290.0850.0350.258
19970.2810.0950.0000.2840.0400.0360.263
19980.1730.1090.0010.3210.0060.0320.358
19990.1470.1120.0010.3520.0340.0340.320
20000.2070.1080.0010.3470.0620.0320.244
20010.1470.1230.0040.3790.0620.0330.253
20020.1520.1140.0070.3740.0520.0540.247
20030.1760.1180.0080.3430.0590.0560.240
20040.2020.1150.0080.3430.0630.0870.182
20050.2130.1140.0080.3230.0680.1000.174
20060.3510.0820.0050.2590.0530.0790.171
20070.3450.0850.0040.2190.0540.0910.201
20080.3480.1060.0040.2590.0650.0900.129
20090.2610.1320.0060.2550.0580.1010.187
20100.3120.1060.0040.2770.0620.0990.139
20110.1970.1120.0040.3390.0800.1350.132
20120.1790.1380.0040.3370.0670.1360.139
20130.1870.1470.0070.3230.0640.1250.146
20140.1750.1460.0080.2970.0570.1380.178
20150.1960.1740.0140.2750.0540.1470.139
20160.1730.1950.0170.3270.0540.0970.137
20170.2220.1820.0180.2960.0700.0810.132
20180.2260.1760.0150.3210.0740.0770.111

Source: National Statistical Office, https://www.1212.mn/.

Government revenue growth
YearIncome taxSocial security contributionsProperty taxesTaxes on domestic goods & servicesTaxes on foreign tradeOther taxesNon-tax revenue
19921.1170.2770.0000.4363.9171.0100.247
19934.1942.1250.0173.5313.5402.4971.986
19940.1271.2103.9180.3810.1462.1732.711
19950.5151.5120.8000.4390.2690.6811.094
1996−0.0600.172−0.1740.3250.4540.6330.073
19970.3730.1500.7580.700−0.3680.3980.395
1998−0.3380.2272.0640.215−0.828−0.0190.469
1999−0.0590.1430.2460.2214.9730.178−0.009
20000.8980.299−0.0360.3241.4750.2590.025
2001−0.1290.3954.9490.3380.2110.2640.271
20020.1230.0080.9510.073−0.0900.7680.061
20030.3470.1990.3720.0650.3280.1940.128
20040.4770.2590.2490.2850.3701.017−0.022
20050.2390.1650.1020.1090.2740.3480.120
20061.6710.1710.0920.3020.2650.2850.595
20070.3600.4340.1950.1670.4220.5860.628
20080.1640.4290.1140.3650.3740.140−0.261
2009−0.3110.1490.213−0.095−0.1760.0320.336
20100.8740.2570.2380.7010.6670.5410.163
2011−0.1450.4290.2420.6580.7450.8480.287
20120.0450.4240.2790.145−0.0300.1640.213
20130.2730.2971.0050.1690.1650.1160.279
2014−0.0070.0500.139−0.029−0.0680.1670.291
20150.0630.1320.725−0.120−0.0980.012−0.258
2016−0.1090.1320.2100.2050.025−0.332−0.004
20170.5460.1240.2530.0880.5600.0010.160
20180.2930.2270.0780.3780.3320.2140.071
Covariance matrix of government revenue
COVAR (X)X1X2X3X4X5X6X7
X10.76920.2684−0.25380.50330.57070.33910.2542
X20.26840.22660.10350.24150.17750.22690.2444
X3−0.25380.10351.4036−0.0608−0.38650.13620.3116
X40.50330.2415−0.06080.44100.40420.29340.2267
X50.57070.1775−0.38650.40421.78300.30590.0925
X60.33910.22690.13620.29340.30590.39250.3184
X70.25420.24440.31160.22670.09250.31840.4095
4. Numerical Results

In this section, we implement the Markowitz model for Mongolian economy. We examine government budget revenue structure which depends on seven types of tax and nontax revenues.

Variable x i is the weight of i-th tax revenue in the portfolio. The Mongolian government budget consists of the following revenues such as income tax, social security contributions, property taxes, taxes on domestic goods and services, taxes on foreign trade, other taxes and non-tax revenues. Table 4 shows the initial values of variables as well as the optimal solution of problem (1)-(4) found by the conditional gradient method on MATLAB.

Thus, the government should take into account these results in fiscal policy decision making.

Solution
NameInitial valueOptimal valueChange
Income tax0.2550.227−2.8%
Social security contributions0.1180.115−0.3%
Property taxes0.0050.0181.3%
Taxes on domestic goods & services0.2960.194−10.2%
Other taxes0.0710.1477.6%
Non-tax revenue0.1920.2606.8%
5. Conclusion

We have tested the Markowitz model on Mongolian economic data in order to define optimal structure of the government revenue which consists of 7 components. Since the variance minimization problem was convex quadratic, for solving the problem we have applied the conditional gradient method coded in MATLAB. The numerical solution was obtained. In the same way, we can consider the problem of maximizing the government return subject to variance constraint. But it will be discussed in the next paper.

Acknowledgements

This work was supported by the research grant P2018-3588 of National University of Mongolia.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Ankhbayar, Ch., Lkhagvajav, B., Tungalag, N. and Enkhbat, R. (2019) Application of Markowitz Model to Mongolian Government Budget. iBusiness, 11, 42-50. https://doi.org/10.4236/ib.2019.113004

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