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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.

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 [

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 [

Sharpe’s Capital Asset Pricing Model (CAPM) [

Considering the equity markets in perspective, Fernholzs Stochastic Portfolio Theory [

Portfolio optimization problems have been studied in [

[

[

[

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 [

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).

X_{1} | X_{2} | X_{3} | X_{4} | X_{5} | X_{6} | X_{7} | |
---|---|---|---|---|---|---|---|

Year | Income tax | Social security contributions | Property taxes | Taxes on domestic goods & services | Taxes on foreign trade | Other taxes | Non-tax revenue |

1991 | 0.358 | 0.099 | 0.001 | 0.301 | 0.041 | 0.013 | 0.187 |

1992 | 0.427 | 0.071 | 0.000 | 0.243 | 0.113 | 0.015 | 0.131 |

1993 | 0.493 | 0.049 | 0.000 | 0.245 | 0.114 | 0.011 | 0.087 |

1994 | 0.372 | 0.073 | 0.000 | 0.227 | 0.088 | 0.024 | 0.217 |

1995 | 0.336 | 0.109 | 0.000 | 0.194 | 0.066 | 0.024 | 0.270 |

1996 | 0.280 | 0.113 | 0.000 | 0.229 | 0.085 | 0.035 | 0.258 |

1997 | 0.281 | 0.095 | 0.000 | 0.284 | 0.040 | 0.036 | 0.263 |

1998 | 0.173 | 0.109 | 0.001 | 0.321 | 0.006 | 0.032 | 0.358 |

1999 | 0.147 | 0.112 | 0.001 | 0.352 | 0.034 | 0.034 | 0.320 |

2000 | 0.207 | 0.108 | 0.001 | 0.347 | 0.062 | 0.032 | 0.244 |
---|---|---|---|---|---|---|---|

2001 | 0.147 | 0.123 | 0.004 | 0.379 | 0.062 | 0.033 | 0.253 |

2002 | 0.152 | 0.114 | 0.007 | 0.374 | 0.052 | 0.054 | 0.247 |

2003 | 0.176 | 0.118 | 0.008 | 0.343 | 0.059 | 0.056 | 0.240 |

2004 | 0.202 | 0.115 | 0.008 | 0.343 | 0.063 | 0.087 | 0.182 |

2005 | 0.213 | 0.114 | 0.008 | 0.323 | 0.068 | 0.100 | 0.174 |

2006 | 0.351 | 0.082 | 0.005 | 0.259 | 0.053 | 0.079 | 0.171 |

2007 | 0.345 | 0.085 | 0.004 | 0.219 | 0.054 | 0.091 | 0.201 |

2008 | 0.348 | 0.106 | 0.004 | 0.259 | 0.065 | 0.090 | 0.129 |

2009 | 0.261 | 0.132 | 0.006 | 0.255 | 0.058 | 0.101 | 0.187 |

2010 | 0.312 | 0.106 | 0.004 | 0.277 | 0.062 | 0.099 | 0.139 |

2011 | 0.197 | 0.112 | 0.004 | 0.339 | 0.080 | 0.135 | 0.132 |

2012 | 0.179 | 0.138 | 0.004 | 0.337 | 0.067 | 0.136 | 0.139 |

2013 | 0.187 | 0.147 | 0.007 | 0.323 | 0.064 | 0.125 | 0.146 |

2014 | 0.175 | 0.146 | 0.008 | 0.297 | 0.057 | 0.138 | 0.178 |

2015 | 0.196 | 0.174 | 0.014 | 0.275 | 0.054 | 0.147 | 0.139 |

2016 | 0.173 | 0.195 | 0.017 | 0.327 | 0.054 | 0.097 | 0.137 |

2017 | 0.222 | 0.182 | 0.018 | 0.296 | 0.070 | 0.081 | 0.132 |

2018 | 0.226 | 0.176 | 0.015 | 0.321 | 0.074 | 0.077 | 0.111 |

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

Year | Income tax | Social security contributions | Property taxes | Taxes on domestic goods & services | Taxes on foreign trade | Other taxes | Non-tax revenue |
---|---|---|---|---|---|---|---|

1992 | 1.117 | 0.277 | 0.000 | 0.436 | 3.917 | 1.010 | 0.247 |

1993 | 4.194 | 2.125 | 0.017 | 3.531 | 3.540 | 2.497 | 1.986 |

1994 | 0.127 | 1.210 | 3.918 | 0.381 | 0.146 | 2.173 | 2.711 |

1995 | 0.515 | 1.512 | 0.800 | 0.439 | 0.269 | 0.681 | 1.094 |

1996 | −0.060 | 0.172 | −0.174 | 0.325 | 0.454 | 0.633 | 0.073 |

1997 | 0.373 | 0.150 | 0.758 | 0.700 | −0.368 | 0.398 | 0.395 |

1998 | −0.338 | 0.227 | 2.064 | 0.215 | −0.828 | −0.019 | 0.469 |

1999 | −0.059 | 0.143 | 0.246 | 0.221 | 4.973 | 0.178 | −0.009 |

2000 | 0.898 | 0.299 | −0.036 | 0.324 | 1.475 | 0.259 | 0.025 |

2001 | −0.129 | 0.395 | 4.949 | 0.338 | 0.211 | 0.264 | 0.271 |

2002 | 0.123 | 0.008 | 0.951 | 0.073 | −0.090 | 0.768 | 0.061 |

2003 | 0.347 | 0.199 | 0.372 | 0.065 | 0.328 | 0.194 | 0.128 |

2004 | 0.477 | 0.259 | 0.249 | 0.285 | 0.370 | 1.017 | −0.022 |

2005 | 0.239 | 0.165 | 0.102 | 0.109 | 0.274 | 0.348 | 0.120 |

2006 | 1.671 | 0.171 | 0.092 | 0.302 | 0.265 | 0.285 | 0.595 |
---|---|---|---|---|---|---|---|

2007 | 0.360 | 0.434 | 0.195 | 0.167 | 0.422 | 0.586 | 0.628 |

2008 | 0.164 | 0.429 | 0.114 | 0.365 | 0.374 | 0.140 | −0.261 |

2009 | −0.311 | 0.149 | 0.213 | −0.095 | −0.176 | 0.032 | 0.336 |

2010 | 0.874 | 0.257 | 0.238 | 0.701 | 0.667 | 0.541 | 0.163 |

2011 | −0.145 | 0.429 | 0.242 | 0.658 | 0.745 | 0.848 | 0.287 |

2012 | 0.045 | 0.424 | 0.279 | 0.145 | −0.030 | 0.164 | 0.213 |

2013 | 0.273 | 0.297 | 1.005 | 0.169 | 0.165 | 0.116 | 0.279 |

2014 | −0.007 | 0.050 | 0.139 | −0.029 | −0.068 | 0.167 | 0.291 |

2015 | 0.063 | 0.132 | 0.725 | −0.120 | −0.098 | 0.012 | −0.258 |

2016 | −0.109 | 0.132 | 0.210 | 0.205 | 0.025 | −0.332 | −0.004 |

2017 | 0.546 | 0.124 | 0.253 | 0.088 | 0.560 | 0.001 | 0.160 |

2018 | 0.293 | 0.227 | 0.078 | 0.378 | 0.332 | 0.214 | 0.071 |

COVAR (X) | X_{1} | X_{2} | X_{3} | X_{4} | X_{5} | X_{6} | X_{7} |
---|---|---|---|---|---|---|---|

X_{1} | 0.7692 | 0.2684 | −0.2538 | 0.5033 | 0.5707 | 0.3391 | 0.2542 |

X_{2} | 0.2684 | 0.2266 | 0.1035 | 0.2415 | 0.1775 | 0.2269 | 0.2444 |

X_{3} | −0.2538 | 0.1035 | 1.4036 | −0.0608 | −0.3865 | 0.1362 | 0.3116 |

X_{4} | 0.5033 | 0.2415 | −0.0608 | 0.4410 | 0.4042 | 0.2934 | 0.2267 |

X_{5} | 0.5707 | 0.1775 | −0.3865 | 0.4042 | 1.7830 | 0.3059 | 0.0925 |

X_{6} | 0.3391 | 0.2269 | 0.1362 | 0.2934 | 0.3059 | 0.3925 | 0.3184 |

X_{7} | 0.2542 | 0.2444 | 0.3116 | 0.2267 | 0.0925 | 0.3184 | 0.4095 |

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.

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

Name | Initial value | Optimal value | Change |
---|---|---|---|

Income tax | 0.255 | 0.227 | −2.8% |

Social security contributions | 0.118 | 0.115 | −0.3% |

Property taxes | 0.005 | 0.018 | 1.3% |

Taxes on domestic goods & services | 0.296 | 0.194 | −10.2% |

Taxes on foreign trade | 0.063 | 0.040 | −2.3% |

Other taxes | 0.071 | 0.147 | 7.6% |

Non-tax revenue | 0.192 | 0.260 | 6.8% |

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.

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

The authors declare no conflicts of interest regarding the publication of 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