Ilona V. Tregub, ScD., Professor


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1 Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation Abstact In this pape a simulation model of the fomation of an effective investment potfolio had estimated fo the diffeent conditions and states of the economy on the basis of selected stocks of lage and eliable companies with stable ates of eanings in the Russian and Ameican makets. Futhe wok is aimed at assessing the method of simulation time inteval duing which compiled a potfolio is efficient. In the Russian Fedeation duing the last 10 yeas, measues wee taken to impove the pension system. Howeve, as pat of the pension efom is not achieved longtem financial stability and balance of the Pension Fund. Funded pilla of the pension system equies substantial modenization, especially as it elates to the financial sustainability of pension funds and the ceation of an effective system of investment, as well as expanding the list of financial institutions and instuments. In this aticle we ae going to constuct a potfolio of the pension fund based on estimates futue pofitability of financial assets. Potfolio theoy was initiated by economist Hay Makowitz and imagined MP in Moden potfolio theoy (MP) is a theoy of investment which attempts to maximize potfolio expected etun fo a given amount of potfolio isk, o equivalently minimize isk fo a given level of expected etun, by caefully choosing the popotions of vaious assets. Although MP is widely used in
2 pactice in the financial industy and seveal of its ceatos won a Nobel memoial pize fo the theoy. MP is a mathematical fomulation of the concept of divesification in investing, with the aim of selecting a collection of investment assets that has collectively lowe isk than any individual asset. hat this is possible can be seen intuitively because diffeent types of assets often change in value in opposite ways. Moe technically, MP models an asset's etun as a nomally distibuted function (o moe geneally as an elliptically distibuted andom vaiable), defines isk as the standad deviation of etun, and models a potfolio as a weighted combination of assets, so that the etun of a potfolio is the weighted combination of the assets' etuns. By combining diffeent assets whose etuns ae not pefectly positively coelated, MP seeks to educe the total vaiance of the potfolio etun. MP also assumes that investos ae ational and makets ae efficient. In othe wods we take this model because it is a theoy on how iskavese investos can constuct potfolios to optimize o maximize expected etun based on a given level of maket isk, emphasizing that isk is an inheent pat of highe ewad. he futue values of the assets will ae estimated using the Shape Model. he Shape Model is based on the method of linea egession analysis to link the two andom vaiables. he independent vaiable X and the dependent vaiable Y connected by a linea expession. In the Shape Model independent vaiable is the maket index. Fo the dependent vaiable takes the pofitability of the financial instument. A fomal statement of the Shape Model is given below Whee i is pofitability of secuity i fo given peiod of time, I is pofitability fo maket indices I fo the same peiod, ε ii is the distubance tem, α ii, β i1 ae the coefficients.
3 o achieve the goal wee selected stocks of lage, liquid and eliable companies with stable income to the Russian and Ameican makets fo the peiod fom 2006 to 2012 able 1. List of Assets he Russian Maket (GAZP) (GMKN) (LKOH) (SIBN) (AFL) he U.S. Maket Apple Inc. (AAPL) Cisco Systems, Inc. (CSCO) Google Inc. (GOOG) Micosoft Copoation (MSF) CocaCola (KO) he basic assumptions of the model ae following. 1. Investos ae assessing investment potfolios based on the expected etun and standad deviation fo the peiod of owneship. 2. hee is a iskfee inteest ate at which the investo can lend (invest), o take the money. 3. axes and tansaction costs ae insignificant. 4. Investos ae neve ovesatuated. When you select a potfolio they choose the one that gives the highest expected etuns (at equal isk). 5. Investos do not want to isk it. hey will select a potfolio that has the lowest standad deviation (with equal expected etuns.) he expected etun of the potfolio is п 5 = E( Rп ) = i= 1 ν ii = ν
4 Potfolio isk is defined by the fomula σ п 5 i i. j= 1 = ν ν ρ σ σ j ij i j = ν V ν whee ρ ij is coelation coefficient between andom vaiables R i and R, V is j covaiance matix of the andom vaiables R i. In matix fom, the poblem can be witten as ν ν V ν min ( I ) = 0 * 0 whee 0 is iskfee ate of etun of the asset (fo the study will take it at 4%), *  given the expected ate of etun (equal to 12%), I = (1,...,1) is column vecto of dimension 5. he fomula fo detemining the effective inteest of the fom * 0 1 ν = V ( 0 I) 1 ( 0 I ) V ( 0 I ) he effective potions of shaes fo the potfolio ae calculated by this fomula. Fo the investment of pension fund potfolios is impotant to know the peiod duing which the potfolio will be composed effective. o detemine this peiod wee consideed diffeent options fo the shaes fo each asset and calculate the mean etun and isk. Results fo some peiods ae pesented in able 2 fo the taining sample and in able 3 fo the contol sample.
5 able 2. he esults on the leaning sample Peiod of time Pofit Risk Russia U.S. Russia U.S. Вefoe the Сisis 12% 12% Duing the Сisis 12% 12% Afte the Сisis 12% 12% able 3. he esults fo the contol sample Peiod of time Risk Russia Daily Pofit Вefoe the Сisis the Leaning Sample U.S. Annual Daily Pofitabiliy Peiod of time Risk Pofit Annual Pofitabiliy 12 months % 12% 12 months % 12% the Сontol Sample 1 month % % 1 month % 2.891% 2 months % % 2 months % 0.850% 3 months % % 3 months % 6.586% 6 months % % 6 months % 7.501% 9 months % % 9 months % 9.665% 12 months % % 12 months % 8.270% Duing the Сisis the Leaning Sample 12 months % 12% 12 months % 12% the Сontol Sample 1 month % 5.971% 1 month % % 2 months % % 2 months % % 3 months % % 3 months % % 6 months % % 6 months % % 9 months % % 9 months % % 12 months % % 12 months % %
6 Afte the Сisis the Leaning Sample 12 months % 12% 12 months % 12% the Сontol Sample 1 month % % 1 month % 0.85% 2 months % 9.76% 2 months % 10.46% 3 months % 3.49% 3 months % 9.89% 6 months % 0.14% 6 months % 11.65% 9 months % 1.08% 9 months % 11.93% 12 months % 4.00% 12 months % 9.28% As can be seen fom these esults, in both cases, a potfolio designed fo the contol sample was efficient. It should be noted that if an investo wants to get the same ate of etun, which he used in the constuction of the potfolio using the taining set, it must ceate a potfolio of U.S. stocks and keep it 6 months. In compiling the potfolio at 12 months in the U.S. investo gets a lowe yield (9.283%). Based on the esults, we can daw the following conclusions: Investing in U.S. stocks, to obtain the desied yield (12%) duing the cisis and afte the cisis, the investo will have to take on moe isk than investing in Russian equities. In Russia befoe the cisis inceased the expected etun fo peiods 1,3,6 and 9 months (an incease of 5% to 37%). Only fo a peiod of 12 months, this indicato deceased (1%  looming cisis). he isk fo the entie contol sample deceased. his is due pimaily to the gowth of the entie economy of Russia ( yeas). heefoe, the seies in question do not pass the test fo stationay. Howeve, the potfolio emained effective thoughout the contol sample. In the United States befoe the cisis thee was a decline, both isk and expected etun. As well as educing yield was low (5% to 4%), the pepaed potfolio still emains effective. hus, the aticle discussed the constuction of
7 efficient potfolio can be used fo economic gowth fo the peiod of 9 months in Russia, and up to a yea in the U.S. Duing the cisis, nealy all occasions potfolio was ineffective. hee was a negative etun of 19% to 12%. hus, duing the cisis potfolio can only be used fo puchase of shaes fo 1 month in the U.S. maket. In the postcisis peiod, almost all of the time seies wee stationay. In almost all cases the Russian maket potfolio was ineffective because instability has affected the Russian economy. he evese situation was obseved in the United States. It is impotant to note that fo a peiod of 6 and 9 months the potfolio etun was 12% pe annum, that is the same as in the taining set. hus, if an investo wants to get the same ate of etun, which he used in constucting the potfolio, the taining set, it must ceate a potfolio of U.S. stocks and keep it 6 o 9 months. he potfolio must be ceated in a steadystate peiod.
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