OPTIMIZATION PROBLEM OF FOREIGN RESERVES

Transcription

1 Advanced Math. Models & Applications Vol.2, No.3, 27, pp OPIMIZAION PROBLEM OF FOREIGN RESERVES Ch. Ankhbayar *, R. Enkhbat, P. Oyunbileg National University of Mongolia, Ulaanbaatar, Mongolia Abstract. In this work, we propose non-portfolio optimization problem with one stochastic factor in continuous time. Based on Merton s investment model [9], we construct a mathematical model for foreign reserves of Mongolia. We reduce it to econometric model and estimate statistical parameters. Keywords: Merton model, Hamilton-Jacobi Bellman equation, Bernoulli equation, the first-order autocorrelation. AMS Subject Classification: 37N4. Corresponding Author: Ankhbayar Chuluunbaatar, National University of Mongolia, University district, 2646, Ulaanbaatar, Mongolia, ankhaa.22@gmail.com Manuscript received: 6 June 27. Introduction here is a lot of research papers on strategic asset allocations but less attention has been paid foreign reserves allocation. Particularly, the notion strategic asset allocation was introduced in Brennan et al. [3] to describe the portfolio optimization problem with time-varying returns and long-term investor objectives. In general, the problem of long-term investments is a well-established research field introduced by Samuelson [] and Merton [8], respectively. In the continuous time, using stochastic optimal control theory, Merton [8] has developed the problem of lifetime portfolio selection under uncertainty. In [8] important financial economic principles were established, but there are no explicit results for portfolio choice problems. Merton s paper [9] highlights the difficulties in solving complex cases of assets dynamics with stochastic factors. Advances in numerical techniques and the growth of computing power led to the development of numerical solutions to multiperiod portfolio optimization problems, which are solved by a discrete state approximation [3,4]. However, the use of numerical dynamic programming is very often restricted to few factors, due to the fact that the algorithms use excessive computation time and become numerically unreliable for high dimensions. herefore, closed-form solutions of the Merton model in continuous time with two or three stochastic factors are given in [5, 6, ]. In [2], Bielecki et al. present a closed-form solution to the portfolio optimization problem in continuous time for multiple assets and multiple factors with an infinite time horizon. Under the assumption of the uncorrelated residual of the asset prices and factors, they find the optimal portfolio allocation decision for many assets and many factors. In [2], general analytical solutions Merton s type 259

2 ADVANCED MAH. MODELS & APPLICAIONS, V.2, N.3, 27 consumption-investment problem were obtained. Brownian motion with a white noise was examined in [2]. Based on [7], we suppose non-portfolio problem in the continuous time with one stochastic factor controls optimal value of foreign reserves when the exchange rates fluctuate. 2. Merton s type consumption-investment problem In our case the investor lives from to time ; his wealth at time t is denoted by. he investor starts with a known initial wealthx. At time tthe investor must choose what fraction of his wealth to consume c t and what fraction to invest in the riskless and risky portfolio. Let s assume that F t, c t is the period utility function for consumption at time t, and Φ, X is the boundary condition. hen the optimization problem for the investor is max u,u,c E F t, c t dt + Φ, X, () where E is the expectation operator, is the planning horizon. Suppose there are two assets.. he investor can invest money in the bank at the deterministic short rate of interest r, i.e, the investor has access to the risk free asset B with db t = rb t dt. (2) he investor can invest in a risky asset with price process S, where we assume that the S-dynamics are given by a standard Black-Scholes model ds t = αs t dt + σs t dw, (3) where α: drift function, σ: diffusion coefficient, W t : standard Brownian motion. If we assume that, at time t the investor owns n, n 2 units of asset, then the total wealth is determined by = n t B t + n 2 t S t. (4) he portfolio n, n 2 remains unchanged over the time interval [t, t + t[, and assuming the consumption pattern is constant in interval t, t + t in the same way as for portfolio selection, the budget equation becomes d = n t db t + n 2 t ds t c t dt (5). Let u t = n t B t, u t = n 2 t S t, (6) be the share of wealth in assets, with u t + u t =. (7) hen the budget equation (5) can be written as d = u t r + u t α dt c t dt + u t σ dw t. (8) Now, we may formally state the consumer s utility maximization problem a stochastic optimal control problem max u,u,c E F t, c t dt + Φ, X, d = u t r + u t α dt c t dt + u t σ dw t, (9) 26

3 Ch. ANKHBAYAR et al.: OPIMIZAION PROBLEM OF FOREIGN RESERVES X = x, c t, t, u t + u t =, t, where X is a state process, u, u and c are control variables. 3. Foreign reserves problem In problem (.9), we assume the following.. here is only one stochastic factor described by the risky asset. 2. Social welfare is increasing due to foreign reserves. 3. Changes of foreign reserves is not constant. here are u t =,u t = w t for (), F t, c t = e δt γ R t c t = βr t for (3), and Φ, X = from (4). According to assumptions, (.9) can be written as follows max w,r E e δt R t γ dt, for (2), d = w t α dt βr t dt + w t σ dw t, () X = x, R t, t, w t, t. he Hamilton-Jacobi-Bellman (HJB) equation has the following form V + sup t R,w R e δt R γ t + wαx V V βr + X X 2 w2 σ 2 X 2 2 V X 2 =. () Let R and w be solution problem (). Assuming an interior solution, the first order conditions are γr γ = βe δt V X, (2) αv X w =. (3) Xσ 2 V XX aking into account assumption (2), we have V t, X = e δt hx γ. he boundary conditions require that Φ, X =. Also, we can get the following equations V = t e δt h X γ δe δt hx γ, (4) V X = γe δt hx γ, (5) 26

4 ADVANCED MAH. MODELS & APPLICAIONS, V.2, N.3, 27 2 V X 2 = γ γ e δt hx γ 2. (6) Substituting (5) into (2), (6) and (3), we get R t, X = βh t / γ X, (7) w t, X = α γ σ 2. (8) If we substitute the expressions (4) and (7)-(8) into the HJB equation, we obtain: X γ h t + Ah t + Bh t γ/ γ =, (9) where A and B are the constants given by A = α 2 γ δ, 2 γ σ 2 B = γ β γ/ γ. If we search yin the following formy = h / γ, we get We can easily see that y t = y B A A e γ t + B. (2) A lim y t = B t A. Now we can find the optimal value of foreign reserves using the expression (2) R t, X = β / γ X A. (2) B From here we find expected foreign reserves: E R = β / γ A E X. (22) B 4. Estimation of econometric model From (.), a stochastic differential equation is written as d R t = w t αdt β dt + w t σdw t. In order to construct econometric model, assume that w t = w and wσdw t = u t. hen we have the following econometric equation: dln = wα β R t + u. t First-order autocorrelation is: u t = ρu t + e t. where ρ can be computed ρ = Corollary. If s, t are positive, then Cov wσdw t,wσdw s Var wσdw t. (23) 262

5 Ch. ANKHBAYAR et al.: OPIMIZAION PROBLEM OF FOREIGN RESERVES Cov dw t, dw s = Var dw t δ s t for the differential process, where δ s t is continuous Dirac delta function. If we set δ as δ s t =, then ρ becomes ρ = wσ Cov dw t, dw s w 2 σ 2 = Var dw t δ s t = Var dw t wσ Var dw t wσ. Corollary 2. Let R t be a foreign reserves with an exponential growth R t = τ R eγ t. If objective function is hen, there exists τ such τ > δ. Proof follows from max w,r E R max w,r E e r it R γ e τt dt E R = β A γ E X = β γ B e δt R t γ dt >. = Rγ τ δ e τ δ >. If τ δ >, the objective function is positive which proves the assertion. Changes in (.8) can be written: w αw w = γ σw 2. From the above, we have γ = αw σw 2 = αwρ2. Optimal level of foreign reserves is α 2 γ δ 2 γ σ 2 γ β γ γ E X = = β γ 2 α γ σ 2 αγ δ E X = β γ αwγ τ E X Computational results For our econometric model, we used observations of foreign reserves and exchange rate for 32 months during he first-order autocorrelation estimation gives the following values: d = R t AR. If we look at estimation results, all parameters of the equation are statistically significant. he quarterly average growth of foreign reserves is τ γ =

6 ADVANCED MAH. MODELS & APPLICAIONS, V.2, N.3, 27 Also, expected exchange rate level is E X = hus, we final all parameters as follows wα =.82, β =.72, ρ =.957. Dependent Variable: (USD-USD(-2))/USD(-2) Method: Least Squares Sample (adjusted): 26M 26M2 Included observations: 32 after adjustments Convergence achieved after 8 iterations Variable Coefficient Std. Error t-statistic Prob. C RES/USD AR() R-squared.9833 Mean dependent var.5939 Adjusted R-squared.9687 S.D. dependent var.885 S.E. of regression.3324 Akaike info criterion Sum squared resid.423 Schwarz criterion Log likelihood Hannan-Quinn criteria F-statistic Durbin-Watson stat Prob(F-statistic). Inverted AR Roots.96 If we compute then γ = =.8333, τ = =.55 and expected the optimal value of foreign reserves is obtained by 2.5 billion USD. = 6. Conclusion E R = β γ αwγ τ E X = We solved non-portfolio optimization problem with one stochastic factor in continuous time. his problem reduced to an econometric model with significant statistical parameters. Using this result, we evaluated optimal value of foreign reserves of Mongolia. 264

7 Ch. ANKHBAYAR et al.: OPIMIZAION PROBLEM OF FOREIGN RESERVES References. Batkhurel, С. (2). he Optimal level of foreign reserves in Mongolia. Mongolbank, Research working paper, 6, Bielecki,.R., Pliska, S.R., Sherries, M. (2). Risk-sensitive asset allocation. J. Econ. Dynam. Control, 24, Brennan, M.J., Schwartz, E.S., Lagnado, R. (997). Strategic asset allocation. J. Econ. Dynam. Control, 2, Brennan, M.J., Schwartz E.S. (999). he use of reasury bill futures in strategic asset allocation programs. In: W.. Ziemba and J.M. Mulvey (Eds.), Worldwide Asset and Liability Modelling, Chapter, Cambridge University Press, Cambridge, UK. 5. Campbell, J.Y., Chacko, G., Rodriguez, J., Viceira, L.M. (23). Strategic asset allocation in a continuous-time var model. J. Econ. Dynam. Control, 28, Haugh, M.B., Lo, A.W. (2). Asset allocation and derivatives. Quant. Finan.,, Nyamsuren, D., Batsukh, s. (22). Merton s type portfolio optimization problem in thefinite-horizon case with HARA utility function and proportional transaction costs, explicit solution, International Journal of Mathematical Archive, 3(), Merton, R.C. (969). Lifetime portfolio selection under uncertainty, he continuous case. Rev. Econ. Stat., 5, Merton, R.C. (97). Optimum consumption and portfolio rules in a continuous-time model. J. Econ. heory, 3, Munk, C., Sørensen, C., Vinther,.N. (23). Dynamic asset allocation under mean reverting returns, stochastic interest rates, and inflation uncertainty. Proceedings of the 3th Annual Meeting of the European Finance Association, Glasgow.. Samuelson, P.A. (969). Lifetime portfolio selection by dynamic stochastic programming. Rev. Econ. Stat., 5,