Faculty of Computational Mathematics and Cybernetics
The Department of Operations Research
(Dr. T. Belyankina, Head of section, Insurance Control Department, Ministry of Finance;
Dr. P. Vasina, Consultant CARANA Ltd.)
Introduction to Econometrics. The cross-sectional and time-series data. Samples. Explanatory and explained variables. Two-dimension linear regression model. The ordinary least square method. Classical Normal Linear regression Model. The concepts of homoscedasticy, heteroscedasticity, serial correlation. Gauss-Markov Theorem. A concept of the best linear unbiased estimator. The estimation of the error variance. Necessary statistical distributions. Statistical characteristics of regression coefficients. Confidence intervals for regression coefficients. The determination coefficient (R2). The maximum likelihood method. The multiply regression model. The ordinary least square method (OLS). Statistical characteristics of the OLS-estimators. The determination coefficient for multiply regression model. The adjusted determination coefficient. Testing the statistical hypotheses. Chow Test. The problem of multicollinearity. Dummy variables. Model specification. The Generalized Least Squares Method. Heteroscedasticity. Heteroscedasticity tests. Time correlation. Autoregressive model. Durbin-Watson test. Introduction to time-series analysis. The forecasting. Systems of regression equations.
The econometric modeling of time series
(Dr. T. Belyankina, Head of section, Insurance Control Department, Ministry of Finance;Dr. P. Vasina, Consultant CARANA Ltd.)
The course aims to provide students with basic models and methods of econometric modeling of financial time series. Such kind of data is typical for economics, its analysis is of great practical interest and has some specifics. Econometric modeling of time series serves to draw out economical (natural) laws from empirical observations, to make forecasts, to interpret data and to verify the hypotheses correspondent to economic time series. The course considers in detail the major methods of econometric modeling of time series, in particular modeling by stochastic processes, modeling of stationary and non-stationary time series, the spectrum analysis of time series. The course studies different aspects of existing methods, their properties, adequacy, and application to forecasting. The necessary background is the basic knowledge of linear algebra, mathematical analysis, probability theory, and mathematical statistics.
Course outline: The stochastic processes. The stochastic difference equations. Modeling of stationary time series (autoregressive moving average models). Models of time series which include conditional heteroscedasticity. The spectrum analysis of time series. Modeling of non-stationary time series.
(Professor I. Pospelov, Dr. I. Pospelova, CMC MSU and Computing Center of RAS )
In this course, the foundations of mathematical description of economy are studied. The course can be divided into two parts. The first part is devoted to main macroeconomic concepts, indexes, indicators, and relations. In particular, monetary policy and national budget are studied in detail. In the second part, mathematical description of economic relations and macroeconomic models are studied.
(A.A. Vasin, Professor of CMC MSU and New Economic School, Full member of RANS)
I. Normal form games. Condition for the Nash equilibrium existence. Mixed strategies. Computation of Nash equilibrium. Dominance elimination and Nash equilibrium. Positional games with complete information. SPE. Utility function. Models of adaptive behaviour.
II. Models of one-good market. Cost functions. Supply function and demand function. Competitive equilibrium principle. Conditions of the perfect competition. Optimal strategy of monopoly. Welfare theorems .
III. Bertrand model of imperfect competition. Residual demand functions. Relation of NE to the Cournot outcome and CE. Nash equilibrium: necessary and sufficient conditions. Cournot oligopoly. Nash equilibrium and competitive equilibrium. Taxes. The problem of tax optimization under a given state budget. Model of tax inspection organization. Optimal audit strategy. Model with corruption. Model of stimulation of auditors.
Mathematical models of imperfect competition and tax optimization
(A.A. Vasin, Professor of CMC MSU and New Economic School, Full member of RANS)
I. Models of imperfect competition. Cournot oligopoly. Competition via supply functions. Relation of NE to the Cournot outcome and CE. Bertrand-Edgeworth oligopoly. The sets of surviving prices for different rationing rules. Successive setting: prices after quantities. Condition of equivalence to the Cournot oligopoly. Quantities after prices. Equivalence to the model of price competition. Duopoly with variable prices. Adjustment of production capacities. Correspondence of SPE to the monopoly price.
II. Tax optimization under tax evasion. Social welfare optimization. Welfare theorems. MEBs of taxes. Income tax optimization. Participation and penalty constrains. Evasion proofness. Taxation of firms. Presumptive and s a les taxes. The optimal auditing rule under given tax rates. Optimality of the linear presumptive tax dependent on the production capacity.
III. Models with corrupted auditors. The optimal behavior of agents and net tax revenue depending on the tax authority strategy. The optimal enforcement strategy. A model with firing of corrupted inspectors. Optimal system of premiums for inspectors. Impact of random mistakes by taxpayers and audit costs for inspectors on the optimal tax enforcement strategy.
(Denisov D.V., Professor of CMC MSU, Director of Russian Society of Actuaries)
I. The Utility theory. Utility function. Decision marking in insurance. Two principles: expected payoff and expected utility. Optimal insurance. Optimality of stop-loss insurance.
II. Individual risk model. Methods for computation of total claim distribution. Normal approximation for this distribution.
III. Collective risk model. Properties of compound Poisson and negative binomial distributions. Recurrent formula for distribution of aggregate claim in case of discrete individual claims. Methods of approximation of compound Poisson distribution.
IV. The ruin theory. Here we consider the formula for probability of ruin in continues and discrete cases. The exact formula for the exponential distributions and the mixture of exponential distributions.
V. Approximation of distribution of the sum of independent claims by compound Poisson distribution. The theory of reinsurance for compound Poisson distribution. Optimality of the stop-loss reinsurance.
Game theory and operations research
(V.V. Morozov, Professor of CMC MSU)
Zero-sum two-player game and its solution. Condition of the existence of a saddle point. Computation of all saddle points. Minimax and maximin strategies existence conditions. Existence of a saddle point of convex-concave function. Zero-sum game in mixed strategies. Von Neumann theorem for matrix games. The properties of optimal mixed strategies. Dominance of rows and columns in matrix games. Graphical method for solution of 2xn and mx2 matrix games. Matrix games and linear programming. Iterative method of matrix games solution. Games with complete information. Zermelo theorem on games with complete information. Nash equilibrium existence condition. Cournot model of duololy. Properties of mixed equilibrium in two-player finite game. Computation of mixed equilibrium. Hierarchical games. Stackelberg equilibrium.
(G.A. Belyankin, Professor of CMC MSU, Head of Actuarial and Investment Dpt.,Ost-West Allianz)
Interest Rate and Discount Rate. Present Value. Individual Risk models. Survival Distributions and Mortal ity Tables. Analytical Laws of Mortality. Death Benefits Payable at the End of the Year of Death. Death Benefits Payable at Moment of Death. Continuous Life Annuities, Annual Annuities and Annuities m-thly payments. Single Premiums. Annual Premium. Premiums with m-thly payments. Premium Calculation for Different Insurance Programs. Benefit Reserves. Calculation of Reserves for Different Insurance Programs. Recursion Relations for Benefit reserves. Expenses in Insurance Contracts. Expense Allowances. Proportional and Non-Proportional Expenses. Gross Premium Calculation. Zillmerisation. Cash Values. Insurance Options. Fully variable life insurance. Calculation of Premium and Resererve for Fully variable life insurance.
(G.A. Belyankin, Professor of CMC MSU, Head of Actuarial and Investment Dpt.,Ost-West Allianz)
Product Pricing. Analysis of Mortality, Expenses, Withdrawals and Financial Factors. Policy Signature. Policy Profile. Profit testing. Solvency requirements. Interest Rate as Random Value. Investment Strategy. Asset Liability Match. Determining Bonus Distribution Policy. Product Business Plan. Assigned Capital. Embedded Value. Gross Reserves. Deferred Acquisition Cost. Actuarial Basis. Unit Linked Contracts. Variable Life and Universal Life Contracts.
Dependent Lifetime Models. Multiple Life Functions. Special Two-Life Annuities. Multiple Decrement Models. Lump sum Benefits. Disability Benefits. Waiver of Premium Benefits. Theory of Pension Funding. Basic Functions for Retired Lives. Basic Functions for Active Lives
(Denisov D.V., Professor of CMC MSU, Director of Russian Society of Actuaries)
I. Introduction. Optimization problems: setting and classification. Conditions for existence of a global solution.
II. Linear programming. Linear programming theory. Basic methods.
III. Nonlinear programming. Elements of convex analysis. First-order optimality conditions for optimization problems with convex feasible sets. First- and second-order optimality conditions for unconstrained optimization problems, for purely equality-constrained problems (the Lagrange principle), and for mixed-constrained problems (the Karush-Kuhn-Tucker condition).
(Izmailov A.F., Professor of CMC MSU, Senior researcher of Computing Center of RAS)
I. Introduction. Optimization methods: classification. Types of convergence. Convergence rate estimates. Stopping criteria. Algorithms for one-dimensional problems.
II. Unconstrained optimization methods. Descent methods. The Newton method, quasi-Newton methods. Conjugate direction methods. Methods of order zero.
III. Constrained optimization methods. Methods for problems with simple constraints (gradient projection methods, conditional gradient methods, conditional Newton methods). Feasible direction methods. Methods for purely equality-constrained problems (Newton-type methods for Lagrange optimality system, quadratic penalty method, augmented Lagrangians and exact penalty functions).
(Izmailov A.F., Professor of CMC MSU, Senior researcher of Computing Center of RAS)
I. Contemporary optimization methods. Sequential quadratic programming. Equivalent reformulations of the Karush-Kuhn-Tucker optimality system, elements of nonsmooth analysis, the generalized Newton method. Identification of active constraints, error bounds. Penalties and augmented Lagrangians for mixed-constrained problems.
II. Globalization strategies. Linesearch. Trust-region methods. Pathfollowing methods. Globalization of sequential quadratic programming methods.
III. Methods for nonsmooth convex optimization. Elements of subdifferential calculus. Lagrangian relaxation. Subgradient methods. Cutting plane methods. Bundle methods.
IV. Special optimization problems. Methods for quadratic programming problems. Interior point methods for linear programming problems.
(M.R. Davidson, Professor of CMC MSU, Consultant CARANA Ltd.)
The course gives a thorough treatment of mathematical theory of interest. It includes both financial and economical aspects of the theory of interest. The interrelationship between assets and liabilities and quantitative methods used to manage this relationship are studied. The techniques of duration analysis, immunization and scenario testing are considered.
The first part of the course takes a mostly deterministic approach to interest rates and yield curves. Discounted cash flow analysis and capital budgeting as decision tools are given particular emphasis. The second part discusses Capital Asset Pricing Model as well as some approaches to pricing modern non-linear financial instruments.
The course contains a large number of exercises and real world examples.