Магистерское и послевузовское образование

1. Algebraic characterization of basic solution of linear program.
2. Simplex method.
3. Bland’s rule.
4. Other forms of linear program.
5. Dual linear program.
6. Duality theorem.
7. Dual simplex method.
8. Simplex method for linear program with bilateral constraints.
9. Solution of transport problem.
10. Branch and borders method for discreet problem.
11. Balash’s method for boolean program.
12. Dynamic programming.

Lecturer –V.V. Morozov, Professor of CMC MSU.
The 3rd term..
Intensity of the course – 2 hour a week, 1 lecture
Requirements for the course - exam

The main purpose of the course is to develop basic concepts of Actuarial Mathematics and apply them to the study of prospective loss functions, reserves calculations, multiple life functions and multiple decrement models. This course is modified. The main modifications concern the theory of prospective loss functions and reserves. The decomposition of prospective loss functions is studied in concern with cash flows in different kinds of insurance: term insurance, endowment insurance, pure endowment and so on. Moreover, the recursive formulas are studied not only for reserves, but for variations of prospective loss functions. Especially in continuous case, many interesting examples of prospective loss functions are considered. In these examples the decomposition of prospective loss functions is studied with help of differential equation of reserves. In reserve theory the problem of recalculation of the insured sum is examined for the case of changing police condition. The problem of gross premiums calculation is studied with accounted inflation and different kinds of bonuses. The last part of the course discusses prospective loss functions in the general case where the premiums are not constant.

Lecturer –A.A. Belolipeckii., Professor of CMC MSU; Denisov D.V., Professor of CMC MSU, Director of Russian Society of Actuaries
The 1st term..
Intensity of the course – 2 hour a week, 1 lecture 
Requirements for the course - test

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).

Lecturer – Izmailov A.F., Professor of CMC MSU, Senior researcher of Computing Center of RAS.
The 1st term..
Intensity of the course – 2 hour a week, 1 lecture 
Requirements for the course - exam

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.

Lecturer – M. Davidson, Professor of CMC MSU, Consultant CARANA Ltd.
The 1st term..
Intensity of the course – 2 hour a week, 1 lecture
Requirements for the course - test

In the first part of the course we present a modern approach to modeling economic systems. This approach combines a methodology developed for complex systems modeling with achievements of the modern economic theory. The course consists of several sections, in which we describe a language of material balances that present a basis both for economic statistic and for the most of macroeconomic models. Further for better understanding of essence and problem of constructing economic models we consider main ideas of the economic theory and basic economic concepts, phenomena, and economic interactions. Then we investigate in details an example of a one-product dynamic model of real economy that includes such economic agents as a producer, a consumer, a trader, a bank, a government and a state bank. This part of the course ends with discussion of different economic models used practically for estimation of Russian economy evolution in various historical periods.
In the second part we present main approaches to description of productive relations and consumer behavior. We start with classical static multi-sector production models and their dynamic analogues. We consider models of Leontief and von Neumann types. Also we describe approaches for analysis of behavior of dynamic systems. Then we study element of the demand theory. After that we turn to production functions studied both in classical macroeconomic form and in the form of the Houthakker-Johansen model. The course ends with the description of new information techniques specially designed for support of economic modeling process.

Lecturer – Professor I. Pospelov, Dr. I. Pospelova, CMC MSU and Computing Center of RAS
The 1st and 2nd terms..
Intensity of the course – 2 hour a week, 1 lecture
Requirements for the course – test, exam

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.

Lecturer – Dr. P. Vasina, Consultant CARANA Ltd
The 2nd term..
Intensity of the course – 4 hour a week, lectures and exercises 
Requirements for the course – test, exam

I. Normal form games. Condition of the of 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. Walrasian equilibrium optimality.
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 financing budget. Model of tax inspection organization. Optimal audit strategy. Model with corruption. Model of stimulation of auditors.

Lecturer – A.A. Vasin, Professor of CMC MSU and New Economic School, Corr. member of RANS; V.V. Morozov,, Professor of CMC MSU
The 3rd term..
Intensity of the course – 4 hour a week, lectures
Requirements for the course – exam

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).

Lecturer – Izmailov A.F., Professor of CMC MSU, Senior researcher of Computing
Center of RAS
The 3rd term..
Intensity of the course – 2 hour a week, lectures
Requirements for the course – test

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.

Lecturer – Dr. T. Belyankina, Head of section, Insurance Control Department,
Ministry of Finance; Dr. P. Vasina, Consultant CARANA Ltd.
The 4th term..
Intensity of the course – 4 hour a week, lectures
Requirements for the course – test, exam

Mathematical models of imperfect competition and tax optimization

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 soles 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.

Lecturer - A.A. Vasin, Professor of CMC MSU and New Economic School, Corr. member of RANS

Risk theory

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.

Lecturer - Denisov D.V., Professor of CMC MSU, Director of Russian Society of Actuaries

Actuarial Math II

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

Lecturer - G. Belyankin, Professor of CMC MSU, Head of Actuarial and Investment Dpt., Ost-West Allianz

Advanced chapters of optimization methods

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.

Lecturer - Izmailov A.F., Professor of CMC MSU, Senior researcher of Computing
Center of RAS.

The econometric modeling of time series

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.

Lecturers - Dr. T. Belyankina, Head of section, Insurance Control Department,
Ministry of Finance; Dr. P. Vasina, Consultant CARANA Ltd.

The 2nd, 3rd and 4th terms.
Intensity of the course – 2 hour a week, lectures
Requirements for the course – exam