Portfolio analysis

elective courses

Statistics 1000-116bST

Statistics 1000-116bST

Statistics course

Classroom
Remote learning

The course will present economic basis and mathematical technique required in selecting optimal investments under uncertainty. The topics discussed will include: solutions to the classical Markowitz problem for risky assets, risky assets with riskless asset, both with short sale constraints and without that constraints; risk measures Value-at-Risk and Conditional-Value-at-Risk, their properties and applications in portfolio optimization.

1. Decision problem under uncertainty: preference relations, von Neumann-Morgenstern utility functions, coefficients of risk aversion, examples of utility functions.

2. Classical Markowitz problem without short sale constrains: expected rate of return and portfolio risk, optimal portfolio of risky assets, preference function, optimal portfolios for preference function, equivalence of solutions for that two approaches, portfolio frontier and efficient frontier, optimal portfolio with riskless asset, tangent portfolio, two fund theorem, efficient frontier as maximizer of the Sharpe ratio.

3. Capital asset pricing model (CAPM): orthogonal portfolios, efficient market, market equilibrium, CAPM, market portfolio and relation to tangent portfolio, CAPM theorem, capital market line, security market line.

4. Optimal portfolios with short sale constraints: optimization problem and its solution (proof of existence), differentiability of portfolio frontier except a finite number of points, optimal portfolios with riskless asset and short sale constraints.

5. Optimal portfolio estimators: estimation of model parameters – mean and variance of returns, portfolio weights estimators – computations for one risky asset, portfolio weights estimators for many risky assets.

6. Safety first investments: Roy, Telsar and Kataoka criteria, coherent risk measures, VaR and CVaR, CVaR as coherent risk measure, VaR for normal distributions as coherent risk measure, problems with VaR for general distributions, VaR and CVaR in portfolio optimization.

E. J. Elton, M. J. Gruber – Modern Portfolio Theory and Investment Analysis, Wiley 1981.

G. P. Szegö – Portfolio Theory with Application to Bank Asset Management, Academic Press

1980.

R. A. Haugen – Modern Investment Theory, Prentice Hall 1984.

J-L. Prigent – Portfolio Optimization and Performance Analysis, Chapman and Hall 2007.

Knowledge ans skills:

1. understand the decision problem under uncertainty, understand the following concepts: expected rate of return, preference relations, von Neumann-Morgenstern utility functions, coefficients of risk aversion;

2. know classical Markowitz problem, understand the following concepts: portfolio frontier and efficient frontier, optimal portfolio with riskless asset, tangent portfolio;

3. understand efficient frontier for the Markowitz model with short-sale, with short-sale and a riskless asset, understand relations between efficient frontiers for these models, carry on computations for simple models (2 risky assets);

4. understand efficient frontier for the Markowitz model without short-sale;

5. understand the difference between the model with maximized expected return and the model with maximized the Sharpe ratio;

6. understand the problem of model estimation and the effect of using estimators instead of exact parameters;

7. know capital asset pricing model (CAPM) and the following concepts: efficient market, market equilibrium, market portfolio and relation to tangent portfolio, two fund theorem;

8. know concepts of coherent risk measure and examples of such a measure, know under which conditions VaR is a coherent risk measure, understand an effect of using VaR and CVaR in portfolio optimization.

Competence:

1. understand the position of portfolio analysis in modeling capital markets;

2. understand an interplay between asset management and mathematical analysis of portfolio optimization.

The grade is based on the result of the final exam.