A Quantitative Risk Optimization Of Markowitz Model
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This thesis investigates assumptions of the Markowitz model and evaluates alternative measures for risk-adjusted return. It analyzes Swedish large cap stock returns and finds evidence against the normal distribution assumption. The Sharpe ratio is found to be unreliable due to extreme events. Modified Sharpe ratios that incorporate higher moments like skewness and kurtosis provide more stable measures of portfolio performance over time. Monthly returns best replicate future portfolio performance when considering risk and return, as they experience less variation than daily or weekly returns. Incorporating skewness into the model slightly improves performance estimation for future periods relative to the traditional Markowitz approach.
A Quantitative Risk Optimization Of Markowitz Model
- 1. A Quantitative Risk Optimization of Markowitz Model An Empirical Investigation on Swedish Large Cap List MASTER THESIS IN MATHEMATICS /APPLIED MATHEMATICS By Amir Kheirollah Oliver Bjärnbo Supervisor Lars Pettersson
- 2. Agenda Short Description of Markowitz Model and its Assumptions Questions to Investigate: Normality of the Asset Returns; ”Is it a myth?” Is Sharpe Ratio Still Reliable? If not so, Why? Extreme events and Sharpe ratio. Traditional Sharpe VS. Modified Sharpe ratio. Historical VS. Future Portfolios Which Time Series to Use? The Higher Statistical Moments in Construction of the Model, Are They Positive Factors in Decision Making?
- 3. Short Description of Markowitz Model Harry Markowitz & Portfolio Selection Assumptions of Modern Portfolio Theory Mean and Variance Analysis Mathematics of Markowitz Model Diversification in Markowitz Model Capitial Allocation Line and Efficient Frontier The Sharpe Ratio Skewness & Kurtosis
- 4. Assumptions of Modern Portfolio Theory Investors seek to maximize the expected return of total wealth. All investors have the same expected single period investment horizon. All investors are risk-adverse, that is they will only accept a higher risk if they are compensate with a higher expected return. Investors base their investment decisions on the expected return and risk. All markets are perfectly efficient.
- 5. Harry Markowitz & Portfolio Selection/Risk & Reward Reward is defiened by expected return of investment where risk is the weighted average of the standard deviation of those individual components of investment. Balancing Risk & Rewards Vs. Consumer Preferences;
- 6. Mathematics of Markowitz Model Markowitz Model Maximizes: With respect to the constraint, Where,
- 7. Diversification in Markowitz Model 1. Weights Sum to One: 2. Portfolio’s Expected Return: 3. The Objective is: The maximum expected return in its risk class. The minimum risk at its level of expected return. 4. The Portfolio Risk 5. The Capital Allocation Line (CAL)
- 8. Capitial Allocation Line (CAL) and Efficient Frontier The efficient frontier is convex as a result of risk and return characteristics of the portfolio, which changes in a non-linear fashion as its components’ weighting are changed. CAL: The line connecting a risk free asset with the Optimal Market Portfolio (M).
- 9. The Sharpe Ratio A measurement for risk-adjusted return by William F. Sharpe The use is to measure portfolio performance, a higher Sharpe ratio implies a higher risk-adjusted return for the portfolio. At equilibrium, the Sharpe ratio of the portfolio and the market portfolio are equal; A well-diversified portfolio has a slope close to that of the market.
- 10. Skewness & Kurtosis Skewness is a parameter that describes asymmetry in a random variable’s probability distribution. In probability theory Kurtosis is the measure of peakedness of the probability distribution of a real valued random variable.
- 11. Normality of the Asset Returns; ”Is it a myth?” The Jarque Bera Test is a goodness-of-fit measure; it measures the departure of data set from normality. It is based on three parameters Skewness, Kurtosis and Variance. Where, The result of JB-Test The study on 5 different data time series resulted in strong rejection of normality hypothesis for daily, weekly and monthly asset returns, as well as good level of confidence for rejection of normality for quarterly data time series.
- 12. Normality of the Asset Returns; ”Is it a myth?” The nomal Probability plot; Are the data normally distributed? What is the nature of the departure from the normality?
- 13. Normality of the Asset Returns; ”Is it a myth?” Frequency Distribution of Assets’ Return This plotting system let us observe closely any deviation from a normally distributed sample. The peakedness and deviation from the expected mean is traceable. The tails are clearly supporting the argument of non-Gaussian distribution, as the top of the distributions confirm it.
- 14. Normality of the Asset Returns; ”Is it a myth?” QQ-Plot for Assets’ Return The Empirical Investigation on 4 types of data time series showed deviation from normality and high Kurtosis, also the slight skewed pattern is also observable.
- 15. Is Sharpe Ratio Still Reliable? If not so, Why? Standard Deviation of Assets’ return is a parameter of Sharpe ratio which is based on normally distributed assumption. This is the source of dilemma. The Extreme Losses on two distributions with equal standard deviation.
- 16. Is Sharpe Ratio Still Reliable? If not so, Why? The effect of Extreme Losses
- 17. Is Sharpe Ratio Still Reliable? If not so, Why? Adjustments to Risk Incorporating Higher Moments’ Effects where, MVaR, accounts for the effect of Skewness and Kurtosis.
- 18. Is Sharpe Ratio Still Reliable? If not so, Why? Traditional Sharpe VS. Modified Sharpe ratio
- 19. Is Sharpe Ratio Still Reliable? If not so, Why? The traditional Sharpe ratio shows a higher change in compare to the modifiede version. The portfolio with positive skewness shows a consistent development for both types of ratios, since it considers skewness as a crucial parameter for portfolio construction. Bear in mind that Skewness is a parameter for MVaR. The portfolio with Kurtosis parameter shows the lowest change in ratios while it has a low level of diversification. It is not an investment alternative in this case, but the development is obviously interesting for an investment manager.
- 20. Historical VS. Future Portfolios & Which Time Series to Use? The development of return for different portfolios.
- 21. Historical VS. Future Portfolios & Which Time Series to Use? The best portfolio performance regarding return is difficult to determine unless we consider the risk factor. All portfolios show almost equal results for return. The lowest change in portfolios regarding time horizon belongs to the portfolio with Kurtosis parameter, where the level of diversifications is low. The lowest change which also incorporates the diversification is the traditional Markowitz model for weekly data set. The returns calculated for both ratios, traditional vs. modified Sharpe are almost equal.
- 22. Historical VS. Future Portfolios & Which Time Series to Use? The development of risk for different portfolios.
- 23. Historical VS. Future Portfolios & Which Time Series to Use? Compared with other data time series, the monthly portfolios shows a stable change in the risk parameter. The lowest change can be interpreted as stability of the portfolio. Due to the size of time horizons and diversification, the standard deviation of portfolios calculated for daily and weekly portfolios is higher than monthly and quarterly. Generally the lowest risk change with respect to diversification belongs to monthly time series.
- 24. The Higher Statistical Moments in Construction of the Model, Are They Positive Factors in Decision Making? The concept of diversification on portfolio selection showed its importance in the mean-variance optimisation approach, due to the balancing of risk and reward. Incorporating higher statistical moments in decision-making has shown both weaknesses and strengths. The incorporation of Skewness has shown slightly better effect on the mean-variance optimisation compared to future portfolios. The data set which replicated best for the future portfolios was the monthly time series. It showed a moderate accurate estimate of the future, when risk and return was taken into account. In general the traditional Sharpe model showed an inconsistent estimation compared with modified version when two time periods collated. This was mainly due to extreme events.
- 25. The End Thank You for Your Attendance!