ii
Table of Contents
List of figures ............................................................................................................................ iii
List of tables .............................................................................................................................. iv
List of formulae .......................................................................................................................... v
List of abbreviations .................................................................................................................. vi
1. Introduction ............................................................................................................................ 1
1.1 Selection of the stocks ...................................................................................................... 1
1.2 Selection of historical data ............................................................................................... 1
1.3 The stock simulator platform ............................................................................................ 1
2. Modern Portfolio Theory of Markowitz (1952) ..................................................................... 2
2.1 Calculation of daily returns, variance, standard deviation, and Sharpe ratio ................... 2
2.2 The variance-covariance matrix ....................................................................................... 3
2.3 Optimizing the Sharpe ratio .............................................................................................. 4
2.4 Plotting the efficient frontier ............................................................................................ 5
3. Risk management – Put options ............................................................................................. 6
4. Assessment of the 20-days simulation: Overview ................................................................. 8
4.1 Assessment of the 20-days simulation: Assumption on normal distribution ................... 8
4.2 Assessment of the 20-days simulation: Jensen’s Alpha ................................................. 10
5. Time-Series analysis ............................................................................................................ 10
6. Improving Markowitz’s approach – Model Fit Assessment ................................................ 11
Appendix .................................................................................................................................. 13
References ................................................................................................................................ 17
, iii
List of figures
Figure 1: Drawing the efficient frontier.….....…………..……………………………….…….6
Figure 2: Put option’s payoff diagram………..……………………………………….….…....7
Figure 3: Impact of put options on maximum losses…...……...………………………………8
Figure 4: Assumed normal distribution graph………………………………………..………..9
Figure 5: Upward trend of Apple’s adjusted daily closing prices (24 days forecast)………...11
, iv
List of tables
Table 1: Average daily returns …………………………………………………………………2
Table 2: Daily standard deviation…………….………………………………………………...2
Table 3: Daily Sharpe ratio……………………………………………………………………..3
Table 4: Variance-covariance matrix……………………………………………………….….3
Table 5: Equally-weighted portfolio……………………………………………………...……4
Table 6: Optimized portfolio for Sharpe ratio…………………………………………………5
Table 7: Scenarios for plotting the efficient frontier…………………………………….….…5
Table 8: Comparing daily adjusted closing prices……………………...…..………………....8
Table 9: Summary of the simulation assessment…………………………………………...…9
Table 10: Jensen’s Alpha findings for the simulated four-stock portfolio………….....……..10
Table 11: Comparing Markowitz 1952 versus Markowitz 2.0……………………………….12
Table of Contents
List of figures ............................................................................................................................ iii
List of tables .............................................................................................................................. iv
List of formulae .......................................................................................................................... v
List of abbreviations .................................................................................................................. vi
1. Introduction ............................................................................................................................ 1
1.1 Selection of the stocks ...................................................................................................... 1
1.2 Selection of historical data ............................................................................................... 1
1.3 The stock simulator platform ............................................................................................ 1
2. Modern Portfolio Theory of Markowitz (1952) ..................................................................... 2
2.1 Calculation of daily returns, variance, standard deviation, and Sharpe ratio ................... 2
2.2 The variance-covariance matrix ....................................................................................... 3
2.3 Optimizing the Sharpe ratio .............................................................................................. 4
2.4 Plotting the efficient frontier ............................................................................................ 5
3. Risk management – Put options ............................................................................................. 6
4. Assessment of the 20-days simulation: Overview ................................................................. 8
4.1 Assessment of the 20-days simulation: Assumption on normal distribution ................... 8
4.2 Assessment of the 20-days simulation: Jensen’s Alpha ................................................. 10
5. Time-Series analysis ............................................................................................................ 10
6. Improving Markowitz’s approach – Model Fit Assessment ................................................ 11
Appendix .................................................................................................................................. 13
References ................................................................................................................................ 17
, iii
List of figures
Figure 1: Drawing the efficient frontier.….....…………..……………………………….…….6
Figure 2: Put option’s payoff diagram………..……………………………………….….…....7
Figure 3: Impact of put options on maximum losses…...……...………………………………8
Figure 4: Assumed normal distribution graph………………………………………..………..9
Figure 5: Upward trend of Apple’s adjusted daily closing prices (24 days forecast)………...11
, iv
List of tables
Table 1: Average daily returns …………………………………………………………………2
Table 2: Daily standard deviation…………….………………………………………………...2
Table 3: Daily Sharpe ratio……………………………………………………………………..3
Table 4: Variance-covariance matrix……………………………………………………….….3
Table 5: Equally-weighted portfolio……………………………………………………...……4
Table 6: Optimized portfolio for Sharpe ratio…………………………………………………5
Table 7: Scenarios for plotting the efficient frontier…………………………………….….…5
Table 8: Comparing daily adjusted closing prices……………………...…..………………....8
Table 9: Summary of the simulation assessment…………………………………………...…9
Table 10: Jensen’s Alpha findings for the simulated four-stock portfolio………….....……..10
Table 11: Comparing Markowitz 1952 versus Markowitz 2.0……………………………….12
