GLOBAL STOCK MARKET RETURNS

1. GLOBAL STOCK MARKET RETURNS AM: 0121 Yusuf Kazi 12/10/06

2. PROBLEM Basic Markowitz Portfolio Problem • Find optimum portfolio of risky securities by minimizing risk as measured by variance. • The variables are the weights of the securities in the portfolio. • Other Possible Restrictions: Required growth rates and upper bounds on weights of securities. • By diversifying stocks one removes stock-specific risk. • However you are still subject to market risk if the whole market crashes.

3. PROBLEM Choice of Securities • One way to get around market specific risk is to invest in different markets around the world. • I used stock market indices from around the world as the securities. • Developed markets such as the U.S. and London offer steady but relatively safe growth. • Developing markets offer rapid growth but considerable more risk.

4. PROBLEM Why Choose Indices? • Most developed markets are efficient. The marginal investor can not easily beat the market. • Developed markets may be less efficient but the effort and cost of finding deals will generally make the process difficult. • Therefore it makes sense to invest in a broad market index where ever possible as it will be hard to beat the market. • This passive strategy saves on transaction costs and has been shown to beat the majority of mutual funds.

5. FORMULATION Objective Function • The problem is non-linear. • The objective function is: Minimize Var (P) where P=Portfolio of securities. • Let: • xi = Return on index of stock market i • wi= Weight of security i • where i= 1,2…22 • Var (P) = w1w1Cov(x1,x1) + w1w2Cov(x1,x2) + … + w1w22 (x1,x22) + w2w1Cov(x2,x1) + w2w2Cov(x1,x2) + … + w2w22(x2,x22)+ . . . w22w1Cov(x22,x1) + w22w2Cov(x1,x2) + … + w22w22(x22,x22).

6. FORMULATION Constraints • The investor must be fully invested: • w1 + w2 + … + w22 = 1 • The following are optional constraints: • Upper Bounds on wi <= 0.25 • Desired Growth Rates: w1x1 + w2x2 + … w22x22 = g, where g = desired growth rate. • We can also remove the possibility of short-selling by having: • wi >=0 for i=1,2,…22.

7. DATA Source • The Data was collected from Yahoo finance and consisted of the opening and closing values of 22 indices from December 1997 to November 2006. • This should theoretically satisfy the need for certainty in the values we calculate from this data. • From this data, the monthly return was calculated for each month and the covariances as required by the objective function. • Cov (x1,x2) = ∑ (x1i – E(x1))(x2i – E(x2))

8. DATA Table 1 – Stock Market Returns and Risk

9. DATA Table 2 – Covariance Matrix

10. DATA Table 2 – Covariance Matrix Continued

11. DATA Table 2 – Covariance Matrix Continued

12. LINGO CODE LINGO Model • MODEL: • ! GENPRT: Generic Markowitz portfolio Weights < 0.25 and g = 1.015; • SETS: • ASSET/1..22/: RATE, UB, X; • COVMAT( ASSET, ASSET): V; • ENDSETS • DATA: • ! The data; • ! Expected growth rate of each asset; • RATE = 1.0162 1.0187 1.0177 1.0044 1.0077 1.0064 1.0115 1.0183 1.0082 1.0004 1.0066 1.0140 1.0016 1.0124 1.0050 1.0066 1.0062 1.0024 1.0041 1.0026 1.0145 1.0108; • ! Upper bound on investment in each; • UB = .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25;

13. LINGO CODE LINGO Model • ! Covariance matrix; • V = 0.0144 0.0051 0.0054 0.0017 0.0013 0.0036 0.0023 0.0036 0.0036 0.0013 0.0049 0.0035 0.0046 0.0022 0.0011 0.0018 0.0028 0.0020 0.0016 0.0014 0.0008 0.0021 • 0.0051 0.0097 0.0048 0.0028 0.0019 0.0040 0.0030 0.0035 0.0026 0.0026 0.0041 0.0034 0.0035 0.0023 0.0018 0.0031 0.0040 0.0029 0.0024 0.0023 0.0014 0.0026 • 0.0054 0.0048 0.0056 0.0021 0.0015 0.0037 0.0025 0.0025 0.0028 0.0016 0.0039 0.0031 0.0029 0.0018 0.0014 0.0020 0.0028 0.0023 0.0016 0.0017 0.0009 0.0018 • 0.0017 0.0028 0.0021 0.0019 0.0009 0.0020 0.0011 0.0015 0.0015 0.0011 0.0021 0.0022 0.0015 0.0010 0.0013 0.0019 0.0024 0.0019 0.0015 0.0014 0.0004 0.0011 • 0.0013 0.0019 0.0015 0.0009 0.0009 0.0012 0.0010 0.0010 0.0009 0.0009 0.0014 0.0016 0.0010 0.0007 0.0007 0.0010 0.0013 0.0011 0.0008 0.0008 0.0004 0.0007 • 0.0036 0.0040 0.0037 0.0020 0.0012 0.0055 0.0017 0.0017 0.0029 0.0014 0.0045 0.0034 0.0027 0.0014 0.0013 0.0021 0.0025 0.0023 0.0016 0.0016 0.0005 0.0017 • 0.0023 0.0030 0.0025 0.0011 0.0010 0.0017 0.0052 0.0017 0.0015 0.0016 0.0022 0.0023 0.0018 0.0009 0.0007 0.0011 0.0014 0.0014 0.0007 0.0008 0.0011 0.0014 • 0.0036 0.0035 0.0025 0.0015 0.0010 0.0017 0.0017 0.0079 0.0030 0.0020 0.0031 0.0040 0.0016 0.0019 0.0017 0.0018 0.0022 0.0020 0.0019 0.0014 0.0009 0.0013 • 0.0036 0.0026 0.0028 0.0015 0.0009 0.0029 0.0015 0.0030 0.0066 0.0007 0.0039 0.0027 0.0031 0.0009 0.0006 0.0013 0.0020 0.0016 0.0009 0.0010 0.0008 0.0009 • 0.0013 0.0026 0.0016 0.0011 0.0009 0.0014 0.0016 0.0020 0.0007 0.0029 0.0015 0.0028 0.0015 0.0010 0.0006 0.0013 0.0016 0.0013 0.0011 0.0010 0.0006 0.0011 • 0.0049 0.0041 0.0039 0.0021 0.0014 0.0045 0.0022 0.0031 0.0039 0.0015 0.0063 0.0035 0.0029 0.0016 0.0015 0.0021 0.0025 0.0025 0.0017 0.0016 0.0006 0.0015 • 0.0035 0.0034 0.0031 0.0022 0.0016 0.0034 0.0023 0.0040 0.0027 0.0028 0.0035 0.0102 0.0033 0.0017 0.0017 0.0025 0.0029 0.0030 0.0022 0.0021 0.0007 0.0014 • 0.0046 0.0035 0.0029 0.0015 0.0010 0.0027 0.0018 0.0016 0.0031 0.0015 0.0029 0.0033 0.0059 0.0015 0.0010 0.0017 0.0025 0.0019 0.0013 0.0011 0.0007 0.0011 • 0.0022 0.0023 0.0018 0.0010 0.0007 0.0014 0.0009 0.0019 0.0009 0.0010 0.0016 0.0017 0.0015 0.0025 0.0014 0.0013 0.0018 0.0016 0.0014 0.0012 0.0006 0.0009 • 0.0011 0.0018 0.0014 0.0013 0.0007 0.0013 0.0007 0.0017 0.0006 0.0006 0.0015 0.0017 0.0010 0.0014 0.0023 0.0020 0.0023 0.0023 0.0017 0.0014 0.0004 0.0008 • 0.0018 0.0031 0.0020 0.0019 0.0010 0.0021 0.0011 0.0018 0.0013 0.0013 0.0021 0.0025 0.0017 0.0013 0.0020 0.0031 0.0035 0.0030 0.0021 0.0018 0.0006 0.0014 • 0.0028 0.0040 0.0028 0.0024 0.0013 0.0025 0.0014 0.0022 0.0020 0.0016 0.0025 0.0029 0.0025 0.0018 0.0023 0.0035 0.0048 0.0036 0.0025 0.0022 0.0007 0.0019 • 0.0020 0.0029 0.0023 0.0019 0.0011 0.0023 0.0014 0.0020 0.0016 0.0013 0.0025 0.0030 0.0019 0.0016 0.0023 0.0030 0.0036 0.0035 0.0023 0.0019 0.0005 0.0014 • 0.0016 0.0024 0.0016 0.0015 0.0008 0.0016 0.0007 0.0019 0.0009 0.0011 0.0017 0.0022 0.0013 0.0014 0.0017 0.0021 0.0025 0.0023 0.0023 0.0015 0.0005 0.0010 • 0.0014 0.0023 0.0017 0.0014 0.0008 0.0016 0.0008 0.0014 0.0010 0.0010 0.0016 0.0021 0.0011 0.0012 0.0014 0.0018 0.0022 0.0019 0.0015 0.0016 0.0004 0.0009 • 0.0008 0.0014 0.0009 0.0004 0.0004 0.0005 0.0011 0.0009 0.0008 0.0006 0.0006 0.0007 0.0007 0.0006 0.0004 0.0006 0.0007 0.0005 0.0005 0.0004 0.0020 0.0005 • 0.0021 0.0026 0.0018 0.0011 0.0007 0.0017 0.0014 0.0013 0.0009 0.0011 0.0015 0.0014 0.0011 0.0009 0.0008 0.0014 0.0019 0.0014 0.0010 0.0009 0.0005 0.0034 ;

14. LINGO CODE LINGO Model • ! Desired growth rate of portfolio; • GROWTH = 1.015; • ENDDATA • ! The model; • ! Min the variance; • [VAR] MIN = @SUM( COVMAT( I, J): • V( I, J) * X( I) * X( J)); • ! Must be fully invested; • [FULL] @SUM( ASSET: X) = 1; • ! Upper bounds on each; • @FOR( ASSET: @BND( 0, X, UB)); • ! Desired value or return after 1 period; • [RET] @SUM( ASSET: RATE * X) >= GROWTH; • END

15. SOLUTIONS Table 3 – Solutions with no weight restrictions

16. SOLUTIONS Table 4 – Solutions with maximum weight of 25%

17. SOLUTIONS Table 5 – Solutions of Variance and Growth Rates

18. SENSITIVITY ANALYSIS Variable That Do Not Enter Solution • In all of the reports, I saw that the reduced cost of the variables not entering the solution is of the magnitude of 10-2 or 10-3. • This means that if we change them a little bit, we will get a very large change in the variance. Therefore they are sensitive variables. • However as they don’t enter the solution this does not concern us too much.

19. SENSITIVITY ANALYSIS Variable That Do Enter Solutions Without Weights • Variables entering the solution generally have a reduced cost of the order of 10-6 or 10-7. • Therefore changing these variables would have a very minimal effect on the standard deviation. • Therefore they are relatively insensitive and slight deviations will not throw off our results. • In fact some even have a reduced cost of zero, such as Egypt in many of the solutions.

20. SENSITIVITY ANALYSIS Variable That Do Enter Solutions With Weights • Occasionally some of the variables are fairly significant with orders of magnitude between 10-2 and 10-4. • Therefore we are a little more restricted when it comes to asset allocation when we impose weight restrictions as well because our variables are more sensitive in general.

21. CONCLUSION Basic Patterns • The exercise had many predictable patterns: • Increasing desired growth rate increased risk. • Adding weight restrictions increased risk. • However in all the solutions, the weights and choice of indices was surprising. • One might have expected the major stock markets of the world such as the U.S., London or Tokyo to play a more prominent role.

22. CONCLUSION Mutual Fund Theorem • This result seems to disprove a basic theorem in economics called the mutual fund theorem. • In essence it states that given the same information investors should all pick the same portfolio of risky assets. • An Investor might mix this with different amounts of risk-free assets such as U.S. treasury bills according to their risk preferences but the weights of the portfolio of risky assets should be identical. • If this condition holds then the market capitalization of each asset as a percentage of the entire market capitalization should reflect its weight in any portfolio. • This is the basis of Markowitz’s idea that everyone should hold the market portfolio. • As we know, major financial markets such as those in the U.S., France, Germany, London or Tokyo easily dwarf the other markets in this study according to market capitalization. • If Markowitz was correct and our results are right, this should not be the case. Most of the capital should be in Egypt of Australia.

23. CONCLUSION Possible Source of Discrepancies • Firstly there are many arguments against the mutual fund theorem and Markowitz’s ideas on portfolios; however that is beyond the scope of this project. • One major issue is data. Although I got a fair span of time, covering some major economic events such as the dot-com boom and bust, the Asian financial crisis etc., more data over a longer time period might have given different results and been more accurate. • Other risk factors: Many investors may prefer more developed markets because of the regulations and liquidity that make them safer options. This is not reflected in the variance.