1 / 23

GLOBAL STOCK MARKET RETURNS

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

mary-arnold
Download Presentation

GLOBAL STOCK MARKET RETURNS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related