GLOBAL STOCK MARKET RETURNS
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Transcript of GLOBAL STOCK MARKET RETURNS
GLOBAL STOCK MARKET RETURNS
AM: 0121Yusuf Kazi12/10/06
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.
Basic Markowitz Portfolio Problem
PROBLEM
• 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.
Choice of Securities
PROBLEM
• 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.
Why Choose Indices?
FORMULATION
• 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).
Objective Function
FORMULATION
• 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.
Constraints
DATA
• 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))
Source
DATA
Table 1 – Stock Market Returns and Risk
Stock Market Monthly % Return Variance Standard Deviation Equivalent Annual Return
Buenos Aires 1.62% 0.0144 12.00% 21.27%
Sao Paulo 1.87% 0.0097 9.86% 24.90%
Mexico 1.77% 0.0056 7.48% 23.43%
U.S. 0.44% 0.0019 4.36% 5.41%
Australia 0.77% 0.0009 3.05% 9.64%
Hong Kong 0.64% 0.0055 7.42% 7.96%
Bombay 1.15% 0.0052 7.23% 14.71%
Jakarta 1.83% 0.0079 8.89% 24.31%
Kuala Lumpur 0.82% 0.0066 8.15% 10.30%
Tokyo 0.04% 0.0029 5.37% 0.48%
Singapore 0.66% 0.0063 7.93% 8.21%
Seoul 1.40% 0.0102 10.12% 18.16%
Taiwan 0.16% 0.0059 7.68% 1.94%
Vienna 1.24% 0.0025 5.00% 15.94%
Brussels 0.50% 0.0023 4.75% 6.17%
Paris 0.66% 0.0031 5.59% 8.21%
Germany 0.62% 0.0048 6.92% 7.70%
Amsterdam 0.24% 0.0035 5.94% 2.92%
Switzerland 0.41% 0.0023 4.83% 5.03%
London 0.26% 0.0016 3.97% 3.17%
Egypt 1.45% 0.002 4.45% 18.86%
Tel Aviv 1.08% 0.0034 5.87% 13.76%
DATA
Buenos Aires Sao Paulo Mexico U.S. Australia Hong Kong Bombay Jakarta
Buenos Aires 0.0144
Sao Paulo 0.0051 0.0097
Mexico 0.0054 0.0048 0.0056
U.S. 0.0017 0.0028 0.0021 0.0019
Australia 0.0013 0.0019 0.0015 0.0009 0.0009
Hong Kong 0.0036 0.004 0.0037 0.002 0.0012 0.0055
Bombay 0.0023 0.003 0.0025 0.0011 0.001 0.0017 0.0052
Jakarta 0.0036 0.0035 0.0025 0.0015 0.001 0.0017 0.0017 0.0079
Kuala Lumpur 0.0036 0.0026 0.0028 0.0015 0.0009 0.0029 0.0015 0.003
Tokyo 0.0013 0.0026 0.0016 0.0011 0.0009 0.0014 0.0016 0.002
Singapore 0.0049 0.0041 0.0039 0.0021 0.0014 0.0045 0.0022 0.0031
Seoul 0.0035 0.0034 0.0031 0.0022 0.0016 0.0034 0.0023 0.004
Taiwan 0.0046 0.0035 0.0029 0.0015 0.001 0.0027 0.0018 0.0016
Vienna 0.0022 0.0023 0.0018 0.001 0.0007 0.0014 0.0009 0.0019
Brussels 0.0011 0.0018 0.0014 0.0013 0.0007 0.0013 0.0007 0.0017
Paris 0.0018 0.0031 0.002 0.0019 0.001 0.0021 0.0011 0.0018
Germany 0.0028 0.004 0.0028 0.0024 0.0013 0.0025 0.0014 0.0022
Amsterdam 0.002 0.0029 0.0023 0.0019 0.0011 0.0023 0.0014 0.002
Switzerland 0.0016 0.0024 0.0016 0.0015 0.0008 0.0016 0.0007 0.0019
London 0.0014 0.0023 0.0017 0.0014 0.0008 0.0016 0.0008 0.0014
Egypt 0.0008 0.0014 0.0009 0.0004 0.0004 0.0005 0.0011 0.0009
Tel Aviv 0.0021 0.0026 0.0018 0.0011 0.0007 0.0017 0.0014 0.0013
Table 2 – Covariance Matrix
DATA
Kuala Lumpur Tokyo Singapore Seoul Taiwan Vienna Brussels Paris Germany
Kuala Lumpur 0.0066
Tokyo 0.0007 0.0029
Singapore 0.0039 0.0015 0.0063
Seoul 0.0027 0.0028 0.0035 0.0102
Taiwan 0.0031 0.0015 0.0029 0.0033 0.0059
Vienna 0.0009 0.001 0.0016 0.0017 0.0015 0.0025
Brussels 0.0006 0.0006 0.0015 0.0017 0.001 0.0014 0.0023
Paris 0.0013 0.0013 0.0021 0.0025 0.0017 0.0013 0.002 0.0031
Germany 0.002 0.0016 0.0025 0.0029 0.0025 0.0018 0.0023 0.0035 0.0048
Amsterdam 0.0016 0.0013 0.0025 0.003 0.0019 0.0016 0.0023 0.003 0.0036
Switzerland 0.0009 0.0011 0.0017 0.0022 0.0013 0.0014 0.0017 0.0021 0.0025
London 0.001 0.001 0.0016 0.0021 0.0011 0.0012 0.0014 0.0018 0.0022
Egypt 0.0008 0.0006 0.0006 0.0007 0.0007 0.0006 0.0004 0.0006 0.0007
Tel Aviv 0.0009 0.0011 0.0015 0.0014 0.0011 0.0009 0.0008 0.0014 0.0019
Table 2 – Covariance Matrix Continued
DATA
Amsterdam Switzerland London Egypt Tel Aviv
Amsterdam 0.0035
Switzerland 0.0023 0.0023
London 0.0019 0.0015 0.0016
Egypt 0.0005 0.0005 0.0004 0.002
Tel Aviv 0.0014 0.001 0.0009 0.0005 0.0034
Table 2 – Covariance Matrix Continued
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;
LINGO CODE
• ! 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 ;
LINGO Model
LINGO CODE
• ! 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
LINGO Model
SOLUTIONS
Table 3 – Solutions with no weight restrictions
No Restrictions Growth Rate = 1.012 Growth Rate = 1.015
Country Weight Country Weight Country Weight
Australia 65.57% Mexico 0.42% Mexico 15.52%
Vienna 1.55% Australia 30.89% Jakarta 10.63%
Brussels 7.25% Jakarta 4.32% Vienna 12.64%
Egypt 22.30% Vienna 14.47% Egypt 57.57%
Tel Aviv 3.62% Egypt 42.52% Tel Aviv 3.64%
Tel Aviv 7.38%
SOLUTIONS
No Other Restrictions Growth Rate = 1.012 Growth Rate = 1.015
Country Weight Country Weight Country Weight
Australia 25.00% Mexico 2.84% Mexico 23.99%
Kuala Lumpur 1.87% Australia 25.00% Jakarta 16.26%
Tokyo 6.36% Bombay 1.85% Vienna 25.00%
Vienna 4.04% Jakarta 5.88% Egypt 25.00%
Brussels 10.64% Vienna 25.00% Tel Aviv 9.75%
London 18.93% Egypt 25.00%
Egypt 25.00% Tel Aviv 14.43%
Tel Aviv 8.16%
Table 4 – Solutions with maximum weight of 25%
SOLUTIONS
Portfolio Variance Standard Deviation Monthly Growth Rate
No Restrictions 0.00076 0.027 1.0093
G= 1.2% 0.00096 0.031 1.012
G=1.5% 0.00155 0.039 1.015
W<0.25 0.00086 0.029 1.0081
G=1.2% & W<0.25 0.00106 0.032 1.012
G=1.5% & W<0.25 0.00186 0.043 1.015
Table 5 – Solutions of Variance and Growth Rates
SENSITIVITY ANALYSIS
• 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.
Variable That Do Not Enter Solution
SENSITIVITY ANALYSIS
• 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.
Variable That Do Enter Solutions Without Weights
SENSITIVITY ANALYSIS
• 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.
Variable That Do Enter Solutions With Weights
CONCLUSION
• 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.
Basic Patterns
CONCLUSION
• 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.
Mutual Fund Theorem
CONCLUSION
• 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.
Possible Source of Discrepancies