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