case 2

Introduction

In this case, 16 stocks were selected to find out the minimum variance portfolio. Hang

Seng Industry Classification System would be a good tool to select stocks according to their

different industries, it is a system designed for the Hong Kong Stock market, covering 11

industries and 28 sectors.

4 stocks from materials industry, 3 stocks from industrial Goods industry, 3 stocks from

the Consumer Services industry and 4 stocks from the Properties and Construction industry were

selected to form a portfolio in order to determine the minimum variance of the portfolio.

For a minimum variance portfolio, it is a portfolio where the investor invest their money

to buy the stocks according to the portfolio weights of different selected stocks, at which the

investor is in the lowest risk situation when compare to other portfolio weights of the same group

of stocks.

Methodology

` First, to compute the minimum variance portfolio, some basic information are needed,

using Bloomberg, the end-of-month closing price from February 2012 to February 2104 of the

selected stocks could obtained.

After finding out the end-of-month closing price of the selected stocks, rate of return of

the stocks could be obtained by using the formula below,

where is the rate of return at time t.

Using the following formula, annualized mean, annualized variance and annualized

covariance of the selected stock could be determined,

1

where is the rate of return at time t, M is the total number of period , n is the number of

period per year.

After finding the annualized mean, variance and covariance, the annualized mean vector

and the covariance matrix of the rate of return could be obtained, where the annualized mean

vector of rate of return is in the form as below,

where is the annualized mean of rate of return of the stock.

The form of the covariance matrix of the rate of return of the selected stocks is shown below,

2

where is the covariance of the stock rate of return and the rate of return, if

i = j, then represents the variance of the stock rate of return.

By the Markowitz portfolio theory, which developed by Harry Markowitz, who derived

the expected rate of return for a portfolio of assets and an expected risk measure. This model

shower that the variance of the rate of return was a meaningful measure of portfolio risk under a

reasonable set of assumptions. This model derived the formulas for computing the variance of a

portfolio not only indicated the importance of diversifying the investments to reduce the total

risk of a portfolio, but also showed how to effectively diversify. The Markowitz solution can be

found by using the method of Lagrange Multiplier, then the minimum variance set of the

portfolio could be determined, where the minimum variance set would be shown as below,

where represents the portfolio weight of the stock.

The mean ) and standard deviation ( ) of the annual rate of return of this minimum

variance portfolio then could be calculated

3

.

Then the mean-standard deviation diagram could be computed when setting two different

level of expected rate of return, which contained the minimum variance set, finally the efficient

portfolio with minimum variance could be determined.

In the above case, only one constraint exist, which is the sum of the 16 portfolio weights

(wi) need to equal to one. Now adding one more constraint to the Lagrange Multiplier, setting the

expected rate of return of the portfolio equal to the second highest annual rate of return among

16 selected stocks, which means adding the following constraints,

4

Using the two fund theorem, the weights of different selected stocks then could be calculated

where w1i represents the weightings of the ith stock in the portfolio 1, w2i is the weightings of the

ith stock in the portfolio 2

Select 16 stocks in 4 given industries

Using the Hang Seng Industry Classification system, the 16 stocks were selected as follow,

Materials Stock Code

1 297 Sinofert

2 347 Angang Steel

3 1208 MMG

Inductrial Goods

1 148 Kingboard Chem

2 316 OOIL

3 566 Hanergy Solar

4 658 C Transmission

Consumer Services

1 66 MTR Corporation

2 293 Cathay Pac. Air

3 308 China Travel HK

4 753 Air China

5

5 69 Shangri-La Asia

Properties & Construction

1 1 Cheung Kong

2 12 Henderson Road

3 17 New World Dev

4 119 Poly Property

Find the end-of-month closing price

The end-of-price of the 16 selected stocks from February 2012 to February 2014 were

attached in the Appendix A.

Annualized mean vector and covariance matrix

By using the formula stated in the methodology part, the annualized mean vector of the

annual rate of return can be calculated and the result is shown below,

6

The covariance matrix can also be computed according to the formula below,

where M represents the total number of period, n represents the number of period per year.

The covariance matrix of the annual rate of return of these 16 selected stocks was attached in the

Appendix B.

Find the minimum variance set and the mean-standard deviation

diagram

To find out the minimum variance set, two set of solution that have the minimum

variance with different level of expected return need to be considered as to apply the two fund

theorem.

7

Setting the two expected return as 0.002 and 0.1, two different set of solutions including

the weightings of each of the 16 selected stocks, the portfolio mean and standard deviation of the

rate of return could be obtained by using the Langrage Multiplier and the formula stated above.

In this case, rather than one constraint, two constraints were subjected to minimize the

variance of the portfolio,

where in this case would equal to 0.002 and 0.1 respectively in order to find out two set of

solutions according to the different expected rate of return.

8

Find out and then put them equal to zero, there would have 18 equations and 18

unknowns, after that, transformed the 18 equations into the form ,

V is the var-cov(r) matrix

R is the annualized mean vector,

9

Result could be computed and the solutions of the two portfolios with different expected rate of

return could be found in the Appendix C.

Two sets of solution could be computed, using the result of the weightings of the 26 selected

stocks in two different expected rate of return, the mean and standard deviation of the portfolio could

be obtained, letting the portfolio with expected rate of return 0.002 be portfolio A and the one with

expected rate of return 0.1 be portfolio B, the table below could be constructed,

portfolio A portfolio Bvariance 0.007930757 0.008376584S.D 0.089054796 0.091523679expected rate of return 0.002 0.1

covariance 0.00786807

where the covariance is computed using the equation below,

XA is the matrix of weightings of the 16 selected stocks in portfolio A, XB represented that of

portfolio B. V is the variance covariance matrix of the rate of return of the 16 selected stocks.

Consider portfolio A as asset A and portfolio B as asset B, then using the two assets case,

the minimum variance set can be calculated and the diagram could be computed. Let

and be the rate of return of the asset A and B respectively, has the mean of and the

variance of ; has the mean and variance of and respectively. The covariance of

and is .

Let x and 1-x be the portfolio weight of asset A and asset B respectively. The rate of

return of the portfolio and the mean of the rate of return are,

10

and the variance is,

Different portfolio weights involving the two portfolios mentioned above are used with the mean and

standard deviation of the rate of return of portfolios that are listed in the following table,

weight of portfolio A portfolio standard deviation portfolio mean

-0.5 0.095015251 0.149-0.4 0.094206088 0.1392-0.3 0.093451058 0.1294-0.2 0.092751482 0.1196-0.1 0.092108624 0.1098

0 0.091523679 0.10.1 0.090997764 0.09020.2 0.090531908 0.08040.3 0.090127041 0.07060.4 0.08978399 0.06080.5 0.089503464 0.0510.6 0.089286054 0.04120.7 0.089132221 0.03140.8 0.089042294 0.02160.9 0.089016467 0.01181 0.089054796 0.002

1.1 0.089157199 -0.00781.2 0.089323455 -0.01761.3 0.089553207 -0.02741.4 0.08984597 -0.03721.5 0.09020113 -0.047

After finding the portfolio standard deviation and mean with different weighting of portfolio A

and B, the mean-standard deviation diagram could be obtained by the minimum variance sets in

the table above,

11

From the diagram above, the minimum-variance set has bullet shape, there is a special

point having the minimum variance which called minimum-variance point, which is shown by

the cross in the above diagram. Using this minimum-variance point, the efficient portfolio could

be obtained by using the two fund theorem.

The curve in the above mean-standard deviation diagram defined by nonnegative

mixtures of two assets A and B lies within the triangular region shown below which defined by

the two original assets A and B and point on the vertical axis of height is

12

Point on the vertical axis of height:

By substituting the value of , the vertical axis of height is 0.05033

Using the two fund theorem,

By solving the equation, x = 0.50684 and using the equation stated before, the efficient portfolio

could be calculated easily, the mean, variance and standard deviation of the portfolio could also

be found out, the mean of this portfolio is equal to 0.05033 and the variance is 0.008008.

Stock AllocationSinofert -0.00612Angang Steel -0.137344MMG 0.1544809Kingboard Chem -0.199923OOIL -0.088097Hanergy Solar 0.0077333C Transmission 0.0806506MTR Corporation 0.6629117Cathay Pac. Air 0.1650581China Travel HK 0.3799111Air China 0.0911574Shangri-La Asia -0.150657Cheung Kong 0.289253Henderson Road 0.1659749New World Dev -0.378129Poly Property -0.036862expected rate of return 0.05033Variance 0.0080078

13

Determine the efficient portfolio

To find the solution of the Markowitz model, Lagrange Multiplier is a good method to

solve this problem, by setting the condition and the constraints,

where wi is the portfolio weight of the ith stock of the 16 selected stocks.

Then using the method of Lagrange Multiplier, the equation below could be obtained,

After setting the equation above, by finding and , then put these 17 equations equal to

zero,

where i=1,2,3,….,16

14

Using as an example, it can be used to prove the 17 equations can be expressed in a matrix

form.

which is the same as

For i=1,2,3,…16, same expression could be obtained. When combining these 16 equation to

matrix form like the one above, the below result would be obtained,

Also, for the , could also be transformed to matrix form

15

By combing all the matrix above, the matrix form of these 17 equations were shown as below,

V is the var-cov(r) matrix

is in the form of , where

By using the property of matrix, value of matrix x could be found out easily,

16

By using the var-cov(r) matrix obtained above, the weightings of the 16 selected stocks with the

minimum variance could be found out eventually, the result was shown below,

The mean and the standard deviation of the portfolio can be calculated using the formula stated

in the methodology part,

.

17

The mean and variance of the annual rates of return of this minimum variance set is 0.01276 and

0.0079739 respectively.

Stock AllocationSinofert 0.0182314Angang Steel -0.166412MMG 0.1938072Kingboard Chem -0.185961OOIL -0.091607Hanergy Solar -0.00247C Transmission 0.0702031MTR Corporation 0.5955472Cathay Pac. Air 0.1582968China Travel HK 0.4394392Air China 0.0542938Shangri-La Asia -0.152109Cheung Kong 0.2865231Henderson Road 0.1878331New World Dev -0.387335Poly Property -0.018281expected rate of return 0.0127551Variance 0.0079239

The efficient portfolio can also be estimated by the mean-standard deviation diagram

shown above. For the diagram above, the minimum-variance point is (0.089016467, 0.0118),

which means the expected rate of return and standard deviation of the efficient portfolio is

0.0118 and 0.089016467 respectively. At this level of mean, by checking the table in the

previous page, the weight of portfolio A is 0.9, which means x=0.9, by the two-fund theorem,

two efficient funds can be established so that any efficient portfolio can be duplicated, in terms

of mean and variance, as a combination of these two. On other words, all investor seeking

efficient portfolios need only in combinations of these two funds.

18

is the weighting of the ith stock of the 16 selected stocks in the efficient portfolio,

and are weightings of the ith stock of the selected stocks in the portfolio A and portfolio

B which have found already in the previous part.

The efficient portfolio would be,

with the mean equal to 0.118 and variance equal to 0.0079239 which used the annualized mean

vector and the variance-covariance matrix of the rate of return to compute.

Find the efficient portfolio when given the target expected rate of

return

This time setting the target expected rate of return as the 2nd highest annualized mean of

rate of return of the 16 selected stocks, by using the same method as before, the efficient

portfolio could be calculated easily,

19

By checking the annualized mean vector of the rate of return of the selected stocks, the

target expected rate of return is 0.03491, using two fund theorem,

Then the weightings of the stocks in this efficient portfolio are,

Stock AllocationSinofert 0.0038748Angang Steel -0.149275MMG 0.170622Kingboard Chem -0.194192OOIL -0.089537Hanergy Solar 0.0035456C Transmission 0.0763625MTR Corporation 0.6352626Cathay Pac. Air 0.162283China Travel HK 0.4043438Air China 0.0760271Shangri-La Asia -0.151253Cheung Kong 0.2881326Henderson Road 0.1749464New World Dev -0.381908Poly Property -0.029235expected rate of return 0.0349078Variance 0.0159061

the portfolio mean and variance are 0.0349078 and 0.0159061 respectively.

Conclusion

From the result calculated above, a portfolio which consists of a basket of stocks has the

lowest variance compare to any single stock in the basket. In the case above, the portfolio

consists of the 16 selected stocks can construct a efficient minimum variance portfolio has a

relatively lower variance when compare to the 16 stocks, which means the portfolio would have

20

a lower risk for investor to invest comparing to the singe stocks. Moreover, when more and more

stocks are added into the portfolio, the variance of the minimum efficient portfolio would be

lower and lower, the more the stocks in the portfolio, the lower the variance, which implied a

lower risk could be obtained. This phenomenon called diversification. Investor always pretend to

invest in an assets which has a lower risks, which mean a relatively lower variance portfolio, by

diversification, choosing a portfolio is a better choice than buying any one of the selected stocks

in the portfolio individually.

Appendix

Appendix A.

End-of-month stock price of the 16 selected stocks.

21

Materials

Month Price Ln((St)/(St-1)) Price Ln(St/St-1) Price Ln(St/St-1)Feb-12 2.31 6.04 4.45Mar-12 1.89 -0.200670695 4.99 -0.190968102 3.74 -0.173818485Apr-12 1.69 -0.1118483 5.31 0.062155925 3.96 0.057158414

May-12 1.34 -0.232058915 4.45 -0.176687739 3.33 -0.173271721Jun-12 1.19 -0.118716307 4.22 -0.053068968 3.23 -0.030490167Jul-12 1.58 0.28347154 4 -0.053540767 2.94 -0.094072556

Aug-12 1.49 -0.058648727 3.79 -0.053928342 2.89 -0.017153079Sep-12 1.52 0.019934215 4 0.053928342 2.97 0.027305451Oct-12 1.64 0.075985907 4.65 0.150572858 3.08 0.036367644

Nov-12 1.65 0.006079046 4.88 0.048278 3.02 -0.019672766Dec-12 1.88 0.130496489 5.68 0.151806013 3.21 0.061014106Jan-13 1.93 0.026248226 5.74 0.010507978 3.14 -0.022048137Feb-13 1.92 -0.005194817 5.22 -0.094961808 3.43 0.088337461Mar-13 1.97 0.025708357 4.25 -0.205578419 2.85 -0.185241267Apr-13 1.68 -0.159239749 4.57 0.072594222 2.33 -0.201450727

May-13 1.78 0.057819571 4.22 -0.079678077 2.14 -0.085062439Jun-13 1.3 -0.3142491 3.81 -0.102205939 2.03 -0.052770036Jul-13 1.21 -0.071743905 4.3 0.120985834 1.87 -0.082097362

Aug-13 1.21 0 4.88 0.126530197 1.75 -0.066322643Sep-13 1.29 0.064021859 4.6 -0.059088916 1.74 -0.005730675Oct-13 1.26 -0.023530497 4.7 0.021506205 1.73 -0.005763705

Nov-13 1.33 0.054067221 5.54 0.164431992 1.82 0.050715093Dec-13 1.26 -0.054067221 5.76 0.038942974 1.64 -0.104140259Jan-14 1.14 -0.100083459 4.94 -0.153572144 1.6 -0.024692613Feb-14 1.18 0.034486176 4.87 -0.014271394 1.32 -0.192371893

Annualized mean -0.335866543 -0.107655037 -0.60763618Variance 0.190245855 0.151077819 0.090606679

Sinofert Angang Steel MMG

22

Industrial goods

Month Price Ln((St)/(St-1))Price Ln(St/St-1) Price Ln(St/St-1) Price Ln(St/St-1)Feb-12 24 53.35 0.28 5.14Mar-12 22.625 -0.05899834 55.25 0.034994363 0.201 -0.331484695 4.15 -0.213945Apr-12 18.125 -0.22176329 53 -0.04157643 0.216 0.0719735 3.76 -0.098689

May-12 13.967 -0.2605948 42 -0.2326223 0.205 -0.052268429 3.2 -0.161268Jun-12 12.433 -0.11634318 37.6 -0.11066557 0.227 0.101940038 2.4 -0.287682Jul-12 13.367 0.072434755 44.2 0.161720739 0.209 -0.082615766 2.09 -0.138305

Aug-12 14.133 0.055723505 41.4 -0.06544391 0.213 0.018957914 2.38 0.1299364Sep-12 15.517 0.093423709 42.75 0.032088315 0.211 -0.009434032 2.28 -0.042925Oct-12 19.208 0.213390662 49 0.136451103 0.208 -0.014320054 2.67 0.157903

Nov-12 19.167 -0.00213681 49.2 0.004073325 0.27 0.260883879 2.58 -0.034289Dec-12 22.917 0.178688943 50.2 0.020121403 0.35 0.259511195 3.02 0.1574674Jan-13 21.333 -0.07162382 54.35 0.079429587 0.395 0.12095261 3.1 0.0261453Feb-13 19.708 -0.07923052 54.65 0.005504601 0.48 0.194900339 3.89 0.227007Mar-13 18.417 -0.06775049 52.45 -0.04108888 0.495 0.030771659 3.65 -0.063682Apr-13 17.583 -0.04634162 46.1 -0.12904738 0.56 0.123379021 3.75 0.0270287

May-13 16.7 -0.05152381 48.95 0.059986419 0.51 -0.093526058 4 0.0645385Jun-13 15.98 -0.04407078 50.1 0.023221639 0.6 0.162518929 3.53 -0.124996Jul-13 17.06 0.065398602 43.05 -0.15165878 0.64 0.064538521 3.5 -0.008535

Aug-13 16.86 -0.01179259 42.7 -0.00816331 0.75 0.15860503 3.3 -0.058841Sep-13 19.94 0.167783812 45.55 0.064611703 1.4 0.624154309 3.3 0Oct-13 20.4 0.022807136 40.05 -0.12868195 1.32 -0.0588405 4.16 0.2315926

Nov-13 20.8 0.019418086 41 0.023443393 1.22 -0.078780878 3.89 -0.067106Dec-13 20.25 -0.02679819 38.95 -0.05129329 0.79 -0.434573192 4.21 0.0790535Jan-14 17.44 -0.14938838 32.4 -0.18412035 1.1 0.331032513 4.69 0.1079699Feb-14 17.02 -0.0243773 38.15 0.163367335 1.08 -0.018349139 5.76 0.2055049

Annualized mean -0.17183235 -0.16767411 0.674963358 0.0569422Variance 0.158384282 0.131229386 0.535522251 0.235358

Kingboard Chem OOIL Hanergy Solar C Transmission

23

Consumer Serivces

Month Price Ln((St)/(St-1)) Price Ln((St)/(St-1)) Price Ln((St)/(St-1)) Price Ln((St)/(St-1)) Price Ln((St)/(St-1))Feb-12 27.6 15.4 1.54 5.86 19.4Mar-12 27.8 0.007220248 14.38 -0.068529157 1.59 0.0319516 5.38 -0.085461229 16.98 -0.133236885Apr-12 27.6 -0.007220248 13.16 -0.088656426 1.56 -0.019048195 5.63 0.045421068 16.48 -0.029888656

May-12 25.05 -0.096941945 11.98 -0.093943333 1.45 -0.073122265 4.74 -0.172072306 15.26 -0.076912499Jun-12 26.45 0.054382331 12.46 0.039284921 1.43 -0.013889112 4.54 -0.043110124 14.78 -0.03196011Jul-12 27.1 0.02427757 12.84 0.030041785 1.42 -0.007017573 5.49 0.190001243 15.3 0.034577913

Aug-12 27.8 0.025502293 12.64 -0.01569891 1.33 -0.065477929 4.67 -0.161769184 14.52 -0.052325819Sep-12 29.4 0.055958654 12.62 -0.001583532 1.4 0.051293294 4.88 0.043986148 15.04 0.035186309Oct-12 30.3 0.030153038 14.04 0.106627541 1.5 0.068992871 5.5 0.119602872 15 -0.002663117

Nov-12 30.95 0.021225287 13.62 -0.030371098 1.51 0.006644543 5.19 -0.058014395 15 0Dec-12 30.5 -0.014646316 14.22 0.043110124 1.59 0.051624365 6.55 0.232731352 15.44 0.028911343Jan-13 32 0.048009219 15.06 0.057392798 1.7 0.066894235 6.64 0.013646914 18.36 0.173212841Feb-13 32 0 14.48 -0.039273835 1.63 -0.042048236 6.28 -0.055741983 18.08 -0.01536803Mar-13 30.85 -0.036599152 13.28 -0.086509243 1.51 -0.076470364 6.9 0.094151431 15.2 -0.173510927Apr-13 32 0.036599152 13.64 0.026747508 1.58 0.045315196 6.28 -0.094151431 15 -0.013245227

May-13 30.7 -0.041473248 14.48 0.059761735 1.52 -0.038714512 6.42 0.022048137 14.38 -0.042211849Jun-13 28.6 -0.070855937 13.56 -0.065644104 1.45 -0.047146778 5.58 -0.140229341 13.4 -0.070583645Jul-13 28.85 0.008703275 14.36 0.057322281 1.42 -0.020906685 5.25 -0.0609607 12.2 -0.093818755

Aug-13 29.25 0.013769581 13.3 -0.076682528 1.46 0.027779564 4.95 -0.0588405 11.94 -0.021541844Sep-13 30.7 0.048383081 15.2 0.133531393 1.52 0.040273899 5.25 0.0588405 12.84 0.07267119Oct-13 30.05 -0.021399994 15.38 0.011772536 1.5 -0.013245227 5.29 0.007590169 14.2 0.100676666

Nov-13 30.15 0.003322262 16.42 0.06543214 1.66 0.101352494 6.02 0.129269013 14.84 0.044084273Dec-13 29.35 -0.026892377 16.4 -0.001218769 1.63 -0.018237588 5.79 -0.038954968 15.12 0.018692133Jan-14 27.45 -0.066926378 16.1 -0.018462063 1.48 -0.096537927 5.06 -0.134765808 12.9 -0.158791059Feb-14 28.15 0.025181187 15.8 -0.018809332 1.62 0.090384061 5.06 0 12.96 0.00464038

annualized mean 0.009865791 0.012821215 0.025321866 -0.07339156 -0.201702688Variance 0.020360748 0.047597365 0.036636108 0.137862289 0.07529346

MTR Corporation Cathay Pac. Air China Travel HK Air China Shangri-La Asia

24

Properties and construction

Month Price Ln((St)/(St-1))Price Ln(St/St-1)Price Ln(St/St-1)Price Ln(St/St-1)Feb-12 113.4 44.273 10.68 4.95Mar-12 100.3 -0.12276 38.955 -0.12797 9.33 -0.13514 3.61 -0.31568Apr-12 103.2 0.028503 40.182 0.031012 9.67 0.035793 4.06 0.1174752

May-12 89.5 -0.14243 35.545 -0.12262 8.36 -0.14557 3.77 -0.074108Jun-12 94.6 0.055419 38.773 0.086925 9.01 0.074877 4.15 0.0960333Jul-12 102.1 0.076295 41.091 0.058065 9.93 0.097225 4.07 -0.019465

Aug-12 105.5 0.032758 43.364 0.05384 9.64 -0.02964 3.84 -0.058171Sep-12 113.7 0.074852 50.727 0.156829 12.02 0.220651 4.16 0.0800427Oct-12 114.5 0.007011 48.818 -0.03836 11.98 -0.00333 4.71 0.1241728

Nov-12 118.3 0.032649 50.182 0.027557 12.28 0.024733 5.36 0.1292761Dec-12 119 0.0059 49.727 -0.00911 12.02 -0.0214 6.06 0.1227458Jan-13 127.2 0.066637 50.727 0.01991 14.26 0.170886 5.95 -0.018319Feb-13 120.6 -0.05328 49.045 -0.03372 14.28 0.001402 5.54 -0.071397Mar-13 114.6 -0.05103 48.273 -0.01587 13.14 -0.0832 4.91 -0.120721Apr-13 116.8 0.019015 51.091 0.056736 13.54 0.029987 5.41 0.0969752

May-13 109.8 -0.0618 49.727 -0.02706 12.4 -0.08795 5.26 -0.028118Jun-13 105.2 -0.0428 46.3 -0.07141 10.74 -0.14372 4.19 -0.22743Jul-13 109 0.035485 48.4 0.044358 11.34 0.054361 4.21 0.0047619

Aug-13 110.7 0.015476 45.5 -0.06179 10.88 -0.04141 4.69 0.1079699Sep-13 118.1 0.064708 47.9 0.051403 11.66 0.069238 4.66 -0.006417Oct-13 121.2 0.02591 45.95 -0.04156 10.74 -0.08219 4.75 0.0191292

Nov-13 122.6 0.011485 45.35 -0.01314 10.52 -0.0207 4.54 -0.045218Dec-13 122.4 -0.00163 44.25 -0.02455 9.79 -0.07192 4.14 -0.092231Jan-14 114.9 -0.06323 41.8 -0.05696 9.7 -0.00924 3.71 -0.109664Feb-14 121.6 0.056675 43.5 0.039865 10.04 0.034451 3.55 -0.044084

Annualized mean 0.034908 -0.00881 -0.0309 -0.16622Variance 0.042929 0.052015 0.101629 0.1562426

Cheung Kong Henderson Road New World Dev Poly Property

25

Appendix B

Variance covariance matrix for the annualized rate of return of the 16 selected stocks

0.1902 0.0592 0.0374 0.1028 0.0953 0.0287 0.0697 0.0258 0.0484 0.0311 0.1306 0.0553 0.0470 0.0382 0.0637 0.07010.0592 0.1511 0.0517 0.0795 0.0257 0.0115 0.0491 0.0242 0.0376 0.0477 0.0606 0.0557 0.0472 0.0307 0.0458 0.10960.0374 0.0517 0.0906 0.0376 0.0157 0.0805 0.0397 0.0087 0.0204 0.0079 0.0372 0.0334 0.0197 0.0143 0.0313 0.04770.1028 0.0795 0.0376 0.1584 0.0739 0.0492 0.0694 0.0279 0.0579 0.0383 0.0812 0.0412 0.0409 0.0301 0.0357 0.04510.0953 0.0257 0.0157 0.0739 0.1312 -0.0107 0.0226 0.0230 0.0249 0.0410 0.0787 0.0403 0.0276 0.0158 0.0337 0.00260.0287 0.0115 0.0805 0.0492 -0.0107 0.5355 0.0429 0.0159 0.0366 -0.0026 0.0058 0.0198 0.0387 0.0386 0.0717 0.09770.0697 0.0491 0.0397 0.0694 0.0226 0.0429 0.2354 -0.0006 0.0276 0.0087 0.0211 0.0373 0.0131 0.0001 0.0021 0.03190.0258 0.0242 0.0087 0.0279 0.0230 0.0159 -0.0006 0.0204 0.0153 0.0176 0.0157 0.0197 0.0227 0.0233 0.0346 0.02630.0484 0.0376 0.0204 0.0579 0.0249 0.0366 0.0276 0.0153 0.0476 0.0201 0.0376 0.0319 0.0228 0.0198 0.0305 0.02950.0311 0.0477 0.0079 0.0383 0.0410 -0.0026 0.0087 0.0176 0.0201 0.0366 0.0353 0.0325 0.0195 0.0122 0.0262 0.02590.1306 0.0606 0.0372 0.0812 0.0787 0.0058 0.0211 0.0157 0.0376 0.0353 0.1379 0.0421 0.0313 0.0239 0.0389 0.05440.0553 0.0557 0.0334 0.0412 0.0403 0.0198 0.0373 0.0197 0.0319 0.0325 0.0421 0.0753 0.0368 0.0230 0.0450 0.04750.0470 0.0472 0.0197 0.0409 0.0276 0.0387 0.0131 0.0227 0.0228 0.0195 0.0313 0.0368 0.0429 0.0395 0.0522 0.04980.0382 0.0307 0.0143 0.0301 0.0158 0.0386 0.0001 0.0233 0.0198 0.0122 0.0239 0.0230 0.0395 0.0520 0.0612 0.04870.0637 0.0458 0.0313 0.0357 0.0337 0.0717 0.0021 0.0346 0.0305 0.0262 0.0389 0.0450 0.0522 0.0612 0.1016 0.06640.0701 0.1096 0.0477 0.0451 0.0026 0.0977 0.0319 0.0263 0.0295 0.0259 0.0544 0.0475 0.0498 0.0487 0.0664 0.1562

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Appendix C.

Portfolio A Portfolio B

Stock AllocationSinofert 0.0252015Angang Steel -0.174732MMG 0.2050635Kingboard Chem -0.181965OOIL -0.092612Hanergy Solar -0.00539C Transmission 0.0672127MTR Corporation 0.5762655Cathay Pac. Air 0.1563616China Travel HK 0.4564779Air China 0.0437423Shangri-La Asia -0.152524Cheung Kong 0.2857417Henderson Road 0.1940896New World Dev -0.389971Poly Property -0.012962expected rate of return 0.002Variance 0.0079308


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case 2

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