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1. Introduction
In face of huge quantities of securities in stock market, deciding which security portfolio tend to bring higher returns is annoying. This research will give inventors a rough expression that how to allocate portfolio withMarkowitz’s modern theory. First part will illustrate several instruments in deducing Markowitz efficient frontier, containing basic kinds of return and risk of single stock and portfolio. Meanwhile, the benefit of diversification will be well demonstrated.
2. Primary assumptions
l Every inventor is a rational person with an objective to expected mean return with minimum risks. Every investor pursues maximum value.
l With respect to yield of stocks, transaction expense is taken no account. It will simplify the process of allocating portfolio.
l Suppose market is efficient without arbitrage behavior.
2.1Portfolio construction
In order to illustrate the Markowitz’s modern portfolio theory, three stocks concerned with three totally different sectors have been chosen. One of the stocks is combined with the most popular corporation British American Tobacco, whose major business is related to producing tobacco as well as distributing cigarettes. The second security is related to Capital Shopping Centres Group PLC, a global shopping centre owning 14 regional branches. Due to its significant in size, PLC shares a steady growth rate of 0.1% and a low variance of 2.8% during past 9 years. Associated British Foods plc also enjoys great popularity in the field of food retailing. With various Bata of three stocks ,it is possible to hedge risk. The efficient frontier does not include zero dispersion assets, so the treasury bond is not taking into account.
2.2A single security’s yield and risk
As sufficient historical data is provided on Yahoo Finance, it is convenience to employ statistics calculating all the characteristic of stocks of BATS.L, CSCG.L, ABF.L.
Mean return gives an average return an inventor expected to earn before the event. In the case of BATS.L, monthly mean return from 2003/7/1 to 2012/11/1 is calculated as the average of all the twelve monthly returns.
Sample mean return monthly= = =4.49%
Annually nominal return for BATS.L is easily calculated as the monthly mean return (4.49% ) multiplied by 12. High yield is an appealing factor in choosing stocks.
=4.49%×12=53.88%
Table1 . historical data and rate of return of three stocks
Date 
BATS.L 
CSCG.L 
ABF.L 
BATS.L 
CSCG.L 
ABF.L 

Historical data 


Rate of return 


2012/11/1 
3074.5 
335.6 
1394 
0.00162893 
0.007807808 
0.006498 
2012/10/1 
3069.5 
333 
1385 
0.034596635 
0.031630472 
0.074476 
2012/9/3 
3179.5 
322.79 
1289 
0.0369529 
0.031387847 
0.02644 
2012/8/1 
3301.5 
333.25 
1324 
0.028256071 
0.048747482 
0.05498 
2012/7/2 
3397.5 
317.76 
1255 
0.048125868 
0.000314802 
0.02106 
2012/6/1 
3241.5 
317.66 
1282 
0.059833252 
0.042738971 
0.079125 
2012/5/1 
3058.5 
304.64 
1188 
0.031813865 
0.025775504 
0.02543 
2012/4/2 
3159 
312.7 
1219 
0.002697984 
0.019318823 
0.00082 
2012/3/1 
3150.5 
318.86 
1220 
0.493736784 
0.003904908 
0.018364 
2012/2/1 
2109.14 
320.11 
1198 
0.089133657 
0.029391903 
0.039029 
2012/1/3 
1936.53 
310.97 
1153 
0.045329508 
0.034910809 
0.041554 
*Data collected from yahoo finance
2.1.2Risk
In the field of portfolio selection, variance and probability of an adverse outcome are three bestknown mathematical definitions of risk( XiaoxiaHuang, 2008).
BATS.L deviation from mean in Jua12 is (0.73%4.49% =14.14%
The variance of the expected return of asset I is demonstrated as
Standard Deviationcan be calculated as the positive square root of sample variance.
By the means of formula, the standard deviation of BATS.L is 14.76%.
Through this method, the variance of the expected return of three enterprise is below.
Table2. mean, variance & volatility of three enterprises.


BAST.L 
CSCT.L 
ABS.L 
Mean 

0.04492732 
0.000113586 
0.009821 
Variance 

0.021793294 
0.000863006 
0.00141 
Volatility 
0.147625519 
0.029376971 
0.037554 
Respectively, mean return and variance of three enterprises show the tendency that high risk deserves high revenue.
2.2The characteristics of portfolio
2.2.1Expected return of portfolio
For example, BATS.L counts 90%, CSCG.L counts 5%, ABF.L counts 5%, the expected return is 4.09%.
The expected return of portfolio is a simple linear combination of individual asset whose weight isthe value of a single asset of the percentage of the total value. Considering three enterprise’s historical data, different weight of each stock results in different revenue and risk of portfolio. The approach to deciding weight of individual stock is what MeanVariance methodology illustrates.
2.2.2Risk of portfolio
Covariancestands for the correlation coefficient of every part of two assets. When covariance is a negative number, it makes sense to include these two assets in a portfolio because of risk diversification. Specifically, suppose stock A and B have a negative covariance, when market suffers a recession, stock A reached its lowest degree while stock B kicks its highest degree, then theunsystematic risk may reduce.
Covariance is formulated as
Table3covariance between each two stocks

BATS.L 
CSCG.L 
ABF.L 
BATS.L 

0.2092506 
0.93373536 
CSCG.L 
0.2092506 

0.319402078 
ABF.L 
0.093373536 
0.319402078 

*Original data is collected from yahoo finance
The covariance between BATS.L and CSCG.L is 0.02093. A positive covariance refers to the positive correlation between them,indicating the unsystematic risk will not be diversified.
Variance of portfolio
Variance is a measure of how far a set of numbers is spread out and it also reflects the risk of assets. The variance of portfolio depends on both the weight of individual asset and the covariance of each twoasset group. This is a representative of portfolio risk. If BATS.L counts 90%, CSCG.L counts 5%, ABF.L counts 5%, the variance of portfolio will be 1.76%. From the calculation, we can find the variance of the portfolio, which is diversified with three assets, is much less than the variance of each single asset. Thus, riskaverse investor will diversify with portfolios at least some extent or completely.
2.3.0Caculating
Employing instruments below, a series of outcomes on BATS.L, CSCG.L & ABF.L emerge.
2.4.1Use combined means and portfolio variance for threeasset portfolio to construction feasible sets.
Table4portfolio return and variance of different weight of three stocks.
BATS.L 
CSCG.L 
ABF.L 
Portfolio Return 
Portfolio Variance 
100% 
0% 
0% 
0.04492732 
0.021793294 
90% 
5% 
5% 
0.040931301 
0.017636294 
90% 
1% 
9% 
0.041319585 
0.017669092 
90% 
9% 
1% 
0.040543017 
0.017604491 
90% 
2% 
8% 
0.041222514 
0.017660799 
90% 
8% 
2% 
0.040640088 
0.017612349 
90% 
3% 
7% 
0.041125443 
0.017652569 
90% 
7% 
3% 
0.040737159 
0.017620268 
90% 
4% 
6% 
0.041028372 
0.0176444 
90% 
6% 
4% 
0.04083423 
0.01762825 
(More details is in Appendix I)
Through table 4, it is obvious that as the growing percentage of BATS.L combining with the reducing percentage accordingly in portfolio returns go up while portfolio variances go up, too. Because of positive correlation coefficient between BATS.L and CSCG.L, portfolio variance is not lower than each single asset.
Chart1
Applying statistics in the table to formulate a pattern, feasible set appears. The spot in the most left side along line is Minimum Variance Portfolio. Relatively, the other part below the point of Minimum Variance Portfolio is not efficient comparably. To verify this is making a vertical line from 0.001 point in horizontal axis and observing two crossing points. They possess same risk but the upper point enjoys higher rate of return. Hence, as a rational person, optimum portfolio one selects is along feasible line upper the spot of Minimum Variance Portfolio. It assumes that there is a portfolio which gives both maximum expected return and minimum variance, and it commends this portfolio to the investors (Markowitz, 1952)
Every two assets possess a positive covariance, which does not contribute to a much low rate of portfolio risk.
From Chart1, the pattern also comes from the statistics from table 4. Similarly, the value of BATS.L in portfolio goes down from 100% to 0%, while the values of CSCG.L & ABT.L go up at an inconsistent rate from 0%.
In conclusion, feasible set of portfolio represents all the possible combination of risky assets with different weights.
The pattern of Markowitz efficient frontier lies on the left boundary of feasible set and upper the spot ofMinimum Variance Portfolio. Every spot in the Markowitz efficient frontier satisfies the desire to every inventor who has a preference for the same level in return with the lowest rate in risk as well as the same level in risk with the highest rate in return..
2.4.3How to select optimal portfolio
Indifferent curve is a line in the return and risk plane, which provides the same utility to investors via a series of different returns and risks. It is a proper level of measurement of inventors’ utilities.
In the absence of riskless asset, a correspondence can be established between the safetyfirst criterion and expected utility maximization results in concave indifferent curves in the meanstandard deviation space (David H. Pyle and Stephen J. Turnovsky)
2.5.1Indifference curve
Provided that inventors are risk aversion (refer to assumption I), indifference curve exhibits a shape with an increasing slope. Indifference curve has many characteristics, such as two curves standing for different utility will not cross. The curve owning a longer distance from horizontal axis provides inventors higher utility.

2.5.2Choosing optimal portfolio for twoasset portfolio
Chart 2
Pink line and orange line represents indifferent curve indicating the utility of investor while the blue line shows feasible set of twoasset portfolio. Then the optimal portfolio, that is to say, the portfolio bringing inventors most contents, just located in the crossing point of two lines.(indifferent curve varies from people to people and depends on one’s risk tolerance) Consequently, all kinds of indifferent curve emerge in the market. Different person has distinct portfolio allocation.
2.5.3Choosing optimal portfolio for threeasset portfolio
Chart 3
Why threeasset portfolio’s feasible set looks like an umbrella not a single curve? Simply, weight plays a significant role in deciding the two shapes. Taking two assets into consideration, as the weight of one security floats, the other’s weight is certainty.
On the other hand, thinking over a scene with three securities, it is impossible to determine the other two securities as the weight of one security floats up and down.
2.5.4A summary on Markowitz’s modern portfolio theory
‘My 1952 article on portfolio selection proposed expected (mean) return(E) and variance of return（V）of the portfolio as a whole as criteria for portfolio selection should figure out Markowitz efficient frontier according to mean returns and variances of every single security and invest his assets based on efficient frontier. In addition, for the purpose ofminimizeunsystematic risk, investors had better to diversify assets over different sectors. One of the methods to distinguish the interdependence among various stocks is calculate covariance between each two assets.
No riskless securities includeMarkowitz devoted all his energy to allocating risk assets, regardless of riskfree stocks, the fundamental of market. In the following research, it is true that allocating part of investment in riskfree stocks does improve efficient frontier both in increasing yields and decreasing risk.
Too strong assumptionsAll the theory is processing under the assumption that prohibiting securities and riskfree lending. Practically, the assumption is too hard to reach, particularly, in the market with rapid pace of international lending. The operating principle should be that, to the extent that reliable information is available, it should be included as part of the optimization procedure (Michaud, 1989).
Is not adapt to single stockMarkowitz efficient frontier is only applied to select security portfolio. In terms of other fields, individual stock, for example, it is at a loss.
Unsuitable for specific evaluation When talking about utility function as well as indifference curve, investors lacking a sufficient application to measure one’s own utility. Thus, allocating each asset accurately is an annoying question. In portfolio selection problem, the expected return, risk, liquidity etc. cannot be predicted precisely. The investor generally makes his portfolio decision according to his experience and his economic wisdom (Rupak, 2011).
3.0Conclusion
This article lays emphasis on allocating optimal portfolio under the assumption that every investor is a risk aversion man, which deciding the location of efficient frontier. It stretches by the means ofthree stocks influenced by completely different factors. BAST.L has a good match to the assumption because of its high return and steady risk. In spite of this, the portfolio with most utilities can be found out by putting indifferent curve and efficient frontier together. The crossing point is optimal portfolio owning proper allocation. Last part points out several limitations on Markowitz’s modern theory. However, for more than three assets, the general approach has been to display qualitative results in terms of graphs (Merton, 1972).
4.0Reference
Markowitz. (1999) ‘An Early History of Portfolio Theory’, Financial Analysts Journal, vol.55, no.4, pp516
Michaud. (1989) ‘The Markowitz Optimization Enigma: Is ‘Optimized’ optimal?’ , Financial Analysts Journal, vol.45, no.1, pp3142
Merton. (1972) ‘An Analytic Derivation of the Efficient Portfolio Frontier’, Cambridge Journal [Online] Available at: http://journals.cambridge.org/action/displayAbstract;jsessionid
Markowitz. (1952) ‘Portfolio Selection’, The Journal of Finance, vol.7, no.1, pp7791
Rupak. (2011) ‘Fuzzy mean–variance–skewness portfolio selection models by interval analysis’, Computers& Mathematics with application, vol.61, no.1, pp126137
Steuer. (2007) ‘Randomly generating portfolioselection covariance matrices with specified distributional characteristics’, European Journal of Operational Research, vol.177, no.3, pp16101625
XiaoxiaHuang. (2008) ‘Portfolio selection with a new definition of risk’, European Journal of operational research, vol.186, no.1 pp351357
David H. Pyle and Stephen J. Turnovsky, ‘SafetyFirst and Expected Utility Maximization in MeanStandard Deviation Portfolio Analysis’ The Review of Economics and Statistics Vol. 52, No. 1 (Feb., 1970), pp. 7581
Appendix I
BATS.L 
CSCG.L 
ABF.L 
Portfolio Return 
Portfolio Variance 
100% 
0% 
0% 
0.04492732 
0.021793294 
90% 
5% 
5% 
0.040931301 
0.017636294 
90% 
1% 
9% 
0.041319585 
0.017669092 
90% 
9% 
1% 
0.040543017 
0.017604491 
90% 
2% 
8% 
0.041222514 
0.017660799 
90% 
8% 
2% 
0.040640088 
0.017612349 
90% 
3% 
7% 
0.041125443 
0.017652569 
90% 
7% 
3% 
0.040737159 
0.017620268 
90% 
4% 
6% 
0.041028372 
0.0176444 
90% 
6% 
4% 
0.04083423 
0.01762825 
90% 
0% 
10% 
0.041416656 
0.017677446 
90% 
10% 
0% 
0.040445946 
0.017596696 
95% 
1% 
4% 
0.043074917 
0.019669317 
95% 
4% 
1% 
0.042783704 
0.019644658 
95% 
2% 
3% 
0.042977846 
0.019661035 
95% 
3% 
2% 
0.042880775 
0.019652815 
95% 
0% 
5% 
0.043171988 
0.019677661 
95% 
5% 
0% 
0.042686633 
0.019636562 
85% 
1% 
14% 
0.039564254 
0.015784286 
85% 
14% 
1% 
0.038302331 
0.01568119 
85% 
2% 
13% 
0.039467183 
0.015775983 
85% 
13% 
2% 
0.038399402 
0.015688747 
85% 
3% 
12% 
0.039370112 
0.015767741 
85% 
12% 
3% 
0.038496473 
0.015696367 
85% 
4% 
11% 
0.039273041 
0.015759562 
85% 
11% 
4% 
0.038593544 
0.015704049 
85% 
5% 
10% 
0.03917597 
0.015751445 
85% 
10% 
5% 
0.038690615 
0.015711793 
85% 
15% 
0% 
0.03820526 
0.015673694 
85% 
0% 
15% 
0.039661325 
0.015792652 
80% 
10% 
10% 
0.036935283 
0.013942309 
80% 
20% 
0% 
0.035964573 
0.013867558 
80% 
0% 
20% 
0.037905993 
0.014023277 
75% 
25% 
0% 
0.033723886 
0.012178286 
75% 
0% 
25% 
0.036150661 
0.012369321 
75% 
10% 
15% 
0.035179951 
0.012288245 
75% 
15% 
10% 
0.034694596 
0.012250038 
74% 
5% 
21% 
0.03531424 
0.012011054 
74% 
21% 
5% 
0.033761104 
0.011889255 
74% 
6% 
20% 
0.035217169 
0.012002976 
74% 
20% 
6% 
0.033858175 
0.011896401 
74% 
7% 
19% 
0.035120098 
0.011994959 
74% 
19% 
7% 
0.033955246 
0.011903609 
74% 
8% 
18% 
0.035023027 
0.011987005 
74% 
18% 
8% 
0.034052317 
0.01191088 
74% 
9% 
17% 
0.034925956 
0.011979113 
74% 
17% 
9% 
0.034149388 
0.011918213 
74% 
11% 
15% 
0.034731814 
0.011963515 
74% 
15% 
11% 
0.03434353 
0.011933065 
70% 
0% 
30% 
0.034395329 
0.010830784 
70% 
30% 
0% 
0.0314832 
0.010605879 
70% 
20% 
10% 
0.032453909 
0.010674632 
70% 
10% 
20% 
0.033424619 
0.0107496 
65% 
5% 
30% 
0.032154642 
0.009366244 
65% 
30% 
5% 
0.029727868 
0.009182437 
65% 
10% 
25% 
0.031669288 
0.009326375 
65% 
25% 
10% 
0.030213223 
0.009216091 
65% 
15% 
20% 
0.031183933 
0.00928806 
65% 
20% 
15% 
0.030698578 
0.009251298 
65% 
0% 
35% 
0.032639997 
0.009407667 
65% 
35% 
0% 
0.029242513 
0.009150338 
62% 
11% 
27% 
0.030519018 
0.008520047 
62% 
27% 
11% 
0.028965882 
0.008403798 
62% 
12% 
26% 
0.030421947 
0.008512315 
62% 
26% 
12% 
0.029062953 
0.008410598 
62% 
13% 
25% 
0.030324876 
0.008504646 
62% 
25% 
13% 
0.029160024 
0.008417459 
62% 
14% 
24% 
0.030227805 
0.008497038 
62% 
24% 
14% 
0.029257095 
0.008424383 
60% 
20% 
20% 
0.028943246 
0.007943384 
60% 
5% 
35% 
0.030399311 
0.008058492 
60% 
35% 
5% 
0.027487181 
0.007842261 
60% 
15% 
25% 
0.029428601 
0.007980199 
60% 
25% 
15% 
0.028457891 
0.007908122 
60% 
10% 
30% 
0.029913956 
0.008018569 
60% 
30% 
10% 
0.027972536 
0.007874414 
60% 
40% 
0% 
0.027001826 
0.007811661 
60% 
0% 
40% 
0.030884666 
0.00809997 
50% 
50% 
0% 
0.022520453 
0.005484902 
50% 
0% 
50% 
0.027374002 
0.005830833 
50% 
30% 
20% 
0.024461873 
0.005604627 
50% 
20% 
30% 
0.025432582 
0.005673813 
50% 
10% 
40% 
0.026403292 
0.005749215 
50% 
15% 
35% 
0.025917937 
0.005710737 
50% 
35% 
15% 
0.023976518 
0.005572365 
50% 
25% 
25% 
0.024947227 
0.005638443 
50% 
45% 
5% 
0.023005808 
0.005512503 
50% 
5% 
45% 
0.026888647 
0.005789247 
50% 
40% 
10% 
0.023491163 
0.005541657 
40% 
50% 
10% 
0.019009789 
0.003676359 
40% 
10% 
50% 
0.022892629 
0.003941539 
40% 
40% 
20% 
0.019980499 
0.003733331 
40% 
20% 
40% 
0.021921919 
0.00386592 
40% 
5% 
55% 
0.023377984 
0.003981679 
40% 
55% 
5% 
0.018524434 
0.003650204 
40% 
15% 
45% 
0.022407274 
0.003902953 
40% 
25% 
35% 
0.021436564 
0.003830442 
40% 
35% 
25% 
0.020465854 
0.003764147 
40% 
45% 
15% 
0.019495144 
0.003704068 
40% 
30% 
30% 
0.020951209 
0.003796518 
30% 
45% 
25% 
0.015984481 
0.002357311 
30% 
25% 
45% 
0.0179259 
0.002484118 
30% 
40% 
30% 
0.016469836 
0.002386682 
30% 
30% 
40% 
0.017440545 
0.002450086 
30% 
35% 
35% 
0.01695519 
0.002417607 
30% 
50% 
20% 
0.015499126 
0.002329494 
30% 
20% 
50% 
0.018411255 
0.002519705 
30% 
5% 
65% 
0.01986732 
0.002635789 
30% 
65% 
5% 
0.014043061 
0.002255366 
30% 
15% 
55% 
0.01889661 
0.002556846 
30% 
55% 
15% 
0.015013771 
0.00230323 
30% 
10% 
60% 
0.019381965 
0.00259554 
30% 
60% 
10% 
0.014528416 
0.002278521 
30% 
70% 
0% 
0.013557706 
0.002233764 
30% 
0% 
70% 
0.020352675 
0.002677591 
20% 
5% 
75% 
0.016356656 
0.001751576 
20% 
75% 
5% 
0.009561687 
0.001327987 
20% 
10% 
70% 
0.015871302 
0.00171122 
20% 
70% 
10% 
0.010047042 
0.001348143 
20% 
15% 
65% 
0.015385947 
0.001672417 
20% 
65% 
15% 
0.010532397 
0.001369853 
20% 
20% 
60% 
0.014900592 
0.001635168 
20% 
60% 
20% 
0.011017752 
0.001393117 
20% 
25% 
55% 
0.014415237 
0.001599472 
20% 
55% 
25% 
0.011503107 
0.001417934 
20% 
30% 
50% 
0.013929882 
0.001565331 
20% 
50% 
30% 
0.011988462 
0.001444306 
20% 
35% 
45% 
0.013444527 
0.001532744 
20% 
45% 
35% 
0.012473817 
0.001472231 
20% 
40% 
40% 
0.012959172 
0.001501711 
10% 
0% 
90% 
0.013331348 
0.001371061 
10% 
90% 
0% 
0.004594959 
0.000852466 
10% 
5% 
85% 
0.012845993 
0.001329041 
10% 
85% 
5% 
0.005080314 
0.000868068 
10% 
80% 
10% 
0.005565669 
0.000885225 
10% 
10% 
80% 
0.012360638 
0.001288576 
10% 
15% 
75% 
0.011875283 
0.001249665 
10% 
75% 
15% 
0.006051024 
0.000903935 
10% 
70% 
20% 
0.006536379 
0.000924199 
10% 
20% 
70% 
0.011389928 
0.001212308 
10% 
70% 
20% 
0.006536379 
0.000924199 
10% 
25% 
65% 
0.010904573 
0.001176504 
10% 
65% 
25% 
0.007021734 
0.000946017 
10% 
30% 
60% 
0.010419218 
0.001142255 
10% 
60% 
30% 
0.007507089 
0.00096939 
10% 
35% 
55% 
0.009933863 
0.001109559 
10% 
55% 
35% 
0.007992444 
0.000994316 
10% 
40% 
50% 
0.009448508 
0.001078417 
10% 
45% 
45% 
0.008963154 
0.001048829 
10% 
50% 
40% 
0.008477799 
0.001020796 
0% 
100% 
0% 
0.000113586 
0.000863006 
0% 
0% 
100% 
0.009820684 
0.001410312 
0% 
5% 
95% 
0.009335329 
0.001368184 
0% 
95% 
5% 
0.000598941 
0.000875609 
0% 
10% 
90% 
0.008849974 
0.001327611 
0% 
90% 
10% 
0.001084296 
0.000889766 
0% 
15% 
85% 
0.00836462 
0.001288591 
0% 
85% 
15% 
0.001569651 
0.000905477 
0% 
80% 
20% 
0.002055005 
0.000922742 
0% 
20% 
80% 
0.007879265 
0.001251125 
0% 
25% 
75% 
0.00739391 
0.001215213 
0% 
75% 
25% 
0.00254036 
0.000941561 
0% 
30% 
70% 
0.006908555 
0.001180855 
0% 
70% 
30% 
0.003025715 
0.000961933 
0% 
35% 
65% 
0.0064232 
0.001148051 
0% 
65% 
35% 
0.00351107 
0.00098386 
0% 
40% 
60% 
0.005937845 
0.001116801 
0% 
60% 
40% 
0.003996425 
0.00100734 
0% 
45% 
55% 
0.00545249 
0.001087105 
0% 
55% 
45% 
0.00448178 
0.001032375 










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