One of cardinal theories for constructing successful portfolio is Modern Portfolio Theory ( MPT ) . The attack is used in existent universe and has influenced the behavior of practicians. In 1952 Harry Markowitz, the male parent of MPT, published a paper named “ Portfolio Selection ” , where he showed the footing of constructing portfolio: how to plot efficient frontier of the portfolio, found out the expected rate of return and effect degree of hazard and happen optimum portfolio ( Haugen, 1997 ) .

## Diversification

Professionals do non urge investors to keep a individual plus and rede keeping group of assets that is called portfolio to hold higher return and lower hazard ( Elton et al, 2007 ) . Harmonizing to Modern Portfolio Theory the portfolio has to incorporate a sensible mix of stocks from different industries following the rule of variegation.

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Table 1 shows that there have been collected 20 stocks for the 61 month period of the companies from different industries from London Stock Exchange. This fiscal market was chosen because it is planetary one, one of the most international and successful markets ( hypertext transfer protocol: //www.londonstockexchange.com/about-the-exchange/company-overview/company-overview.htm ) .

Table 1. Collected companies

## No.

## Name of the Company

## Industry sector

1.

AVIVA

Life insurance

2.

WOLSELEY

Support services

3.

LAND SECURITIES GROUP

Real estate investing trusts

4.

AVEVA GROUP

Software and computing machine services

5.

TELECITY GROUP

Software and computing machine services

6.

AMLIN

Nonlife insurance

7.

HIKMA PHARMACEUTICALS

Pharmaceutical and biotechnology

8.

MONDI

Forests and paper

9.

PREMIER FOODS

Food manufacturers

10.

ROTORK

Industrial technology

11.

DUNELM GROUP

General retail merchants

12.

MITCHELLS & A ; Butlers

Travel and leisure

13.

RIGHTMOVE

Media

14.

BARR ( AG )

Beverages

15.

FIBERWEB

Support services

16.

KARELIAN DIAMOND RES.

Mining

17.

KBC ADVANCED TECHS.

Oil Equipment and Servicess

18.

RIO TINTO

Mining

19.

NIGHTHAWK ENERGY

Oil and Gas Producers

20.

NORTHBRIDGE INDL.SVS.

Industrial technology

The aggregation was based on taking different stocks from different market sectors to construct for the client good diversified portfolio, i.e. investor can be certain that he/she will hold higher return and lower hazard on the portfolio, but variegation can non extinguish the hazard, it can be merely decreased ( Figure 1 ) . The remained hazard is called nondiversifiable hazard and influenced by market broad hazard beginnings. But variegation can non salvage investings if there are fortunes like crisis that impact all the companies ( Bodie, Kane and Marcus, 2007 ) .

Figure 1. Portfolio Diversification

## Hazard, return and Sharpe ratio

Tax return means how much investor expects to have on keeping the investing. ( Keown, Martin, Petty and Scott, 2008 )

Hazard is uncertainness in having the return that comes from concern rhythm, rising prices, involvement rate and exchange rates. ( Bodie et al, 2007 )

Asset has a hazard that is measured by standard divergence that is a historical step of volatility that shows to investor how hazardous the plus and if it worths investing in it. ( Keown, Martin, Petty and Scott, 2008 )

Collected stocks have different returns and different criterion divergence, hence all stocks will travel otherwise on the market ( see Appendix A ) .

There are the undermentioned findings about each person stock harmonizing to Figure 2 and Appendix A:

DUNELM GROUP has the highest annualised return of 37.7 % in comparing with other stocks, but meanwhile it has the high criterion divergence in sum of 40.42 % .

PREMIER FOODS, on the contrast, has the lowest return of -12.7 % in comparing with other stocks and the standard divergence is really high of 93.59 % .

AMLIN stock has the lowest standard divergence of 26.36 % among 20 gathered stocks and annualised return for this stock is 5.80 %

KARELIAN DIAMOND RES. stock is the riskiest among the gathered stocks and have standard divergence of 103.29 % with the return -6.65 %

Figure 2. Annualised mean, Annualised criterion divergence

Sharpe ratio is extra step that helps investor to measure the hazard of the investing. Sharpe ratio shows the risk-adjusted return per unit hazard and uses capital market line as a benchmark. The high Sharpe ratio means that the market is outperformed by plus, low step means the underperformance. ( Haugen, 1997 ) .

Figure 3. Expected Sharpe ratio

Harmonizing to Figure 3 and informations in Appendix B it follows that DUNELM GROUP has the highest Sharpe ratio in comparing with other stocks in portfolio and equal to 0.899. Hence, this stock has an extra return to the Client that he expected to have for the extra volatility of keeping the hazardous stock over a riskless plus.

On the reverse, LAND SECURITIES GROUP stock has the lowest Sharpe ratio among other 20 stocks in the portfolio in sum of -0.264, hence the Client will non have an extra return for the extra volatility of keeping this stock.

## Correlation Matrix

It is of import to understand the inclination of two stocks to travel together, how they correlate to each other. Inclination is measured by correlativity coefficient that shows how stocks can be combined to make a riskless portfolio. The correlativity coefficient is precise step that can be merely between minus 1 and plus 1 and it cancels out the units of measuring ( Bringham and Houstan, 2001 ) .

There have been made computations based on the correlativity coefficient. Harmonizing to Appendix B there are findings about correlativity between stocks:

Stock ‘s correlativity with itself is equal to 1.

stocks AVIVA and LAND SECURITIES GROUP have the highest positive correlativity equal to 0.6718, that means that if AVIVA stock will travel and will travel down or up, the LAND SECURITIES GROUP will travel in lockstep either

Stockss NIGHTHAWK ENERGY and NORTHBRIDGE INDL.SVS. are negatively correlated to each other and have the minimal correlativity intending equal to -0.0752, effect when one of the stocks goes down, the other stock will travel up and vise versa

Stockss PREMIER FOODS & A ; BARR ( AG ) and HIKMA PHARMACEUTICALS & A ; KARELIAN DIAMOND RES. are negatively correlated to each other every bit good

The mean correlativity between 20 stocks is 0.2454

## Portfolio and Minimum Variance Frontier

Collection of investing assets is investor ‘s portfolio. Portfolio has its return and its hazard. Standard divergence of portfolio is lower than standard divergence of leaden norm of each stock of portfolio ( Bodie et al, 2002 ) . Meanwhile expected return of portfolio is measured as leaden norm of the expected returns of the assets that build up the portfolio ( Francis et al, 2002 ) . The hazard on portfolio is a complex step that depends on if return on single plus moves in unison or in different way with other single assets, i.e. when one plus has negative return, another 1 has a positive return ( Edwin, 2003 ) .

There are 20 gathered stocks that can be combined with different proportions to make different portfolios. Minimum discrepancy frontier is plotted after computation of return ( X axis ) and hazard ( Y axis ) . The portfolio with the lowest degree of hazard is planetary minimal discrepancy portfolio. Efficient frontier of hazardous assets is a portion of the minimal discrepancy frontier that lies above planetary minimal discrepancy portfolio ( Maginn, 2007 ) . This portion provides the best risk-return combinations and hence there are campaigners for optimum portfolio. The underside of minimal discrepancy frontier is inefficient ( Bodie et al, 2011 ) .

The risk-return chances available to investor can be seen on minimal discrepancy frontier. All single assets lie on the right inside the frontier, at least when there is a short sale allowed in the building of hazardous portfolios. The tangency between minimal discrepancy frontier and Capital Market Line is optimum hazardous portfolio with limitation of short sale and non short sale. ( Bodie, Kane and Marcus 2011 )

There are can be two sorts of limitations in plotting efficient frontier: with short-selling and without short-selling. Graph in Appendix E shows that no short sale minimal discrepancy frontier is located inside minimal discrepancy frontier with allowed short sale. Therefore, efficient frontier with short sale has higher retun for the same degree of hazard than with no short sale. ( Bodie et al, 2011 ) .

The easiest manner to depict the short sale process is to demo it on the illustration. There is a stock that investor A holds and its current monetary value is 100 $ per portion, but in the terminal of the twelvemonth expected value of the stock will be 95 $ per portion and will be paid dividend of 3 $ . Investor A wants to go on to keep the stock, but investor B can borrow the stock for one twelvemonth with the limitation that the investor A will be no worse off imparting investor B the stock. The investor B sells the stock and acquire 100 $ in hard currency. When there is a clip for dividend payment, investor B makes payment to investor A in sum of dividend of 3 $ as the stock has already been sold. Therefore investor B has hard currency escape in sum of dividend -3 $ . In the terminal of twelvemonth investor B has to return the stock and buys the stock in sum of 95 $ ( Edwin et al, 2007 ) .

## Markowitz ‘s average discrepancy efficient frontier with no short gross revenues allowed

There are some instances when portfolio director is non allowed to take short place. The Table 2 shows the undermentioned informations:

Portfolio 1 has minimum return of -12,7 % in comparing with other portfolios and degree of hazard of 93,59 %

Portfolio 11, on the reverse, has maximal return of 37,7 % among portfolios and standard divergence of 40,42 %

Portfolio 3 is portfolio with minimal hazard as standard divergence is equal to 15,96 % , meanwhile return is 13,93 %

Portfolio 8 is optimum portfolio as it tangency the efficient frontier. The return of the tangency portfolio is 26,86 % with the hazard of 20,28 %

As there are annualized standard divergence and mean return for each portfolio, so there can be plotted Markowitz ‘s average discrepancy frontier with short sale limitation ( Appendix C ) . Investor choose portfolio depend on his hazard averseness and willingness of acquiring high return. But tangency portfolio is the most preferred pick as its return equal to 26,86 % , meanwhile standard divergence is 20,28 % . Hence, for rather high return there is non really high degree of hazard.

## Table 2. Markowitz ‘s minimal discrepancy chance set

## A

## Annualised criterion divergence

## Annualised mean return

Portfolio 1

0.935877

-0.126969

Portfolio 2

0.208946

-0.001918

Portfolio 3

0.159618

0.139340

Portfolio 4

0.160018

0.153895

Portfolio 5

0.167485

0.195618

Portfolio 6

0.178417

0.224197

Portfolio 7

0.193259

0.253401

Portfolio 8

0.202827

0.268619

Portfolio 9

0.274434

0.329228

Portfolio 10

0.345959

0.360719

Portfolio 11

0.404176

0.377044

## Markowitz ‘s average discrepancy efficient frontier with short gross revenues allowed

The instance when when portfolio director is allowed to sell plus that does non belong to to him is called short-selling. Short sale process is utile when investor expects that the return on the plus will be negative ( Edwin et al, 2007 ) .

Table 3 shows that:

Portfolio 1 is portfolio with minimal return in comparing with others built portfolios with return of -54,60 % and standard divergence of 314,98 %

Portfolio 4 is portfolio with minimal hazard among other portfolios and have standard divergence of 16,01 % and return of 14,43 %

Portfolio 7 is tangency portfolio that has the optimum degree of return of 54,85 % for hazard in sum of 22,56 %

Portfolio 11 has the maximal return in comparing with other portfolios in sum of 163,48 % with the degree of hazard in sum of 141,51 %

## Table 3. Markowitz ‘s minimal discrepancy chance set

## A

Annualised criterion divergence

Annualised mean return

Portfolio 1

3.419756469

-0.575999864

Portfolio 2

0.48392641

-0.387290245

Portfolio 3

0.162214398

0.018149257

Portfolio 4

0.144302698

0.160100351

Portfolio 5

0.153210403

0.268241794

Portfolio 6

0.189097688

0.425760887

Portfolio 7

0.225563612

0.541852932

Portfolio 8

0.337455484

0.795856323

Portfolio 9

0.463751081

1.01219647

Portfolio 10

0.685635557

1.252191589

Portfolio 11

1.415093502

1.63477923

Harmonizing to return and hazard at that place have been plotted Markowitz ‘s average discrepancy frontier ( Appendix D ) . Choice of portfolio depends on the investor ‘s penchant of hazard and return. But investor is advised to take optimum portfolio as it has the best combination of hazard ( 22,56 % ) and return ( 54,19 % ) in comparing with other portfolios.

If we analyze frontier with short sale limitation and without short sale limitation so we find out that portfolios with allowed short merchandising have much higher return, but higher hazard, meanwhile portfolios without short-selling has lower return and hence lower hazard.

## Question 2

## a ) Capital Market Line

Figure 4. Capital Market Line

“ Capital Market Line is “ a capital allotment line provided by the market index portfolio ” ( Bodie, Kane and Marcus, 2011, p.G-2 ) .

Assuming that:

the market portfolio has the average return of 25 % and standard divergence of 17 % and

the hazard free rate has the average return of 1.37 % and standard divergence of 0.50 %

the Sharpe ratio is equal to 0.66 by utilizing the undermentioned expression:

The Capital Market Line is used in the Capital Asset Pricing Model to tag the rate of return of tangency portfolio. The rate of return depends on the hazard free rate of return ( 1.37 % in the Figure ) and the standard divergence ( 0.5 % in the Figure ) for tangency portfolio.

## B ) Indifference curves for three investors

Assuming that there are following expected public-service corporation maps for investors:

Investor 1 E ( U1 ) = E ( R ) – 0.02Var ( R )

Investor 2 E ( U2 ) = E ( R ) – 0.10Var ( R )

Investor 3 E ( U3 ) = E ( R ) – 1.00Var ( R )

Harmonizing to Maginn ( 2007 ) the investor ‘s both willing and able to take the hazard depends on hazard antipathy of the investor.

There is the undermentioned expression for step hazard averseness of investor

## U = E ( R ) – A? A * Discrepancy,

Where:

U is an public-service corporation value

A is the degree of investor ‘s hazard averseness

The higher public-service corporation level the higher hazard that investor will take if make investing in the plus.

Hence, Investor 3 is the most risk averse among all investors as he has the highest A coefficient and hence the lowest degree of the public-service corporation. On the contrast, investor 1 is less risk averse as he has the lowest A degree coefficient and the highest public-service corporation degree.

Hence, the graph for three investors:

Figure 5. Indifference curves for three investors

Indifference curves for investors shows the expected degree of public-service corporation. The up portion of the curve is more preferred for investors as there is higher return ( Haugen, 1997 ) . “ The name “ indifference curves ” is used because the curves are constructed so that everyplace along the same curve the investor is assumed to be happy ” ( Elton, Gruber, Brown, Goetzmann, 2007, pp. 5-6 ) .

## degree Celsius ) Capital Market Line and indifference curves for three investors

Optimal portfolio for three investors will be different as the indifference curves tangent the Capital Market Line in different points. Optimal portfolio for each investor depends on his hazard averseness.

As we know indifference curves from inquiry 2b and Capital Market Line from 2a, so we can unite the information in one graph and happen optimum portfolio for each curve and for each investor ( Appendix F )

Hence,

For the investor 1 the optimum portfolio has the expected return of 10 % and standard divergence of 15 % ,

For the investor 2 the optimum portfolio has the expected return of 18 % and standard divergence of 27 % ,

For the investor 3 the optimum portfolio has the expected return of 25 % and standard divergence of 38 % .

The pick of optimum portfolio for each investor depends on what hazard he can accept. The high hazard will be rewarded with higher return, but there can be losingss every bit good.