One of fundamental theories for building successful portfolio is Modern Portfolio Theory (MPT). The approach is used in real world and has influenced the behaviour of practitioners. In 1952 Harry Markowitz, the father of MPT, published a paper named "Portfolio Selection", where he showed the basis of building portfolio: how to plot efficient frontier of the portfolio, found out the expected rate of return and consequence level of risk and find optimal portfolio (Haugen, 1997).
Diversification
Professionals do not recommend investors to hold a single asset and advise holding group of assets that is called portfolio to have higher return and lower risk (Elton et al, 2007). According to Modern Portfolio Theory the portfolio has to contain a reasonable mix of stocks from different industries following the principle of diversification.
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 financial market was chosen because it is global one, one of the most international and successful markets (http://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 investment trusts
4.
AVEVA GROUP
Software and computer services
5.
TELECITY GROUP
Software and computer services
6.
AMLIN
Nonlife insurance
7.
HIKMA PHARMACEUTICALS
Pharmaceutical and biotechnology
8.
MONDI
Forests and paper
9.
PREMIER FOODS
Food producers
10.
ROTORK
Industrial engineering
11.
DUNELM GROUP
General retailers
12.
MITCHELLS & 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 Services
18.
RIO TINTO
Mining
19.
NIGHTHAWK ENERGY
Oil and Gas Producers
20.
NORTHBRIDGE INDL.SVS.
Industrial engineering
The collection was based on choosing different stocks from different market sectors to build for the client well diversified portfolio, i.e. investor can be sure that he/she will have higher return and lower risk on the portfolio, but diversification cannot eliminate the risk, it can be only reduced (Figure 1). The remained risk is called nondiversifiable risk and influenced by market wide risk sources. But diversification cannot save investments if there are circumstances like crisis that impact all the companies (Bodie, Kane and Marcus, 2007).
Figure 1. Portfolio Diversification
Risk, return and Sharpe ratio
Return means how much investor expects to receive on holding the investment. (Keown, Martin, Petty and Scott, 2008)
Risk is uncertainty in receiving the return that comes from business cycle, inflation, interest rate and exchange rates. (Bodie et al, 2007)
Asset has a risk that is measured by standard deviation that is a historical measure of volatility that shows to investor how risky the asset and if it worths investment in it. (Keown, Martin, Petty and Scott, 2008)
Collected stocks have different returns and different standard deviation, therefore all stocks will move differently on the market (see Appendix A).
There are the following findings about each individual stock according to Figure 2 and Appendix A:
DUNELM GROUP has the highest annualised return of 37.7 % in comparison with other stocks, but meanwhile it has the high standard deviation in amount of 40.42 %.
PREMIER FOODS, on the contrast, has the lowest return of -12.7 % in comparison with other stocks and the standard deviation is very high of 93.59 %.
AMLIN stock has the lowest standard deviation of 26.36 % among 20 collected stocks and annualised return for this stock is 5.80 %
KARELIAN DIAMOND RES. stock is the riskiest among the collected stocks and have standard deviation of 103.29 % with the return -6.65 %
Figure 2. Annualised mean, Annualised standard deviation
Sharpe ratio is additional measure that helps investor to evaluate the risk of the investment. Sharpe ratio shows the risk-adjusted return per unit risk and uses capital market line as a benchmark. The high Sharpe ratio means that the market is outperformed by asset, low measure means the underperformance. (Haugen, 1997).
Figure 3. Expected Sharpe ratio
According to Figure 3 and data in Appendix B it follows that DUNELM GROUP has the highest Sharpe ratio in comparison with other stocks in portfolio and equal to 0.899. Hence, this stock has an additional return to the Client that he expected to receive for the additional volatility of holding the risky stock over a risk-free asset.
On the contrary, LAND SECURITIES GROUP stock has the lowest Sharpe ratio among other 20 stocks in the portfolio in amount of -0.264, therefore the Client will not receive an additional return for the additional volatility of holding this stock.
Correlation Matrix
It is important to understand the tendency of two stocks to move together, how they correlate to each other. Tendency is measured by correlation coefficient that shows how stocks can be combined to create a riskless portfolio. The correlation coefficient is precise measure that can be only between minus 1 and plus 1 and it cancels out the units of measurement (Bringham and Houstan, 2001).
There have been made calculations based on the correlation coefficient. According to Appendix B there are findings about correlation between stocks:
Stock's correlation with itself is equal to 1.
stocks AVIVA and LAND SECURITIES GROUP have the highest positive correlation equal to 0.6718, that means that if AVIVA stock will move and will go down or up, the LAND SECURITIES GROUP will move in lockstep either
Stocks NIGHTHAWK ENERGY and NORTHBRIDGE INDL.SVS. are negatively correlated to each other and have the minimum correlation meaning equal to -0.0752, consequence when one of the stocks goes down, the other stock will go up and vise versa
Stocks PREMIER FOODS & BARR (AG) and HIKMA PHARMACEUTICALS & KARELIAN DIAMOND RES. are negatively correlated to each other as well
The average correlation between 20 stocks is 0.2454
Portfolio and Minimum Variance Frontier
Collection of investment assets is investor's portfolio. Portfolio has its return and its risk. Standard deviation of portfolio is lower than standard deviation of weighted average of each stock of portfolio (Bodie et al, 2002). Meanwhile expected return of portfolio is measured as weighted average of the expected returns of the assets that build up the portfolio (Francis et al, 2002). The risk on portfolio is a complex measure that depends on if return on individual asset moves in unison or in different direction with other individual assets, i.e. when one asset has negative return, another one has a positive return (Edwin, 2003).
There are 20 collected stocks that can be combined with different proportions to create different portfolios. Minimum variance frontier is plotted after calculation of return (X axis) and risk (Y axis). The portfolio with the lowest level of risk is global minimum variance portfolio. Efficient frontier of risky assets is a part of the minimum variance frontier that lies above global minimum variance portfolio (Maginn, 2007). This part provides the best risk-return combinations and therefore there are candidates for optimal portfolio. The bottom of minimum variance frontier is inefficient (Bodie et al, 2011).
The risk-return opportunities available to investor can be seen on minimum variance frontier. All individual assets lie on the right inside the frontier, at least when there is a short sale allowed in the construction of risky portfolios. The tangency between minimum variance frontier and Capital Market Line is optimal risky portfolio with restriction of short sale and not short sale. (Bodie, Kane and Marcus 2011)
There are can be two kinds of restrictions in plotting efficient frontier: with short-selling and without short-selling. Graph in Appendix E shows that no short sale minimum variance frontier is located inside minimum variance frontier with allowed short sale. Therefore, efficient frontier with short sale has higher retun for the same level of risk than with no short sale. (Bodie et al, 2011).
The easiest way to describe the short sale procedure is to show it on the example. There is a stock that investor A holds and its current price is 100 $ per share, but in the end of the year expected value of the stock will be 95$ per share and will be paid dividend of 3$. Investor A wants to continue to hold the stock, but investor B can borrow the stock for one year with the restriction that the investor A will be no worse off lending investor B the stock. The investor B sells the stock and get 100$ in cash. When there is a time for dividend payment, investor B makes payment to investor A in amount of dividend of 3$ as the stock has already been sold. Therefore investor B has cash outflow in amount of dividend -3$. In the end of year investor B has to return the stock and buys the stock in amount of 95$ (Edwin et al, 2007).
Markowitz's mean variance efficient frontier with no short sales allowed
There are some cases when portfolio manager is not allowed to take short position. The Table 2 shows the following data:
Portfolio 1 has minimum return of -12,7 % in comparison with other portfolios and level of risk of 93,59%
Portfolio 11, on the contrary, has maximum return of 37,7% among portfolios and standard deviation of 40,42 %
Portfolio 3 is portfolio with minimum risk as standard deviation is equal to 15,96%, meanwhile return is 13,93%
Portfolio 8 is optimal portfolio as it tangency the efficient frontier. The return of the tangency portfolio is 26,86% with the risk of 20,28%
As there are annualized standard deviation and average return for each portfolio, then there can be plotted Markowitz's mean variance frontier with short sale restriction (Appendix C). Investor choose portfolio depend on his risk averseness and willingness of getting high return. But tangency portfolio is the most preferable choice as its return equal to 26,86%, meanwhile standard deviation is 20,28%. Hence, for quite high return there is not very high level of risk.
Table 2. Markowitz's minimum variance opportunity set
Annualised standard deviation
Annualised average 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 mean variance efficient frontier with short sales allowed
The case when when portfolio manager is allowed to sell asset that does not belong to to him is called short-selling. Short sale procedure is useful when investor expects that the return on the asset will be negative (Edwin et al, 2007).
Table 3 shows that:
Portfolio 1 is portfolio with minimum return in comparison with others built portfolios with return of -54,60 % and standard deviation of 314,98%
Portfolio 4 is portfolio with minimum risk among other portfolios and have standard deviation of 16,01% and return of 14,43%
Portfolio 7 is tangency portfolio that has the optimal level of return of 54,85% for risk in amount of 22,56%
Portfolio 11 has the maximum return in comparison with other portfolios in amount of 163,48 % with the level of risk in amount of 141,51 %
Table 3. Markowitz's minimum variance opportunity set
Annualised standard deviation
Annualised average 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
According to return and risk there have been plotted Markowitz's mean variance frontier (Appendix D). Choice of portfolio depends on the investor's preference of risk and return. But investor is advised to choose optimal portfolio as it has the best combination of risk (22,56%) and return (54,19%) in comparison with other portfolios.
If we analyze frontier with short sale restriction and without short sale restriction then we find out that portfolios with allowed short selling have much higher return, but higher risk, meanwhile portfolios without short-selling has lower return and therefore lower risk.
QUESTION 2
a) Capital Market Line
Figure 4. Capital Market Line
"Capital Market Line is "a capital allocation line provided by the market index portfolio" (Bodie, Kane and Marcus, 2011, p.G-2).
Assuming that:
the market portfolio has the mean return of 25% and standard deviation of 17% and
the risk free rate has the mean return of 1.37% and standard deviation of 0.50%
the Sharpe ratio is equal to 0.66 by using the following formula:
The Capital Market Line is used in the Capital Asset Pricing Model to mark the rate of return of tangency portfolio. The rate of return depends on the risk free rate of return (1.37% in the Figure ) and the standard deviation (0.5 % in the Figure) for tangency portfolio.
b) Indifference curves for three investors
Presuming that there are following expected utility functions 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)
According to Maginn (2007) the investor's both willing and able to take the risk depends on risk aversion of the investor.
There is the following formula for measure risk averseness of investor
U = E(R) - ½ A * Variance,
Where:
U is an utility value
A is the level of investor's risk averseness
The higher utility level the higher risk that investor will take if make investment in the asset.
Hence, Investor 3 is the most risk averse among all investors as he has the highest A coefficient and therefore the lowest level of the utility. On the contrast, investor 1 is less risk averse as he has the lowest A level coefficient and the highest utility level.
Hence, the graph for three investors:
Figure 5. Indifference curves for three investors
Indifference curves for investors shows the expected level of utility. The up part of the curve is more preferable for investors as there is higher return (Haugen, 1997). "The name "indifference curves" is used because the curves are constructed so that everywhere along the same curve the investor is assumed to be happy" (Elton, Gruber, Brown, Goetzmann, 2007, pp. 5-6).
c) 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 risk averseness.
As we know indifference curves from question 2b and Capital Market Line from 2a, then we can combine the data in one graph and find optimal portfolio for each curve and for each investor (Appendix F)
Hence,
For the investor 1 the optimal portfolio has the expected return of 10% and standard deviation of 15%,
For the investor 2 the optimal portfolio has the expected return of 18% and standard deviation of 27%,
For the investor 3 the optimal portfolio has the expected return of 25% and standard deviation of 38%.
The choice of optimal portfolio for each investor depends on what risk he can accept. The high risk will be rewarded with higher return, but there can be losses as well.
Appendix A "Annualised means, annualised standard deviations and Sharpe ratio"
Appendix B "Correlation Matrix"
Appendix C "Markowitz's mean variance efficient frontier and Capital Market Line with short sales allowed"
Appendix D "Markowitz's mean variance efficient frontier and Capital Market Line with no short sales allowed"
Appendix E "Markowitz mean varience efficient frontier with short sale restriction and without short sale restriction"
Appendix F "Capital Market Line and indifference curves for three investors"
