PORTFOLIO ANALYSIS AND INVESTMENT PAIlecture07 投资组合分析.docx

1、PORTFOLIO ANALYSIS AND INVESTMENT PAIlecture07 投资组合分析Lecture 7:The MV approach: the case of n assets, feasible portfolio set, efficient portfolio frontier, optimum portfolio, diversification, idiosyncratic and systematic risk, lending and borrowing opportunities and the capital market line, separati

2、on fund theorem, portfolio theory benefitsThe Mean-Variance Approach - The case of n assets (illustration) The risk return illustration of individual shares from the London Stock Exchange shows that there exist shares that are superior to others, in the sense that they award investors with higher re

3、turns and lower risk. However, if we excluded the case of shares that underperform, then there exists a positive relationship between risk and return, which means that shares with higher returns embed also higher risk.Figure 2.10 London Stock ExchangeSuppose, we are investigating the case of six sha

4、res (simulation) with the following mean and standard deviation, respectively:Figure 2.11 Return and Standard deviation of 6 sharesstockreturnSt.dev15% 7%26% 9%39%14%44% 7%53% 6%67%13% From this figure we observe that there is a trend according to which, higher risk is awarded with higher returns. S

5、uppose now, that there exist 15 investors, each of which constructs a portfolio consisting of these shares based on his informational set and his risk tolerance. The following figure illustrates the share weights that each investor has utilized in order to construct his portfolio (i.e. the 1st inves

6、tor has invested most of his money on the 3rd stock, while the 12th on the 5th stock).Figure 2.12 Simulated portfolios portfolio weightsFeasible portfolio set It is obvious that the construction of a portfolio is a more complicated issue, since there exist, many shares (not only six). In the followi

7、ng figure we have illustrated the 15 portfolios risk-return relationship, as well as a dashed curve. The area below the dashed curve represents the set of all feasible risk-return combinations (all possible portfolios) and is called feasible portfolio set (FPS). As it is obvious, the FPS contains al

8、so the 15 investors choices. Thus, investors based on their expectations, construct portfolios that contribute to the formulation of the required returns which would compensate them for the risk they undertake. As it is obvious, there exist shares the returns of which do not account for the high ris

9、k level they embed, sufficiently.Figure 2.13 Feasible Portfolio SetEfficient Portfolio Frontier Rational investors, who are risk averters, would prefer portfolios the return of which is maximized for a specific level of risk, or inversely, would prefer portfolios the risk of which is minimized for a

10、 specific target return. Thus, rational investors choices are represented by the north- west points of figure 2.13.In the case of the 6 shares we could construct 20 different portfolios with respect to the components of the portfolio. However, in each case of the 20 portfolios there are many differe

11、nt combinations with respect to the share weights an investor is willing to apply. In the case we investigate n shares we may construct many portfolios of different size and for each of these portfolios there exist many other choices regarding the share weights. All possible combinations of n shares

12、 that formulate a portfolio of size k (kn) are equal to:Thus, a reasonable question is whether an investor, should consider all these combinations before constructing his portfolio. The Modern Portfolio Theory answers this problem, since investors should not investigate all possible portfolios, but

13、instead only those portfolios that for a specific level of risk offer the maximum return, or inversely, those portfolios that for a specific target return embed the lower level of risk. The portfolios that follow this property are called efficient, and the combination of all these efficient portfoli

14、os, forms the efficient portfolio frontier (EPF).Thus, rational investors would formulate a new set of portfolios, the Efficient Portfolio Frontier, which is derived by the feasible portfolio set and satisfies the following two principles:- for a specific target portfolio return, embeds the lower le

15、vel of risk (west)- for a specific level of risk, offers the highest portfolio return (north)More specifically, investors in order to reach the EPF should move in the north-west direction on the risk-return illustration of the feasible portfolio set. All portfolios that belong to the EPF contain the

16、 highest level of the ratio return/risk and consequently represent the optimum choices for investors.Figure 2.14 Feasible Portfolio Set and the Efficient Portfolio Frontier In the following figure (Figure 2.15) we observe the EPF and two other choices, points K and K. K is not a feasible portfolio,

17、while K is not efficient, since there exist other portfolios (i.e. M, N) that for the same portfolio return embed lower risk, or for the same level of risk they offer higher returns, respectively. Thus, rational investors that are risk averters will never choose a portfolio in the area below the EPF

18、. The finally decision of the investor will be chosen so as to maximize his satisfaction.Figure 2.15 Efficient Portfolio Frontier - EPFThe Efficient Portfolios of our analysis consist only of equities, i.e. financial products that embed risk. Thus, the slope of the tangency of the EPF curve represen

19、ts the extra risk that investors are willing to undertake for a marginal increment on their portfolios returns. All portfolios on the EPF have the maximum return/risk ratio.Optimum portfolioAs it is know the utility function of a risk averter expressed on the first two moments is convex, and every i

20、nvestor is willing to maximize his utility.Figure 2.16 Risk averter profile (convex function) By consideration of the utility function in the investigation of a portfolio choice, there exist a portfolio on the EPF for which investors utility is maximized. This portfolio (A) is called optimum, since

21、it offers the investor the maximum expected utility among all EPF portfolios and is identified at the point at which the slope of the utility function is equal to the slope of the EPF, as shown on Figure 2.17.Figure 2.17 Optimum PortfolioHowever, the utility function is a locus that expresses the in

22、dividual investors expectations and as a result the optimum portfolio should be different among investors. For a more risk averter investor (U1) the optimum portfolio is A while for a less risk averter investor (U2) the optimum portfolio is B, as illustrated below:Figure 2.18 Optimum portfolios for

23、two risk averter investors (with different risk tolerance)- As we have already seen the concavity of the EPF depends on the correlation structure of the equity returns, inversely, and as a result lower correlation structure of asset returns would maximize investors satisfaction.DiversificationThe po

24、rtfolio variance is given by the following equation:Alternatively, by isolating the elements of the main diagonal of the var-covariance matrix we get:Assuming that all shares have the same variance (2), that each pair of shares has the same covariance (cov) and that the all shares in the portfolio h

25、ave the same weight (w=1/n) then we get:This relationship implies that as the size of the portfolio is increased (n) the contribution of individual shares on the portfolio variance, is decreased, since the portfolio variance depends mainly on the covariance between shares returns.Figure 2.19 Portfol

26、io size and diversification As the number of shares on the portfolio is increased the idiosyncratic risk of each share is omitted (diversified), while the systematic risk is not affected.- idiosyncratic risk:It is the part of the risk of a share, due to the specific characteristics of the correspond

27、ing listed firm, such as the management, the technological factors, the equipment, the sector and many other factors and is called idiosyncratic or specific or non systematic risk. When a share is included in the market portfolio, part of the shares risk is going to be omitted given that the correla

28、tion coefficient of the returns of the share and the market portfolio is low. This portion of share risk is not of interest when dealing with well diversified portfolios, in the sense that unanticipated losses from one share on the portfolio are hedged by the gains of another one.- systematic riskIt

29、 is the part of the risk of a share, due to the covariance between the portfolios shares. More specifically, it is the part of the risk that is undiversifiable because it is associated with many financial and macroeconomic variables (the national or international political regime, the inflation, the

30、 monetary policy, the tax policy, the level of interest rates, the investors expectations about future economic states) and is called systematic or market risk. This part of the risk is undiversifiable and all investors should undertake this when including this share to their portfolios. Thus, finan

31、cial markets do reward investors only for the systematic risk of shares, since the specific risk could be eliminated in a well diversified portfolio. In a developed and complete financial market, investors should consider only the systematic risk in the formulation of their portfolios, because only

32、this part of shares risk is not omitted in a well diversified portfolio.(total risk) = (systematic risk) + (specific risk)Lending and Borrowing opportunities and the Capital Market LineSuppose now, that an investor could invest on equity (shares) and riskless assets, such as government bonds (rf). The fixed income securities embed no risk which means that their standard deviation as well as their correlation or covariance with other random variables is equal with zero.Now, s