![Depending on degree of acceptable risk, investors expected return from a portfolio would be different for a risk
averse and risk seeker investor. As shown below the indifference curve for a risk averse investor would be more
steeper:
Indifference curves will always have a positive slope because as the risk increases, the expected return of the
investor also rises.
Indifference curves are the investors’ expectations & there may not be portfolios available to satisfy his
expectations. Investors have to choose among the available efficient portfolios that lie on efficient frontier.
Optimal Portfolio for an investor would be the portfolio that corresponds to the point of tangency between Efficient
Frontier and the indifference curve of the investor.
Limitations of Markowitz Model:
Major contributions to modern portfolio theory were done by Markowitz. He was the one who first introduced the
‘portfolio approach’ to investment management. For Markowitz justification of diversification, he was awarded
Nobel Prize in 1990.
But the analysis as done by Markowitz requires large number of data inputs and involves lengthy & complex
computations. For 50 asset portfolio, the number of data inputs required are 2n + {[n(n-1)]/2} i.e. 1225 data inputs.
Thus, the identification of efficient portfolios on the basis of Markowitz’s Model has found little use in practical
applications, though it is theoretically sound.
P
Portfolio
Return
(%)
A
B
P
Portfolio
Return
(%)
A
B
Indifference curve of
risk averse investor
Indifference curve of
risk seeker investor
P
Portfolio
Return
(%)
A
B
E
F
Indifference curve of
an investor
Efficient Frontier
Optimal Portfolio](https://image.slidesharecdn.com/modernportfoliotheory-231016094728-88a8dbde/85/MODERN-PORTFOLIO-THEORY-docx-6-320.jpg)
![Sharpe Single Index Model:
William Sharpe developed simplifies variant of Markowitz model. This model reduces the data inputs and the
computational requirements considerably and thus overcomes the practical difficulties encountered by investors in
application of Markowitz model.
Sharpe single index model has its name because it relies on a single index that represents the security market.
Instead of comparing each security with every other available security, each security is compared only with market
index. Sharpe suggested that this gives reasonably accurate information about security and it is needless to study the
relationship of each security with every other security.
Risk & Return of Individual Security:
Note that α & β of security can be obtained through regression analysis of the historical data.
Risk & Return of portfolio:
(Note: Here, risk represents variance. Standard deviation is square root of variance.)
Advantages of Sharpe’s Single Index Model:
If there are ‘N’ securities, number of inputs required in Sharpe’s Model are reduced from 2n + {[n(n-1)]/2} to only
(3n+2) i.e. for a 50 asset portfolio, the number of data inputs are reduced from 1225 to 152.
Even optimising the portfolio is simpler in Sharp’s single index model as compared to Markowitz model.
Capital Market Line:
Markowitz portfolio theory deals with portfolio of risky assets. He devised a method for locating the efficient frontier
that contains efficient portfolios of risky assets. According to Markowitz, every portfolio on the efficient frontier
dominates every other portfolio in the feasible region. In other words, portfolios on efficient frontier curve will offer
highest returns (at a particular level of risk) than any other portfolio.
But in real world, apart from risky asset, risk free assets are also available for investment and investor also in invest
in such assets. When an investor combines a risk free asset along with a risky portfolio, investor can earn a return
more than the efficient portfolio at the same level of risk.
Return from a security = α of security +(β of security * Market Return)
Total Risk = Systematic Risk + Unsystematic Risk
Total Risk = [βi2
* m
2
] + ei2
Portfolio Return = α of Portfolio + (β of Porfolio * Market Return)
Total Risk = Systematic Risk + Unsystematic Risk
Total Risk = [βp2
* m
2
] + Ʃ ei2
*Wi2](https://image.slidesharecdn.com/modernportfoliotheory-231016094728-88a8dbde/85/MODERN-PORTFOLIO-THEORY-docx-7-320.jpg)
MODERN PORTFOLIO THEORY.docx
- 1. MODERN PORTFOLIO THEORY Portfolio Management consists of the following components: (a) Security Analysis (b) Portfolio Analysis Portfolio Analysis consists of the following three stages: (a) Construction of an optimal portfolio (b) Periodical evaluation of the portfolio (c) Revision of the portfolio as and when required Modern portfolio theory helps investors in constructing optimal portfolios. TRADITIONAL PORTFOLIO THEORY: In early days, portfolio analysis was done on the basis of traditional portfolio theory. The traditional theory was based on the assumption that investment can be made by analysing the individual risk-return characteristics of individual securities. So it gives a general implication that spread the risk by not putting risk gets reduced through diversification. The interrelationship between the different securities (i.e. correlation coefficient) in the portfolio was not given the required attention. It also failed to quantify the portfolio risk & return & it doesn’t give a way to optimise a portfolio. MODERN PORTFOLIO THEORY (I) MARKOWITZ MODEL: In 1952, Harry Markowitz proposed the modern portfolio theory approach to investing. The basic concept behind modern portfolio theory is that securities in a portfolio of investments cannot be selected individually based on merits of individual securities alone. According to MPT, what is important is to see how the price of each security in the portfolio changes relative to the change in the price of other securities in the portfolio. It attempts to build the most efficient portfolio by combining securities of different risk-return characteristics. The portfolio selection based on this method is known as Markowitz Model. Markowitz Model marks a beginning to modern portfolio theory. Assumptions of Markowitz Portfolio Theory: (a) Investors are risk averse. (b) Investors have full facts about all the securities in the market (i.e. market is efficient) (c) Standard deviation or variance of returns offer correct estimate of the risk involved in securities. (d) Investors take investment decision only based on two factors, viz. the expected return & the expected risk. (e) Investors have homogenous expectations of risk & return. (f) Investors have identical time horizon of investments.
- 2. Correlation & Portfolio Risk: The risk-return relationship of the portfolio of two securities A & B for different correlation coefficients are plotted in the figure above. When r = +1 Securities have a perfect positive correlation between their returns. In this case, portfolio risk is simply weighted average of risk of both securities. This means there is no risk reduction due to diversification. Here, returns would move along the line AB. When r= -1 Securities have a perfect negative correlation between their returns. In this case there are different risk levels as represented by 2 straight lines AC & BC. The interesting point to note here is that risk gets totally nullified for a particular combination of securities. This combination is represented by point C in the diagram above. When r= + 0.5 Securities have an imperfect correlation between their returns. In this case, portfolio returns would lie above straight line AB. It will lie on curve AB. This makes clear that whenever security returns are less than perfectly correlated, the investors gain through diversification. But here the question arises that can we reduce the risk to zero by adding more securities in the portfolio? Answer to this is obviously ‘No’. This is because practically, securities having perfect negative correlation are rare to find. Also diversification only helps you to eliminate only unsystematic risk, systematic risk however is a non- diversifiable risk. A B C P Portfolio Return (%) r= +1 r= +0.5 P Portfolio Return (%) Unsystematic Risk Systematic Risk Unsystematic Risk = Diversifiable Risk Sytematic Risk = Non-diversifiable Risk
- 3. Efficient Frontier: An investor can combine the different securities available in different proportions to suit his tastes and preferences. If all securities available in market are combined in various groups and in varying proportions, we will get a very large number of portfolios. All such portfolios put together is called the Feasible set of portfolio (also called as portfolio opportunity set) . It means that theoretically these are the portfolios that can be constructed by combining the available securities. The feasible set will contain both efficient and inefficient portfolios. Efficient Portfolios are those: (a) that offer maximum return for a given level of risk or (b) that have the minimum level of risk for a given return Thus, there will be many efficient portfolios in market and all these efficient portfolio lies on Efficient Frontier Curve. How to identify efficient frontier of various two assets portfolios? Let us choose two securities A & B for which correlation = + 0.5. The risk-return relationship of portfolio of two securities is reproduced in the following diagram: Point A corresponds to a portfolio with all its investments in security A and point B corresponds to a portfolio with all its investments in security B. The points on curve between A & B correspond to portfolios of securities A and B with different combination of weights for the two securities. Let there are two other securities C&D having a different risk-return relationship. The diagram below shows these risk-return characteristics: Curve AB : represents risk-return relationship of portfolio containing securities ‘A’ & ‘B’ Curve CD : represents risk-return relationship of portfolio containing securities ‘C’ & ‘D’ P Portfolio Return (%) B A P Portfolio Return (%) B A D C
- 4. This way you can plot risk-return relation of various securities in market in the diagram. If the envelope curve (as shown below) is drawn that contains all best possible combinations. Such a curve containing the best possible combinations of securities is referred to as the efficient frontier. Efficient Frontier for a three assets portfolio: We have drawn an envelope curve above i.e. efficient frontier curve for a set of portfolios by combining two securities. This concept of building portfolios and drawing an envelope curve can be extended to portfolios with more than two securities. Let us assume three securities A, B & C. The risk-return relation of all these securities can be plotted as follows: In the above diagram, Point A, B & C represents 100% investment in each of the stocks A, B & C. Curve AB represents two asset portfolios with varying proportions of stock A and B in portfolio. Curve AC represents two asset portfolios of stock A & C. P Portfolio Return (%) Efficient Frontier P Portfolio Return (%) A B C D P Portfolio Return (%) A B C
- 5. Point D in the diagram above represents a two assets portfolio of stocks A & B. If security C is included in this portfolio, we will find curve DC representing a portfolio with three asset portfolio with securities A, B & C. If we plot the varying proportions of investments in this diagram, we will get the following curves: In the above diagram, curve CX represents the efficient frontier for the three security portfolio containing securities A, B & C. In this manner we can draw efficient frontier for portfolio containing any number of securities. Efficient Frontier of n assets portfolio: If we plot all possible combinations of a portfolio starting from two securities to n securities, number of curves would be large & we will get an entire region as feasible region for investment: The curve CX of the feasible region represents Efficient Frontier of n asset portfolio. Each point on curve CX is an efficient portfolio that offers maximum return at a given level of risk & has lowest risk at a given level of return. Optimal Portfolio: All the portfolios that lie on efficient frontier are efficient portfolios and rational investors opt to invest only in such efficient portfolios. The investor will select a particular portfolio on efficient frontier depending upon the degree of risk that investor prefers to bear. Investor Indifference Curve: P Portfolio Return (%) A B C X Efficient Frontier P Portfolio Return (%) Feasible Region C X Efficient Frontier
- 6. Depending on degree of acceptable risk, investors expected return from a portfolio would be different for a risk averse and risk seeker investor. As shown below the indifference curve for a risk averse investor would be more steeper: Indifference curves will always have a positive slope because as the risk increases, the expected return of the investor also rises. Indifference curves are the investors’ expectations & there may not be portfolios available to satisfy his expectations. Investors have to choose among the available efficient portfolios that lie on efficient frontier. Optimal Portfolio for an investor would be the portfolio that corresponds to the point of tangency between Efficient Frontier and the indifference curve of the investor. Limitations of Markowitz Model: Major contributions to modern portfolio theory were done by Markowitz. He was the one who first introduced the ‘portfolio approach’ to investment management. For Markowitz justification of diversification, he was awarded Nobel Prize in 1990. But the analysis as done by Markowitz requires large number of data inputs and involves lengthy & complex computations. For 50 asset portfolio, the number of data inputs required are 2n + {[n(n-1)]/2} i.e. 1225 data inputs. Thus, the identification of efficient portfolios on the basis of Markowitz’s Model has found little use in practical applications, though it is theoretically sound. P Portfolio Return (%) A B P Portfolio Return (%) A B Indifference curve of risk averse investor Indifference curve of risk seeker investor P Portfolio Return (%) A B E F Indifference curve of an investor Efficient Frontier Optimal Portfolio
- 7. Sharpe Single Index Model: William Sharpe developed simplifies variant of Markowitz model. This model reduces the data inputs and the computational requirements considerably and thus overcomes the practical difficulties encountered by investors in application of Markowitz model. Sharpe single index model has its name because it relies on a single index that represents the security market. Instead of comparing each security with every other available security, each security is compared only with market index. Sharpe suggested that this gives reasonably accurate information about security and it is needless to study the relationship of each security with every other security. Risk & Return of Individual Security: Note that α & β of security can be obtained through regression analysis of the historical data. Risk & Return of portfolio: (Note: Here, risk represents variance. Standard deviation is square root of variance.) Advantages of Sharpe’s Single Index Model: If there are ‘N’ securities, number of inputs required in Sharpe’s Model are reduced from 2n + {[n(n-1)]/2} to only (3n+2) i.e. for a 50 asset portfolio, the number of data inputs are reduced from 1225 to 152. Even optimising the portfolio is simpler in Sharp’s single index model as compared to Markowitz model. Capital Market Line: Markowitz portfolio theory deals with portfolio of risky assets. He devised a method for locating the efficient frontier that contains efficient portfolios of risky assets. According to Markowitz, every portfolio on the efficient frontier dominates every other portfolio in the feasible region. In other words, portfolios on efficient frontier curve will offer highest returns (at a particular level of risk) than any other portfolio. But in real world, apart from risky asset, risk free assets are also available for investment and investor also in invest in such assets. When an investor combines a risk free asset along with a risky portfolio, investor can earn a return more than the efficient portfolio at the same level of risk. Return from a security = α of security +(β of security * Market Return) Total Risk = Systematic Risk + Unsystematic Risk Total Risk = [βi2 * m 2 ] + ei2 Portfolio Return = α of Portfolio + (β of Porfolio * Market Return) Total Risk = Systematic Risk + Unsystematic Risk Total Risk = [βp2 * m 2 ] + Ʃ ei2 *Wi2
- 8. In the above diagram, a concave curve AC represents efficient frontier curve of risky securities. Point B is any risky portfolio on efficient frontier. Rf represents a risk free government security. Straight Line DB is capital allocation line representing the manner in which capital is allocated in risk free security and risky portfolio B. Point D represents entire amount invested in Risk free asset & Point B represents entire amount invested in risky portfolio B. Any point on this straight DB line gives a higher return than any point on curve AB at the same level of risk. Since there are many efficient portfolios that lie on AC, it is possible to draw many capital allocation lines from point D (as shown in the above diagram). All the portfolios that lie on straight line DM dominate all other portfolios. Point M is the market portfolio & straight line DM is Capital Market Line. CML is an efficient set of risk-free and risky securities and it shows risk-return trade-off in the market equilibrium. Lending & Borrowing Portfolio: P Portfolio Return (%) A B C D Efficient Frontier Rf P Portfolio Return (%) A B C Efficient Frontier M D Rf Capital market Line P Portfolio Return (%) A B Efficient Frontier Rf Capital market Line M D E C
- 9. All rational investors would choose a portfolio along capital market line. A risk averse investor may create a lending portfolio on CML i.e. he may invest a portion of his funds in risk-free asset & remaining funds in Risky Market Portfolio M. On the other hand, risk seeker investor may create a borrowing portfolio i.e. he may invest all his owned funds in Portfolio M & may borrow at risk free rate and invest those additional funds also in risky portfolio M. Slope of CML: Slope of CML is also known as reward to variability ratio. It describes the best price at a given level of risk. The slope of CML is calculated using the following formula: If the slope of CML is 1.5, then investors are expecting 1.5 units of return for every 1 unit increase in risk. P Portfolio Return (%) Rf Capital market Line C Slope of CML Slope, = E(R m ) – R f m