Portfolio Theory

1. Portfolio Theory The Benefits of Diversification

2. Overview • Originally proposed by Harry Markowitz in 1950s, was the first formal attempt to quantify the risk of a portfolio and develop a methodology for determining the optimal portfolio. • Harry Markowitz was the first person to show quantitatively why and how diversification reduces risk. • In recognition of his seminal contributions in this field, he was awarded the Nobel Prize in Economics in 1990.

3. Portfolio Expected Return • The expected return of a portfolio is a weighted average of the expected returns of the individual securities in the portfolio: • Note that the weight of a security represents the proportion of portfolio vvalue.

4. Portfolio Risk • Measured by Standard Deviation or Variance as in case of individual securities. • The standard deviation of a portfolio is not a weighted average of the standard deviations of the individual securities. • The riskiness of a portfolio depends on both the riskiness of the securities, and the way that they move together over time (correlation) • This is because the riskiness of one asset may tend to be canceled by that of another asset

5. Measurement of Co-movements in security returns • Co-movements between the returns of securities are measured by covariance (an absolute measure) and coefficient of correlation (a relative measure) • Covariance reflects the degree to which the returns of two securities vary or change together. • A positive covariance means that the returns of the two securities move in the same direction • A negative covariance implies that the returns of the two securities move in opposite direction. • Correlation coefficient vary between -1.0 and + 1.0

6. Measurement of Portfolio Risk – 2 securities case

7. Matrix Method

8. Matrix Method

9. Portfolio Risk – n security case

10. Matrix Method – n securities

11. Matrix Method – n securities

12. Efficient Frontier – 2 Security Case • Suppose an investor is evaluating two securities A and B, Security ASecurity B • Expected Return 12% 20% • Standard Deviation 20% 40% • Coef. Of Correlation - 0.20 • The investor can combine securities A and B in a portfolio in a number of ways by changing the proportion of funds allocated to them.

13. Portfolios of Investor – Infinite Choices Plotting the options described above, we get the following exhibit:

14. Portfolio Options • The effect of diversification can be seen by comparing the curved line between points A and B with the straight line between A and B. • The straight line represents the risk return possibilities by combining A and B if the correlation coef. Between two had been 1. • Since the curved line is always to the left of the straight line, the diversification effect is illustrated in the figure.

15. Portfolio Options…….. • Portfolio 3 represents the minimum variance portfolio (MVP) or more accurately, the minimum std. deviation portfolio. • The investor considering portfolio of A and B faces an opportunity set or feasible set represented by curved line AB. • By choosing an appropriate mix between the two securities, the investor can achieve any point on the curved line.

16. Portfolio Options…….. • The curve bends backwards between points A and 3 (MVP), meaning that, for a portion of the feasible set, the std. deviation decreases although expected return increases. • This happens due to diversification effect of including security B in the portfolio (corr. Coef being negative) • For some length, increasing the proportion of B, curve bends backwards. • After that, std deviation increases with return. Technically, a backward bend occurs when r ≤ 0; it may or may not occur when r > 0

17. Portfolio Options…….. • No investor would like to invest in a portfolio whose expected return is less than that of MVP. • Although, the entire curve from A to B is feasible, investors would consider only the segment from 3 to B. • This is called the efficient set or the efficient frontier. • Points lying along the efficient frontier are called efficient portfolios.

18. Efficient frontier for diff degrees of r….. • Risk can even be reduced to zero, by choosing the weights (wA and wB ) suitably in case where r = -1.0 • This can be done by replacing r = -1.0 in portfolio variance equation and equating the same to zero.

19. Efficient frontier for diff degrees of r • The feasible frontier under various degree of Coefficient of Correlation is delineated in the figure

20. Efficient Frontier for the n-Security case • In a multi-security case, the collection of all the possible portfolios is represented by the broken egg shaped region, referred to as the feasible region, shown in the figure. • The number of possible portfolios in that region is virtually endless.

21. Efficient Frontier for the n-Security case….. • What really matters to the investor is the northwest boundary of the feasible region (AFX) • Referred to as efficient frontier, this boundary contains all the efficient portfolios.

22. Efficient portfolio: • A portfolio is efficient if and only if there is no alternative with: • The same E(Rp) and a lower σp • The same σp and higher E(Rp) • A higher E(Rp) and a lower σp

23. Efficient Frontier for the n-Security case….. • A portfolio like Z is inefficient because portfolio like B and D, among others, dominate it. • The efficient frontier is the same for all investors because portfolio theory is based on the assumption that investors have homogeneous expectations.

24. How can this set of efficient portfolios be actually obtained from the innumerable portfolio possibilities that lie before the investor ? • The set of efficient portfolios may be determined with the help of graphical analysis, or calculus analysis, or quadratic programming analysis.

25. Risk-Return Indifference Curve • All points lying on an indifference curve provide the same level of satisfaction. • In general, the steeper the slope of the indifference curve, the greater the degree of risk aversion. • Each person has a map of indifference curves. • The indifference cureves Ip and Iq represent the risk return trade offs of two investors P and Q (both are risk averse) • Q is more risk averse than P

26. Utility Indifference Curves • Points A and B, offer the same level of satisfaction • Points R and S offer the same level of satisfaction. • The level of satisfaction increases as one moves leftward in the indifference curve map. • The indifference curve Ip2 represents a higher level of satisfaction as compared to the indifference curve Ip1…….

27. Optimal Portfolio • Given the efficient frontier and the risk-return indifference curve, the optimal portfolio is found at the point of tangency between the efficient frontier and utility indifference curve. • This point represents the highest level of utility the investor can reach.

28. Optimal Portfolio…. • Two investors P and Q, confronted with the same efficient frontier, but having different indifference curves are shown to achieve their highest utility at points P* and Q* respectively

29. Riskless Lending and Borrowing • So far we have assumed that all the securities on the efficient set are risky. • Suppose that investors can also lend and borrow money at risk free rate Rf • Risk-free asset has zero correlation with all the points in the feasible region of risky portfolios. • So a combination of Rf and any point in the feasible region of risky securities will be represented by a straight line.

30. Lending and Borrowing Opportunity • Consider point Y, a portfolio of risky securities. • Investors can combine Rf and Y and reach any point along straight line from Rf to Y and even beyond – to go beyond they have to leverage. • Lets refer this line as I

31. Although the investor can reach any point on line I, no point on this line is optimal. Contd… • Consider line II which runs from Rf to S and beyond. • Line II is tangent to the efficient set of risky securities, so it provides the investor the best possible opportunities. • Line II dominates line I – for that matter, it dominates any other line between Rf and any point in the feasible region of risky securities.

32. With the opportunity of lending and borrowing, the efficient frontier changes. It is no longer AFX, it becomes Rf SG Contd… • For every point on AFX, there is at least one point on Rf SG which is superior. • Compared to point C on AFX, D on Rf SG offers higher expected return for the same standard deviation. • Compared to point B on AFX, E on Rf SG offers higher expected return for the same standard deviation.

33. Every investor would do well to choose some combination of Rf and S. Contd… • A conservative investor may choose point like U – weighing Rf more in his portfolio (lending) • An aggressive investor may choose point like V – weighing S more in his portfolio (borrowing)

34. Contd…. Risk Free Borrowing Risk free Lending

35. Steps: Portfolio Selection • Identification of S, the optimal portfolio of risky securities. • Choice of a combination of Rf and S, depending on one’s risk attitude. This is the import of celebrated Separation Theorem, first enunciated by James Tobin, a Nobel Laureate in Economics.

36. The Single Index Model • Markowitz model is highly information sensitive. • If there are n securities, the Markowitz model requires n expected returns, n variance terms and n(n-1)/2 covariance terms. [Total inputs required = n(n+3)/2] • The problem of estimating a large number of covariance terms becomes intractable, particularly for the institutional investors. • Until the Markowitz model is simplified in terms of covariance inputs, it can scarcely become an operational tool. • In his seminal contribution, Markowitz had suggested that an index, to which securities are related, may be used for the purpose of generating the covariance terms.

37. Contd… • Taking a cue from Markowitz, William Sharpe developed the single index model, which expresses the returns on each security as a function of the return on a broad market index as follows: Ri = ai + bi RM + ei Where, Ri = Return on security I RM = Return on market index ai = constant return bi = measure of the sensitivity of the security i’s return to the return on the market index (popularly called beta) ei = error term

38. Contd… • The estiamtes of ai (the constant term) and bi (the slope term) may be obtained by regressing the return on security i on the corresponding return on the market index. • Note that the single-index model is based on the following assumptions: • The error term ei has an expected value of zero and a finite variance. • The error term is not correlated with the return on the market portfolio: COV (ei , RM ) = 0 • Securities are related only through their common response to the return on the market index. This implies that the error term for security i is not correlated with the error term of any other security: COV (ei , ej ) = 0

39. Generating the inputs to the Markowitz Model • The single index model helps immensely in obtaining the following inputs required for applying the Markowitz model: • The expected return of each security • The variance of return on each security, and • The covariance of returns between each pair of securities. • The following equations may be used for the purpose: E(Ri ) = ai + bi E(RM ) Var (Ri ) = bi2 [Var(RM )] + Var (ei ) Cov (Ri , Rj ) = bibjVar (RM )

40. Calculation of the Single Index Model • Single index model requires only (3n+2) number of inputs. [ai , bi , Var (ei ) for each security and E(RM ) and Var (RM )] • William Sharpe found considerable similarity between the efficient portfolios generated by single index model and the Markowitz model. • Subsequent studies have also found that the single index model performs well. • Since the single index model simplifies considerably the input requirements and performs fairly well, it represents a major practical advance in portfolio analysis.