An Introduction to Portfolio Management

1. An Introduction to Portfolio Management Fin 825

2. Greedy & Risk Aversion Greedy: Given a choice between two assets with equal level of risk, greedy investors will select the asset with the higher level of risk. Risk Averse: Given a choice between two assets with equal rates of return, risk averse investors will select the asset with the lower level of risk.

3. Implications for the investment process All investors are risk averse? Yes. All investors are risk averse? Yes/No, risk preference may depends on amount of money involved - risking small amounts, but insuring large losses Since most investors are risk averse, there is a positive relationship between expected return and expected risk.

4. Covariance between Returns of Two Assets For two assets, i and j, the covariance of rates of return is a measure of the degree to which two variables “move together” relative to their individual mean values over time. Covariance is defined as: Covij = E{[Ri - E(Ri)][Rj - E(Rj)]}

5. Covariance and Correlation Covariance between two assets can be derived from their standard deviations and the correlation coefficient using the following formula:

6. Markowitz portfolio optimization Required inputs: Expected returns of all securities in the portfolio Standard deviations of all securities in the portfolio Covariance(s) (or correlation coefficient) among entire set of securities in the portfolio With 100 assets, 4,950 correlation estimates

7. Portfolio Expected Return Formula

8. Portfolio Standard Deviation Formula

9. Returns Distribution for Two Perfectly Negatively Correlated Stocks (r = -1.0) and for Portfolio WM

10. Returns Distributions for Two Perfectly Positively Correlated Stocks (r = +1.0) and for Portfolio MM’

11. Combining Stocks with Different Returns and Risk Case Correlation Coefficient Covariance a +1.00 .0070 b +0.50 .0035 c 0.00 .0000 d -0.50 -.0035 e -1.00 -.0070

12. Portfolio Risk-Return Plots for Different Weights

13. Portfolio Risk-Return Plots for Different Weights

14. Portfolio Risk-Return Plots for Different Weights

15. Portfolio Risk-Return Plots for Different Weights

16. Portfolio Risk-Return Plots for Different Weights

17. Concept of Diversification Combining different assets in a portfolio to reduce overall risks. The lower the correlation between assets, the lower the overall portfolio risk produced. Combining two assets with perfectly negative correlation (correlation coefficient of -1) could reduce the portfolio standard deviation to zero

18. Correlation Coefficient Correlation coefficient is a standardized covariance. It varies from -1 to +1.

19. The Efficient Frontier The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk Frontier will be portfolios of investments rather than individual securities

20. Efficient Frontier for Alternative Portfolios

21. The Efficient Frontier and Investor Utility An individual investor’s utility curve specifies the trade-offs he is willing to make between expected return and risk The optimal portfolio results in the highest utility possible for a given investor It lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility

22. Selecting an Optimal Risky Portfolio

23. Example: P8-4 You are considering two assets with the following characteristics: E(R1)=.15, E(?1)=.10, W1=.5 E(R2)=.20, E(?2)=.20, W1=.5 Compute the mean and standard deviation of two portfolios if r1,2=0.4 and –0.60, respectively.

24. Solution E(RP)=.5 x (.15) + .5 x (.20)= .175 If r1,2=0.4, If r1,2=-0.6, ?p=0.08062

25. An Introduction to Asset Pricing Models

26. Risk-Free Asset An asset with no risk. Zero variance and zero correlation with all other assets Provides the risk-free rate of return (RFR) Will lie on the vertical axis of a portfolio graph The combination of risk-free asset and any risky asset or portfolio will always have a linear relationship between expected return and risk.

27. Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier

28. Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier

29. The Market Portfolio Portfolio M lies at the point of tangency, it has the highest slope of trade-off between expected return and risk. All investors will want to invest in Portfolio M and borrow or lend to be somewhere on the CML Therefore this portfolio must include ALL RISKY ASSETS in proportion to their market values. M is a completely diversified portfolio, which means that all the unique risk of individual assets is diversified away

30. Systematic Risk Only systematic risk remains in the market portfolio, M Systematic risk is the variability in all risky assets caused by macroeconomic variables Systematic risk is measured by the standard deviation of returns of the market portfolio

31. Examples of Macroeconomic Factors Affecting Systematic Risk Variability in growth of money supply Interest rate volatility Inflation Fiscal and Monetary policy changes War and political events

32. Portfolio Standard Deviations

33. Portfolio Diversification

34. Portfolio Diversification

35. The CML and the Separation Theorem The CML leads all investors to invest in the M portfolio (the investment decision) The decision to borrow or lend to obtain a point on the CML is based on individual risk preferences (the financing decision) Tobin refers to this separation of the investment decision from the financing decision as the Separation Theorem

36. CML and the Separation Theorem

37. The Capital Asset Pricing Model: Expected Return and Risk The existence of a risk-free asset resulted in capital market line (CML) that became the relevant frontier An asset’s covariance with the market portfolio (systematic risk) is the relevant risk measure Systematic risk can be used to determine an appropriate expected rate of return on a risky asset

38. Graph of Security Market Line

39. The Security Market Line (SML) The equation for the risk-return line is

40. Plot of Estimated Returnson SML Graph

41. Calculating Systematic Risk: The Characteristic Line

42. Scatter Plot of Rates of Return

43. Arbitrage Pricing Theory (APT) Assumptions: - Capital markets are perfectly competitive. - Investors always prefer more wealth to less wealth with certainty. - The stochastic process generating asset returns can be represented as a K factor model. As a result of the criticisms of CAPM regarding its implementation, along with its many assumptions, the academic community set forth to develop an alternative asset pricing theory. The Arbitrage Pricing Theory (APT), developed by Stephen Ross in the early 1970’s, is reasonably intuitive and requires only limited assumptions. APT requires that only three simple assumptions be made. One is that capital markets are perfectly competitive. Another is that investors always prefer more wealth to less wealth with certainty. Finally, APT assumes that the stochastic process generating security returns can be represented by a K factor model. As a result of the criticisms of CAPM regarding its implementation, along with its many assumptions, the academic community set forth to develop an alternative asset pricing theory. The Arbitrage Pricing Theory (APT), developed by Stephen Ross in the early 1970’s, is reasonably intuitive and requires only limited assumptions. APT requires that only three simple assumptions be made. One is that capital markets are perfectly competitive. Another is that investors always prefer more wealth to less wealth with certainty. Finally, APT assumes that the stochastic process generating security returns can be represented by a K factor model.

44. Assumptions do not Required: - Quadratic utility function. - Normally distributed security returns. - A market portfolio that contains all risky assets and is mean-variance efficient. Arbitrage Pricing Theory (APT) While it is appealing that APT requires few assumptions, it is also important to recognize the assumptions that APT does not require that are required by the CAPM. These include: APT does not require that investors have quadratic utility functions, there is no restriction regarding the shape of the returns distribution, and and it is not necessary to identify a market portfolio that contains all risky assets and is mean-variance efficient. While it is appealing that APT requires few assumptions, it is also important to recognize the assumptions that APT does not require that are required by the CAPM. These include: APT does not require that investors have quadratic utility functions, there is no restriction regarding the shape of the returns distribution, and and it is not necessary to identify a market portfolio that contains all risky assets and is mean-variance efficient.

45. Ri = E(Ri) + bi1d1 + bi2d2 + ... + bikdk +Îi for i= 1 to n where: Ri = return on asset i during a specified time period E (Ri)= expected return for asset i bik = reaction in asset i’s returns to movements in the common factor k dk= a common factor k with a zero mean that influences the returns on all assets Îi = a unique effect on asset i’s return that is completely diversifiable in large portfolios and has a mean of zero n = number of assets Return Generating Process The returns generating process is a stochastic process described by the K factor model shown here. The terms dk is the multiple factors (K common factors) expected to have an impact on the returns of all assets (for example, inflation, growth in GNP, major political upheavals, or changes in interest rates). The bik terms determine how each asset reacts to this common factor. If the factor K were interest rates, then a stock that was interest rate sensitive might have a large response term. A non-interest sensitive stock would likely have a small b coefficient. The Îi terms will be diversified away because they represent firm-unique influences. The returns generating process is a stochastic process described by the K factor model shown here. The terms dk is the multiple factors (K common factors) expected to have an impact on the returns of all assets (for example, inflation, growth in GNP, major political upheavals, or changes in interest rates). The bik terms determine how each asset reacts to this common factor. If the factor K were interest rates, then a stock that was interest rate sensitive might have a large response term. A non-interest sensitive stock would likely have a small b coefficient. The Îi terms will be diversified away because they represent firm-unique influences.

46. E(Ri) = l0 + l1bi1, + l2bi2 + ... + l kbik where: l0 = the expected return on an asset with zero systematic risk where l0 = E(R0) l1 = the risk premium related to each of the common factors bi = the pricing relationship between the risk premium and asset i Expected Return for Any Asset Like CAPM, it is assumed that the unique effects (Îi) are independent and will be diversified away in a large portfolio. The APT assumes that, in equilibrium, the return on a zero-investment, zero-systematic-risk portfolio is zero when the unique effects are diversified away. This assumption and some theory from linear algebra imply that the expected return on any asset can be described by the relationship shown here.Like CAPM, it is assumed that the unique effects (Îi) are independent and will be diversified away in a large portfolio. The APT assumes that, in equilibrium, the return on a zero-investment, zero-systematic-risk portfolio is zero when the unique effects are diversified away. This assumption and some theory from linear algebra imply that the expected return on any asset can be described by the relationship shown here.

47. Factors: ?1 = Changes in the rate of inflation ? 2 = percent growth in industrial production l1 = 0.01, the risk premium associated with ?1 l2 = 0.015, the risk premium associated with ? 2 l0 = 0.04, rate of return on a zero-systematic-risk asset 2 Assets, 2-Factor Model To illustrate how APT can be applied, consider the two stock, two-factor example shown here. Notice that asset G is more sensitive to both sources of systematic risk than asset F. Therefore, asset G should be the riskier asset. To illustrate how APT can be applied, consider the two stock, two-factor example shown here. Notice that asset G is more sensitive to both sources of systematic risk than asset F. Therefore, asset G should be the riskier asset.

48. Response Coefficients (B) for Assets F & G bF1 = response of asset F to changes in the rate of inflation (0.5) bF2 = response of F to changes in level of industrial production (1.25) bG1 = response of asset G to changes in rate of inflation (1.75) bG2 = response of G to changes in level of industrial production (2.00) 2 Assets, 2-Factor Model (Cont.)

49. E(RF) = 0.04 + (0.01)(0.5) + (0.015)(1.25) = 0.06375, or 6.38% E(RG) = 0.04 + (0.01)(1.75) + (0.015)(2.00) = 0.0875, or 8.75% E(Ri) = l0 + l1bi1 + l2bi2 The results here indicates that asset G has a higher required rate of return than F. If the price of either of these assets don’t reflect these returns we would expect investors to enter into arbitrage trading whereby they would sell overpriced assets short and use the proceeds to purchase the underpriced assets until the relevant prices were corrected. Given these linear relationships, it should be possible to find an asset or a combination of assets with equal risk to the mispriced asset, yet with a higher return.The results here indicates that asset G has a higher required rate of return than F. If the price of either of these assets don’t reflect these returns we would expect investors to enter into arbitrage trading whereby they would sell overpriced assets short and use the proceeds to purchase the underpriced assets until the relevant prices were corrected. Given these linear relationships, it should be possible to find an asset or a combination of assets with equal risk to the mispriced asset, yet with a higher return.

50. APT and CAPM Compared APT applies to well diversified portfolios and not necessarily to individual stocks With APT it is possible for some individual stocks to be mispriced - not lie on the SML APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio APT can be extended to multifactor models