Chapter 12: Choosing an Investment Portfolio

1. Chapter 12: Choosing an Investment Portfolio • Objective • To understand the theory of personal • portfolio selection in theory • and in practice

2. Chapter 12: Contents • The process of personal portfolio selection • The trade-off between expected return and risk • Efficient diversification with many risky assets

3. The Concept of ‘Portfolio’ • A person’s wealth portfolio includes • Assets: stocks, bonds, shares in unincorporated business, houses or apartments, pensions benefits, insurance policies, etc. • Liabilities: student loans, auto loans, home mortgages, etc.

4. Portfolio Selection • A study of how people should invest their wealth optimally • A process of trading off risk and expected return to find the best portfolio of assets and liabilities • Narrow and broad definitions: • How much to invest in stocks, bonds, and other securities • Whether to buy or rent one’s house • What types and amounts of insurance to purchase • How to manage one’s liabilities • How much to invest in one’s human capital

5. Portfolio Selection • Although there are some general rules for portfolio selection that apply to virtually everyone, there is no single portfolio or portfolio strategy that is best for everyone.

6. The Life Cycle • In portfolio selection, the best strategy depends on an individual’s personal circumstances (family status, occupation, income, wealth). • Illustrations • Young couple: buy a house and take out a mortagage loan / older couple: sell house and invest in assets provding a steady stream of income. • Investing in stock market: Chang (30, a security analyst) / Obi (30, an English teacher). • Buying insurance policies: Miriam (a parent with dependent children) / Sanjiv (a single person with no dependents).

7. Time Horizon • In formulating a plan for portfolio selection, you begin by determining your goals and time horizons. • Planning horizon: the total length of time for which one plans • Decision horizon: the length of time between decisions to revise the portfolio • Trading horizon: the minimum time interval over which investors can revise their portfolios / its determination and impacts • Investment strategy & trading horizon: portfolio insurance or dynamic portfolio strategy.

8. Risk Tolerance • A major determinant of portfolio choices • It is influenced by such characteristics as • age, family status, job status, wealth, and • other attributes that affect a person’s ability to maintain his standard of living in the face of adverse movements in the market value of his investment portfolio

9. Professional Asset Managers • Investment advisors & “finished products” from a financial intermediary • Specialization, information and cost advantages

10. The Trade-off between Expected Return and Risk • The objective is to find the portfolio which offers investors the highest expected rate of return for the degree of risk they are willing to tolerate. • Two step process: • find the optimal combination of risky assets. • mix this optimal risk-asset with the riskless asset.

11. Riskless Asset • A security that offers a perfectly predictable rate of return in terms of the unit of account selected for the analysis and the length of the investor’s decision horizon. • For example, if the U.S dollars is taken as the unit of account and the decision horizon is half a year, the riskless rate is the interest rate on U.S Treasury bills maturing after half a year.

12. Rates of Return on Risky Assets • Required return depends on the risk of the investment. • Greater the risk, greater the return • Risk premium

17. Portfolio of n risky assets Measuring Portfolio Return • Ii : the initial investment in asset i (if Ii <0, short selling) • wi: the proportion of the portfolio investing in asset I • ri : the rate of return on asset I • rp: the rate of return on the portfolio

18. Short Selling • Ik < 0 : short selling (borrowing) asset k

19. : the expected value ofri • : the standard deviation of ri • : the correlation between riand rj Mean and Variance of Portfolio Return

20. Variance with 2 Securities • Variance with 3 Securities

21. An Example: A Portfolio of BM and FM • Suppose you invest $6000 in Bristol-Myers at an expected return of 15%, and $4000 in Ford Motor at an expected return of 21%. • The standard deviation of the return on BM’s stock is 18.6%, while the standard deviation of the return on FM is 28%. • The correlation between the returns is 0.4.

22. Expected Return (%) ● 40% F M 60% BM ● Bristol-Myers Portfolios of BM and FM Ford Motor Standard Deviation (%)

23. Portfolios of Two Correlated Common Stock • Two common stock with these statistics: • mean return 1 = 0.15 • mean return 2 = 0.10 • standard deviation 1 = 0.20 • standard deviation 2 = 0.25 • correlation of returns = 0.90 • initial price 1 = $57.25 • initial price 2 = $72.625

25. Portfolio of Two Securities 0.25 Efficient Portfolio 0.20 Security 1 Sub-optimal Portfolio 0.15 Expected Return Security 2 0.10 Minimum Variance Portfolio 0.05 0.00 0.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29 Standard Deviation Is one “better”?

26. Formula for Minimum Variance Portfolio

27. s.t. Portfolio Selection with n Risky Assets Harry Markowitz (1952): Portfolio Selection, Journal of Finance

28. Solution：

29. where

30. Expected Return (%) Standard Deviation (%) Efficient frontier: the set of portfolios offering the highest expected return for any given standard deviation. • Portfolio of many risky assets efficient frontier minimum-variance portfolio

31. Combining the Riskless Asset and a Single Risky Asset: An illustration • Let’s suppose that you have $100,000 to invest. • You are choosing between a riskless asset with a interest of 6% per year and a risky asset with an expected rate of return of 14% per year and a standard deviation of 20%. • How much of your $100,000 should you invest in the risky asset?

32. Mean and Standard Deviation

33. 0.16 0.14 S 0.12 J 0.1 H Expected Return 0.08 R inefficient G 0.06 F 0.04 0.02 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Standard Deviation The Risk-Return Trade-off Line

34. where Combining the Riskless Asset and a Single Risky Asset • We know something special about the portfolio, namely that security 2 is riskless, so σ2 = 0, and σp becomes

36. 100% Risky Long risky and short risk-free CML • Long both risky • and risk-free • 100% Risk-less

37. Sharpe Ratio Risk Premium • The slope measure the extra expected return the market offers for each extra risk a investor is willing to bear

38. Achieving a Target Expected Return • To find the portfolio corresponding to an expected rate of return of 0.11 per year, we substitute 0.11 for E(rp) and solve for w1. • Thus, the portfolio mix is 62.5% risky asset and 37.5% riskless asset.

39. Portfolios of the Riskless Security and Two Risky Securities • The riskless security and two risky securities with the following statistics: • riskless rate of return rf = 0.06 • mean return 1 = 0.14 • mean return 2 = 0.08 • standard deviation 1 = 0.20 • standard deviation 2 = 0.15 • correlation of returns = 0

40. 0.16 0.14 S 0.12 T ◆ 0.1 Tangent Portfolio Expected Return E 0.08 R 0.06 0.04 0.02 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Standard Deviation The Optimal Combination of the Three Securities

41. Formula for Tangent Portfolio = s = E ( r ) 0 . 12154 0 . 14595 T T

42. Efficient Trade-off Line • New efficient trade-off line: • Compare the old trade-off line connecting points F and S. • Clearly the investor is better off.

43. Achieving a Target Expected Return • The investment criterion is to generate a 10% expected rate of return. • Thus, the portfolio mix is 35% riskless asset and 65% tangent portfolio, namely 45% risky security 1 and 20% risky security 2.

44. Selecting the Preferred Portfolio • It is important to note that in finding the optimal combination of risky assets, we do not need to know anything about investor preferences. • There is always a particular optimal portfolio of risky assets that all risk-averse investors who share the same forecasts of rates of return will combine with the riskless asset to reach their most-preferred portfolio.

45. Return Low Risk High Return High Risk High Return Low Risk Low Return High Risk Low Return Risk The Rationale for Portfolio Selection

46. Expected Return (%) rf Standard Deviation (%) • Portfolio of many risky assets and the riskless asset Short sell Efficient frontier Tangent Portfolio

47. Expected Return T Standard Deviation Efficient Frontier • The jelly fish shape contains all possible combinations of risk and return: The feasible set. • The red line constitutes the efficient frontierof portfolios of risky assets: Highest return for given risk. • The tangent portfolio T is the optimal portfolio of risky assets that all risk-averse investors will combine with the riskless asset. Two-Fund Separation Theorem (Tobin, 1958)

48. Theory & Practice • The static mean-variance model & elementary theory of mutual fund financial intermediation. • Dynamic versions integrating intertemporal optimization of the life-cycle consumption-saving decisions with the allocation of those savings among alternative investments & a richer theory for the role of securities and financial intermediation. • Optimal combination of assets & optimal hedging portfolio more tailored to the needs of different clienteles.