Understanding Return, Risk, and Risk Aversion in Investment Decision-Making

1. Chapter 5 Concepts and Issues: Return, Risk and Risk Aversion

2. Chapter Summary • Objective: To introduce key concepts and issues that are central to informed decision making • Determinants of interest rates • The historical record • Risk and risk aversion • Portfolio risk

3. Factors Influencing Rates • Supply • Households • Demand • Businesses • Government’s Net Supply and/or Demand • Central Bank Actions

4. Interest Rates Supply r1 r0 Demand Q0 Funds Q1 Level of Interest Rates

5. Real vs. Nominal Rates Fisher effect: Approximation R = r + i or r = R - i Example: r = 3%, i = 6% R = 9% = 3%+6% or r = 3% = 9%-6% Fisher effect: Exact or Numerically:

6. Rates of Return: Single Period HPR = Holding Period Return P0 = Beginning price P1 = Ending price D1 = Dividend during period one

7. Rates of Return: Single Period Example Ending Price = 48 Beginning Price = 40 Dividend = 2

8. Characteristics of Probability Distributions 1) Mean: most likely value 2) Variance or standard deviation 3) Skewness * If a distribution is approximately normal, the distribution is described by characteristics 1 and 2

9. s.d. s.d. r Symmetric distribution Normal Distribution

10. Measuring Mean: Scenario or Subjective Returns Subjective returns ‘s’ = number of scenarios considered pi = probability that scenario ‘i’ will occur ri = return if scenario ‘i’ occurs

11. Numerical example: Scenario Distributions E(r) = (.1)(-.05)+(.2)(.05)...+(.1)(.35) E(r) = .15 = 15%

12. Measuring Variance or Dispersion of Returns Subjective or Scenario Distributions Standard deviation = [variance]1/2 = s Using Our Example: s2=[(.1)(-.05-.15)2+(.2)(.05- .15)2+…] =.01199 s= [ .01199]1/2 = .1095 = 10.95%

13. Summary Reminder • Objective: To introduce key concepts and issues that are central to informed decision making • Determinants of interest rates • The historical record • Risk and risk aversion • Portfolio risk

14. Annual HPRsCanada, 1957-2001

15. Annual HP Risk Premiums and Real Returns, Canada

16. Annual HPRsUS, 1926-1999

17. Annual HP Risk Premiums and Real Returns, US

18. Summary Reminder • Objective: To introduce key concepts and issues that are central to informed decision making • Determinants of interest rates • The historical record • Risk and risk aversion • Portfolio risk

19. p = .6 W1 = 150; Profit = 50 1-p = .4 W2 = 80; Profit = -20 Risk - Uncertain Outcomes W = 100 E(W) = pW1 + (1-p)W2 = 122 s2 = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2 s2 = 1,176 and s = 34.29%

20. p = .6 W1 = 150 Profit = 50 Risky Investment W2= 80 Profit = -20 1-p = .4 Risk Free T-bills Profit = 5 Risky Investments with Risk-Free Investment 100 Risk Premium = 22-5 = 17

21. Risk Aversion & Utility • Investor’s view of risk • Risk Averse • Risk Neutral • Risk Seeking • Utility • Utility Function U = E ( r ) – .005 A s2 • A measures the degree of risk aversion

22. T-bill = 5% Risk Aversion and Value: The Sample Investment U = E ( r ) - .005 A s2 = 22% - .005 A (34%) 2 Risk AversionAUtility High 5 -6.90 3 4.66 Low 1 16.22

23. Expected Return 4 2 3 1 Variance or Standard Deviation Dominance Principle • 2 dominates 1; has a higher return • 2 dominates 3; has a lower risk • 4 dominates 3; has a higher return

24. Utility and Indifference Curves • Represent an investor’s willingness to trade-off return and risk Example (for an investor with A=4):

25. Expected Return Increasing Utility Standard Deviation Indifference Curves

26. Summary Reminder • Objective: To introduce key concepts and issues that are central to informed decision making • Determinants of interest rates • The historical record • Risk and risk aversion • Portfolio risk

27. Portfolio Mathematics:Assets’ Expected Return Rule 1 : The return for an asset is the probability weighted average return in all scenarios.

28. Portfolio Mathematics:Assets’ Variance of Return Rule 2: The variance of an asset’s return is the expected value of the squared deviations from the expected return.

29. Portfolio Mathematics: Return on a Portfolio Rule 3: The rate of return on a portfolio is a weighted average of the rates of return of each asset comprising the portfolio, with the portfolio proportions as weights. rp = w1r1 + w2r2

30. Portfolio Mathematics:Risk with Risk-Free Asset Rule 4: When a risky asset is combined with a risk-free asset, the portfolio standard deviation equals the risky asset’s standard deviation multiplied by the portfolio proportion invested in the risky asset.

31. Portfolio Mathematics:Risk with two Risky Assets Rule 5: When two risky assets with variances s12 and s22 respectively, are combined into a portfolio with portfolio weights w1 and w2, respectively, the portfolio variance is given by: