Risk and Return

1. Risk and Return Riccardo Colacito

2. Roadmap • Rates of Return • Holding Period Return • Arithmetic and Geometric Averages • Annual Percentage Rate and Effective Annual Rate • Summary Statistics of rates of return • Probability Distribution • Expected Return • Variance, Covariance and Standard Deviation • Other properties • Historical record of Bills, Bonds, and Stocks • Risk premia from 1926-2003? • Inflation and Real Rates of Return

3. Holding Period Return

4. Rates of Return: Single Period Example Ending Price = 24 Beginning Price = 20 Dividend = 1 HPR = ( 24 - 20 + 1 )/ ( 20) = 25%

5. Roadmap • Rates of Return • Holding Period Return • Arithmetic and Geometric Averages • Annual Percentage Rate and Effective Annual Rate • Summary Statistics of rates of return • Probability Distribution • Expected Return • Variance, Covariance and Standard Deviation • Other properties • Historical record of Bills, Bonds, and Stocks • Risk premia from 1926-2003? • Inflation and Real Rates of Return

6. Returns Using Arithmetic and Geometric Averaging Arithmetic ra = (r1 + r2 +... rn) / n ra = (.10 + .25 - .20 + .25) / 4 = .10 or 10% Geometric rg = [(1+r1) (1+r2) .... (1+rn)]1/n - 1 rg = [(1.1) (1.25) (.8) (1.25)]1/4 - 1 = (1.5150) 1/4 -1 = .0829 = 8.29%

7. Roadmap • Rates of Return • Holding Period Return • Arithmetic and Geometric Averages • Annual Percentage Rate and Effective Annual Rate • Summary Statistics of rates of return • Probability Distribution • Expected Return • Variance and Standard Deviation • Other properties • Historical record of Bills, Bonds, and Stocks • Risk premia from 1926-2003? • Inflation and Real Rates of Return

8. Quoting Conventions • Annual Percentage Rate APR = (periods in year) X (rate for period) • Effective Annual Rate EAR = ( 1+ rate for period)Periods per yr – 1 • Example: monthly return of 1% APR = 1% X 12 = 12% EAR = (1.01)12 - 1 = 12.68%

9. Roadmap • Rates of Return • Holding Period Return • Arithmetic and Geometric Averages • Annual Percentage Rate and Effective Annual Rate • Summary Statistics of rates of return • Probability Distribution • Expected Return • Variance, Covariance and Standard Deviation • Other properties • Historical record of Bills, Bonds, and Stocks • Risk premia from 1926-2003? • Inflation and Real Rates of Return

10. Probability distribution • Definition: list of possible outcomes with associated probabilities • Example:

11. Probability distribution: figure

12. Normal distribution

13. Notation • Let p(i) denote the probability with which state i occurs • Then • p(1)=0.1 • p(2)=0.2 • p(3)=0.4 • p(4)=0.2 • p(5)=0.1

14. Roadmap • Rates of Return • Holding Period Return • Arithmetic and Geometric Averages • Annual Percentage Rate and Effective Annual Rate • Summary Statistics of rates of return • Probability Distribution • Expected Return • Variance, Covariance and Standard Deviation • Other properties • Historical record of Bills, Bonds, and Stocks • Risk premia from 1926-2003? • Inflation and Real Rates of Return

15. Expected Return S E ( r ) = p ( s ) r ( s ) s Definition: • p(s) = probability of a state • r(s) = return if a state occurs • 1 to s states

16. Numerical Example E(r) = (.1)(-2) + (.2)(-1) + (.4)(0) + (.2)(1) + (.1)(2) = 0

17. Roadmap • Rates of Return • Holding Period Return • Arithmetic and Geometric Averages • Annual Percentage Rate and Effective Annual Rate • Summary Statistics of rates of return • Probability Distribution • Expected Return • Variance, Covariance and Standard Deviation • Other properties • Historical record of Bills, Bonds, and Stocks • Risk premia from 1926-2003? • Inflation and Real Rates of Return

18. Why do we need the variance? • Two variables with the same mean. • What do we know about their dispersion?

19. Measuring Variance or Dispersion of Returns S 2 Variance = p ( s ) [ r - E ( r )] s s Standard deviation = variance1/2 Why do we take squared deviations?

20. Numerical example Var = .1 (-2-0)2 + .2 (-1-0)2 + .4 (0-0)2 + .2 (1-0)2 + .1 (2-0)2 = 1.2 Std dev= (1.2)1/2 = 1.095

21. One important property of variance and standard deviation • Let w be a constant Var(wxr) = w2x Var(r) • Similarly Std Dev(wxr) = w x Std Dev(r)

22. Covariance: Preliminaries • Covariance • The extent at which two assets tend to move together • Can be positive or negative • Correlation • Same idea of covariance, but bounded between -1 and 1

23. Covariance: definition

24. Correlation: definition

25. Correlation (cont’d)

26. Other properties - -

27. Correlation=-1

28. Correlation=+1

29. Correlation=0

30. Roadmap • Rates of Return • Holding Period Return • Arithmetic and Geometric Averages • Annual Percentage Rate and Effective Annual Rate • Summary Statistics of rates of return • Probability Distribution • Expected Return • Variance, Covariance and Standard Deviation • Other properties • Historical record of Bills, Bonds, and Stocks • Risk premia from 1926-2003? • Inflation and Real Rates of Return

31. 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

32. Skewed Distribution: Large Negative Returns Possible Median Negative Positive r

33. Skewed Distribution: Large Positive Returns Possible Median Negative r Positive

34. Roadmap • Rates of Return • Holding Period Return • Arithmetic and Geometric Averages • Annual Percentage Rate and Effective Annual Rate • Summary Statistics of rates of return • Probability Distribution • Expected Return • Variance, Covariance and Standard Deviation • Other properties • Historical record of Bills, Bonds, and Stocks • Risk premia from 1926-2003? • Inflation and Real Rates of Return

35. Risk premium • An expected return in excess of that of a risk free rate • Example • The expected return on the S&P500 is 9% • The return on a 1-month T-bill is 3% • The risk premium is 6% (9%-3%)

36. Annual Holding Period ReturnsFrom Table 5.3 of Text Geom. Arith. Stan. Series Mean% Mean% Dev.% World Stk 9.41 11.17 18.38 US Lg Stk 10.23 12.25 20.50 US Sm Stk 11.80 18.43 38.11 Wor Bonds 5.34 6.13 9.14 LT Treas 5.10 5.64 8.19 T-Bills 3.71 3.79 3.18 Inflation 2.98 3.12 4.35

37. Risk Premia Arith. Stan. Series Mean% Dev.% World Stk 7.37 18.69 US Lg Stk 8.46 20.80 US Sm Stk 14.64 38.72 Wor Bonds 2.34 8.98 LT Treas 1.85 8.00

38. Figure 5.1 Frequency Distributions of Holding Period Returns

39. Figure 5.2 Rates of Return on Stocks, Bonds and Bills

40. Roadmap • Rates of Return • Holding Period Return • Arithmetic and Geometric Averages • Annual Percentage Rate and Effective Annual Rate • Summary Statistics of rates of return • Probability Distribution • Expected Return • Variance, Covariance and Standard Deviation • Other properties • Historical record of Bills, Bonds, and Stocks • Risk premia from 1926-2003? • Inflation and Real Rates of Return

41. Real vs. Nominal Rates • Notation: • R=nominal return • i =inflation rate • r =real return • Exact relationship • Approximate relationship • Example R = 9%, i = 6%: what is r?

42. Figure 5.4 Interest, Inflation and Real Rates of Return