1. Investments, 8 th edition Bodie, Kane and Marcus Slides by Susan Hine McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved. CHAPTER 5 Learning About Return and Risk from the Historical Record

2. 5-2 Factors Influencing Interest Rates In macroeconomics, it's determined by the following but very hard to predict Supply –Households Demand –Businesses Government’s Net Supply and/or Demand –Federal Reserve Actions

3. 5-3 Real and Nominal Rates of Interest Nominal interest rate –Growth rate of your money Real interest rate –Growth rate of your purchasing power If R is the nominal rate and r the real rate and i is the inflation rate:

4. 5-4 Equilibrium Real Rate of Interest Determined by: –Supply –Demand –Government actions –Expected rate of inflation There are as many interests as securities but we talk as if there were a unique interest rate

5. 5-5 Figure 5.1 Determination of the Equilibrium Real Rate of Interest

6. 5-6 Equilibrium Nominal Rate of Interest As the inflation rate increases, investors will demand higher nominal rates of return If E(i) denotes current expectations of inflation for the coming period, then we get the Fisher Equation: Implies R predicts E(i) when r is stable.

7. 5-7 Taxes and the Real Rate of Interest Tax liabilities are based on nominal income –Given a tax rate (t), nominal interest rate (R), after-tax interest rate is R(1-t) –Real after-tax rate is:

8. 5-8 Comparing Rates of Return for Different Holding Periods Zero Coupon Bond with $100 face value, maturity of T years, price of P(T) r f (T), rate of return per period T is

9. 5-9 Formula for EARs and APRs How to compare returns of different horizons Effective annual rate (EAR) e.g. r f (.5)=2.71 1+EAR=(1.0271) 2 e.g. r f (25)=329.18 (1+EAR) 25= 4.2918

10. 5-10 Formula for EARs and APRs So, generally, Effective annual rate (EAR)

11. 5-11 Formula for EARs and APRs Often, short term investment (T<1) rate is given by Annual percentage rate (APR) -if there are n compounding periods a year and r f (T) is a per period rate, APR=nr f (T) or r f (T)=TxAPR e.g. APR=9% credit card charge for 90 day payment: r f (90/365)=90/365 xAPR

12. 5-12 Formula for EARs and APRs e.g. 6 month zero-coupon bond of r f (0.5)=2.71%: APR=2.71x2 For investment of time length T, there are 1/T compounding periods a year. So 1+EAR=(1+r f (T)) 1/T =(1+TxAPR) 1/T

13. 5-13 Formula for EARs and APRs Comparison of EAR and APR for r f (.25) 3mo, 6mo, 9 mo APR 1+r f (.25), 1+2r f (.25), 1+3r f (.25) EAR 1+r f (.25), (1+r f (.25)) 2, (1+r f (.25)) 3

14. 5-14 Table 5.1 Annual Percentage Rates (APR) and Effective Annual Rates (EAR)

15. 5-15 Formula for EARs and APRs How EAR(E) and APR(A) get different as T gets smaller. Fix E. Then 1+E=(1+r f (T)) 1/T =(1+A/(1/T)) 1/T =(1+1/(1/(AT))) 1/T =((1+1/(1/(AT))) 1/(AT) ) A For different Y, we have different A For T tending to 0, we write corresponding A as r cc. Then the right side goes to

16. 5-16 Formula for EARs and APRs e rcc =exp(r cc ). Given EAR, when compounding period is very very small, corresponding APR is r cc. That is, 1+EAR= e rcc r cc is called continuously compounded annual rate

17. 5-17 Formula for EARs and APRs Obviously, we have the relation ln(1+EAR)=r cc where ln is the natural logarythm. Also given a councinuous compounding rate r cc, the total return for any period S is (1+EAR) S= (e rcc ) s

18. 5-18 Bills and Inflation, 1926-2005 Entire post-1926 history of annual rates: –www.mhhe.com/bkmwww.mhhe.com/bkm Average real rate of return on T-bills for the entire period was 0.72 percent Real rates are larger in late periods

19. 5-19 Table 5.2 History of T-bill Rates, Inflation and Real Rates for Generations, 1926-2005

20. 5-20 Figure 5.2 Interest Rates and Inflation, 1926-2005

21. 5-21 Figure 5.3 Nominal and Real Wealth Indexes for Investment in Treasury Bills, 1966-2005

22. 5-22 Risk and Risk Premiums HPR = Holding Period Return P 0 = Beginning price P 1 = Ending price D 1 = Dividend during period one assumed dividend is paid at the end. Rates of Return: Single Period

23. 5-23 Ending Price =48 Beginning Price = 40 Dividend = 2 assumed dividend is paid at the end. HPR = (48 - 40 + 2 )/ (40) = 25% Rates of Return: Single Period Example

24. 5-24 Expected returns p(s) = probability of a state r(s) = return if a state occurs s = state Expected Return and Standard Deviation

25. 5-25 StateProb. of Stater in State 1.1-.05 2.2.05 3.4.15 4.2.25 5.1.35 E(r) = (.1)(-.05) + (.2)(.05)… + (.1)(.35) E(r) =.15 Scenario Returns: Example

26. 5-26 Standard deviation = [variance] 1/2 Variance: Var =[(.1)(-.05-.15) 2 +(.2)(.05-.15) 2 …+.1(.35-.15) 2 ] Var=.01199 S.D.= [.01199] 1/2 =.1095 Using Our Example: Variance or Dispersion of Returns risk premium=expected HPR - risk free rate excess return=actual rate of return – risk free asset

27. 5-27 Time Series Analysis of Past Rates of Return Expected Returns and the Arithmetic Average We assume as if each p(s) carries equal probability, that is, 1/n

28. 5-28 Geometric Average Return (1+g) n =TV = Terminal Value of the Investment g= geometric average rate of return

29. 5-29 Geometric Variance and Standard Deviation Formulas Variance = expected value of squared deviations When eliminating the bias, Variance and Standard Deviation become:

30. 5-30 The Reward-to-Volatility (Sharpe) Ratio Sharpe Ratio for Portfolios = Risk Premium SD of Excess Return

31. 5-31 Figure 5.4 The Normal Distribution

32. 5-32 If the return is normal Very convenient Symmetric The portfolio is normal if each security is. Need only two parameters to specify, mean and variance The question is, does return follow normal distribution?

33. 5-33 Figure 5.5A Normal and Skewed Distributions (mean = 6% SD = 17%)

34. 5-34 Figure 5.5B Normal and Fat-Tailed Distributions (mean =.1, SD =.2)

35. 5-35 Deviation from normality Measure of asymmetry Measure of fat tails -3 because in formal distribution, the first term is 3.

36. 5-36 Figure 5.6 Frequency Distributions of Rates of Return for 1926-2005

37. 5-37 Table 5.3 History of Rates of Returns of Asset Classes for Generations, 1926- 2005

38. 5-38 Table 5.4 History of Excess Returns of Asset Classes for Generations, 1926- 2005

39. 5-39 Figure 5.7 Nominal and Real Equity Returns Around the World, 1900-2000

40. 5-40 Figure 5.8 Standard Deviations of Real Equity and Bond Returns Around the World, 1900-2000

41. 5-41 Figure 5.9 Probability of Investment Outcomes After 25 Years with A Lognormal Distribution

42. 5-42 Terminal Value with Continuous Compounding  When the continuously compounded rate of return on an asset is normally distributed, the effective rate of return will be lognormally distributed  The Terminal Value will then be:

43. 5-43 Figure 5.10 Annually Compounded, 25-Year HPRs from Bootstrapped History and A Normal Distribution (50,000 Observation)

44. 5-44 Figure 5.11 Annually Compounded, 25-Year HPRs from Bootstrapped History(50,000 Observation)

45. 5-45 Figure 5.12 Wealth Indexes of Selected Outcomes of Large Stock Portfolios and the Average T-bill Portfolio

46. 5-46 Table 5.5 Risk Measures for Non-Normal Distributions