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Understanding Historical Rates of Return and Risk

Explore the factors and influences on rates of return and risk from historical data. Learn about supply and demand dynamics, Federal Reserve actions, interest rates, taxes, risk premiums, and more. Discover how to compare rates of return, analyze past rates, and calculate risk measures effectively.

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Understanding Historical Rates of Return and Risk

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  1. CHAPTER 5 Learning About Return and Risk from the Historical Record

  2. Factors Influencing Rates Supply Households Demand Businesses Government’s Net Supply and/or Demand Federal Reserve Actions

  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. Equilibrium Real Rate of Interest Determined by: Supply Demand Government actions Expected rate of inflation

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

  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, then we get the Fisher Equation:

  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. Comparing Rates of Return for Different Holding Periods Zero Coupon Bond

  9. Example 5.2 Annualized Rates of Return

  10. Formula for EARs and APRs

  11. Table 5.1 Annual Percentage Rates (APR) and Effective Annual Rates (EAR)

  12. Bills and Inflation, 1926-2005 Entire post-1926 history of annual rates: www.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

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

  14. Figure 5.2 Interest Rates and Inflation, 1926-2005

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

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

  17. Ending Price = 48 Beginning Price = 40 Dividend = 2 HPR = (48 - 40 + 2 )/ (40) = 25% Rates of Return: Single Period Example

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

  19. Scenario Returns: Example StateProb. of State r 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

  20. Variance: Variance or Dispersion of Returns Standard deviation = [variance]1/2 Using Our Example: Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2…+ .1(.35-.15)2] Var= .01199 S.D.= [ .01199] 1/2 = .1095

  21. Time Series Analysis of Past Rates of Return Expected Returns and the Arithmetic Average

  22. Geometric Average Return TV = Terminal Value of the Investment g= geometric average rate of return

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

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

  25. Figure 5.4 The Normal Distribution

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

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

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

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

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

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

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

  33. Figure 5.9 Probability of Investment Outcomes After 25 Years with A Lognormal Distribution

  34. 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:

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

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

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

  38. Table 5.5 Risk Measures for Non-Normal Distributions

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