CHAPTER 5 Learning About Return and Risk from the Historical Record Factors Influencing Rates • Supply – Households • Demand – Businesses • Government’s Net Supply and/or Demand – Federal Reserve Actions 5-2 Figure 5.1 Determination of the Equilibrium Real Rate of Interest 5-3 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: R r E (i) 5-4 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: R(1 t ) i (r i)(1 t ) i r (1 t ) it 5-5 Comparing Rates of Return for Different Holding Periods Zero Coupon Bond 100 rf (T ) 1 P(T ) 5-6 Formula for EARs and APRs 1 EAR {1 r f (T ) }T 1 1 (1 EAR ) APR T T 5-7 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 5-8 Table 5.2 History of T-bill Rates, Inflation and Real Rates for Generations, 1926-2005 5-9 Figure 5.2 Interest Rates and Inflation, 1926-2005 5-10 Figure 5.3 Nominal and Real Wealth Indexes for Investment in Treasury Bills, 1966-2005 5-11 Risk and Risk Premiums Rates of Return: Single Period P 1 P0 D1 HPR P0 HPR = Holding Period Return P0 = Beginning price P1 = Ending price D1 = Dividend during period one 5-12 Rates of Return: Single Period Example Ending Price = Beginning Price = Dividend = 48 40 2 HPR = (48 - 40 + 2 )/ (40) = 25% 5-13 Expected Return and Standard Deviation Expected returns E (r ) p ( s )r ( s ) s p(s) = probability of a state r(s) = return if a state occurs s = state 5-14 Scenario Returns: Example State 1 2 3 4 5 Prob. of State .1 .2 .4 .2 .1 r in State -.05 .05 .15 .25 .35 E(r) = (.1)(-.05) + (.2)(.05)… + (.1)(.35) E(r) = .15 5-15 Variance or Dispersion of Returns Variance: p( s ) r ( s ) E (r ) 2 2 s 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 5-16 Time Series Analysis of Past Rates of Return Expected Returns and the Arithmetic Average 1 n E (r ) s 1 p( s)r ( s) s 1 r ( s ) n n 5-17 Geometric Average Return TV n (1 r1 )(1 r2 ) x x (1 rn ) TV = Terminal Value of the Investment g TV 1/ n 1 g= geometric average rate of return 5-18 Variance and Standard Deviation Formulas • Variance = expected value of squared deviations 2 n 1 2 r ( s) r n s 1 • When eliminating the bias, Variance and Standard Deviation become: n 1 r (s) r n 1 j 1 2 5-19 The Reward-to-Volatility (Sharpe) Ratio Risk Premium Sharpe Ratio for Portfolios = SD of Excess Return 5-20 Figure 5.4 The Normal Distribution 5-21 Figure 5.6 Frequency Distributions of Rates of Return for 1926-2005 5-22 Table 5.3 History of Rates of Returns of Asset Classes for Generations, 1926- 2005 5-23 Table 5.4 History of Excess Returns of Asset Classes for Generations, 1926- 2005 5-24 Figure 5.7 Nominal and Real Equity Returns Around the World, 1900-2000 5-25 Figure 5.8 Standard Deviations of Real Equity and Bond Returns Around the World, 1900-2000 5-26