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History of Interest Rates and Risk Premiums Chapter 5 McGraw-Hill/Irwin 5-1 Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Factors Influencing Rates Supply Households Demand Businesses Government’s Net Supply and/or Demand Federal Reserve Actions 5-2 »Level of Interest Rates Interest Rates Supply r1 r0 Demand Q0 Q1 Funds 5-3 Real vs. Nominal Rates Fisher effect: Approximation nominal rate = real rate + inflation premium R = r + i or r = R - i Example r = 3%, i = 6% R = 9% = 3% + 6% or 3% = 9% - 6% Fisher effect: Exact r = (R - i) / (1 + i) 2.83% = (9%-6%) / (1.06) Empirical Relationship: Inflation and interest rates move closely together 5-4 Financial Return Total return: the total gain or loss experienced on an investment over a given period of time Components of the total return Income stream from the investment Capital gain or loss due to changes in asset prices Total return can be expressed either in dollar terms or in percentage terms. 5-5 Cash Flow Time Line Pt-1 cash payments Pt +-----------------------------------------------------+ Time t-1 Time t Pt = Price at time t (today) Capital Gain = Pt – Pt-1 = Price today – Price last period Dollar Return = Cash Payments + Capital Gain = Cash Payments + Pt – Pt-1 5-6 Example 5.1 You purchased a stock last year for $25. It has paid $1 in dividends and is not worth $21. What is your Dollar Return? 5-7 Example 5.2 You bought an 11% coupon bond one year ago for $1125. You can sell that bond today for $1100. What is your Dollar Return? 5-8 Holding Period Return (hpr) or Percentage Return This is the most common way to express the gains or losses over a period It is the $Return relative to the amount invested CashPmt Pt Pt 1 hpr %Return Pt 1 1+hpr is often called the wealth relative 5-9 Example 5.1 (Revised) You purchased a stock last year for $25. It has paid $1 in dividends and is not worth $21. What is your Dollar Return? What is your hpr? 5-10 Example 5.2 Revised You bought an 11% coupon bond one year ago for $1125. You can sell that bond today for $1100. What is your Dollar Return? What is your hpr? 5-11 Measuring Wealth over Time Year hpr $1 Investment $1 Investment each year 1 15% 1*(1.15) = 1.15 Vt-1(1+it) = Vt 1*(1.15)=1.15 (1+Vt-1)(1+it) = Vt 2 -10% 1.15*(0.90) =1.035 (1+1.15)*(0.90) = 1.935 3 13% 1.035*1.13 = 1.1696 (1+1.935)*(1.13) = 3.3166 5-12 Arithmetic Average Return Add the individual hpr’s and divide by the number of years 15% 10% 13% x 6.00% 3 5-13 Geometric Rate of Return Multiply by the wealth relatives, raise to the 1/N power and subtract 1 Is the constant rate of wealth building over time that results in the observed future value g [(1 r1 )(1 r2 )(1 r3 )...(1 rN )] [(1.15)(0.90)(1.13)] 1 3 1 N 1 1 0.0536 5.36% By Financial Calculator: P/YR=1 I/YR(FV=1.1696, PV=-1, N=3) = 5.36% 5-14 IRR from a Constant Investment P/YR=1 t CF 0 -1 1 -1 2 -1 3 3.3166 •Press IRR = 5.10% 5-15 Value of $1 Invested in Equities, Treasury Bonds and Bills, 1900 - 2003 100,000 $ 15,579 10,000 1,000 Equities Bonds Bills Inflation $148 100 61 $22 10 1 1900 1920 1940 1960 1980 2000 2003 Year 5-16 Geometric Return Calculation A $1 investment in Large Stocks (with dividends reinvested) was worth 15579 after 103 years. The geometric mean return can be computed as (use P/Yr = 1) I/YR(FV=15579, PV=-1, N=103)=9.83% Stocks I/YR(FV=148, PV=-1, N=76)=4.97% Long US Bond I/YR(FV=61, PV=-1, N=76)=4.07% US TBill I/YR(FV=22, PV=-1, N=76)=3.05% Inflation 5-17 Geometric Real Rates of Return To compute the long run real rate of return one can divide the ending value of the investment by the ending value of the inflation figure to determine the purchasing power of the investment. Then compute the return using “Real Dollars” Real value of the Large Stocks at end of period is 15579/22 = 708.14 I/YR(FV=708.14, PV=-1, N=103)=6.58% I/YR(FV=6.73, PV=-1, N=103)=1.87% Long US Bond I/YR(FV=2.73, PV=-1, N=103)=1.00% TBill 5-18 Arithmetic versus Geometric Returns (1900-2003) Nominal Returns Arithmetic Avg Nominal Returns Geometric Real Returns Arithmetic Avg Real Returns Geometric Stocks Bonds Bills 11.7 5.2 4.1 9.8 5.0 4.1 8.5 2.3 1.1 6.6 1.9 1.0 5-19 Annual Holding Period Returns (1926-2002) Geom. Series Mean% Sm Stk 11.6 Lg Stk 10.0 LT Gov 5.4 T-Bills 3.8 Inflation 3.1 Arith. Mean% 17.7 12.0 5.7 3.8 3.1 Stan. Dev.% 39.3 20.6 8.2 3.2 4.4 5-20 Stocks Annual Returns for Securties Bonds T-Bills 50% 40% Return 30% 20% 10% 0% -10% -20% -30% 1980 1985 1990 1995 2000 Year 5-21 IRR From a Constant Annual Investment (Jan 1, each year) through Oct 2002 20% 15% 10% 5% 0% -5% -10% -15% Stocks LongGovBond -20% Tbills 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991 1990 1989 1988 1987 1986 1985 1984 1983 1982 1981 1980 -25% Year of First Investment 5-22 IRR From a Constant Annual Investment (Jan 1, each year) through Dec 2003 30% Stocks LongGovBond 25% Tbills 20% 15% 10% 5% 0% 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991 1990 1989 1988 1987 1986 1985 1984 1983 1982 1981 1980 -5% Year of First Investment 5-23 IRR From a Constant Annual Investment (Jan 1, each year) through Dec 2004 20% 18% Stocks LongGovBond 16% Tbills 14% 12% 10% 8% 6% 4% 2% 2003 2004 1999 2000 2001 2002 1995 1996 1997 1998 1990 1991 1992 1993 1994 1986 1987 1988 1989 1982 1983 1984 1985 1980 1981 0% Year of First Investment 5-24 Dollar Gain per Dollar Invested From a Constant Annual Investment (Jan 1, each year) through Dec 2004 6.00 Stocks 5.00 LongGovBond Tbills 4.00 3.00 2.00 1.00 2003 2004 2000 2001 2002 1997 1998 1999 1994 1995 1996 1991 1992 1993 1988 1989 1990 1985 1986 1987 1982 1983 1984 1980 1981 0.00 Year of First Investment 5-25 Take home message If you have a short holding period, stocks are very risky, but from a longer term perspective they have provided the best returns both recently and historically Investment is not about saving money for the future, its about earning money from the money you invested so that most of your portfolio is from the earning of that portfolio and not from your deposits into that fund 5-26 Percentage Returns on Bills, Bonds, and Stocks, 1900 - 2003 Nominal (%) Asset Class Average Best Year Real (%) Worst Year Average Best Year Worst Year Bills 4.1 14.7 0.0 1.1 19.7 -15.1 Bonds 5.2 40.4 -9.2 2.3 35.1 -19.4 Stocks 11.7 57.6 -43.9 8.5 56.8 -38 Difference between average return of stocks and bills = 7.6% Difference between average return of stocks and bonds = 6.5% Risk premium: the difference in returns offered by a risky asset relative to the risk-free return available 5-27 Why are Treasury Bills considered risk free? If the government default on Treasury Bills, your last concern will be the money you might have earned on the TB When you buy a Treasury Bill, you purchase it at a discount and redeem it at par, so you know when you buy it, what your return will be If you buy a stock, you don’t know what you will sell it for, or what dividends it will pay; thus, it is risky The yield on Treasury Bills, is generally taken to be the risk free return 5-28 Distribution of Historical Stock Returns, 1900 - 2003 Histogram of Nominal Returns on Equities 1900-2003 <-30 -30 to -20 to -10 to 0 to -20 -10 0 10 10 to 20 20 to 30 to 30 40 40 to 50 >50 Percent return in a given year Probability distribution for future stock returns is unknown. We can approximate the unknown distribution by assuming a normal distribution. 5-29 Variability of Stock Returns Normal distribution can be described by its mean and its variance. A Normal Distribution is symmetric around the mean Variance (2) - the expected value of squared deviations from the mean N Variance 2 ( R R) t 1 2 t N 1 Units of variance (%-squared) - hard to interpret, so calculate standard deviation, a measure of volatility equal to square root of 2 5-30 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 fully characterized by its mean and standard deviation 5-31 The Normal Distribution 5-32 Volatility of Asset Returns Asset Class Equities Bonds Bills Nominal Returns Real Returns Average(%) Std. Dev. (%) Average(%) Std. Dev. (%) 11.7 5.2 4.1 20.1 8.2 2.8 8.5 2.3 1.1 20.4 10 4.7 Asset classes with greater volatility pay higher average returns. Average return on stocks is more than double the average return on bonds, but stocks are 2.5 times more volatile. 5-33 Risk Premiums and Real Returns (1926-2002) Risk Series Premiums% Sm Stk 13.9 Lg Stk 9.3 LT Gov 1.9 T-Bills --Inflation --- Real Returns% 14.6 8.9 2.6 0.7 --- 5-34 Mean Scenario or Subjective Returns Subjective returns s E (r ) p s r s 1 ps = probability of a state rs = return if a state occurs 1 to s states 5-35 Scenario or Subjective 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 5-36 Variance or Dispersion of Returns Subjective or Scenario p s r s E r 2 s 2 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.= sqrt[ .01199] = .1095 5-37