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R(1 t ) i (r i)(1 t ) i r (1 t ) it
• The after-tax real rate of return falls as the inflation rate rises.
INVESTMENTS | BODIE, KANE, MARCUS
INVESTMENTS | BODIE, KANE, MARCUS
5-9
Example 5.2 Annualized Rates of Return
INVESTMENTS | BODIE, KANE, MARCUS
5-10
Equation 5.7 EAR
• EAR definition: percentage increase in funds invested over a 1-year horizon
σ2 = .25(.31 - 0.0976)2+.45(.14 - .0976)2 + .25(-0.0675 - 0.0976)2 + .05(-.52 .0976)2 = .038
• Example STD calculation: .038 .1949
INVESTMENTS | BODIE, KANE, MARCUS
INVESTMENTS | BODIE, KANE, MARCUS
E(r) = (.25)(.31) + (.45)(.14) + (.25)(-.0675) + (0.05)(-0.52) E(r) = .0976 or 9.76%
INVESTMENTS | BODIE, KANE, MARCUS
5-20
Variance and Standard DeviNVESTMENTS | BODIE, KANE, MARCUS
5-7
Taxes and the Real Rate of Interest
• Tax liabilities are based on nominal income – Given a tax rate (t) and nominal interest rate (R), the Real after-tax rate is:
5-17
Rates of Return: Single Period Example
Ending Price = Beginning Price = Dividend = 110 100 4
HPR = (110 - 100 + 4 )/ (100) = 14%
INVESTMENTS | BODIE, KANE, MARCUS
5-23
Geometric Average Return
TVn (1 r 1 r2 )...( 1 rn ) 1 )(
TV = Terminal Value of the Investment
g TV
1/ n
1
g= geometric average rate of return
5-22
Time Series Analysis of Past Rates of Return
The Arithmetic Average of rate of return:
1 n E (r ) s 1 p( s)r ( s) s 1 r ( s ) n
n
INVESTMENTS | BODIE, KANE, MARCUS
5-14
Bills and Inflation, 1926-2009
• Moderate inflation can offset most of the nominal gains on low-risk investments. • A dollar invested in T-bills from1926–2009 grew to $20.52, but with a real value of only $1.69. • Negative correlation between real rate and inflation rate means the nominal rate responds less than 1:1 to changes in expected inflation.
INVESTMENTS | BODIE, KANE, MARCUS
5-15
Figure 5.3 Interest Rates and Inflation, 1926-2009
INVESTMENTS | BODIE, KANE, MARCUS
5-16
Risk and Risk Premiums
Rates of Return: Single Period
5-18
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
Risk Premium SD of Excess Returns
INVESTMENTS | BODIE, KANE, MARCUS
5-27
The Normal Distribution
• Investment management is easier when returns are normal.
n
^ 2
_ 2
INVESTMENTS | BODIE, KANE, MARCUS
5-25
Geometric Variance and Standard Deviation Formulas
• When eliminating the bias, Variance and Standard Deviation become:
P 1 P0 D1 HPR P0
HPR = Holding Period Return P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one
INVESTMENTS | BODIE, KANE, MARCUS
5-12
Table 5.1 APR vs. EAR
INVESTMENTS | BODIE, KANE, MARCUS
5-13
Table 5.2 Statistics for T-Bill Rates, Inflation Rates and Real Rates, 1926-2009
INVESTMENTS | BODIE, KANE, MARCUS
• 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: • Nominal rate = real rate + inflation forecast
– Federal Reserve Actions
INVESTMENTS | BODIE, KANE, MARCUS
5-3
Real and Nominal Rates of Interest
• Nominal interest • Let R = nominal rate: Growth rate of rate, r = real rate your money and I = inflation rate. Then: • Real interest rate: Growth rate of your r R i purchasing power
INVESTMENTS | BODIE, KANE, MARCUS
5-24
Geometric Variance and Standard Deviation Formulas
• Estimated Variance = expected value of squared deviations
1 r s r n s 1
1 EAR 1 rf T
1 T
INVESTMENTS | BODIE, KANE, MARCUS
5-11
Equation 5.8 APR
• APR: annualizing using simple interest
1 EAR APR
T
T
1
INVESTMENTS | BODIE, KANE, MARCUS
5-8
Rates of Return for Different Holding Periods
Zero Coupon Bond, Par = $100, T=maturity, P=price, rf (T)=total risk free return
100 rf (T ) 1 P(T )
– Standard deviation is a good measure of risk when returns are symmetric. – If security returns are symmetric, portfolio returns will be, too. – Future scenarios can be estimated using only the mean and the standard deviation.
R i r 1 i
