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CHAPTER 5CHAPTER 5 Learning About Learning About Return and Risk Return and Risk from the from the Historical RecordHistorical Record
5-2
Factors Influencing Rates
• Supply
– Households
• Demand
– Businesses
• Government’s Net Supply and/or Demand
– Federal Reserve Actions
5-4
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
5-5
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:
(1 ) ( )(1 ) (1 )R t i r i t i r t it
5-8
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
5-12
Risk and Risk Premiums
P
DPPHPR0
101
HPR = Holding Period Return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one
Rates of Return: Single Period
5-13
Ending Price = 48
Beginning Price = 40
Dividend = 2
HPR = (48 - 40 + 2 )/ (40) = 25%
Rates of Return: Single Period Example
5-14
Expected returns
p(s) = probability of a stater(s) = return if a state occurss = state
Expected Return and Standard Deviation
( ) ( ) ( )s
E r p s r s
5-15
State Prob. of State r in State 1 .1 -.052 .2 .053 .4 .154 .2 .255 .1 .35
E(r) = (.1)(-.05) + (.2)(.05)… + (.1)(.35)E(r) = .15
Scenario Returns: Example
5-16
Standard deviation = [variance]1/2
Variance:
Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2…+ .1(.35-.15)2]Var= .01199S.D.= [ .01199] 1/2 = .1095
Using Our Example:
Variance or Dispersion of Returns
22 ( ) ( ) ( )s
p s r s E r
5-17
Time Series Analysis of Past Rates of Return
n
s
n
ssr
nsrsprE
11)(
1)()()(
Expected Returns and the Arithmetic Average
5-18
Geometric Average Return
1 2(1 )(1 ) (1 )nnr r rx xTV
TV = Terminal Value of the Investment
1/1 TVg n
g= geometric average rate of return
5-19
Variance and Standard Deviation Formulas
• Variance = expected value of squared deviations
• When eliminating the bias, Variance and Standard Deviation become:
22
1
1( )
n
s
r s rn
2
1
1( )
1
n
j
r s rn
5-20
The Reward-to-Volatility (Sharpe) Ratio
Sharpe Ratio for Portfolios =Risk PremiumSD of Excess Return