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McGraw-Hill/Irwin 2008 The McGraw-Hill Companies, Inc., All Rights Reserved.
Risk and Return: Pastand Prologue
CHAPTER 5
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5.1 RATES OF RETURN
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Rates of Return: Single PeriodExample
Ending Price = 24
Beginning Price = 20Dividend = 1
HPR = ( 24 - 20 + 1 )/ ( 20) = 25%
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Measuring Investment Returns OverMultiple Periods
May need to measure how a fundperformed over a preceding five-yearperiod
Return measurement is more ambiguousin this case
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Rates of Return: Multiple PeriodExample Text (Page 128)
Data from Table 5.1
1 2 3 4
Assets(Beg.) 1.0 1.2 2.0 .8
HPR .10 .25 (.20) .25
TA (Before
Net Flows 1.1 1.5 1.6 1.0
Net Flows 0.1 0.5 (0.8) 0.0
End Assets 1.2 2.0 .8 1.0
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Returns Using Arithmetic andGeometric Averaging
Arithmetic
ra = (r1 + r2 + r3 + ... rn) / n
ra = (.10 + .25 - .20 + .25) / 4= .10 or 10%
Geometric
rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1rg = {[(1.1) (1.25) (.8) (1.25)]}
1/4 - 1
= (1.5150) 1/4 -1 = .0829 = 8.29%
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Dollar Weighted Returns
Internal Rate of Return (IRR) - thediscount rate that results in present valueof the future cash flows being equal to the
investment amountConsiders changes in investment
Initial Investment is an outflow
Ending value is considered as an inflow
Additional investment is a negative flow
Reduced investment is a positive flow
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Dollar Weighted AverageUsing Text Example (Page 128)
Net CFs 1 2 3 4
$ (mil) - 0.1 -0 .5 0.8 1.0
2 3 4
0.1 0.5 0.8 1.01.0 4.17%
1 (1 ) (1 ) (1 )IRR IRR IRR IRR
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Quoting Conventions
APR = annual percentage rate
(periods in year) X (rate for period)
EAR = effective annual rate
( 1+ rate for period)Periods per yr- 1
Example: monthly return of 1%
APR = 1% X 12 = 12%EAR = (1.01)12 - 1 = 12.68%
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Scenario Analysis and ProbabilityDistributions
1) Mean: most likely value
2) Variance or standard deviation
3) Skewness
* If a distribution is approximately normal,the distribution is described by
characteristics 1 and 2
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rSymmetric distribution
Normal Distribution
s.d. s.d.
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rNegative Positive
Skewed Distribution: Large NegativeReturns Possible
Median
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rNegative Positive
Skewed Distribution: Large PositiveReturns Possible
Median
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Subjective returns
p(s) = probability of a state
r(s) = return if a state occurs
1 to s states
Measuring Mean: Scenario orSubjective Returns
1
( ) ( ) ( )S
t
E r p s r s
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Measuring Variance or Dispersion ofReturns
Subjective or Scenario
2
1
( ) ( ) ( ) ( )s
t
Var r p s r s E r
( ) ( )SD r Var r
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Measuring Variance or Dispersion ofReturns
Using Our Example:
Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2]
Var= .01199
S.D.= [ .01199] 1/2 = .1095 or 10.95%
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Risk Premiums and Risk Aversion
Degree to which investors are willing to commitfunds
Risk aversion
If T-Bill denotes the risk-free rate, rf, andvariance, , denotes volatility of returns then:
The risk premium of a portfolio is:
2
P
( )P fE r r
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Risk Premiums and Risk Aversion
To quantify the degree of risk aversion withparameter A:
Or:
21( )2
P f PE r r A
2
( )1
2
P f
P
E r rA
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The Sharpe (Reward-to-Volatility) Measure
portfolio risk premium
standard deviation of portfolio excess return
( )P f
P
S
E r r
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5.3 THE HISTORICAL RECORD
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Annual Holding Period ReturnsFrom Table 5.3 of Text
Geom. Arith. Stan.
Series Mean% Mean% Dev.%
World Stk 9.80 11.32 18.05
US Lg Stk 10.23 12.19 20.14
US Sm Stk 12.43 18.14 36.93
Wor Bonds 5.80 6.17 9.05
LT Treas. 5.35 5.64 8.06T-Bills 3.72 3.77 3.11
Inflation 3.04 3.13 4.27
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Annual Holding Period Excess ReturnsFrom Table 5.3 of Text
Risk Stan. Sharpe
Series Prem. Dev.% Measure
World Stk 7.56 18.37 0.41US Lg Stk 8.42 20.42 0.41
US Sm Stk 14.37 37.53 0.38
Wor Bonds 2.40 8.92 0.27LT Treas 1.88 7.87 0.24
Fi 5 1 F Di t ib ti f H ldi
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Figure 5.1 Frequency Distributions of HoldingPeriod Returns
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Figure 5.2 Rates of Return on Stocks,Bonds and T-Bills
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Figure 5.3 Normal Distribution withMean of 12% and St Dev of 20%
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Table 5.4Size-Decile Portfolios
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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%
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Real vs. Nominal Rates
Fisher effect:
2.83% = (9%-6%) / (1.06)
11 or:
1
1
Rr
i
R ir
i
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Figure 5.4 Interest, Inflation and RealRates of Return
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Possible to split investment fundsbetween safe and risky assets
Risk free asset: proxy; T-bills
Risky asset: stock (or a portfolio)
Allocating Capital
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Allocating Capital
Issues Examine risk/ return tradeoff
Demonstrate how different degrees of risk
aversion will affect allocations between riskyand risk free assets
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The Risky Asset:Text Example (Page 143)
Total portfolio value = $300,000
Risk-free value = 90,000
Risky (Vanguard and Fidelity) = 210,000
Vanguard (V) = 54%
Fidelity (F) = 46%
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The Risky Asset:Text Example (Page 143)
210,0000.7(risky assets, portfolio )
300,000
90,000
1 0.3(risk-free assets)300,000
y P
y
Vanguard 113,400/300,000 = 0.378
Fidelity 96,600/300,000 = 0.322
Portfolio P 210,000/300,000 = 0.700
Risk-Free Assets F 90,000/300,000 = 0.300
Portfolio C 300,000/300,000 = 1.000
Calc lating the E pected Ret rn
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rf= 7% rf= 0%
E(rp) = 15% p = 22%
y = % in p (1-y) = % in rf
Calculating the Expected ReturnText Example (Page 145)
E t d R t f C bi ti
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E(rc) = yE(rp) + (1 - y)rf
rc = complete or combined portfolio
For example, y = .75
E(rc) = .75(.15) + .25(.07)= .13 or 13%
Expected Returns for Combinations
Figure 5 5 Investment Opportunity
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Figure 5.5 Investment OpportunitySet with a Risk-Free Investment
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C bi ti With t L
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c = .75(.22) = .165 or 16.5%
If y = .75, then
c = 1(.22) = .22 or 22%
If y = 1
c = 0(.22) = .00 or 0%
If y = 0
Combinations Without Leverage
Using Leverage with
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Using Leverage withCapital Allocation Line
Borrow at the Risk-Free Rate and invest instock
Using 50% Leverage
rc = (-.5) (.07) + (1.5) (.15) = .19
c = (1.5) (.22) = .33
Ri k A i d All ti
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Risk Aversion and Allocation
Greater levels of risk aversion lead tolarger proportions of the risk free rate
Lower levels of risk aversion lead to larger
proportions of the portfolio of risky assetsWillingness to accept high levels of risk forhigh levels of returns would result in
leveraged combinations
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5.6 PASSIVE STRATEGIES AND THE
CAPITAL MARKET LINE
T bl 5 5 A R t f R t
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Table 5.5 Average Rates of Return,Standard Deviation and Reward to
Variability
C t d B fit f P i I ti
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Costs and Benefits of Passive Investing
Active strategy entails costs
Free-rider benefit
Involves investment in two passiveportfolios
Short-term T-bills
Fund of common stocks that mimics a broad
market index