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8/14/2019 Risk, return and Opp.Cost of Capital
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7- 1
Introductionto Risk, Return, and
the Opportunity Cost of Capital
Chapter 10
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Topics Covered
w 75 Years of Capital Market History
w Measuring Risk
w Portfolio Risk
w Beta and Unique Risk
w Diversification
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7- 3
The Value of an Investment of $1 in 1926
Source: Ibbotson Associates
0.1
10
1000
1925 1940 1955 1970 1985 2000
S&P
Small Cap
Corp Bonds
Long Bond
T Bill
Index
Year End
1
6,402
2,587
64.1
48.9
16.6
Assuming reinvestment of all dividend and interest
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0.1
10
1000
1925 1940 1955 1970 1985 2000
S&P
Small Cap
Corp Bonds
Long Bond
T Bill
Source: Ibbotson Associates
Index
Year End
1
660
267
6.6
5.0
1.7
Real returns
The Value of an Investment of $1 in 1926
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Rates of Return 1926-2000
Source: Ibbotson Associates
-60
-40
-20
0
20
40
60
26 30 35 40 45 50 55 60 65 70 75 80 85 90 95
2000
Common Stocks
Long T-BondsT-Bills
Year
PercentageR
eturn
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Average Rates of Return (1926-2000)
Portfolio
Average AnnualRate of Return
Average RiskPremium (ExtraReturn vs. TreasuryBills)
Nominal Real
Treasury Bills 3.9 0.8 0
Government Bonds 5.7 2.7 1.8
Corporate Bonds 6.0 3.0 2.1
Common Stocks (S&P 500)
13.0 9.7 9.1Small Firm Common Stocks 17.3 13.8 13.4
Figures are in percent per year.
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Average vs. Compounded Returns
w If the cost of capital is estimated fromhistorical returns or risk premiums, usearithmetic averages, not compounded
annual rates of return.
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Average Market Risk Premia (1900-2000)
4.35.1
6 6.1 6 .1 6.5 6.77.1 7.5
88.5 9.9
9.9 10 11
0
12
3
45
67
8
9
10
11
DenBel
Can
Swi
Spa
UK Ir
e
Neth
USA
Swe
Aus
Ger
Fra
Jap It
Risk premium, %
Country
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Evaluating Cost of Capital
w For estimating the cost of capital, can we usehistorical market returns?
w Likely Candidates:
Market return, rm Risk-free rate plus risk premium
w Is the expected future risk premium the same
as the historic risk premium?
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Measuring Risk
Variance Expected squared deviation from expected
return, denoted by 2.When it is estimated from a sample ofobserved returns,
mean return is taken as the expected return.
Standard Deviation The square root of variance, denotedby .
=
=N
t
tt rrN
r
1
22 )(1
1)(
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Measuring Risk
1 1 2
4
1311
13 12 13
3201
23456789
10111213
-50to-40
-40to-30
-30to-20
-20to-10
-10to0
0to10
10to20
20to30
30to40
40to50
50to60
Return %
# of YearsHistogram of Annual Stock Market ReturnsHistogram of Annual Stock Market Returns
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Annual st.dev. and variances(U.S. 1926-2000)
Portfolio StandardDeviation Variance
Treasury Bills 3.2 10.1
Government Bonds 9.4 88.7
Corporate Bonds 8.7 75.5
Common Stocks (S&P 500) 20.2 406.9Small Firm Common Stocks 33.4 1118.4
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Does volatility remain constant?
Period(NYSE)
Market St.Dev.( m)1926-1930 21.7
1931-1940 37.8
1941-1950 14.0
1951-1960 12.1
1961-1970 13.0
1971-1980 15.8
1981-1990 16.5
1991-2000 13.4
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Volatility Across Markets
Market Standard DeviationFrance 21.5
Switzerland 19.0Finland 43.2
Japan 18.2
Argentina 34.3
September 1996-August 2001Percent per year
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Volatility of Individual Securities
Stock Standard Deviation
Amazon 110.6
Boeing 30.9Coca-Cola 31.5
Dell Computer 62.7
Exxon Mobil 17.4
General Electric 26.8
General Motors 33.4
McDonalds 27.4Pfizer 29.3
August 1996-July 2001
Percent per year
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Volatility of Market vs.Individual Stocks
w Individual Stocks are much more variablethan the market index.
wWhy doesnt the volatility of the marketportfolio reflect the average variability of
its components, individual stocks?
w Diversification reduces variability.
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Measuring Risk
Diversification- Strategy designed to reduce risk byspreading the portfolio across many investments.
Unique Risk- Risk factors affecting only that firm.Also called diversifiable risk.
Market Risk- Economy-wide sources of risk thataffect the overall stock market. Also calledsystematic risk.
7
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Measuring Risk
0
5 10 15
Number of Securities
Portfo
lio
standard
deviation
7
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Measuring Risk
0
5 10 15
Number of Securities
Portfo
lio
standard
deviation
Market risk
Unique
risk
7
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Measuring Risk
==n
i
ii rx1
p ..r
Where:xi=Fraction of portfolio in asset i
ri=Rate of return on asset i
Portfolio Return:
7
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Measuring Risk
)(2 2112212
2
2
2
2
1
2
1p
2
xxxx ++=
Where:xi=Fraction of portfolio in asset i
i=standard deviation of asset i
Portfolio Variance:(for a two-asset portfolio)
7
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Portfolio Variance
w The variance of a two stock portfolio is the sum ofthese four boxes
Stock 1 Stock 2
Stock 1
x12 12
x1x2 12=x1x2 12 12
Stock 2x1x2 12=x1x2 12 1 2 x22 22
7
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Covariance between stocks
w Portfolio variance depends on: Variance of the individual stocks (diagonal
boxes)
Covariance between the stocks (off-diagonalboxes)
w Covariance can be expressed as product of :
Individual standard deviations
Correlation Coefficient,
211212 =
7
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Portfolio Risk
Example
Suppose you invest 65% of your portfolio in Coca-Cola and35% in Reebok. Expected return for Coca-Cola stock is 10%and for Reebok, 20%.
The expected dollar return on your CC is 10% x 65% = 6.5%
and on Reebok it is 20% x 35% = 7.0%.
The expected return on your portfolio is 6.5 + 7.0 = 13.50%.
Past standard deviation of returns was 31.5 % for Coca-Colaand 58.5% for Reebok. Assume a correlation coefficient of 0.2.
What is the standard deviation of your portfolio?
7-
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Portfolio Risk
222
2
2
2
211221
211221222
1
2
1
)5.58()35(.x5.585.312.0
35.65.xxReebok
5.585.312.0
35.65.xx)5.31()65(.xCola-Coca
ReebokCola-Coca
=
=
=
=
ExampleSuppose you invest 65% of your portfolio in Coca-Cola and 35% inReebok. The expected dollar return on your CC is 10% x 65% = 6.5%and on Reebok it is 20% x 35% = 7.0%. The expected return on yourportfolio is 6.5 + 7.0 = 13.50%. Assume a correlation coefficient of
0.2.
7-
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Portfolio Risk
ExampleSuppose you invest 65% of your portfolio in Coca-Cola and 35% inReebok. The expected dollar return on your CC is 10% x 65% = 6.5% andon Reebok it is 20% x 35% = 7.0%. The expected return on your portfoliois 6.5 + 7.0 = 13.50%. Assume a correlation coefficient of 0.2.
%31.71,006.1DeviationStandard
1.006,18.5)0.2x31.5x52(.65x.35x
]x(58.5)[(.35)
]x(31.5)[(.65)VariancePortfolio
22
22
==
=+
+
=
7-
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Portfolio Risk
)rx()r(xReturnPortfolioExpected 2211 +=
)xx(2xxVariancePortfolio211221
2
2
2
2
2
1
2
1
++=
7-
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Portfolio Risk
The shaded boxes contain variance terms; the remaindercontain covariance terms.
1
2
3
4
5
6
N
1 2 3 4 5 6 N
STOCK
STOCK
To calculateportfoliovariance addup the boxes
7-
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Individual Securitiesand Portfolio Risk
w Portfolio managers are not interested in thestandard deviations of individual securities,but in the effect that each stock will have
on the risk of their portfolio.w The risk of a well-diversified portfolio
depends on the market risk of the securities
included in the portfolio.w And market risk is measured by .
7-
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Volatility of Individual Securities
Stock
Amazon 3.25 110.6
Boeing 0.56 30.9Coca-Cola 0.74 31.5
Dell Computer 2.21 62.7
Exxon Mobil 0.40 17.4
General Electric 1.18 26.8
General Motors 0.91 33.4
McDonalds 0.68 27.4Pfizer 0.71 29.3
August 1996-July 2001Percent per year
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Beta and Unique Risk
beta
Expected
return
Expected
marketreturn
10%10%- +
-10%+10%
stock
Copyright 1996 by The McGraw-Hill Companies, Inc
-10%
1. Total risk =diversifiable risk +market risk2. Market risk ismeasured by beta,the sensitivity tomarket changes
7-
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Beta and Unique Risk
Market Portfolio- Portfolio of all assets in theeconomy. In practice a broad stock marketindex, such as the S&P Composite, is used
to represent the market.
Beta - Sensitivity of a stocks return to the
return on the market portfolio.
7-
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Beta and Unique Risk
2
m
imiB
=
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