risk return part 1
TRANSCRIPT
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Investment returns
The rate of return on an investment can be calculated asfollows:
(Amount received Amount invested)Return = ________________________
Amount invested
For example, if $1,000 is invested and $1,100 is returnedafter one year, the rate of return for this investment is:
($1,100 - $1,000) / $1,000 = 10%.
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What is investment risk?
Two types of investment risk Stand-alone risk
Portfolio risk Investment risk is related to the probability of
earning a low or negative actual return. The greater the chance of lower than expected or
negative returns, the riskier the investment.
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Probability distributions
A listing of all possible outcomes, and the probability of each occurrence.
Can be shown graphically.
Expected Rate of Return
Rate ofReturn (%)100150-70
Firm X
Firm Y
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Selected Realized Returns,
1926 2004Average StandardReturn Deviation
Small-company stocks 17.5% 33.1%Large-company stocks 12.4 20.3L-T corporate bonds 6.2 8.6L-T government bonds 5.8 9.3
U.S. Treasury bills 3.8 3.1
Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation Edition)2005 Yearbook (Chicago: Ibbotson Associates, 2005), p28.
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Investor attitude towards risk
Risk aversion assumes investors dislike riskand require higher rates of return to encourage
them to hold riskier securities. Risk premium the difference between the
return on a risky asset and a riskless asset,which serves as compensation for investors tohold riskier securities.
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Expected Return, Variance, and Covariance
Consider the following two risky assetworld. There is a 1/3 chance of each stateof the economy, and the only assets are astock fund and a bond fund.
Rate of Retur n Scen ario Pro b ab i l i ty Sto ck Fu n d Bo n d Fu n d Recession 33.3% -7% 17%
Normal 33.3% 12% 7%
Boom 33.3% 28% -3%
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Expected Return
Sto ck Fu n d Bo n d Fu n d
Rate of Sq uared Rate of Sq uared
Scenario Retu rn Dev iati on Retu rn Devi ati onRecession -7% 0.0324 17% 0.0100
Normal 12% 0.0001 7% 0.0000
B o o m 28% 0.0289 -3% 0.0100
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
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Sto ck Fu n d Bo n d Fu n d
Rate of Sq uared Rate of Sq uared
Scenario Retu rn Dev iati on Retu rn Devi ati on
Recession -7% 0.0324 17% 0.0100Normal 12% 0.0001 7% 0.0000
B o o m 28% 0.0289 -3% 0.0100
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
Expected Return
%11)(
%)28(31%)12(3
1%)7(31)(
S
S
r E
r E
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Sto ck Fu n d Bo n d Fu n d
Rate of Sq uared Rate of Sq uared
Scenario Retu rn Devi ati on Retu rn Dev iati on
Recession -7% 0.0324 17% 0.0100Normal 12% 0.0001 7% 0.0000
B o o m 28% 0.0289 -3% 0.0100
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
Variance
0324.%)11%7( 2
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Sto ck Fu n d Bo n d Fu n d
Rate of Sq uared Rate of Sq uared
Scenario Retu rn Devi ati on Retu rn Dev iati on
Recession -7% 0.0324 17% 0.0100Normal 12% 0.0001 7% 0.0000
B o o m 28% 0.0289 -3% 0.0100
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
Variance
)0289.0001.0324(.31
0205.
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Sto ck Fu n d Bo n d Fu n d
Rate of Sq uared Rate of Sq uared
Scenario Retu rn Devi ati o n Retu rn Dev iati onRecession -7% 0.0324 17% 0.0100
Normal 12% 0.0001 7% 0.0000
B o o m 28% 0.0289 -3% 0.0100
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
Standard Deviation
0205.0%3.14
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Covariance
S t o c k B o n d
Scenario Devi ati o n Devi ati o n Pro d u c t W ei g h ted
Recession -18% 10% -0.0180 -0.0060
Normal 1% 0% 0.0000 0.0000
B o o m 17% -10% -0.0170 -0.0057
Sum -0.0117 Covariance -0.0117
Deviation compares return in each state to the expected return.
Weighted takes the product of the deviations multiplied by the probability of that state.
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Correlation
998.0)082)(.143(.
0117.
),(
ba
baCov
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Sto ck Fu n d Bo n d Fu n d
Rate of Sq uared Rate of Sq uared
Scenario Retu rn Devi ati o n Retu rn Dev iati onRecession -7% 0.0324 17% 0.0100
Normal 12% 0.0001 7% 0.0000
B o o m 28% 0.0289 -3% 0.0100
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
The Return and Risk for Portfolios
Note that stocks have a higher expected return than bondsand higher risk. Let us turn now to the risk-return tradeoffof a portfolio that is 50% invested in bonds and 50%
invested in stocks.
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PortfoliosRate of Return
Scenario Stock fund Bond fund Portfol io squared deviation Recession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000
Boom 28% -3% 12.5% 0.0012
Ex pected retu rn 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard Deviation 14.31% 8.16% 3.08%
The rate of return on the portfolio is a weighted average ofthe returns on the stocks and bonds in the portfolio:
S S B B P r wr wr
%)17(%50%)7(%50%5
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Rate of Retu rn Scenario Stock fund Bond fund Portfol io squared deviation Recession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000
Boom 28% -3% 12.5% 0.0012
Ex pected return 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard Deviation 14.31% 8.16% 3.08%
Portfolios
The variance of the rate of return on the two risky assets portfolio is
BS S S B B2
S S 2
B B2
P ) )(w 2(w ) (w ) (w
where BS is the correlation coefficient between the returns
on the stock and bond funds.
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Rate of Retu rn Scenario Stock fund Bond fund Portfol io squared deviation Recession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000Boom 28% -3% 12.5% 0.0012
Expected retu rn 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard Deviation 14.31% 8.16% 3.08%
Portfolios
Observe the decrease in risk that diversification offers.
An equally weighted portfolio (50% in stocks and 50%in bonds) has less risk than either stocks or bonds held
in isolation.
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% in stocks Risk Return
0% 8.2% 7.0%5% 7.0% 7.2%10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%
25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%
50.00% 3.08% 9.00%
55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%
100% 14.3% 11.0%
Portfolo Risk and Return Combinations
5.0%
6.0%
7.0%
8.0%
9.0%
10.0%
11.0%
12.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
P o r t
f o l i o R e
t u r n
The Efficient Set
We can consider other portfolio weights besides50% in stocks and 50% in
bonds
100%bonds
100%stocks
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Portfolo Risk and Return Combinations
5.0%
6.0%
7.0%
8.0%
9.0%
10.0%
11.0%
12.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
P o r t
f o l i o R e
t u r n
% in stocks Risk Return
0% 8.2% 7.0%5% 7.0% 7.2%10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%
25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%50% 3.1% 9.0%55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%
100% 14.3% 11.0%
The Efficient Set
100%stocks
100%bonds
Note that some portfolios arebetter than others. They havehigher returns for the same level of
risk or less.
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Portfolios with Various Correlations
100%bonds
r e
t u r n
100%stocks
= 0.2 = 1.0
= -1.0
Relationship dependson correlationcoefficient
-1.0 < < +1.0 If = +1.0, no risk
reduction is possible If = 1.0, complete
risk reduction is possible
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The Efficient Set for Many Securities
Consider a world with many risky assets; we canstill identify the opportunity set of risk-returncombinations of various portfolios.
r e
t u r n
P
Individual Assets
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The Efficient Set for Many Securities
The section of the opportunity set above theminimum variance portfolio is the efficient frontier.
r e
t u r n
P
minimumvarianceportfolio
Individual Assets
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Riskless Borrowing and Lending
Now investors can allocate their money across the T- bills and a balanced mutual fund.
100%bonds
100%stocks
r f
r e
t u r n
Balancedfund
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Riskless Borrowing and Lending
With a risk-free asset available and the efficientfrontier identified, we choose the capital allocation linewith the steepest slope.
r e
t u r n
P
r f