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Lecture 5
Capital Allocation between theRisky and the Risk-free Asset
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Slide 6-2
w1 = proportion of funds in Security 1w2 = proportion of funds in Security 2
r1 = expected return on Security 1r2 = expected return on Security 2
1wn
1i
i
Two-Security Portfolio: Return
2211P rwrwr
7/31/2019 Lecture 5- Principles of Financial Economics
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Slide 6-3
1
2 = variance of Security 1
2
2 = variance of Security 2
Cov(r1,r2) = covariance of returns forSecurity 1 and Security 2
Two-Security Portfolio: Risk
)r,r(Covww2ww 21212
2
2
2
2
1
2
1
2
p
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Slide 6-4
Range of values for1,2
+ 1.0 > > -1.0
If = 1.0, the securities would be
perfectly positively correlated
If= - 1.0, the securities would be
perfectly negatively correlated
Correlation Coefficients:
Possible Values
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Slide 6-5
1,2 = Correlation coefficient of returns
1 = Standard deviation of returns for
Security 12 = Standard deviation of returns for
Security 2
Covariance
212,121 )r,r(Cov
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Slide 6-6
Possible to split investment fundsbetween safe and risky assets
Risk free asset: proxy; T-bills
Risky asset: stock (or a portfolio)
Allocating Capital Between
Risky & Risk Free Assets
7/31/2019 Lecture 5- Principles of Financial Economics
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Examine risk/return tradeoff
Demonstrate how different degrees of riskaversion will affect allocations betweenrisky and risk free assets
Allocating Capital Between
Risky & Risk Free Assets
7/31/2019 Lecture 5- Principles of Financial Economics
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The Risk-Free Asset
Perfectly price-indexed bond the only
risk free asset in real terms; T-bills are commonly viewed as the
risk-free asset;
Money market funds - the mostaccessible risk-free asset for mostinvestors.
7/31/2019 Lecture 5- Principles of Financial Economics
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Portfolios of One Risky Asset
and One Risk-Free Asset Assume a risky portfolio P defined by :
E(rp) = 15% and p = 22%
The available risk-free asset has:
rf= 7% and
rf= 0%
And the proportions invested:
y% in P and (1-y)% in rf
7/31/2019 Lecture 5- Principles of Financial Economics
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E(rc) = yE(rp) + (1 - y)rf
rc = complete or combined portfolio
If, for example, y = .75E(rc) = .75(.15) + .25(.07)
= .13 or 13%
Expected Returns for
Combinations
7/31/2019 Lecture 5- Principles of Financial Economics
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rf
pc=
Since
y
= 0, then
* Rule 4 in Chapter 5
*
Variance on the Possible
Combined Portfolios
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Possible Combinations
E(r)
E(rp) = 15%
rf= 7%
22%0
P
F
c
E(rc) = 13% C
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CAL (Capital Allocation Line)
E(r)
E(rp) = 15%
rf= 7%
p= 22%0
P
F
) S = 8/22
E(rp) - rf= 8%
7/31/2019 Lecture 5- Principles of Financial Economics
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Greater levels of risk aversion lead tolarger proportions of the risk free rate
Lower levels of risk aversion lead tolarger proportions of the portfolio of riskyassets
Willingness to accept high levels of riskfor high levels of returns would result inleveraged combinations
Risk Aversion and Allocation
7/31/2019 Lecture 5- Principles of Financial Economics
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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
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Borrow at the Risk-Free Rate and invest in
stock Using 50% Leverage
rc
= (-.5) (.07) + (1.5) (.15) = .19
c = (1.5) (.22) = .33
Using Leverage with
Capital Allocation Line
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Borrowing
If investors can borrow at the risk-free rateof rf= 7%, they can construct portfolios that
may be plotted on the CAL to the right ofP.
The leveraged portfolio has a higher
standard deviation than the unleveragedposition in the risky asset.
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Leveraged Position
Suppose the investment budget is $300,000 andthe investor borrows an additional $120,000investing the total available funds in the risky
asset. This is a leveraged position in the risky asset
which is financed in part by borrowing. This reflects a short position in the risk-free
asset. Rather than lending at 7%, the investor borrows
at the rate of 7%.
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Slide 6-19
Effect of Leverage on the Reward-
to-variability ratio The distribution of the portfolio rate of
return still exhibits the same reward-to-
variability ratio. However the borrowing rate is likely to be
higher.
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Slide 6-20
Higher borrowing rate
Assume that the borrowing rate is 9 percent. Calculate the reward-to-variability
ratio if the expected return on the portfolioof risky assets is 15 per cent and itsstandard deviation is 22 per cent.
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Slide 6-21
CAL with Higher Borrowing
RateE(r)
9%
7%) S = .36
) S = .27P
p = 22%
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Slide 6-22
CAL is kinked
With a higher borrowing rate (than the lendingrate) the CAL is kinked at point P.
To the left of P the investor is lending at 7 percent and the slope of the CAL is 0.36
To the right of P, y>1, the investor is borrowingat 9 per cent to finance extra investments in the
risky asset and the slope is 0.27
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Slide 6-23
Indifference Curves and Risk
AversionCertainty equivalent ofportfolio Ps expected return
for two different investors
P
E(r)
rf=7%
A = 4
A = 2
p = 22%
E(rp)=15%
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Slide 6-24
Certainty Equivalent Rate of Return
More risk averse investors have a steeper ICs
Less risk averse investors have flatter ICs
Portfolios utility value is its certainty equivalentrate of return to the investor
The certainty equivalent rate of return of theportfolio is the rate that risk-free investments
would need to offer with certainty to beconsidered equally attractive as the riskyportfolio.
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Slid 6 25
CAL with Risk Preferences
P
E(r)
7%Lender
Borrower
p = 22%