risky-riskfee asset allocation

8/14/2019 Risky-Riskfee Asset Allocation

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Capital Allocation BetweenThe Risky And The Risk-Free

Asset


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Its possible to split investment fundsbetween safe and risky assets.

Risk free asset: proxy; T-bills

Risky asset: stock (or a portfolio)


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Issues

Examine risk/return tradeoff. Demonstrate how different

degrees of risk aversion will

affect allocations between riskyand risk free assets.


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rf = 7% rf= 0%E(rp) = 15% p = 22%

y = % in p (1-y) = % in rf


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E(rc) = yE(rp) + (1 - y)rf

c = complete or combined portfolio

For example, y = .75

E(rc) = .75(.15) + .25(.07)= .13 or 13%


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E(r)

E(rp) = 15%

rf = 7%

22%0

P

F

c

E(rc) = 13%C


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pc =

Since rf

y

= 0, then


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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= (.22) = .00 or 0%

If y = 0


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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


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E(r)

E(rp) = 15%

rf = 7%

p = 22%0

P

F

) S = 8/22

E(rp) - rf = 8%


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E(r)

9

%7%

) S = .36

) S = .27

P

p = 22%


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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 ofrisky assets.

Willingness to accept high levels of riskfor high levels of returns would resultin leveraged combinations.


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U = E ( r ) - .005 A 2

Where

U = utility

E ( r ) = expected return on the asset

or portfolio

A = coefficient of risk aversion

2 = variance of returns


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E(r)

7%

P

Lender

Borrower

p = 22%

The lender has a larger A when

compared to the borrower


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Optimal Risky Portfolios


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Number ofSecurities

St. Deviation

Market Risk

Unique Risk


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Range of values for 1,2+ 1.0 > > -1.0

If = 1.0, the securities wouldbe perfectly positively

correlatedIf = - 1.0, the securitieswould be perfectly negatively

correlated


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2p = W12 12+ W22 12+ 2W1W2

rp = W1r1 +W2r2 + W3r3

Cov(r1r2)

+ W32 32

Cov(r1r

3)

+ 2W1W3

Cov(r2r3)+ 2W2W3


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E(rp) = W1r1 +W2r2

2= w12 12 + w22 22 + 2W1W2 Cov(r1[w1

2 12 + w22 22 + 2W1W2 Cov(r1r2


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= 1

13%

%8

E(r)

St. Dev12% 20%

= .3

= -1

= -1


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The relationship depends oncorrelation coefficient.

-1.0 < < +1.0

The smaller the correlation, the greaterthe risk reduction potential.

If = +1.0, no risk reduction is

possible.


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1

1 2

22 -Cov(r1r2)W

1

=

+ -2Cov(r1r2)W2 = (1 -

W1)

2

2E(r2) = .14 = .20Sec 212= .2

E(r1) = .10 = .15Sec 1

2


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W1=(.2)2 - (.2)(.15)(.2)

(.15)2 + (.2)2 - 2(.2)(.15)(.2)

W1= .6733

W2= (1 - .6733) = .3267


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p = .6733(.10) + .3267(.14) = .1131

p= [(.6733)2(.15)2 + (.3267)2(.2)2 +

2(.6733)(.3267)(.2)(.15)(.2)]1/2

p =[.0171]

1/2

= .1308


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W1=(.2)2 - (.2)(.15)(.2)

(.15)2 + (.2)2 - 2(.2)(.15)(-.3)

W1= .6087

W2= (1 - .6087) = .3913


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p = .6087(.10) + .3913(.14) = .115

p= [(.6087)2(.15)2 + (.3913)2(.2)2

2(.6087)(.3913)(.2)(.15)(-.3)]1/2

p=

[.0102]

1/2

= .1009


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The optimal combinations result inlowest level of risk for a given return.

The optimal trade-off is described asthe efficient frontier.

These portfolios are dominant.


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E(r)

Efficientfrontier

Globalminimum

varianceportfolio

Minimumvariancefrontier

Individualassets

St. Dev.


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The optimal combination becomes linear.

A single combination of risky and riskless

assets will dominate.


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M

E(r)

CAL (Globalminimum variance)

CAL (A)CAL (P)

P

A

F

P P&F A&FM

A

G

P

M


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E(r)

Efficient

frontier ofrisky assets

Morerisk-averseinvestor

U U U

Q

P

S

St. Dev

Less

risk-averseinvestor


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E(r)

FrfA

PQ

B

CAL

St.

Dev

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