Optimal Risk Portfolio

8/6/2019 Optimal Risk Portfolio

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Optimal Risky Portfolios


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Risk Reduction with

Diversification

Number ofSecurities

St. Deviation

Market Risk

Unique Risk


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Risk Reduction with

Diversification
http://../Demo/Diversification%20effect.xlsx


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rp = W1r1 + W2r2W

1= Proportion of funds in Security 1

W2 = Proportion of funds in Security 2

r1 = Expected return on Security 1

r2 = Expected return on Security 21

n

1i

iw

Two-Security Portfolio: Return


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p2 = w1212 + w2222 + 2W1W2 Cov(r1r2)12 = Variance of Security 122 = Variance of Security 2

Cov(r1r2) = Covariance of returns for

Security 1 and Security 2

Two-Security Portfolio: Risk


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1,2

= Correlation coefficient of

returns

Cov(r1r2) = 1,212

1 = Standard deviation ofreturns for Security 12 = Standard deviation ofreturns for Security 2

Covariance


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

If= 1.0, the securities would be perfectlypositively correlated

If= - 1.0, the securities would beperfectly negatively correlated

Correlation Coefficients: Possible

Values


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2p = W1212 + W2212+ 2W1W2

rp = W1r1 + W2r2 + W3r3

Cov(r1r2)

+ W3232

Cov(r1r3)+ 2W1W3

Cov(r2r3)+ 2W2W3

Three-Security Portfolio


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rp = Weighted average of the

n securitiesp2 = (Consider all pairwise

covariance measures)

In General, For An N-Security

Portfolio:


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E(rp) = W1r1 + W2r2

p2 = w1212 + w2222 + 2W1W2 Cov(r1r2)p = [w1212 + w2222 + 2W1W2 Cov(r1r2)]1/2

Two-Security Portfolio


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Portfolios with Different

Correlations

= 1

13%

%8

E(r)

St. Dev12% 20%

= .3

= -1

= -1


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Correlation Effects

The relationship depends on correlation

coefficient.

-1.0 < < +1.0 The smaller the correlation, the greater the

risk reduction potential.

If = +1.0, no risk reduction is possible.
http://../Demo/Portfolio%20Standard%20Deviation%20as%20Correlation%20Changes.xls


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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

Minimum-Variance Combination


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

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

W1 = .6733

W2 = (1 - .6733) = .3267

Minimum-Variance Combination:

= .2


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rp = .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

Risk and Return: Minimum

Variance


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

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

W1 = .6087

W2 = (1 - .6087) = .3913

Minimum - Variance Combination:

= -.3


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rp = .6087(.10) + .3913(.14) = .1157

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

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

p= [.0102]1/2

= .1009

Risk and Return: Minimum

Variance


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Extending Concepts to All

Securities

The optimal combinations result in lowest

level of risk for a given return.

The optimal trade-off is described as the

efficient frontier.

These portfolios are dominant.


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Minimum-Variance Frontier of Risky Assets

E(r) Efficient

frontier

Global

minimum

variance

portfolio Minimumvariance

frontier

Individual

assets

St. Dev.


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Alternative CALs

M

E(r)

CAL (Global

minimum variance)

CAL (A)CAL (P)

P

A

F

P P&F A&FM

A

G

P

M


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Efficient Frontier with Lending & Borrowing

E(r)

Frf

A

P

Q

B

CAL

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Optimal Risk Portfolio