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Elton, Gruber, Brown and Goetzmann Modern Portfolio Theory and Investment Analysis, 6th Edition Solutions to Text Problems: Chapter 8

Problem 1 Given the correlation coefficient of the returns on a pair of securities i and j, the securities’ covariance can be expressed as the securities’ correlation coefficient times the product of their standard deviations: σ ij = ρ ijσ iσ j But if we assume that all pairs of securities have the same constant correlation, ρ * , then the constant-correlation expression for covariance is: CCσ ij = ρ *σ iσ j Given the assumptions of the Sharpe single-index model, the single-index model’s expression for the covariance between the returns on a pair of securites ...view middle of the document...

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which gives:

* I2 = γ 0 + γ 1 × I1 + I2

Substituting the above expression into equation (1) and rearranging we get:

* Ri = ai* + bi*2 × γ 0 + bi*1 + bi*2 × γ 1 × I1 + bi*2 × I2 + bi*3 × I3 + ci

(

) (

)

′ The first term in the above equation is a constant, which we can define as a1 . The coefficient in the second term ′ of the above equation is also a constant, which we can define as bi1 . We can then rewrite the above equation as:

′ ′ * Ri = ai + bi1 × I1 + bi*2 × I2 + bi*3 × I3 + ci

Now define an index I3 which is orthogonal to I1 and I2 as follows:

(2)

* * I3 = θ 0 + θ1 × I1 + θ 2 × I2 + et or I 3 = et = I 3 − (θ 0 + θ 1 × I1 + θ 2 × I 2 )

which gives:

* I 3 = θ 0 + θ 1 × I1 + θ 2 × I 2 + I 3

Substituting the above expression into equation (2) and rearranging we get:

′ ⎞ ⎛ ′ Ri = ⎛ ai + bi 3 × θ 0 ⎟ + ⎜ bi1 + bi 3 × θ1 ⎞ × I1 + bi*2 + bi*3 × θ2 × I2 + bi*3 × I3 + ci ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

In the above equation, the first term and all the coefficients of the new orthogonal indices are constants, so we can rewrite the equation as:

(

)

Ri = ai + bi1 × I1 + bi 2 × I2 + bi 3 × I3 + ci

Problem 3 Recall from the earlier chapter on the single-index model that an expression for the covariance of returns on two securities i and j is:

2 σ ij = β i β j E⎡ Rm − Rm ⎤ + β j E ei Rm − Rm + β i E ej Rm − Rm + E[ei ej ] ⎢ ⎥

⎣

(

)⎦

[(

)]

[ (

)]

The first term contains the variance of the market portfolio, the second two terms contain the covariance of the market portfolio with the residuals and the last term is the covariance of the residuals. Given that one of the model’s assumptions is that the covariance of the market portfolio with the residuals is zero and that, from the problem, the covariance of the residuals equals a constant K, the derived covariance between the two securities is:

2 σ ij = β i β jσ m + K

One expression for the variance of a portfolio is:

Elton, Gruber, Brown and Goetzmann Modern Portfolio Theory and Investment Analysis, 6th Edition Solutions To Text Problems: Chapter 8

2

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2 σ P = ∑ Xi2σ i2 + ∑ ∑ X j Xkσ jk i =1 j =1 k =1 k≠ j

N

N

N

2 2 Recalling that the single-index model’s expression for the variance of a security is σ i2 = β i2σ m + σ ei and substituting that expression and the derived expression for covariance into the above equation and rearranging gives:

2 2 2 2 σ P = ∑ Xi2 β i2σ m + ∑ Xi2σ ei + ∑ ∑ X j Xk β j β kσ m + ∑ ∑ X j Xk K i =1 N i =1 j =1 k =1 k≠ j j =1 k =1 k≠ j N N N N

N

N

N

N

N

N

=

∑∑

i =1 j =1

2 Xi X j β i β jσ m +

∑

i =1

2 Xi2σ ei +

∑∑X X K

j k j =1 k =1 k≠ j

⎛ =⎜ ⎜ ⎝

∑

i =1 2 P

N

⎞⎛ Xi β i ⎟⎜ ⎟⎜ ⎠⎝

2 m

∑

i =1

N

⎞ 2 Xi β i ⎟σ m + ⎟ ⎠

∑

i =1 N

N

2 Xi2σ ei +

∑∑X X K

j k j =1 k =1 k≠ j

N

N

=β σ

+

∑

i =1

N

Xi2

σ

2 ei

⎛ ⎜ + K⎜ ⎜ ⎝

∑∑

j =1

N

⎞ ⎟ X j Xk ⎟ ⎟ k =1 k≠ j ⎠

Problem 4 Using...

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