Modern Portfolio

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 is: 2 SIMσ ij = β i β jσ m

σ im σ jm 2 × 2 × σm 2 σm σm ρ σσ ρ σσ 2 = im 2i m × jm 2j m × σ m σm σm = ρ im ρ jmσ iσ j

=

If the assumptions of both the constant correlation and single-index model hold, then we have CCσ ij = SIMσ ij :

ρ *σ iσ j = ρ imρ jmσ iσ j or ρ * = ρ im ρ jm

This must hold for all pairs of securities, including i and j, i and k and j and k. So we have: ρ * = ρ im ρ jm

ρ * = ρ im ρ km

ρ * = ρ jm ρ km

The only solution to the above set of equations is:

ρ im = ρ jm = ρ km = ρ *

Therefore, for any security i we have:

βi =

ρ* σ im ρ imσ i σ m ρ imσ i = = = ×σ i 2 2 σm σm σm σm

In other words, given that all pairs of securities have the same correlation coefficient and that the Sharpe singleindex model holds, each security’s beta is proportional to its standard deviation, where the proportion is a constant across all securities equal to Problem 2 Start with a general 3-index model of the form:

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

* Set I1 = I1 and define an index I2 which is orthogonal to I1 as follows: * * I 2 = γ 0 + γ 1 × I1 + d t or I2 = dt = I2 − (γ 0 + γ 1 × I1)

ρ* . σm

(1)

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