Unformatted text preview: l
Boeing
Mean r
0.028
0.011
Risk σ
0.145
0.080
Cov(Dell, Boeing)
0.0021
rf, Jan 1st 2012 0.0003 Dell’s average return is greater than Boeing’s – hence, label it risky asset B. The fraction of a portfolio consisting of Dell
and Boeing stocks is:
̅ ̅ { {
̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅̅ }
} { { }
} Using Excel Model Chapter 7.1 we get:
for 3 Investors
Investor A: Investor B: Investor C: 0.55 0.37 0.027 0.020 0.017 0.134 0.093 0.080 Investor A invests 92% of her portfolio money in Dell stocks, with the portfolio return between Dell and Boeing’s
individual returns and portfolio risk lower than either Boeing or Dell’s risk.
Investor B invests 55% of her portfolio money in Dell stocks, with the portfolio return between Dell and Boeing’s
individual returns and portfolio risk lower than either Boeing or Dell’s risk.
33
ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio Allocation in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Investor C invests 37% of her portfolio money in Dell stocks, with the portfolio return between Dell and Boeing’s
individual returns and portfolio risk lower than either Boeing or Dell’s risk.
(n) Suppose an investor wants to construct a portfolio consisting of a risk free asset and two risky assets: Case 3
$X Portfolio Risk Free Asset 2 Risky Assets Fraction (1 – β) Fraction β Risky Asset A Risky Asset B Fraction (1 – f) Fraction f Calculate the optimal fraction of the portfolio in risky assets and the optimal fraction
allocated to risky asset B by solving the problem: of the risky assets amount ̅
Answer:
There are two ways to do this: a faster method relying on intuition and a longer approach by the KT method. We do both
here.
The Fast Approach
This problem combines cases 1 and 2: First, use case 2 to solve for the fraction of the risky assets that will be allocated to asset B (and therefore
(
) to asset A).
Second, compute the return and risk of the two risky assets.
Third, treat the two risky assets as “one risky asset” and use case 1 to solve for the fraction
of of the portfolio
(
) of the portfolio to the risk free asset).
that will be allocated to the 2 risky assets (and therefore 34
ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio Allocation in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Case 3
$X Portfolio
Return = rp Risk = σp
Risky Free Asset 2 Risky Assets Fraction (1 – β) Fraction β Return = rf Risk = σf Return = rr, Risk = σr Risky Asset A First, whatever portion
Of the fraction Risky Asset B Fraction (1 – f) Fraction f of the portfolio is allocated to the two risky assets:
of the portfolio allocated to the two risky assets, the fraction allocated to risky asset B:
̅ { ̅ } {
Of the fraction } of the portfolio allocated to the two risky assets, the fraction allocated to risky asset A:
̅ { ̅ } { } Second:
The return and risk of the fraction of the portfolio allocated to the two risky assets A and B:
(̅ ̅
( ̅) ) ( ) Third:
The fraction of the portfolio allocated to the two risky assets A and B is:
( Where and ) are from step 2. The fraction of the portfolio allocated to the risk free asset is: 35
ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio Allocation in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. (
Where and ) are from step 2.
( Notice that the fraction of the portfolio allocated to assets A and B are, respectively ) and . Don’t do this (yet)
The Long Method:
We want: Now: Here ( )
( is the return of the two risky assets (i.e.
(
( )) so that portfolio return is:
) ) ( ) ( ) The portfolio variance is:
( ( ) ( Now: ) ) ( ) so that: ( Now substitute the variance of the two risky assets
above to get: [( ) ) ( ( ) ) into the equation The UMP becomes: ( ) [( ) ( ) Let’s first solve the problem without the constraints. Differentiate utility with respect to and :
36 ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio Allocation in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. ( ) [( ) ( ) ( ) [ ( ) ( ) ( ) [ ( ) ( ) ( ) Let’s tackle the second FOC: ( [ ) ( ) This looks a lot like the FOC from case 2  they’re almost the same except for the
( { ) term. Carrying on:
} ( { ) } { } { { } { }
}
{ } { {
{ } } } { } Or: { } { } We can substitute this in the 1st FOC:
( ) [( )
(
( Now we would, but won’t, substitute ) ( ) [( ) )
) ( ) ( ( [( ) ( ) )
( ) and solve for .
37 ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio Allocation in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Aj...
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This note was uploaded on 03/20/2014 for the course ECON 204 taught by Professor Ajazhussain during the Fall '09 term at University of Toronto.
 Fall '09
 AJAZHUSSAIN
 Economics, Microeconomics

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