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Unformatted text preview: tion of two random variables and : Applying this formula we get:
[( ) 28
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. (
Denote ) ( ) so that:
( ) ( ( Now, to get a returnsrisk equation we need to compute ) ) for:
( ) To do this, rearrange:
(
( ) ) ( [ ) This is a quadratic equation of the form: The solution is:
√
Thus:
√( Put another way, ) ( ) is a function of portfolio risk:
( As such, we have: ) (
(
√(
( )
)( )
) ( ) )
{ ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio Allocation in ECO 204 (this version 20122013) } 29 University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. This is an equation linking portfolio returns to portfolio risk.
(k) Graph the efficiency frontier for a portfolio where fraction is in Dell stocks and fraction (
for various values of (say
. Interpret this graph. ) is in Boeing stocks Answer:
See Excel Model Chapter 7.1: DELL and BOEING Efficiency Frontier
0.045
0<f<1: Portfolio consisted of Dell and Boeing stocks
s.t. portfolio risk is lower than Dell and Boeing’s risk 0.040 Portfolio return 0.035
0.030 rp=rDell
(B) f=1: Portfolio consisted of
Dell stocks only 0.025 0.020
0.015 rp=rBoeing (A) f=0: Portfolio consisted of Boeing stocks 0.010 only 0.005
0.000
0.000 σp=σBoeing
0.050 σp=σDell 0.100 0.150 0.200 0.250 Portfolio risk Since Dell average returns are greater than Boeing, label Dell “risky asset B” and Boeing “risky asset A”.
This graph was created by plotting portfolio returns vs. portfolio risk for various where:
(
( ) ( )
) (
( ) )
( ) The efficiency frontier allows for a leverage portfolio where
can be invested in Dell stocks. Thus, you have to
)
ensure that whenever
for Dell that (
for Boeing.
The efficiency frontier gives the portfolio returnrisk for various combinations of Dell and Boeing stocks. For example, if
then the portfolio consists of just Boeing stocks and the portfolio return and risk equal Boeing’s return and risk as
shown in the chart point (A).
If
then the portfolio consists of just Dell stocks and the portfolio return and risk equal Dell’s return and risk as
shown in the chart point (B).
It is possible to leverage the portfolio such that portfolio return (and risk) are greater than Dell’s return (the asset with
the higher return). In this scenario
and in the chart any point on curve above point (B) will form a leverage
portfolio.
(l) Suppose an investor wants to construct a portfolio consisting of two risky assets A and B. Calculate the optimal
fraction of the portfolio in risky asset B by solving the problem:
30
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. ̅
Hint: Use the “short method” by substituting expressions for ̅ and . Answer:
We want:
̅
This UMP adds the constraint that the fraction of the portfolio in the risky asset with the higher return (asset B) must be
zero or positive. Recall that:
(
From this we have:
̅
Substitute ̅ ̅ (̅ ̅ ) and ( (̅ ̅ ) ) ̅)
( ) to get: ̅
(̅ ̅ ̅) (̅ ̅ {( ̅) ) {( ( ) ) ( ) }
} Setup the Lagrangian:
(̅ ̅ ̅) (̅ ̅) ̅ ̅ { {(
{( ) (
) ) }
} The FOC and KT condition are:
̅ The 1st KT case is
case whenever: ( ) } (i.e. no money in risky asset B). For this to be a solution we require that ̅
̅ ̅
̅ ( {
{ ( )
) : this will be the }
}
31 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. ̅ { ̅
̅ } ̅
Now { ̅ }
{ ̅ } whenever:
̅ ̅ { } ̅ ̅ { } ̅ ̅ { } The left hand side is the gap between the higher return asset B and the lower return asset A: if this is below a threshold
(that depends on the degree of risk aversion), the investor won’t buy asset B.
The 2nd KT case is For this to be a solution we require that
̅ ̅ ̅ ̅ { ̅ ( { ) ( ̅
̅
̅ } ( ) } { ̅ } ) { ̅ : this will be the case whenever: { ̅ }
{ } ̅ { { } { ̅ }
̅ } Or: { ̅ } {
Now } whenever:
̅ { ̅ } {
̅ }
{ ̅
̅ } } { ̅ }
{ ̅ { ̅ } ̅ }
{ } This is the exact opposite condition as the 1st KT case.
32
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. The solutions are:
Case A: If ̅ ̅ { } then Case B: If ̅ ̅ { } then ̅ ̅̅̅ ̅̅̅ { ̅ { } { }. . } (m) Use the answer for part (l) to construct a portfolio of Dell and Boeing stocks in January 1st, 2012 for
In each case, interpret and calculate the portfolio return and risk. and 2. Answer:
Now:
June 30th 1988  Dec 31st 2011
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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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