Unformatted text preview: portfolio return and risk is between the return and risk of the risk free and risky assets: 20
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. rp rr
rp rf Portfolio of risky and risk free assets
σp σp σr Leveraged Portfolio of Risky Asset Only
(
)
(
(
)
Because cannot be negative, set ( )
) In this case, the investor invests more than her portfolio. How? She borrows (leverages) money and invests her portfolio
plus the borrowed amount in the risky asset (i.e., she borrows
of the portfolio). The leveraged portfolio gross
return and risk is greater than the return and risk of the risky asset. rp Leveraged Portfolio of risky asset only
rp
rr rf
σr σp σp (e) Graph the Capital Allocation Line for a portfolio where fraction is in Dell stocks and fraction (
free asset for various values of (say
. Interpret this graph. ) is in the risk Answer:
Please see Excel Model Chapter 7.1:
21
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. DELL Capital Allocation Line
0.045
0.040 Portfolio return 0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
0.000 0.050 0.100 0.150 0.200 0.250 Portfolio risk Graphing these charts is an excellent Excel exercise: if we were to calculate portfolio returns as:
(
)
Then whenever
we would have (
whenever
(where is in column B) that ( ) which, of course, is impossible. Therefore, we have to tell Excel that
)
: Capital Allocation Line for 3 Month TBills (Jan 1st 2010) and Dell Stocks
βDell Dell Returns TBills Returns σp rp 0 =B297*$C$9 =IF(B297<=1,(1B297)*$C$12,0) =B297*$C$10 =C297+D297 (f) Graph the Capital Allocation Line for a portfolio where fraction is in Boeing stocks and fraction (
free asset for various values of (say
. Interpret this graph. ) is in the risk Answer:
Please see Excel Model Chapter 7.1: 22
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. BOEING Capital Allocation Line
0.030 Portfolio return 0.025 0.020 0.015 0.010 0.005 0.000
0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 Portfolio risk Again, we have to make sure that whenever (where is in column B) that ( ) . See previous question. (g) Suppose an investor wants to construct a portfolio consisting of a risk free asset and a risky asset. Calculate the
optimal fraction of the portfolio in the risky asset by solving the problem:
̅
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 must be zero or positive. Recall that:
(
)
From this we have:
̅
Since risk free returns are guaranteed: ̅ ̅ (̅ ̅) (̅ ) so that:
̅ 23
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. _____________________________________________________________________________________Optional
“proof”: the mean is the sample analog of the expected value. Thus:
( [ Since risk free returns are guaranteed: [ ] )
( )] ( [ ]) ( ) so that: The sample version of this is:
̅
(̅
)
_____________________________________________________________________________________
Substitute ̅ (̅ ) and into the UMP to get:
̅
(̅ ) (̅ ) Setup the Lagrangian:
(̅ ) The FOC and KT condition are:
̅ The 1st KT case is
case whenever: (i.e. no money in the risky asset). For this to be a solution we require that : this will be the ̅
̅
̅
Now whenever:
̅
24 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. ̅
That is, if the risk free asset has the same or greater return than the average risky asset return, the investor should only
invest in the risk free asset.
The 2nd KT case is For this to be a solution we require that : this will be the case whenever: ̅
̅
̅
(̅ Now ) whenever:
(̅ ) ̅
̅
The investor will invest in the risk asset if the average risky asset return is greater than or equal to the risk free assert
return.
The solutions are:
Case A: If ̅ then Case B: If ̅ then ̅ . In this portfolio return and risk are:
(̅ ) and . . In this portfolio return and risk are: (̅ ) (̅ (̅ ) ) (̅ (̅ )
(̅ ) ) (h) Use the answer for part (g) to construct a portfolio of 3month TBills and Dell stocks in January 1st, 2010 for
and 2. In each case, interpret and calculate the portfolio return and risk.
Answer:
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 Fall '09
 AJAZHUSSAIN
 Economics, Microeconomics, Sayed Ajaz Hussain

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