Session 5a
Decision Models -- Prof. Juran
2
OverviewPortfolio Optimization• Parametric Approach
– Array Functions• Scenario Approach
– Put Options– Shorting
Decision Models -- Prof. Juran
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4 Basic Matrix Operations
• Sum Product• Transpose• Matrix Multiplication• Inverting Matrices
Decision Models -- Prof. Juran
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rcrr
c
c
aaa
aaa
aaa
A
21
22221
11211
and
ijii
j
j
bbb
bbb
bbb
B
21
22221
11211
It is conventional to describe the shape of a matrix by listing the number of rows first, and the number of columns second. Matrix A above is an r x c matrix, and matrix B is an i x j matrix.
Decision Models -- Prof. Juran
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F o r m a t r i c e s A a n d B ,
rcrr
c
c
aaa
aaa
aaa
A
21
22221
11211
a n d
ijii
j
j
bbb
bbb
bbb
B
21
22221
11211
S u m p r o d u c t ( A , B ) ijrc babababa 131312121111
Sum Product
Decision Models -- Prof. Juran
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1125
647A and
11210
839B
Sumproduct(A, B) ijrcbabababa 131312121111
111863497
208
Notes: r = i and c = j. In other words, the two matrices must have the same number of rows as each other and the same number of columns as each other. They do not need to be square matrices (where r = c and i = j).
Decision Models -- Prof. Juran
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12345678
A B C D7 4 65 2 11
9 3 810 12 1
208=SUMPRODUCT(A1:C2,A4:C5)
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Transpose
For matrix A,
rcrr
c
c
aaa
aaa
aaa
A
21
22221
11211
rccc
r
r
T
aaa
aaa
aaa
A
21
22212
12111
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Notes:
• If A is an r x c matrix, then must be a c x r matrix.
• A does not need to be a square matrix.
TA
1125
647A
116
24
57TA
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There is an Excel function for this purpose, called TRANSPOSE. This function is one of a special class of functions called array functions. In contrast with most other Excel functions, array functions have two important differences:
• They are entered into ranges of cells, not single cells
• You enter them by pressing Shift+Ctrl+Enter, not just Enter
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Using the spreadsheet above as an example, we start by selecting the entire range A4:C6. Then type into the formula bar =TRANSPOSE(A1:C2)
Press Shift+Ctrl+Enter, and curly brackets will appear round the formula (you can’t type them in).
123456
A B C D E7 4 65 2 11
123456
A B C D E7 4 65 2 11
7 54 26 11
=TRANSPOSE(A1:C2)
Decision Models -- Prof. Juran
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Matrix MultiplicationFor matrices A and B,
rcrr
c
c
aaa
aaa
aaa
A
21
22221
11211
ijii
j
j
bbb
bbb
bbb
B
21
22221
11211
ijrcjrjrircrrircrr
ijcjjicic
ijcjjicic
bababababababababa
bababababababababa
bababababababababa
AB
221122221211212111
222212122222212211221221121
121211121221212111121121111
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In this operation, it is necessary for c = i. However it is not necessary for r = j. In other words, B must have the same number of rows as A has columns, but it is not necessary for B to have the same number of columns as A has rows. The product AB will always be an r x j matrix. Example
2
1
3
0
4
1
A and
25
10B
AB
2*21*0
2*11*4
2*3 + 1*1=
5*2 + 0*0=
5*1 + 0*4=
5*30*1
4
6
7
10
5
15
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Example:
2425262728293031
A B C D E F1 3 0 14 1 5 20 2
mmult15 7
5 610 4
=MMULT(A24:B26,E24:F25)
Remember:
Select the entire range A29:B31 before typing the formula
Press Shift+Ctrl+Enter
We will find this useful in some situations, such as calculating the risk of a stock portfolio.
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Inverting MatricesFirst, define a square matrix Ij as a matrix with j rows and j columns, completely filled with zeroes, except for ones on the diagonal:
100
010
001
I
This special matrix is called the identity matrix.
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Now, for a square matrix A with j rows and j columns, there may exist a matrix called A-inverse (symbolized ) such that:
1A
1AA jI
Not all square matrices can be inverted, a fact that has implications for regression analysis.
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If
85139
124123A
Then
01814.002050.0
01829.001254.01A
Because
1AA
01814.08501829.013902050.08501254.0139
01814.012401829.012302050.012401254.0123
10
01
2I
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123456
A B C D E F G Hcheck
123 124 1 0139 85 0 1
-0.01254 0.018290.02050 -0.01814
=MINVERSE(A2:B3)
=MMULT(A2:B3,A5:B6)
Remember:
• Select the entire range A5:B6 before typing the formula.
• Press Shift+Ctrl+Enter.
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Parametric Approach
Consider a portfolio composed of three stocks. Let the random variable si be the annual return on stock i, with expected value μi, variance 2i, and standard deviation i.
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Parametric Approach
The correlation between si and sj, for any pair of stocks i and j, is ρij.
The covariance between si and sj, for any pair of stocks i and j, is σiσjρij.
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Parametric Approach
The means are: μ1 = 0.14, μ2 = 0.11, μ3 = 0.10.
The variances are: 21s = 0.20, 2
2s = 0.08, 23s = 0.18.
The correlations are: ρ12 = 0.8, ρ13 = 0.7, and ρ23 = 0.9.
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Parametric Approach(a) Determine the minimum-variance portfolio that attains an expected annual return of at least 0.12, with no shorting of stocks allowed.
(b) Draw the efficient frontier for portfolios composed of these three stocks.
(c) Determine the minimum-variance portfolio that attains an expected annual return of at least 0.12, with shorting of stocks allowed.
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Managerial Problem DefinitionDecision Variables
We need to allocate 100% of the available funds to the three stocks.
Objective
We want to minimize risk.
Constraints
The expected return on the portfolio must be at least 12% (or 0.12).
All of the money must be invested.
No shorting is allowed.
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Formulation
Decision Variables
Define xi to be the proportion of the portfolio allocated to stock i.
The decision variables are three proportions: x1, x2, and x3.
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Formulation
T h e e x p e c t e d r e t u r n o n t h e p o r t f o l i o i s :
332211 xxx
T h e v a r i a n c e o f t h e r e t u r n o n t h e p o r t f o l i o i s :
3,1313,2322,12123
23
22
22
21
21
2 222 COVxxCOVxxCOVxxxxx
T h e s t a n d a r d d e v i a t i o n o f t h e r e t u r n o n t h e p o r t f o l i o i s :
3,1313,2322,12123
23
22
22
21
21
2 222 COVxxCOVxxCOVxxxxx
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Formulation
Objective
Minimize Z = 3,1313,2322,12123
23
22
22
21
21 222 COVxxCOVxxCOVxxxxx
Constraints
12.0332211 xxx (1)
0.13
1
i
ix (2)
For all i, 0ix (3)
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Solution Methodology123456789
10111213
14
15161718192021
A B C D E F G H I JStock 1 Stock 2 Stock 3
Mean return 0.140 0.110 0.100Variance of return 0.200 0.080 0.180
StDev of return 0.447 0.283 0.424
Correlations CovariancesStock 1 Stock 2 Stock 3 Stock 1 Stock 2 Stock 3
Stock 1 1.00 0.80 0.70 Stock 1 0.2000 0.1012 0.1328Stock 2 0.80 1.00 0.90 Stock 2 0.1012 0.0800 0.1080Stock 3 0.70 0.90 1.00 Stock 3 0.1328 0.1080 0.1800
Investment decisionStock 1 Stock 2 Stock 3 Total Required
Fractions to invest 1.000 0.000 0.000 1 = 1
Expected portfolio returnActual Required0.140 >= 0.120
Portfolio variance 0.200Portfolio stdev 0.447
=SUM(B14:D14)
=HLOOKUP($G8,$B$1:$D$4,4)*B8*HLOOKUP(H$7,$B$1:$D$4,4)
=SUMPRODUCT(B2:D2,B14:D14)
=SQRT(B20)
{=MMULT(B14:D14,MMULT(H8:J10,TRANSPOSE(B14:D14)))}
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Solution MethodologyThe decision variables are in B14:D14.
Cell B18 keeps track of constraint (1).
Cell E14 keeps track of constraint (2).
We can comply with constraint (3) by selecting “assume nonnegative” in the Solver Options box.
The range H8:J10 uses the HLOOKUP Excel function to calculate the covariances.
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Solution Methodology
Cell B20 uses two of Excel’s matrix functions, MMULT and TRANSPOSE, to calculate the portfolio variance. If you use these functions, you will want to learn more about working with “arrays” in Excel, an advanced topic beyond the scope of this course.
In this case, you need to know not to type in the “curly brackets”; instead, type in the rest of the function and then Ctrl+Shift+Enter. The curly brackets will appear automatically. You will get a #VALUE! Error message if you do this wrong.
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Optimal Solution
123456789
10111213
14
15161718192021
A B C D E F G H I JStock 1 Stock 2 Stock 3
Mean return 0.140 0.110 0.100Variance of return 0.200 0.080 0.180
StDev of return 0.447 0.283 0.424
Correlations CovariancesStock 1 Stock 2 Stock 3 Stock 1 Stock 2 Stock 3
Stock 1 1.00 0.80 0.70 Stock 1 0.2000 0.1012 0.1328Stock 2 0.80 1.00 0.90 Stock 2 0.1012 0.0800 0.1080Stock 3 0.70 0.90 1.00 Stock 3 0.1328 0.1080 0.1800
Investment decisionStock 1 Stock 2 Stock 3 Total Required
Fractions to invest 0.333 0.667 0.000 1 = 1
Expected portfolio returnActual Required0.120 >= 0.120
Portfolio variance 0.103Portfolio stdev 0.321
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Optimal Solution
The least risky way to invest these three stocks while having an expected return of at least 12% is to invest 33.3% in Stock 1 and 66.7% in Stock 2.
This portfolio will have an expected return of 12% and a standard deviation of return of 32.1%.
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Parametric Approach, cont.
(c) Determine the minimum-variance portfolio that attains an expected annual return of at least 0.12, with shorting of stocks allowed.
All we need to do here is remove the nonnegativity constraint and re-run Solver.
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123456789
10111213
14
15161718192021
A B C D E F G H I JStock 1 Stock 2 Stock 3
Mean return 0.140 0.110 0.100Variance of return 0.200 0.080 0.180
StDev of return 0.447 0.283 0.424
Correlations CovariancesStock 1 Stock 2 Stock 3 Stock 1 Stock 2 Stock 3
Stock 1 1.00 0.80 0.70 Stock 1 0.2000 0.1012 0.1328Stock 2 0.80 1.00 0.90 Stock 2 0.1012 0.0800 0.1080Stock 3 0.70 0.90 1.00 Stock 3 0.1328 0.1080 0.1800
Investment decisionStock 1 Stock 2 Stock 3 Total Required
Fractions to invest 0.061 1.754 -0.816 1 = 1
Expected portfolio returnActual Required0.120 >= 0.120
Portfolio variance 0.066Portfolio stdev 0.257
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Optimal Solution
With shorting allowed, The least risky way to invest these three stocks while having an expected return of at least 12% is to sell Stock 3 short in an amount equivalent to 81.6% of the available funds, and invest 6.1% in Stock 1 and 175.4% in Stock 2.
This portfolio will still have an expected return of 12%, but a standard deviation of return of only 25.7% (as opposed to 32.1% with no shorting allowed).
36
• Invest in Vanguard mutual funds under university retirement plan
• No shorting
• Max 8 mutual funds
• Rebalance once per year
• Tools used: • Excel Solver
• Basic Stats (mean, stdev, correl, beta, crude version of CAPM)
Juran’s Lazy Portfolio
Decision Models -- Prof. Juran
37Decision Models -- Prof. Juran
1999 DJ S&P2000 7.8% -10.1%2001 3.9% -13.0%2002 -14.4% -23.4%2003 31.5% 26.4%2004 15.1% 9.0%2005 10.4% 3.0%2006 15.3% 13.6%2007 8.6% 3.5%2008 -41.5% -38.5%2009 45.0% 23.5%2010 17.8% 12.8%2011 -8.5% 0.0%2012 5.6% 13.4%2013 6.0% 29.6%2014 9.6% 8.4%
38Decision Models -- Prof. Juran
$-
$0.50
$1.00
$1.50
$2.00
$2.50
1999 2001 2003 2005 2007 2009 2011 2013
$1 Invested 12/31/1999
DJ
S&P
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Scenario ApproachKate Torelli, a security analyst for Lion Fund, has identified a gold mining stock (ticker symbol GMS) as a particularly attractive investment.
GMS is a highly leveraged company, so it is quite a risky investment by itself.
She would therefore like to hedge the stock purchase — that is, reduce the risk of an investment in GMS stock.
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Scenario Approach
Currently GMS is trading at $100 per share.
Torelli has constructed seven scenarios for the price of GMS stock one month from now.
Scen. 1 Scen. 2 Scen. 3 Scen. 4 Scen. 5 Scen. 6 Scen. 7 Probability 0.05 0.10 0.20 0.30 0.20 0.10 0.05 GMS stock price 150 130 110 100 90 80 70
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Hedging with Put OptionsTorelli called an options trader at a large investment bank for quotes. The prices for three (European-style) put options are shown. Torelli wishes to invest $10 million in GMS stock and put options.
Put Option A Put Option B Put Option C Strike price 90 100 110 Option price $2.20 $6.40 $12.50
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Return on InvestmentEach investment i has a percent return under each scenario j, which we will represent with the symbol rij. For example, under scenario 1 GMS stock has a 50% return, so if r1, 1 represents the return of GMS stock under scenario 1, then r1, 1 = 0.50.
For GMS stock, the return under any scenario j is given by:
0
011 S
SSr
jj
Where S0 is the initial price of the GMS stock, and S1 is the final price.
Decision Models -- Prof. Juran
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Modeling Put OptionsFor a put option i, the return under any scenario j is given by:
Where Ki is the strike price and Ci is the cost of the option.
i
ijiij C
CSKMAXr
0,1
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Scenario Approach, cont.Using these formulas, we can expand Table 1 to include the returns on each possible investment under each scenario.
Scen. 1 Scen. 2 Scen. 3 Scen. 4 Scen. 5 Scen. 6 Scen. 7 Probability 0.05 0.10 0.20 0.30 0.20 0.10 0.05 GMS stock price $150 $130 $110 $100 $90 $80 $70 Return on GMS stock (r1) 50% 30% 10% 0% -10% -20% -30% Return on Option A (r2) -100% -100% -100% -100% -100% 355% 809% Return on Option B (r3) -100% -100% -100% -100% 56% 213% 369% Return on Option C (r4) -100% -100% -100% -20% 60% 140% 220%
Note: We don’t know what the return would be under any scenario unless we know how much money was invested in each of the four instruments.
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Scenario Approach, cont.
The return on the portfolio is represented by the random variable R.
4
1iiixrR
The portfolio return under any scenario j is given by:
4
1iiijj xrR
(Note that we are using r as a percent and R as thousands of dollars.)
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Scenario Approach, cont.
Let Pj represent the probability of scenario j occurring. The expected value of R is given by:
7
1jjjR PR
The standard deviation of the portfolio’s return is given by:
7
1
2
jjRjR PR
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ExampleLet’s say Kate buys $7 million worth of GMS stock, and $1 million worth of each put option. This means that
(x1, x2, x3, x4) = (7000, 1000, 1000, 1000).
Under scenario 3, her return would be 3r
4
13
iii xr
443333223113 xrxrxrxr
100000.1100000.1100000.1700010.0
100010001000700
2300$
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Example, cont.Using the same procedure, it can be shown that for this particular allocation of assets, the seven scenarios would have returns as follows:
Scenario Return 1 500 2 -900 3 -2,300 4 -2,200 5 -538 6 5,670 7 11,878
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Example, cont.Therefore, the expected return on this particular allocation of assets is calculated as follows:
Rm å=
=7
1jjj PR
77665544332211 PRPRPRPRPRPRPR ++++++=
( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )05.0118781.056702.05383.022002.023001.090005.0500
( ) ( ) ( ) ( ) ( ) ( ) ( )5945671086604609025 ++-+-+-+-+=
132-=
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Example, cont.Finally, to calculate the standard deviation of the returns under this particular allocation of assets:
Rs ( )å=
-=7
1
2
jjRj PR m
( ) ( ) ( ) ( ) ( ) ( ) ( ) 72
762
652
542
432
322
212
1 PRPRPRPRPRPRPR RRRRRRR mmmmmmm -+-+-+-+-+-+-=
( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )05.0132118781.013256702.01325383.013222002.013223001.013290005.0132500 2222222 --+--+---+---+---+---+--=
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )05.0120101.058022.04063.020682.021681.076805.0632 2222222 ++-+-+-+-+=
937,211,7307,366,3962,32565,283,1449,940054,59942,19 ++++++=
216,914,12=
594,3=
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Example, cont.
So, in English, if Kate buys $7 million worth of GMS stock, and $1 million worth of each put option, then her return on investment in dollar terms is a random variable with an expected value of about -$132,000 and a standard deviation of about $3,594,000.
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GMS Case, cont.What is the expected return of GMS stock? What is the standard deviation of the return of GMS stock?
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16
A B C D E F GMS price 100
Scenarios for GMS stock in one month Scenario GMS price Probability Return SqDev
1 150 0.05 50% 0.2304 2 130 0.10 30% 0.0784 3 110 0.20 10% 0.0064 4 100 0.30 0% 0.0004 5 90 0.20 -10% 0.0144 6 80 0.10 -20% 0.0484 7 70 0.05 -30% 0.1024
Mean return 0.0200 200,000.00 $ Stdev of return 0.1833 1,833,030.28 $
=SUMPRODUCT(D5:D11,C5:C11)
=SQRT(SUMPRODUCT(E5:E11,C5:C11))
=B13*10000000
=B14*10000000
=(B5-$B$1)/ $B$1 =(D6-$B$13)^2
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GMS Case, cont.
Torelli suspects that a good strategy is to buy one put option A for each share of GMS stock purchased. What are the mean and standard deviation of return for this strategy?
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1234567891011121314151617181920212223
A B C D E F GGMS price 100
Scenarios for GMS stock in one monthScenario GMS price Probability Return Sqdev
1 150 0.05 47% 0.2025882 130 0.10 27% 0.0647213 110 0.20 8% 0.0034474 100 0.30 -2% 0.0015325 90 0.20 -12% 0.0187656 80 0.10 -12% 0.0187657 70 0.05 -12% 0.018765
Put options on GMS stock that expire in one monthOption A B CStrike price 90 100 110Option price $2.20 $6.40 $12.50
Portfolio with one unit of GMS stock and one put AMean 0.0176 176,125.24$ Stdev 0.1559 1,559,430.28$
Portfolio
=(B5-$B$1+IF(B5-$B$15>0,0,$B$15-B5)-$B$16)/ ($B$1+$B$16)
=(D5-$B$19)^2
=SUMPRODUCT(D5:D11,C5:C11)
=SQRT(SUMPRODUCT(E5:E11,C5:C11))
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GMS Case, cont.
Assuming that Torelli's goal is to minimize the standard deviation of the portfolio return, what is the optimal portfolio that invests all $10 million?
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Managerial FormulationDecision VariablesTorelli needs to invest $10 million in some combination of GMS stock and three types of put options.
Objective
Minimize risk (standard deviation of the portfolio’s return).
Constraints
All $10 million must be invested.
No shorting.
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FormulationDecision Variables
The decision variables are four amounts: x1, x2, x3, and x4, representing GMS stock, Put Option A, Put Option B, and Put Option C, respectively.
Objective
Minimize Z =
7
1
2
jjRjR PR
Constraints
000,104
1
i
ix
0ix for all investments i.
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Solution Methodology123456789
1011121314151617181920
21222324
A B C D E F G H I J K LGMS price 100
Scenarios for GMS stock in one monthScenario GMS price Probability GMS Put A Put B Put C Return Sqdev
1 150 0.05 50% -100% -100% -100% 500 3988352 130 0.10 30% -100% -100% -100% -900 5905403 110 0.20 10% -100% -100% -100% -2300 47022444 100 0.30 0% -100% -100% -20% -2200 42785515 90 0.20 -10% -100% 56% 60% -538 1648086 80 0.10 -20% 355% 213% 140% 5670 336630727 70 0.05 -30% 809% 369% 220% 11878 144238735
Put options on GMS stock that expire in one monthOption A B CStrike price 90 100 110Option price $2.20 $6.40 $12.50
Investment decision (thousands of dollars spent on each investment) Return from portfolio ($1000)GMS Put A Put B Put C Total Budget Mean -1327000 1000 1000 1000 10000 = 10000 Stdev 3594
Units of investments purchased (shares for GMS, number of puts for options)GMS Put A Put B Put C
70000 454545 156250 80000
Returns from one unit of each investment PortfolioReturns here are in thousands of dollars
=(B11-$B$1)/ $B$1 =(IF($B11>B$15,0,B$15-$B11)-B$16)/ B$16
=SUMPRODUCT($B$20:$E$20,D11:G11)
=(H11-$K$19)^2
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Optimal Solution1234567891011121314151617181920
21222324
A B C D E F G H I J K LGMS price 100
Scenarios for GMS stock in one monthScenario GMS price Probability GMS Put A Put B Put C Return Sqdev
1 150 0.05 50% -100% -100% -100% 2737 69037742 130 0.10 30% -100% -100% -100% 1039 8634853 110 0.20 10% -100% -100% -100% -660 5914004 100 0.30 0% -100% -100% -20% -302 1690985 90 0.20 -10% -100% 56% 60% 56 28526 80 0.10 -20% 355% 213% 140% 414 926637 70 0.05 -30% 809% 369% 220% 772 438530
Put options on GMS stock that expire in one monthOption A B CStrike price 90 100 110Option price $2.20 $6.40 $12.50
Investment decision (thousands of dollars spent on each investment) Return from portfolio ($1000)GMS Put A Put B Put C Total Budget Mean 1098491 0 0 1509 10000 = 10000 Stdev 795
Units of investments purchased (shares for GMS, number of puts for options)GMS Put A Put B Put C
84913 0 0 120694
Returns from one unit of each investment Portfolio
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Optimal SolutionKate should buy $8,491,000 worth of GMS stock, and $1,509,000 worth of Put Option C.
This portfolio will have an expected one-month return of $109,000 (1.09%) and a standard deviation of $795,000 (7.95%).
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GMS Case, cont.
Suppose that short selling is permitted -- that is, the nonnegativity restrictions on the portfolio weights are removed. Now what portfolio minimizes the standard deviation of return?
Here we simply remove the nonnegativity constraint (by unchecking the box in the Solver Options that says “assume nonnegative”).
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Optimal Solution
1234567891011121314151617181920
21222324
A B C D E F G H I J K LGMS price 100
Scenarios for GMS stock in one monthScenario GMS price Probability GMS Put A Put B Put C Return Sqdev
1 150 0.05 50% -100% -100% -100% 2446 52016742 130 0.10 30% -100% -100% -100% 786 3859783 110 0.20 10% -100% -100% -100% -873 10778104 100 0.30 0% -100% -100% -20% 198 10685 90 0.20 -10% -100% 56% 60% 230 42376 80 0.10 -20% 355% 213% 140% 225 35437 70 0.05 -30% 809% 369% 220% 219 2912
Put options on GMS stock that expire in one monthOption A B CStrike price 90 100 110Option price $2.20 $6.40 $12.50
Investment decision (thousands of dollars spent on each investment) Return from portfolio ($1000)GMS Put A Put B Put C Total Budget Mean 1658297 -8 -665 2376 10000 = 10000 Stdev 718
Units of investments purchased (shares for GMS, number of puts for options)GMS Put A Put B Put C
82972 -3798 -103843 190058
Returns from one unit of each investment Portfolio
Now the nonnegativity conditions for the changing cells is removed, and the investor sells short on the put A and B options. This lowers the standard deviation of the portfolio (and also increases its mean).
Decision Models -- Prof. Juran
64
Optimal Solution
Kate should buy $8,297,000 worth of stock and $2,376,000 worth of Put Option C, and she should short sell $8,000 worth of Put Option A and $665,000 worth of Put Option B.
This portfolio will have an expected monthly return of 1.65% and a standard deviation of 7.18%. In other words, it will be more profitable and less risky than the portfolio without shorting.
Decision Models -- Prof. Juran
65
SummaryPortfolio Optimization• Parametric Approach
– Array Functions• Scenario Approach
– Put Options– Shorting
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