session 5a. decision models -- prof. juran2 overview portfolio optimization parametric approach...

Session 5a

Decision Models -- Prof. Juran

2

OverviewPortfolio Optimization• Parametric Approach

– Array Functions• Scenario Approach

– Put Options– Shorting


3

4 Basic Matrix Operations

• Sum Product• Transpose• Matrix Multiplication• Inverting Matrices


4

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.


5

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


6

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).


7

12345678

A B C D7 4 65 2 11

9 3 810 12 1

208=SUMPRODUCT(A1:C2,A4:C5)


8

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


9

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


10

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


11

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)


12

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


13

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


14

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.


15

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.


16

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.


17

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


18

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.


19

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.


20

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.


21

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.


22

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.


23

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.


24

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.


25

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



26

Formulation

Objective

Minimize Z = 3,1313,2322,12123

23

22

22

21


Constraints

12.0332211 xxx (1)

0.13

1

i

ix (2)

For all i, 0ix (3)


27

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)))}


28

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.


29

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.


30


31

Optimal Solution

123456789

10111213

14

15161718192021



StDev of return 0.447 0.283 0.424




Fractions to invest 0.333 0.667 0.000 1 = 1




32

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%.


33

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.


34

123456789

10111213

14

15161718192021



StDev of return 0.447 0.283 0.424




Fractions to invest 0.061 1.754 -0.816 1 = 1




35

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


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


39

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.


40

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


41

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


42

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.


43

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


44

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.


45

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.)


46

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


47

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$


48

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


49

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-=


50

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=


51

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.


52

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


53

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?


54

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))


55

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?


56

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.


57

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.


58

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


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


59


60

Optimal Solution1234567891011121314151617181920

21222324



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


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


84913 0 0 120694

Returns from one unit of each investment Portfolio


61

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%).


62

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”).


63

Optimal Solution

1234567891011121314151617181920

21222324



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


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


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).


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.


65

SummaryPortfolio Optimization• Parametric Approach

– Array Functions• Scenario Approach

– Put Options– Shorting

session 5a. decision models -- prof. juran2 overview portfolio optimization parametric approach...

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