1. Session 5a
Decision Models Prof. Juran

2. Decision Models -- Prof. Juran
Overview
Portfolio Optimization
Parametric Approach
Array Functions
Scenario Approach
Put Options
Shorting
Decision Models Prof. Juran
Decision Models Prof. Juran

3. 4 Basic Matrix Operations
Sum Product
Transpose
Matrix Multiplication
Inverting Matrices
Decision Models Prof. Juran
Decision Models Prof. Juran

4. Decision Models -- Prof. Juran

5. Decision Models -- Prof. Juran
Sum Product
Decision Models Prof. Juran
Decision Models Prof. Juran

6. Decision Models -- Prof. Juran
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
Decision Models Prof. Juran

7. Decision Models -- Prof. Juran1 2 3 4 5 6 7 8 A B C D 11 9 10 12
208
=SUMPRODUCT(A1:C2,A4:C5)
Decision Models Prof. Juran
Decision Models Prof. Juran

8. Decision Models -- Prof. Juran
Transpose
For matrix A,
Decision Models Prof. Juran
Decision Models Prof. Juran

9. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

10. Decision Models -- Prof. Juran
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
Decision Models Prof. Juran
Decision Models Prof. Juran

11. Decision Models -- Prof. Juran
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).
Decision Models Prof. Juran
Decision Models Prof. Juran

12. Matrix Multiplication
For matrices A and B,
Decision Models Prof. Juran
Decision Models Prof. Juran

13. Decision Models -- Prof. Juran

14. Decision Models -- Prof. Juran

15. Decision Models -- Prof. Juran
Inverting Matrices
First, define a square matrix Ij as a matrix with j rows and j columns, completely filled with zeroes, except for ones on the diagonal:
This special matrix is called the identity matrix.
Decision Models Prof. Juran
Decision Models Prof. Juran

16. Decision Models -- Prof. Juran
Now, for a square matrix A with j rows and j columns, there may exist a matrix called A-inverse (symbolized ) such that:
Not all square matrices can be inverted, a fact that has implications for regression analysis.
Decision Models Prof. Juran
Decision Models Prof. Juran

17. Decision Models -- Prof. Juran

18. Decision Models -- Prof. Juran
Remember:
Select the entire range A5:B6 before typing the formula.
Press Shift+Ctrl+Enter.
Decision Models Prof. Juran
Decision Models Prof. Juran

19. Decision Models -- Prof. Juran
Parametric Approach
Decision Models Prof. Juran
Decision Models Prof. Juran

20. Decision Models -- Prof. Juran
Parametric Approach
Decision Models Prof. Juran
Decision Models Prof. Juran

21. Decision Models -- Prof. Juran
Parametric Approach
The means are: μ
= 0.14, μ
= 0.11, μ
= 0.10. 1 2 3
The variances are: 2 1 s
= 0.20, 2 s
= 0.08, 2 3 s
= 0.18.
The correlations are: ρ
= 0.8, ρ
= 0.7, and ρ
= 0.9. 12 13 23
Decision Models Prof. Juran
Decision Models Prof. Juran

22. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

23. Managerial Problem Definition
Decision 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.
Decision Models Prof. Juran
Decision Models Prof. Juran

24. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

25. Decision Models -- Prof. Juran
Formulation
Decision Models Prof. Juran
Decision Models Prof. Juran

26. Decision Models -- Prof. Juran
Formulation
Decision Models Prof. Juran
Decision Models Prof. Juran

27. Decision Models -- Prof. Juran
Solution Methodology
Decision Models Prof. Juran
Decision Models Prof. Juran

28. Decision Models -- Prof. Juran
Solution Methodology
The 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.
Decision Models Prof. Juran
Decision Models Prof. Juran

29. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

30. Decision Models -- Prof. Juran

31. Decision Models -- Prof. Juran
Optimal Solution
Decision Models Prof. Juran
Decision Models Prof. Juran

32. Decision Models -- Prof. Juran
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%.
Decision Models Prof. Juran
Decision Models Prof. Juran

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.
Decision Models Prof. Juran
Decision Models Prof. Juran

34. Decision Models -- Prof. Juran

35. Decision Models -- Prof. Juran
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).
Decision Models Prof. Juran
Decision Models Prof. Juran

36. Juran’s Lazy Portfolio
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)
Decision Models Prof. Juran 36

37. Decision Models -- Prof. Juran37

38. Decision Models -- Prof. Juran38

39. Decision Models -- Prof. Juran
Scenario Approach
Kate 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.
Decision Models Prof. Juran
Decision Models Prof. Juran

40. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

41. Hedging with Put Options
Torelli 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.
Decision Models Prof. Juran
Decision Models Prof. Juran

42. Decision Models -- Prof. Juran
Return on Investment
Decision Models Prof. Juran
Decision Models Prof. Juran

43. Decision Models -- Prof. Juran
Modeling Put Options
For 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.
Decision Models Prof. Juran
Decision Models Prof. Juran

44. Scenario Approach, cont.
Decision Models Prof. Juran
Decision Models Prof. Juran

45. Scenario Approach, cont.
Decision Models Prof. Juran
Decision Models Prof. Juran

46. Scenario Approach, cont.
Decision Models Prof. Juran
Decision Models Prof. Juran

47. Decision Models -- Prof. Juran
Example
Let’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
Decision Models Prof. Juran
Decision Models Prof. Juran

48. Decision Models -- Prof. Juran
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:
Decision Models Prof. Juran
Decision Models Prof. Juran

49. Decision Models -- Prof. Juran
Example, cont.
Therefore, the expected return on this particular allocation of assets is calculated as follows: å = 7 1 j P R R m 7 6 5 4 3 2 1 P R + = ( ) 05 .
11878 1
5670 2
538 3
2200
2300
900
500 + - = ( )
594
567
108
660
460 90 25 + - =
132 - =
Decision Models Prof. Juran
Decision Models Prof. Juran

50. Decision Models -- Prof. Juran
Example, cont.
Finally, to calculate the standard deviation of the returns under this particular allocation of assets: ( ) å = - 7 1 2 j R P m R s ( ) 7 2 6 5 4 3 1 P R m - + = ( ) 05 .
132
11878 1
5670 2
538 3
2200
2300
900
500 - + = ( ) 05 .
12010 1
5802 2
406 3
2068
2168
768
632 + - =
937 ,
211 7
307
366 3
962 32
565
283 1
449
940
054 59
942 19 + =
216 ,
914 12 =
594 , 3 =
Decision Models Prof. Juran
Decision Models Prof. Juran

51. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

52. Decision Models -- Prof. Juran
GMS Case, cont.
What is the expected return of GMS stock? What is the standard deviation of the return of GMS stock?
Decision Models Prof. Juran
Decision Models Prof. Juran

53. Decision Models -- Prof. Juran
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?
Decision Models Prof. Juran
Decision Models Prof. Juran

54. Decision Models -- Prof. Juran

55. Decision Models -- Prof. Juran
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?
Decision Models Prof. Juran
Decision Models Prof. Juran

56. Managerial Formulation
Decision Variables
Torelli 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.
Decision Models Prof. Juran
Decision Models Prof. Juran

57. Decision Models -- Prof. Juran
Formulation
Decision Models Prof. Juran
Decision Models Prof. Juran

58. Decision Models -- Prof. Juran
Solution Methodology
Decision Models Prof. Juran
Decision Models Prof. Juran

59. Decision Models -- Prof. Juran

60. Decision Models -- Prof. Juran
Optimal Solution
Decision Models Prof. Juran
Decision Models Prof. Juran

61. Decision Models -- Prof. Juran
Optimal Solution
Kate 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%).
Decision Models Prof. Juran
Decision Models Prof. Juran

62. Decision Models -- Prof. Juran
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”).
Decision Models Prof. Juran
Decision Models Prof. Juran

63. Decision Models -- Prof. Juran
Optimal Solution
Decision Models Prof. Juran
Decision Models Prof. Juran

64. Decision Models -- Prof. Juran
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
Decision Models Prof. Juran

65. Decision Models -- Prof. Juran
Summary
Portfolio Optimization
Parametric Approach
Array Functions
Scenario Approach
Put Options
Shorting
Decision Models Prof. Juran
Decision Models Prof. Juran