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Use the scenario approach to determine the minimum-risk portfolio of stocks that yields an expected return of at least 22%, without shorting.
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Overview • Multiple Objective Optimization • Two Dimensions • More than Two Dimensions • Finance and HR Examples • Efficient Frontier • Pre-emptive Goal Programming • Intro to Decision Analysis Decision Models -- Prof. Juran
Scenario Approach Revisited Use the scenario approach to determine the minimum-risk portfolio of these stocks that yields an expected return of at least 22%, without shorting. Decision Models -- Prof. Juran
Using the same notation as in the GMS case, the percent return on the portfolio is represented by the random variable R. In this model, xi is the proportion of the portfolio (i.e. a number between zero and one) allocated to investment i. (In the GMS case, we used thousands of dollars as the units.) Each investment i has a percent return under each scenario j, which we represent with the symbol rij. Decision Models -- Prof. Juran
The portfolio return under any scenario j is given by: Decision Models -- Prof. Juran
Let Pj represent the probability of scenario j occurring. The expected value of R is given by: The standard deviation of R is given by: Decision Models -- Prof. Juran
In this model, each scenario is considered to have an equal probability of occurring, so we can simplify the two expressions: Decision Models -- Prof. Juran
Managerial Formulation Decision Variables We need to determine the proportion of our portfolio to invest in each of the five stocks. Objective Minimize risk. Constraints All of the money must be invested. (1) The expected return must be at least 22%. (2) No shorting. (3) Decision Models -- Prof. Juran
Mathematical Formulation Decision Variables x1, x2, x3, x4, and x5 (corresponding to Ford, Lilly, Kellogg, Merck, and HP). Objective Minimize Z = Constraints (1) (2) For all i, xi ≥ 0 (3) Decision Models -- Prof. Juran
The decision variables are in F2:J2. The objective function is in C3. Cell E2 keeps track of constraint (1). Cells C2 and C5 keep track of constraint (2). Constraint (3) can be handled by checking the “assume non-negative” box in the Solver Options. Decision Models -- Prof. Juran
Conclusions Invest 17.3% in Ford, 42.6% in Lilly, 5.4% in Kellogg, 10.5% in Merck, and 24.1% in HP. The expected return will be 22%, and the standard deviation will be 12.8%. Decision Models -- Prof. Juran
2. Show how the optimal portfolio changes as the required return varies. Decision Models -- Prof. Juran
3. Draw the efficient frontier for portfolios composed of these five stocks. Decision Models -- Prof. Juran
Repeat Part 2 with shorting allowed. Decision Models -- Prof. Juran
GMS Case Revisited 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
Formulation Decision Models -- Prof. Juran
Optimal Solution Decision Models -- Prof. Juran
Efficient Frontier for GMS Decision Models -- Prof. Juran
Parametric Approach Revisited From Session 5a: (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 no shorting of stocks allowed. Decision Models -- Prof. Juran
Formulation Decision Models -- Prof. Juran
Optimal Solution Decision Models -- Prof. Juran
SolverTable Decision Models -- Prof. Juran
SolverTable Output Decision Models -- Prof. Juran
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 non-negativity constraint and re-run SolverTable. Decision Models -- Prof. Juran
Preemptive Goal Programming:Consulting Example The Touche Young accounting firm must complete three jobs during the next month. Job 1 will require 500 hours of work, job 2 will require 300 hours, and job 3 will require 100 hours. At present the firm consists of five partners, five senior employees, and five junior employees, each of whom can work up to 40 hours per month. Decision Models -- Prof. Juran
The dollar amount (per hour) that the company can bill depends on the type of accountant assigned to each job, as shown in the table below. (The "X" indicates that a junior employee does not have enough experience to work on job 1.) Decision Models -- Prof. Juran
All jobs must be completed. • Touche Young has also set the following goals, listed in order of priority: • Goal 1: Monthly billings should exceed $74,000. • Goal 2: At most one partner should be hired. • Goal 3: At most three senior employees should be hired. • Goal 4: At most one junior employee should be hired. Decision Models -- Prof. Juran
Managerial Formulation Decision Variables There are three types of decisions here. First, we need to decide how many people to hire in each of the three employment categories. Second, we need to assign the available human resources (which depend on the first set of decisions) to the three jobs. Finally, since it is not apparent that we will be able to satisfy all of Touche Young’s goals, we need to decide which goals not to meet and by how much. Decision Models -- Prof. Juran
Objective In the long run we want to minimize any negative difference between actual results and each of the four goals. Of course, our optimization methods require that we only have one objective at a time, so we will use a variation of goal programming to solve the problem four times. The approach here will be to treat each of the goals as an objective until it is shown to be attainable, after which we will treat it as a constraint. For example, we will solve the model with the goal of minimizing any shortfall in the $74,000 revenue target. Once we find a solution that has no shortfall, we will solve the problem again, with an added constraint that the shortfall be zero. Decision Models -- Prof. Juran
Constraints The numbers of people hired must be integers. (1) Each project must receive its required number of man-hours. (2) Our model must take into account any difference between the actual performance of the plan and the four targets. (3) We can’t assign people to jobs unless we hire them. (4) Decision Models -- Prof. Juran
Mathematical Formulation Decision Models -- Prof. Juran
Decision Variables xi (three decisions), Aij (nine decisions), δk (up to three decisions) Objective Minimize Z = Constraints All xi are integers. (1) Aij = Rij for all i, j. (2) δ Goals ≠ k = vk for all goals < k (3) for all i. (4) Decision Models -- Prof. Juran
The tk targets are in cells G10:G13. The δk “negative difference” variables will be in B16:B19. We use the range C10:D13 to track all deviations (both positive and negative), and then refer to the “undesirable” one in B16:B19. For example, it is undesirable to have billings under 74,000, so B16 refers to C10. It is undesirable for the number of new partners to be over 1, so B17 refers to D11. The vk “best achieved” variables will be in D16:D19. The xi are in K7:K9. The Aij assignments are in B2:D4. The Rij requirements are in B7:D7. Cells G2:G4 keep track of constraint (4). We constrain B4 to be zero. Decision Models -- Prof. Juran
d 1 3 å £ 40 A x ij i = 1 j First Iteration At first we’ll ignore all of the goals except the billing target of $74,000. Decision Variables (three decisions, cells K7:K9), (nine decisions, cells B2:D4) x A i ij Objective Mini mize Z = (the shortfall, if any, between planned billings and $74,000) Constraints All are integers. (1) x i = for all , . (2) A R i j ij ij = 0. (4) A 11 for all . (5) i Note that constraint (3) doesn’t matter in t his iteration. Also note the balance equation constraint, forcing E10 = G10. Decision Models -- Prof. Juran
We have verified that it is feasible to have billings of $74,000. Decision Models -- Prof. Juran
