1、For office use only T1 _ T2 _ T3 _ T4 _ Team Control Number 46639 Problem Chosen C For office use only F1 _ F2 _ F3 _ F4 _ 2016 MCM/ICM Summary Sheet An Optimal Investment Strategy Model Summary We develop an optimal investment strategy model that appears to hold promise for providing insight into n
2、ot only how to sort the schools according to investment priority, but also identify optimal investment amount of a specific school. This model considers a large number of parameters thought to be important to investment in the given College Scorecard Data Set. In order to develop the required model,
3、 two sub-models are constructed as follows: 1. For Analytic Hierarchy Process (AHP) Model, we identify the prioritized candidate list of schools by synthesizing the elements which have an influence on investment. First we define the specific value of any two elements effect on investment. And then t
4、he weight of each elements influence on investment can be identified. Ultimately, we take the relevant parameters into the calculated weight, and then we get any schools recommended value of investment. 2. For Return On Investment Model, its constructed on the basis of AHP Model. Let us suppose that
5、 all the investment is used to help the students to pay tuition fee.Then we can see optimal investment as that we help more students to the universities of higher return rate. However, because of dropout rate, there will be an optimization investment amount in each university. Therefore, we can chan
6、ge the problem into a nonlinear programming problem. We identify the optimal investment amount by maximizing return-on-investment. Specific attention is given to the stability and error analysis of our model. The influence of the model is discussed when several fundamental parameters vary. We attemp
7、t to use our model to prioritize the schools and identify investment amount of the candidate schools, and then an optimal investment strategy is generated. Ultimately, to demonstrate how our model works, we apply it to the given College Scorecard Data Set. For various situations, we propose an optim
8、al solution. And we also analyze the strengths and weaknesses of our model. We believe that we can make our model more precise if more information are provided. Team # 46639 Page 1 of 17 Contents 1. Introduction 2 1.1 Restatement of the Problem 2 1.2 Our Approach. . 2 2. Assumptions 2 3. Notations 3
9、 4. The Optimal Investment Model 4 4.1 Analytic Hierarchy Process Model 4 4.1.1 Constructing the Hierarchy4 4.1.2 Constructing the Judgement Matrix .5 4.1.3 Hierarchical Ranking.7 4.2 Return On Investment Model8 4.2.1 Overview of the investment strategy 8 4.2.2 Analysis of net income and investment
10、cost. 9 4.2.3 Calculate Return On Investment .11 4.2.4 Maximize the Total Net Income115. Test the Model 12 5.1 Error Analysis12 5.2 Stability Analysis.13 6. Results 13 6.1 Results of Analytic Hierarchy Process.13 6.2 Results of Return On Investment Model14 7. Strengths and Weaknesses 15 7.1 Strength
11、s15 7.2 Weaknesses16 References 16 Appendix A Letter to the Chief Financial Officer, Mr. Alpha Chiang. 17 Team # 46639 Page 2 of 17 1. Introduction 1.1 Restatement of the Problem In order to help improve educational performance of undergraduates attending colleges and universities in the US, the Goo
12、dgrant Foundation intends to donate a total of $100,000,000 to an appropriate group of schools per year, for five years, starting July 2016. We are to develop a model to determine an optimal investment strategy that identifies the school, the investment amount per school, the return on that investme
13、nt, and the time duration that the organizations money should be provided to have the highest likelihood of producing a strong positive effect on student performance. Considering that they dont want to duplicate the investments and focus of other large grant organizations, we interpret optimal inves
14、tment as a strategy that maximizes the ROI on the premise that we help more students attend better colleges. So the problems to be solved are as follows: 1. How to prioritize the schools by optimization level. 2. How to measure ROI of a school. 3. How to measure investment amount of a specific schoo
15、l. 1.2 Our Approach We offer a model of optimal investment which takes a great many factors in the College Scorecard Data Set into account. To begin with, we make a 1 to N optimized and prioritized candidate list of school we are recommending for investment by the AHP model. For the sake that we inv
16、est more students to better school, several factors are considered in the AHP model, such as SAT score, ACT score, etc. And then, we set investment amount of each university in the order of the list according to the standard of maximized ROI. The implement details of the model will be described in s
17、ection 4. 2. Assumptions We make the following basic assumptions in order to simplify the problem. And each of our assumptions is justified. 1. Investment amount is mainly used for tuition and fees. Considering that the income of an undergraduate is usually much higher than a high school students, w
18、e believe that its necessary to help more poor students have a chance to go to college. 2. Bank rates will not change during the investment period. The variation of the bank rates have a little influence on the income we consider. So we make this assumption just to simplify the model. 3. The employm
19、ent rates and dropout rates will not change, and they are different for different schoolsTeam # 46639 Page 3 of 17 4. For return on investment, we only consider monetary income, regardless of the intangible income. 3. Notations We use a list of symbols for simplification of expression.Symbol Definit
20、ion D a set of candidate schools N the number of candidate schools M total investment amount per year jm investment amount for school j jp the tuition and fees for school j jt time duration for school j jn the number of the invested for school j 0b wage base for high school graduates 0a wage base fo
21、r high school graduates r bank rates T the future proceeds of life u expected income of a graduate h expected income of a high school graduate jP net income for school j jQ total income for school j j dropout rates for school j j employment rates for school j jne the number of future employment for
22、school j lu lower limit of investment amount bu upper limit of investment amount Team # 46639 Page 4 of 17 jROI return on investment for school j Ranking vector i Previous weight of factors. i New weight of factors. 4. The Optimal Investment Model In this section, we first prioritize schools by the
23、AHP model (Section 4.1), and then calculate ROI value of the schools (Section 4.2). Ultimately, we identify investment amount of every candidate schools according to ROI (Section 4.3). 4.1 Analytic Hierarchy Process Model In order to prioritize schools, we must consider each necessary factor in the
24、College Scorecard Data Set. For each factor, we calculate its weight value. And then, we can identify the investment necessity of each school. So, the model can be developed in 3 steps as follows: 4.1.1 Constructing the Hierarchy We consider 19 elements to measure priority of candidate schools, whic
25、h can be seen in Fig 1. The hierarchy could be diagrammed as follows: Fig.1 AHP for the investment decisionTeam # 46639 Page 5 of 17 The goal is red, the criteria are green and the alternatives are blue. All the alternatives are shown below the lowest level of each criterion. Later in the process, e
26、ach alternatives will be rated with respect to the criterion directly above it. As they build their hierarchy, we should investigate the values or measurements of the different elements that make it up. If there are published fiscal policy, for example, or school policy, they should be gathered as p
27、art of the process. This information will be needed later, when the criteria and alternatives are evaluated. Note that the structure of the investment hierarchy might be different for other foundations. It would definitely be different for a foundation who doesnt care how much his score is, knows he
28、 will never dropout, and is intensely interested in math, history, and the numerous aspects of study1. 4.1.2 Constructing the Judgement Matrix Hierarchy reflects the relationship among elements to consider, but elements in the Criteria Layer dont always weigh equal during aim measure. In deciders mi
29、nd, each element accounts for a particular proportion. To incorporate their judgments about the various elements in the hierarchy, decision makers compare the elements “two by two”. The fundamental scale for pairwise comparison are shown in Fig 2. Fig 2 Right now, lets see which items are compared.
30、Our example will begin with the six criteria in the second row of the hierarchy in Fig 1, though we could begin elsewhere if we want. The criteria will be compared as to how important they are to the decision Team # 46639 Page 6 of 17 makers, with respect to the goal. Each pair of items in this row
31、will be compared. Fig 3 Investment Judgement Matrix In the next row, there is a group of 19 alternatives under the criterion. In the subgroup, each pair of alternatives will be compared regarding their importance with respect to the criterion. (As always, their importance is judged by the decision m
32、akers.) In the subgroup, there is only one pair of alternatives. They are compared as to how important they are with respect to the criterion. Things change a bit when we get to the alternatives row. Here, the factor in each group of alternatives are compared pair-by-pair with respect to the coverin
33、g criterion of the group, which is the node directly above them in the hierarchy. What we are doing here is evaluating the models under consideration with respect to score, then with respect to Income, then expenditure, dropout rate, debt and graduation rate. The foundation can evaluate alternatives
34、 against their covering criteria in any order they choose. In this case, they choose the order of decreasing priority of the covering criteria. Fig 4 Score Judgement Matrix Fig 5 Expenditure Judgement Matrix Fig 6 Income Judgement Matrix Team # 46639 Page 7 of 17 Fig 7 Dropout Judgement Matrix Fig 8
35、 Debt Judgement Matrix Fig 9 Graduation Matrix 4.1.3 Hierarchical Ranking When the pairwise comparisons are as numerous as those in our example, specialized AHP software can help in making them quickly and efficiently. We will assume that the foundation has access to such software, and that it allow
36、s the opinions of various foundations to be combined into an overall opinion for the group. The AHP software uses mathematical calculations to convert these judgments to priorities for each of the six criteria. The details of the calculations are beyond the scope of this article, but are readily ava
37、ilable elsewhere2345. The software also calculates a consistency ratio that expresses the internal consistency of the judgments that have been entered. In this case the judgments showed acceptable consistency, and the software used the foundations inputs to assign these new priorities to the criteri
38、a: Team # 46639 Page 8 of 17 Fig 10.AHP hierarchy for the foundation investing decision. In the end, the AHP software arranges and totals the global priorities for each of the alternatives. Their grand total is 1.000, which is identical to the priority of the goal. Each alternative has a global prio
39、rity corresponding to its “fit“ to all the foundations judgments about all those aspects of factor. Here is a summary of the global priorities of the alternatives: Fig 11 4.2 ROI Model 4.2.1 Overview of the investment strategy Consider a foundation making investment on a set of N geographically disp
40、ersed colleges and university in the United States, D = 1, 2, 3 N . Then we can select top N schools from the candidate list which has been sorted through analytic hierarchy process. The total investment amount is M per year which is donated by the Goodgrant Foundation. The investment amount is jm f
41、or each school jD , satisfying the following balance constraint: jjDmM (1) We cant invest too much or too little money to one school because we want to help more students go to college, and the student should have more choices. Then the investment amount for each school must have a lower limit lu an
42、d upper limitbu as follows: jlu m bu (2) The tuition and fees is jp , and the time duration is 1,2,3,4jt . To simplify our Team # 46639 Page 9 of 17 model, we assume that our investment amount is only used for freshmen every year. Because a freshmen oriented investment can get more benefits compared
43、 with others. For each school jD , the number of the undergraduate students who will be invested is jn , which can be calculated by the following formula： ,jj jjmn j Dpt(3) Figure12 The foundation can use the ROI model to identify jm and jt so that it can maximize the total net income. Figure1 has s
44、hown the overview of our investment model. We will then illustrate the principle and solution of this model by a kind of nonlinear programming method. 4.2.2 Analysis of net income and investment cost In our return on investment model, we first focus on analysis of net income and investment cost. Obv
45、iously, the future earnings of undergraduate students are not only due to the investment itself. There are many meaning factors such as the effort, the money from their parents, the training from their companies. In order to simplify the model, we assume that the investment cost is the most importan
46、t element and we dont consider other possible influence factors. Then we can conclude that the total cost of the investment is jm for each school jD .Team # 46639 Page 10 of 17 Figure 13 For a single student, the meaning of the investment benefits is the expected earnings in the future. Assuming tha
47、t the student is not going to college or university after graduating from high school and is directly going to work. Then his wage base is 0b as a high school graduate. If he works as a college graduate, then his wage base is 0a . Then we can give the future proceeds of life which is represented symbolically by T and we use r to represent the bank rates which will change over time. We assume that the bank rates will not change during the investment period. Here, we
