Linear Programming Rediscovered
A financial plan is a full analysis of an individual's financial objectives comprising the risk profile, assets, savings and goals (like retirement, buying a house etc.) to devise a detailed investment strategy. At its core it is an asset liability matching problem. On one side you have the current investments and future savings and on the other side are the liabilities or goals that need to be achieved for a comfortable life. A financial plan takes into account all these and the market conditions to develop an optimal investment strategy. Complexities also come in because the character of an investment can differ a lot with the asset allocation strategy and the investment horizon. If done manually there are infinite ways to assign an investment surplus in part to meeting a goal.
A financial plan is a complex mathematical problem. The aim is to minimize the use of existing and future resources (investments) by allocating them in a way that they meet the goals with highest probability. If one frames the problem for any investor, it translates in a system of 500 plus equations, inequalities and constraints. Working on the problem, we realized it a perfect stage for one to use linear programming to solve it. With Matlab as a programing tool we could solve this problem and get the results. The program enabled us to get the investment strategy within a few minutes as compared to hours it was taking us earlier. Linear Programming is a fantastic concept to solve complex problems.
It was then I decided to dig deeper, only to find that linear equations/inequalities was actually a part of high school curriculum and I must have read multiple times during my college education. It was a rediscovery for me.
Linear Programming was put to work in World War II to plan expenditures, reduce costs of the army and to increase losses for the enemy. It was put to actual use in solving industrial problems, in operational research and in computer science with the Simplex algorithm developed by the mathematician George Dantzig. Dantzig is also known as the father of linear programming. He is the scientist who, as the legend goes, mistook an unsolvable statistical problem, written on the class blackboard as a homework assignment and solved it. Linear Programming algorithms also have an Indian angle, with the contribution of Narendra Karmarker in developing the Interior Point method. This was further refined by another Indian mathematician Prof. Sanjay Mehrotra which also made solving the system faster. I used his method to solve the financial planning problem. There are multiple other algorithms available to choose from depending on the character of the problem.
Today Linear Programming is used to solve and optimize large scale problems on supply management, transportation management, portfolio allocation and much more. It is unnoticed and embedded in some of our day to day interactions like courier delivery, traffic control and the investments we make.
Linear Programming is more than just a mathematical construct to solve. It lays a lot of emphasis on framing the problem first. In the center is the objective function that has to be maximized or minimised and around it are the various constraints and boundary conditions. The equalities and non-equalities define dependencies and interplay of each of the variables. The bounds ensure we put threshold boundary conditions. The algorithms look for an optimal solution that satisfies all these conditions.
There is a lot to learn from LP, as a framework, by decision makers in the corporate world. Defining the objective function to any problem is most critical. In today’s demanding corporate world, there can be multiple variables/functions to maximize or minimize i.e. the revenue, return to shareholder, compliance, return on capital, customer satisfaction, market share or even cost. All of them are important objectives and can be framed as an objective function as well as a constraint or a boundary condition. Given that each one of them is a noble objective, one can easily confuse an objective function with a constraint. Not setting the problem-setup right and choosing a wrong objective function can be disastrous. For one you may be solving a wrong problem and it can lead to wrong resource allocation.
The other important learning is in the solution itself. Linear programing aims at an optimal solution and not a perfect solution. Like most problems we aim to solve, a perfect solution may not get discovered in a reasonable period of time. That leaves the optimal solution to bank on. And while we may use our own algorithm to solve the system, linear programming is a beautiful framework to approach any problem with, especially for a financial planner.
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If you are actually using LP for drawing financial plans for customers , then your organization must be ahead of the curve . This requires in-depth understanding of customer needs and goals and that itself is missing most often than not ( defining the problem).