MA 120: Mathematical Modeling for Personal Finance
Course Description
MA120 is an introduction to the applications of mathematics for personal financial decision-making. The course will introduce and apply quantitative techniques involving systems of equations, exponential functions and logarithms, probability, descriptive and inferential statistics, and numerical simulation. Applications will include life cycle decisions about spending and saving; borrowing and repayment; inflation and purchasing power; taxation and government benefits; risk management, insurance and annuities; investments and asset allocation.
Economic Problems and Mathematical Models
The Economic Life-Cycle Model
The economic life-cycle model is the framework for personal financial decision-making, and a half-dozen economists have won the Nobel Prize for work related to the life-cycle (LC) model. The LC model provides a prescription for maximizing your happiness over time, and across good times and bad times. The shorthand for this framework is consumption smoothing, i.e., to spread your economic resources smoothly over your lifetime.
After a brief introduction to the intuition behind the economic LC model, we will begin implementing the LC model as a system of (dozens of) interrelated equations. In this regard, we will also introduce the user of spreadsheets as a way to implement a numerical model in which we can modify inputs and observe the changes in the outputs.
Exponential Growth and Decay
Students will develop an appreciation for the idea of exponential growth (and decay) in terms of the time value of money. In particular, we will use numerical and graphical techniques to illustrate exponential growth of an interest-earning investment, and the exponential decay of an amortizing loan being repaid. We will conclude this section by revising our implementation of the LC model to include the time value of money, and the use of amortization to smooth consumption over time.
Step Functions and Solving Simultaneous Equations
While addressing an individual's relationship with government, we can use quantitative techniques to model marginal taxation as a step function, and implement decision-logic in our spreadsheet programs to model the effect of taxation on labor and asset income. Moreover, the structure of the US tax code creates a circular problem where in decisions about future consumption depend on decisions about current consumption and vice-versa.
Computational Complexity
In the real world, borrowing constraints (i.e., the inability to smooth consumption over time without borrowing) creates an optimization problem whose solution (finding the smoothest level of consumption without borrowing) requires too many possible permutations to solve using traditional computational techniques. We explain the idea of computational complexity and analyzing the running time characteristics of algorithms, to explain why this problem cannot be solved using traditional techniques. Finally, we introduce and use a technique called dynamic programming to solve this problem in a reasonable amount of time.
Probability
Randomness can have an impact on your economic happiness (or unhappiness). Risk management techniques and insurance contracts can provide protection against randomly occurring peril. We discuss probability to help measure the likelihood of such events occurring, and expected values can help us to ascertain the financial magnitude of bad events.
We know that we all must die eventually, but not knowing when that will happen makes personal financial planning quite challenging: should you plan for your expected life-span (e.g., to age 82) or to your maximum possible age (e.g., to age 100)? We will use techniques from actuarial science to estimate expected longevity, and to incorporate this into the LC model. Using these techniques, we can evaluate insurance and annuity products to determine whether and when these should be incorporated into your financial plan.
Descriptive Statistics
Historically, investments in the stock market have provided investors with a high rate of return that has outpaced inflation, albeit with a great number of ups and downs over time. We introduce descriptive statistics in the context of analyzing the historical rate of return on investments, and quantify the degree of random noise (i.e., ups and downs) around the investment signal (i.e., the historical rate of return).
Numerical Simulation
A long-term investor must think about his or her investment as a series of repeated trials, wherein the wealth you have in one year depends on the investment return earned in the previous year. Monte Carlo simulation is a computational technique in which we can run many random trials to simulate possible real outcomes. Using Monte Carlo simulation, we evaluate how investments in the stock market are likely to work out in the long run. Further, we can test the conventional wisdom about how to allocate your investments -- with surprising results the investment management industry doesn't want its customers to know!