Money

Summary: Money has always been one of the subjects of applied mathematics, from interest to currency exchange.

When the earliest people wanted to acquire goods they could not make, grow, or hunt themselves, they exchanged other goods for them. Later, civilizations began to use smaller and more portable objects to represent value: shells, beads, pieces of leather, or shapes made from metal such as iron, among other things. Precious metals and printed paper currency supplanted most of these forms of money, and in the twenty-first century, intangible “digital cash” is exchanged electronically for goods and services.

Money is also a representation of wealth or value and is a basis for measuring economic and financial activity. Whether it is balancing a checkbook, analyzing a complex financial derivative, or anything in between, the mathematics of money is an indispensable tool for understanding and evaluating economic or financial transactions. Money is also multidimensional: value or wealth must be specified not only with respect to its amount, but also according to its time frame and to its country or currency framework. Translation of money and monetary transactions across these dimensions involves mathematical processes and an understanding of financial context, and mathematicians are actively involved in virtually all aspects of its production, management, and study. The first director of the U.S. Mint was David Rittenhouse, a well-known mathematician, inventor, astronomer, and surveyor. Mathematician Marc Fusaro is a research assistant in the Research and Statistics division of the Federal Reserve System. Regarding his career, he has said:

The mathematics in economics, where it is not explicit, is implicit. It underlies the economics everywhere. I can not always identify when I am doing mathematics. However, the thought processes learned in doing mathematics are crucial to economics and help at every step.

Time Value of Money

Money may differ or change in value across one or more dimensions, including, in particular, time. A dollar today is generally not worth the same as a dollar one year from now. The familiar effect of inflation over time is to decrease the value of a unit of money—a dollar bill will typically buy less one year from now than it will buy today. Also, a dollar today can often be invested so that it grows to a greater value a year from now. Another way of looking at this is to ask the question, “How much needs to be invested now, so that an investment account will be worth one dollar one year from now?” If the investment environment involves, as it generally does, positive interest rates or rates of return, the answer would be that an amount less than one dollar would need to be invested now in order to grow to a full dollar one year from now.

The Babylonians appear to have used interest on loans to model time doubling. Clay tablets dating back to about 2000 b.c.e. contain the following example: “given an interest rate of 1/60 per month (no compounding), compute the doubling time.” This situation corresponds to annual interest rate of 12/60=20%. The money would double in five years, which is 100% (growth) divided by 20% (growth per year). Some also cite the Babylonians as the first civilization to use formal banking.

Interest and Interest Rates: The Cost of Money

One of the key issues associated with money is interest, which can be viewed as the cost associated with using money. Interest can be looked at from either side of a financial transaction. An individual earns interest on a savings or money market account or by lending money to someone else; these are examples of asset positions. On the other hand, when someone takes out a loan or otherwise borrows money, that person pays interest to the lender; this is an example of a liability or debt position. So, regardless of the side of the financial transaction, the interest involved in the transaction is the cost, or reward, associated with the use or employment of money.

Interest is the dollar amount of the cost or reward associated with a monetary position. However, it is not really the dollar amount but rather the amount of interest as a proportion of the base or principal money amount that more clearly indicates the cost or reward associated with the transaction. This proportion is expressed as an “interest rate,” and can be represented, for example, by i.

The value of money can change over time. However, time is just one of the dimensions over which the value and cost of money can change. For example, consider the following questions regarding interest rates:

• • A lender is considering loaning money to someone for one year. What interest rate might the lender charge for loaning $1? For loaning $100? For loaning $100,000?
• • A lender is considering loaning $10,000 to someone. What interest rate might the lender charge for a loan of term one month? For a term of one year? For a term of five years?
• • A lender is considering loaning $10,000 for one year to one of three different people. What interest rate might the lender charge to the person who is perceived as the least risky (the one most likely to pay back the loan completely and in a timely fashion)? To the person of middle risk? To the person perceived as most risky?

Precise answers to these questions are not necessary to imagine that, within each set of questions, the answers may potentially be very different. For example, one may require a higher interest when lending a greater quantity of money; one may charge a higher interest rate when lending over a longer term; and one may insist on a higher interest rate when the borrower represents a greater risk. Thus, there are numerous dimensions and contexts in which the cost of money and its use can differ.

Money and Investments

Examining further the phenomenon of the time value of money, it is worth exploring more deeply how and why money can have a different value at one time compared with another. Consider a typical investment situation, which can be characterized as having four parameters: (1) the amount of money initially invested; (2) the interest rate, or the rate of return, which will be earned on the money invested; (3) the period of time over which the money will be invested; and (4) the future, or accumulated, value of the money at the end of the investment period.

As an example, suppose one invests $100 for one year at an effective annual interest rate of 10%. The future value (one year after the initial investment) is then calculated as $100+($100×0.10)=$100×0.10=$110.

This example could also be done in reverse. One could ask what amount, invested now, would yield $110 one year from now, if money can be invested at an effective annual interest rate of 10%. A minor algebraic adjustment to the prior solution yields the answer:

The result of $100 can be referred to as the “present value” (PV) of $110 one year from now—it is the amount obtained when the future value is “discounted back” one year.

The concept of “present value” is one of the most important in all of finance and economics. The present-day equivalent of any set of future cash flows can be determined by “discounting back” each individual future cash flow and summing all of the discounted cash flows together. This discounted sum is the present value of the future cash flows, and—assuming that the interest rate used for discounting is correct—it is essentially the amount of money that, invested now, would replicate those future cash flows. In that sense, a person could be described as being “indifferent” between receiving the future cash flows or receiving an amount now that is equal to the sum of the present values of those future cash flows.

Mathematically, present value can be determined as

where CF_t is the cash flow that will occur t periods from now, and it is the annual effective interest rate appropriate for an investment of t periods.

The above reference to it (an interest rate appropriate for an investment of t periods) suggests that cash flows over different time periods, or with different characteristics, might be associated with different levels of interest rates. Indeed, this is true, and in fact the cost of using money can be different in accordance with the period of time over which an investment is made. Typically, annual interest rates associated with relatively longer term investments are relatively larger than those associated with shorter term investments. This relationship is described formally by the yield curve, or the term structure of interest rates.

Similarly, one of the most critical factors in determining an appropriate interest rate is the level of riskiness inherent in the investment process (or uncertainty in the amount and timing of the future cash flows, if discounting is being performed for present value purposes). In general, riskier cash flows or investment opportunities are associated with higher interest rates. This association is a manifestation of the risk-return relationship, which suggests that taking on greater risk should be compensated by a relatively greater reward.

Bibliography

Biehler, Timothy. The Mathematics of Money: Math for Business and Personal Finance Decisions. New York: McGraw-Hill, 2007.

Broverman, Samuel A. Mathematics of Investment and Credit. Winsted, CT: ACTEX Publications, 2008.

Marquez, Elizabeth, and Paul Westbrook. Teaching Money Applications to Make Mathematics Meaningful. Thousand Oaks, CA: Corwin Press, 2007.