My students, like most people, like money and find the topic of compound interest interesting. After completing a unit on simple, compound and continuous compound interest, one of my students told me that math is useful and interesting after all.
This post will discuss the derivation of the formula for the future value of an investment when interest is compounded continuously, FV = Pert. No prior understanding of the limit concept in calculus is required. I will be using the limit concept, but I will give an informal intuitive explanation of the limit concept as it comes up in the discussion. A recent post discussed an approach for deriving an equation that models exponential growth/decay. Problem (2) in that post showed the derivation of the compound interest formula FV = P(1 + r/k)kt where FV = the future value of the investment account, P = principle or one time lump-sum investment, r = annual percent rate of return expressed as a decimal, k = the number of times per year interest is compounded, and time t = the number of years the principal is invested.
Before I can get to the derivation of the equation FV = Pert, I need to explain what continuous compound interest means. Let’s consider an investment where P = $10,000, average annual rate of return = 7% = 0.07, and the investment collects interest over a period of 20 years. I adopted the standard banking convention rule that 1 year = 360 days. (Whether we use 365 or 360 days in a year makes no significant difference. Apparently banks like 30-day months.) The text box below shows how increasing the number of times per year interest is compounded affects the future value of an investment.
Students immediately notice that there is a point where it makes no difference how often interest is compounded, and they completely understand the difference between simple interest and compound interest. I tell them that the future value of the $10,000 investment, $40,552.00 in this example, represents the upper limit of one’s greed. When interest is compounded more times per year (k approaches infinity), and interest is compounded over smaller and smaller time intervals; say every second, every microsecond, or continuously. No matter what the principal is or the annual interest rate, there is always an upper limit of the future value of an investment, and the upper limit is reached when interest is compounded continuously.
In 1683 in the course of his study of continuous compound interest, Jacob Bernoulli (1654-1705) wanted to find the number that was the limiting value of the expression (1+1/n)^n as n approaches infinity. This is the first time that a number is defined as the limiting value of an expression. Bernoulli determined that this special number is bounded and lies between 2 and 3. In 1748 Leonard Euler (pronounced Oil-er) (1707-1783) published a document in which he named this special number e. He showed that e is the limiting value of the expression (1 + 1/n)n as n approaches infinity, and is approximately equal to 2.718281828459045235. He also gave another definition of e as the limiting value of the infinite sum 1 + 1/1! + 1/2! + 1/3! + . . . . Euler is generally given credit as the first to prove e is an irrational number.
To help you better understand the definition of the irrational number e, I will start by comparing the graphs of functions of the form y = (1 + 1/k)x where k is a fixed constant and the graph of the function y = (1 + 1/x)x. Refer to graphs (A) and (B) and the companion text box below. A quantity that approaches infinity means the quantity gets bigger and bigger without any upper boundary. A quantity that approaches a fixed constant means the quantity gets infinitely close to the fixed constant.
The purpose of the above graphs and the comments in the text box is to demonstrate that a subtle difference in the expressions (1 + 1/k)x and (1 + 1/x)x results in far different limiting values as x approaches ∞. The key result needed in the derivation of the continuous compound interest formula is the fact that e = limiting value of (1 + 1/x)x as x approaches ∞ when x is any positive real number. Considering that the expression (1 + 1/n)n is a rational number for every positive integer n, it is astonishing that the expression (1 + 1/n)n approaches an irrational number as n approaches ∞. I can now show you the derivation of the continuous compound interest formula FV = Pert.
Comments:
• When I did the calculations for compounding every minute and compounding every second with my graphing calculator, I got results that were slightly different than the expected results. When I used double floating point precision real numbers in a computer program, program output agreed with the expected results. We need to constantly remind ourselves that calculator or computer calculations of expressions that involve very large numbers, or require a large number of iterations to arrive at a solution, results may be slightly different than the expected or theoretical value.
• Using problems similar to the examples in this post, I show my students how compound interest works and what continuous compounding of interest means. I have them enter the expressions into their graphing calculator as the lesson progresses. This gives them practice using their calculator and they gain a better understanding and appreciation of what compound interest is all about. They are astonished when I show them $10,000*e.07*20 = $40,552.00.
• For a class of curious or advanced students, it’s not wasted class time to show the derivation of the continuous compound interest formula. Less advanced students are usually content with learning how to use the formula. My handout, Basic Financial Formulas, provides an overview of useful financial formulas that you can use in your classroom.
• The derivation of the continuous compound interest formula is a great opportunity to expose advanced high school algebra, college algebra and pre-calculus students to the limit concept in calculus.
• As mentioned earlier, very term of the sequence an = (1 + 1/n)n is a rational number, but the sequence itself converges to the irrational number e. Most calculus students find this very counterintuitive. What a great opportunity to launch a discussion of any number of related math concepts!
• The constants 0, 1, π, e, and i where i2 = -1 are the five most important constants in mathematics because they are widely used in equations that describe relationships in all branches of mathematics and science. The equation eπi + 1 = 0, which is due to Leonhard Euler, is one of the most interesting and intriguing equations in mathematics. Euler used the symbol e for the irrational constant, and in his honor, e is named Euler’s number.
• Both Bernoulli and Euler were prolific mathematical giants. Much of what is routinely used in mathematics and science can be traced back to the work of these two great men. L’Hospital’s Rule in calculus is due to Bernoulli, not L’Hospital. L’Hospital published the rule, but Bernoulli discovered the rule and gave it, for a fee, to L’Hospital.
Because of limits on food, living space, disease, existing technology, war, and other factors, most populations have limited growth as opposed to unlimited exponential growth which is modeled by the classic exponential growth equation P = P0bt/k. A limited growth population starts growing almost exponentially, but it reaches a critical point in time where its growth rate slows, and the population starts to exponentially and asymptotically approach an upper limit. There are several models that are used to describe limited growth of a population. In my next post, I will discuss the logistic function which was used by the Belgium mathematician Pierre Francois Verhulst (1804-1849) to study limited population growth. The logistic function also has applications in artificial neural networks, biology, chemistry, demography, ecology, economics, biomathematics, geoscience, mathematical psychology, sociology, political science, probability, and statistics.