Interest: money for rent

"If you believe in things you don't understand, YOU SUFFER." Stevie Wonder
$ in the U.S.A.

In 1690, the Massachusetts Bay Colony issued the first paper money in the colonies. In 1796, The Mint, who once considered producing doughnut-shaped coins, settled on the a solid $2.50 gold piece that was to become the standard U.S. coin until the middle 1800’s. But, by the end of the Civil War, nearly 40 % of all U.S. paper currency in circulation was counterfeit. As a result, in 1865, the Secret Service was created under the U.S. Treasury Department and in less than a decade, counterfeiting was sharply reduced. During that decade, the U.S. Treasury created the two-cent coin, the significance here is it was that coin that was the first to bear the motto "In God We Trust". Since 1877, all U.S. currency has been printed by the Bureau of Engraving and Printing, which began as a six person operation in a dark basement of the Department of Treasury. Now, 2,300 Bureau employees work in a twenty-five acre floor space spanning two buildings, where 38 million notes are printed each work day with a face value of approximately $541 million and that comes to eight billion U.S. notes each year, enough to wrap around the equator 30 times. These notes are tough, you could double-fold a U.S. currency note about 4,000 times before it would tear. Almost half of these notes printed are $1 notes. Source: US Federal Reserve and the United States Mint.

Similarly for coins, over half of the coins produced in the U.S. are pennies. The composition of the one cent penny has undergone change upon change in the last 211 years. It was pure copper from 1793 to 1837. Then it was bronze, then copper and nickel, then bronze again, and today it is 97 ½ % copper, 2 ½ % zinc. The actual number of coins produced yearly in this country is roughly, by denomination, 10 billion pennies, 1.3 billion nickels, 2.3 billion dimes, nearly 2 billion quarters, and 30 million half-dollars.

Who’s on this money, aside from an assortment of presidents? Four women have been portrayed on US coins, but Martha Washington is the only woman whose portrait ever appeared on U.S. paper money. While no portraits of African Americans have ever appeared on paper money, commemorative coins were issued bearing the images of George Washington Carver, Booker T. Washington and Jackie Robinson. Paper money did bear the signatures of four African American men who served as Registers of the Treasury and one African American woman who served as Treasurer of the United States. And it is Abraham Lincoln on the penny as the only U.S. president to face to the right, all the other portraits of presidents on U.S. circulating coins face to the left. The reason is less grandiose that one would think. President Lincoln is facing to the right because his likeness on the penny is simply an adaptation of a plaque bearing his resemblance. Source:

Why do we use the symbol of an “S” with a vertical line slashed through it? Many people wonder where the symbol of the dollar sign evolved from, and the answer, though mildly controversial, is the peso. Though adopted by the United States dollar in 1785, the symbol “$” was widely used before that year. From dusting off old manuscripts, one can actually see the Mexican or Spanish "P's" for pesos replaced by an “S” written over the “P”. This "S" gradually evolved as it was written over the "P" into the dollar sign symbol “$” we are all now familiar with.

How important is finance in the media? In the 60 days following President Clinton's State of the Union address, the New York Times ran 27 stories regaling the President's campaign finance misfortunes and 3 stories regarding education reform, the showpiece of his acceptance speech. How important is money in this culture we live? They say the most common gift in the U.S.A. for a newborn is the savings bond. If you think this through, the choice of a savings bond is the perfect gift for the baby, one size fits all, and it grows with the child. When the child becomes school age, if the dog eats their homework, they are out of luck, but if the dog eats their savings bond, it can be replaced. (And by the way, if you had done so in July of 1976, you could have purchased as $ 25 Savings Bond for $ 17.76 instead of the usual price of $ 18.75 thanks to the bank’s celebration of the Bicentennial.) Finally, how important is money on this planet? The number of wars fought between countries that each had at least one McDonald's franchise is zero.

Effective rates, Simple Interest, Compound Interest

All of us see advertisements everyday where a company wants you to purchase their product with credit. They want to give you merchandise even though at this time you simply do not have all the cash. Financing is available.

Interest or ’rent on money’ can be understood and calculated by incorporating some fundamental ideas of mathematics. One of the first observations we need to make is the connection between interest theory and the power of mathematics to succinctly capture numerical patterns. When studying interest we will see rigorously structured patterns and then use the familiar linear and exponential growth models to continue our journey in recognizing mathematics embedded through out our lives.

Most often we think of interest in the form of a financial transaction but this is not always the case. Interest can be in the form of goods or services as well. For example, an artist or craftsman may produce a piece of work and give it to the owner of a studio or workshop in lieu of actual cash payments for shop privileges. Our presentation of the mathematics involved with the theory of interest will focus specifically on monies paid as rent for money borrowed.

While turning the pages of any news paper you will see advertisements from businesses that either want to loan you money, ‘Consolidate all your debt into one low monthly payment: interest only 3.125%/3.32% APR - 30 year fixed 5.375%/5.563% APR’ or want you to invest your money with them ’11.18% Investment opportunity, 3 months 8.32% - 6 months 10.51% - 12 months 11.18% minimum investment of $1000 and all rates are annual yield’ What is APR? and why the fine print?

Lets start our analysis with two interest paying concepts. The first is called an effective rate of interest and the second in called a nominal rate of interest. The effective rate of interest is the actual rate at which interest is paid on principal, P (the money originally invested). We can observe the effective rate using the ratio I/P. Interest payments by themselves are calculated by taking the product of principal the interest rate and the amount of time, . The idea of simple interest follows a linear growth pattern. That is, the original principal is enhanced with interest payments on regular intervals of time and the same amount of interest is added to the principal following each interest period. Algebraically, we see this new quantity as principal plus interest,. Under the simple interest scenario we add the same fixed amount of interest to an account at the end of each interest period. The value of the accounts grow according to the following pattern: , , , and so on until reaching the end of the interest period and the final account balance will be . Lets build our first numerical example on this topic.

Problem One

Under the rules of simple interest how much will an initial principal of $100.00 grow over a period of 1, 2, 3, 4, 5, and 10 years at 3% simple interest paid annually? We have , and . The amount of interest paid at the end of each successive year is . The account value at the end of each year increases by the same fixed amount;, , , , and .

We need to notice two very important things here. First, the structure of each equation is that of a linear model, . Our original principal is our starting value identified with , the amount of interest paid each year is our constant rate of change, time in increments of years occupy the independent variable and finally the end of the year value of the account is represented by the dependent variable . Second, we need to examine the effective rate over time. Our technique of identifying percent changes is of value here. Recall, the percent change formula structure is . How much interest with respect to the amount of money found in our account was actually paid at the end of each year in our first example?

or 3% the first year, or 2.913% the second year, or 2.83% the third year, or 2.752% the fourth year, or 2.679% the fifth year and or 2.362% the tenth year. The effective rate of interest being paid is decreasing. The amount of interest being paid remained constant while the amount of money in our account grew creating the situation where we have a decreasing effective rate. Simple interest provides for a decreasing effective rate.

In ordinary English, we may say since we are accumulating 3 dollars interest per year, the 3 dollars when compared to $ 100, $ 103, $ 106 and so on is a smaller percentage of each amount found in the account each year. Does this seem fair to you? The bank, which may house the money seems to be getting more money each year, but the interest, though constant, in turn is a smaller percentage of that amount as time progresses. Our effective rate ratio I/P shrinks each time interest is paid.

What needs to change in this scenario so that the effective rate does not decrease but remains constant? If you are thinking that interest needs to be paid on previous interest payments into the account you are correct. Situations where interest is paid on previous interest added to the account fall in the area of compound interest. Interest paid on interest earned is the compounding of interest. In our next example we will use the same numbers for our original principal and interest but will employ the use of an exponential growth model to complete the calculations.

Problem Two

Again, starting with the basic concept of interest we can develop an algebraic expression to use when calculating the future value of our account. The exponential model naturally presents itself when we start to analyze this situation. At the completion of the first interest period our account balance has grown from to. Because the time intervals remain constant we do not need to keep track of the variable as we let and our expression becomes . We will need to keep count of the number of times interest is paid and we will use the variable for the number of compounding periods. At the end of the second interest period our balance becomes . The expression used to find the account balance at the end of the third interest period is . Is the pattern obvious? The interest payment will produce a balance of dollars. The structure of this formula is identical to our exponential growth formula . The original principal is our starting value and is our growth rate . Incorporating the same numerical values as used in the previous example the year to year account values under the compound interest scenario are , , , , , and . Examining the effective rate we see the effective rate remains constant. , second year, third year, fourth year , fifth year and the effective rte for the tenth year is . The effective rate of interest paid remains constant through out the life of this account. Why, in this example does the approximate effective rate bounce around the .03 value?

Compound interest provides for a constant effective rate and recall, simple interest provided for a decreasing effective rate. With compound interest, not only is the money found in the bank increasing, the amount of interest paid is increasing and the interest accumulated seems to be a constant percentage of the current account value. This is strikingly different from simple interest, where the interest is constant for each successive year, but it is a smaller percentage of the amount found in the account. So, if you are investing money and the effective rate is simply a measure of the ratio of interest to the money found in the bank, which investing method would you prefer, simple or compound?

In general, we have two distinct methods of accumulating interest. Either by the simple interest process or the compound interest process and there are some occurrences where both scenarios are combined to analyze the growth of interest. Simple interest incorporates a linear growth model and has the characteristic of producing a decreasing effective rate, . As we can see, the percent change formula when applied to the simple interest situation simplifies to a ratio where the denominator will continue to numerically grow with each passing interest payment period. The growth of the denominator and the existence of no change in the numerator of this ratio is what mathematically forces a decreasing effective rate.

Now, when we look at the effective rate of interest produced by the compound interest model we see there is a constant effective rate of interest . Each compounding period has the same interest rate applied to the value of the account. The account grows proportionally at the end of each compounding period.

Example Three

We need a future value of $ 5000 in three years, and can earn 6 ½ % compound interest annually, what must be our present value (or original principle).

We know, so, .

Principle or Present Value = dollars. We have earned $ 5000 – $ 3649.40 = $1,350.60 as rent on our original principal.

Problem Four

Suppose you invest $ 5,000 at 6 ½ % simple interest for 10 years. a) How much is found in the account after 1, 2, 3, 9 and 10 years? b) Find the effective rate of interest for year 1, 2, 3 and 10. c) How much interest is found in the account after 10 years?


a) This investment is based on simple interest, and this formula has a linear structure of , where . Let P = 5000, r = 0.065 and t = 1, then.
At the end of year 1, we have 5000 + (1)325 = 5325

At the end of year 2, we have 5000 + (2)325 = 5650

At the end of year 3, we have 5000 + (3)325 = 5975

Continuing with this process, we have

At the end of year 9, we have 5000 + (9)325 = 7925

At the end of year 10, we have 5000 + (10)325 = 8250

b) The effective rate is found by looking at . Let’s examine this ratio for years 1, 2, 3, 9, and 10. For year 1,

For year 2, . Continuing, for year 3, . For year 9,

And for year 10, the effective rates appear to continue this trend of plummeting fast, and we have c) The amount of interest after 10 years is 10($ 325) = $ 3250

Let’s compare this investment based on simple interest with an investment based on compound interest. We will use the same numbers as we did in Examples 3 and 4, so our comparison is easy to visualize.
Problem Five

Suppose you invest $5,000 at 6 ½ % compound interest for 10 years. a) How much is found in the account after 1, 2, 3, 9 and 10 years? b) Find the effective rate of interest for year 10. c) How much interest is found in the account after 10 years?


a) This investment is based on compound interest, so now we have, which has the exponential structure “”. Let P = 5000, r = 0.065 and n = 1, 2, 3, 9, and 10.

End of year 1, we have which matches identically with the simple interest calculation.

End of year 2, we have = $ 5,671.13 which is more than the $ 5,650 we had in simple interest.

End of year 3, we have = $ 6,039.75 which, again, is more than we had with simple interest, where we had $ 5,975.

End of year 9, we have = $ 8812.85 as compared to the $ 7925 accumulated from the simple interest investment.

End of year 10, we have = $ 9385.69 as compared to the $ 8250 accumulated from the simple interest investment.

b) Again, the effective rate is found by looking at . Let’s examine this ratio for year 10.

which is 6.5 % so it appears the effective rate stays constant, as we would predict.

c) The interest found in the account after 10 years is $9385.69 - $5,000 = $4,385.69 which is considerably more than the interest of $3,250 that was derived from a simple interest bearing account for the same number of years, with the same amount of principle. And this was an investment that was compounded annually, once a year.

Problem Six

The graph below shows two different accounts each with an investment of $100 for 5 years. The horizontal axis represents time measured in years, the vertical axis represents the dollar amount an investment accumulates to after so much time has past. In one account, the $100 is compounded quarterly at 12 % annually, and in the other account, the $100 is compounded quarterly at 9 % annually. Based on this information and the graph below, which statement is the most reasonable?

a)If both accounts continue for 20 years, the 9 % graph would eventually get closer to the 12 % graph.

b)There is no literal difference in the amount in both accounts until after the first year.

c)A small increase in the interest rate makes a substantial difference after 5 years.

d)The difference between the two graphs is constant from year to year.


Let’s explore the reasonableness of each of the choices separately.

a)If both accounts continue for 20 years, the 9 % graph would eventually get closer to the 12 % graph. This does not seem reasonable. Every indiaction is that the difference between the curves should increase as times passes.

b)There is no literal difference in the amount in both accounts until after the first year. This again does not seem reasonable. Since there is a difference between the graphs in the interval 0 < t < 1, and since the money is compounded quarterly, then there should be a difference in the accounts as early as after the first 3 months (1st quarter).

c)A small increase in the interest rate makes a substantial difference after 5 years. This appears quite reasonable. From the graph, there appears to be nearly a $25 difference in the accounts after 5 year, $180 versus $ 155. Since the investment was $100, this is 25 % percent of the original investment, which is substantial.