Interest: money for rent
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 ofmathematics. 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 study of Interest.
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 (the money originally invested). 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 as principal plus interest,. Interest payments by themselves are calculated by taking the product of principal the interest rate and the amount of time, . 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 theinterest period and the final account balance will be . Lets build our first numerical example on this topic.
Example
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 the value, the interest rate is our constant rate of change, time in 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. 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.
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 pervious 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 need to employ the use of an exponential growth model to complete the calculations.
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 . The year to year account values under the compound interest senerio 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.
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.