Computing compound interest and composition of functions

Transcription

1 Computing compound interest and composition of functions In today s topic we will look at using EXCEL to compute compound interest. The method we will use will also allow us to discuss composition of functions. Compound interest is when interest is computed on a regular basis during the year. If interest is computed quarterly, it will be computed four times during the year. Simple interest is interest that is paid only at the end of each year. Starting with a simple interest problem, let s say that we place $ in a savings account with an interest rate of 5%. At the end of the first year we would have our original $ plus interest at 5% or.05 * $ Looking at this from an equation standpoint, we have Factoring out the $ , we have, Since.05 represents the rate, the general formula is Where r = interest rate. Since we won t always deposit $ every time, we can change $ into a variable P which represents principle. With this information, we can begin our EXCEL spreadsheet. Start with labeling Labels are in column A. Values are in column B For 1 + rate, we entered into cell B3 the equation =1 + B2. Thus, if we change the rate, the value in B3 will change accordingly. To find the amount of interest we will have at the end of year 1, we just need to multiply cells B1 times B3

2 In B6, we now have $ in cell B6 which was computed by multiplying cells B3 time B1. With this in EXCEL, we can quickly change with the principle and the rate and determine the effect of different amounts. Try this. Notice in cell B1 and in cell B2 we entered values. Thus, these values are constants (or variables as they are easy to change.) Notice in B3 and B6, we computed the answer using previous cells. When cells are used to compute answers we are using functions. Composition of functions. In the last problem, we have two functions B3 and B6. B3 was computed and use in B6. This is a composition of functions. An example of composition of functions is buying groceries and paying taxes. First we must compute the cost of the groceries and from that value, we must compute taxes. Finally we add cost of groceries with taxes to find the total cost. In this case, finding the cost of the groceries is a function. Since this value is used in finding taxes and finding the total cost, both taxes and total cost are composition of functions. A simple definition for composition of functions, is any time you compute an answer and then use the answer in another function, the second function is a composition of functions. Simple interest over longer periods of time. Let s assume we wish to know how much is in our account after several years. We can do this by making a few changes to our spreadsheet.

3 We have relabeled A6 as year and B6 as amount that would be in the account after the year. In A7 we entered 0 and in B7 we entered B1, by doing this we can change the initial amount by changing B1. In A8 we entered 1 and in B8 we entered the equation =B7 * $B$3. What we are going to do next is highlight A8 and B8, grab the square in the corner and drag down to create year 3, 4, 5 and more. As we drag B7 will become B8 and B9, etc. However, by placing a dollar sign in the cell name, $B$3, these cells will not change as we drag and will remain cell B3. Highlight cells A8 and B8 and drag down. Notice in B20, we have the equation =B19*$B$3. Which means cell B3 is used in all calculations as it should be.

4 When we notice that the values for amount are not in standard $ notation, we can highlight the cells B7 through B20, left click and notice that we can format cells. Click on format cells. This will give us the opportunity to pick the type of value in the cell and the number of decimal places. Click on currency. Choose the number of decimal places to display. In this case, 2. Click on your choice of display Click ok The column should now have a dollar sign in front of all the numbers and only two decimal points showing.

5 One of the reasons for creating the chart in the fashion that we did, is that we can highlight cells A6 through B20 and create the graph of the amount invested as a function of time At this time we can explore by changing the principle (cell B2) and the rate (cell B3) and determine the effect of changing these values. Simple interest rate formula. Can you evaluate the work we have accomplished in EXCEL and from this determine the equation for simple interest with A being the amount in the account, P being the amount started with and r being the rate and t being the time. Compound interest. While simple interest only pays interest at the end of the year, compound interest pays interest periodically throughout the year. If interest is paid quarterly it is paid out every 3 months and interest starts to accrue on the interest as soon as it is paid. In compound interest, we need to add a new variable, n which is the number of compounds in a year. If interest is paid quarterly, n=4, if interest is paid monthly then n=12. In this case the equation changes to { } The curly brackets are not necessary. They highlight that changing the simple interest to compound interest changes into. The value indicates interest rate for the shortened period and raising to the n makes sure we have made all interest payments within one year.

6 We can quickly adjust our EXCEL spreadsheet by adding in A4 the label n and in B4 the number of compounding per year. (start with B4 = 1). Then in A3, change to (1+r/n)^n and B4 into =(1+B2/B4)^B4. In A4, we labeled n In B4 we entered the number of times per year that the compounding occurs. In B3 we replaced the simple interest formula with the compound interest formula for computing growth in a year =(1+B2/B4)^B4 Notice that the amounts did not change. This is because, when n= 1 we have simple interest. Play around with changing the number of compounding and interest rates to see the changes that occur. Extra Practice. In the 1990 s, banks were wooing customers with high interest rates and intriguing compounding deals. But when we mix annual interest rates (APR) with different compounding schemes, it is difficult to determine the best deal. To solve this, banks we told to advertise using annual percent yield (APY). APY is actually the rate you would receive if the amount was computed in simple interest. We have set up the spreadsheet to quickly calculate APY. Consider the fours bank offerings. Which one would you choose. Rate (APR) Compounds per year Bank 1 8% 1 Bank 2 7.9% 2 Bank % 4 Bank % 12 Rate (APY) On first glance Bank 1 has the highest interest rate so they are a good place to start. But wait, Bank 4 compounds 12 times a year and we are making money on our interest much more regularly. So let s go to Bank 4. Hold on, with our spread sheet we can enter the rate into B2 and our compounding times into B4. The result in B3 will be 1 + the APY. Calculate the APY for each bank.

7 Rate (APR) Compounds per Rate (APY) year Bank 1 8% % Bank 2 7.9% % Bank % % Bank % % Hence Bank 3 has the highest APY and is the bank that will return the highest yield.