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MT 1190 – Precalculus Worksheet 2.3 – Compound Interest Name __________________________ Purpose. In this worksheet we examine the calculation of compound interest on an investment. The main purpose of this worksheet is to examine a simple case of exponential growth. Procedure. You are required to complete this worksheet with your lab partner. Review of Percents Review of Exponential Notation Problem. 1. A Patriotic Story of Interest…. One of your ancestors—a lawyer—had been in Philadelphia on July 4, 1776. He became so enthusiastic about the prospects for the new nation and his hopes for its long existence that he invested $1 with a local banker to gain interest at the rate of 3% per year, compounded annually, until July 4, 2001. At that time the entire sum was to go to one of his descendants. The particular formula for who is to receive this money is intricate—as befits a lawyer—but the upshot is that you are the lucky one. What is your initial guess as to how much money will be in the account in the year 2001? Ø Conjecture: Worksheet 2.3 - Compound Interest 2 2. Determining how much money will you receive in 2001? Let's assign symbols to the important quantities in the problem: Let k = the number of years the account has been in existence (as measured by the most recent July 4); Let A[k] = the money in the account on July 4, k years after 1776; We will need the value of A[0], i.e., the amount of money on July 4, 1776. Ø A[0] = _______________________ As a preliminary task, we compute the values A[0], A[1], A[2], A[3], .... These will be the dollar values of the account during the first few years of its existence. Place these values in the following list (remember that the increase in each successive A-value will be three percent of the previous value; in other words, new A-value = 103% of the previous A-value ! ). Use Mathematica or your calculator to find the following values: Ø In the year 1776......A[0] = __________ Ø In the year 1777......A[1] = __________ Ø In the year 1778......A[2] = __________ Ø In the year 1779......A[3] = __________ Ø In the year 1780......A[4] = _________ Since we are interested in the amount of money that will be in the account in the year 2001, to continue computing successive values for A[0],A[1],A[2],A[3],A[4], etc. would be a tedious and time-consuming task. It is certainly in our interest (no pun intended!) to try and formulate a function A[k] that will give us the amount, A, that has accumulated in the account at the end of a period of time, k. Ø Why ? Let's re-express A[0], A[1], A[2],A[3], and A[4] in terms of the initial investment ($1), the interest rate (3%) and the number of times the bank has added interest to the account. Think about what you did to get your values in the preceding section to complete the more general statements below. Ø In the year 1776......A[0] = 1 Ø In the year 1777......A[1] = 1 * (1.03) Ø In the year 1778......A[2] = 1 * (1.03)* (1.03) = __________ Ø In the year 1779......A[3] = ___________ Ø In the year 1780......A[4] = ___________ From these statements, you should see a pattern that gives a formula for A[k], the account's dollar value in the year 1776 plus k years. Place your formula for A[k] below. Ø In the year 1776+k , A[k] = ________________ Ø A[225] = _______________ Ø Does this value surprise you, i.e., was your conjecture from part 1 far off? If so, why? Worksheet 2.3 - Compound Interest 3 4. What happens if the initial investment is doubled? Suppose your ancestor had been twice as enthusiastic and had invested $2. What do you guess your account would be worth in 2001? Ø Conjecture: In order to compute the final balance if the initial investment is doubled, let's follow the same procedures as in Section 2. We must record the value for A[0], then compute the values of A[1], A[2], A[3], A[4],…. These will be the dollar values of the account during the first few years of its existence. Ø In the year 1776.........A[0] = _______________ Ø In the year 1777.........A[1] = _______________ Ø In the year 1778.........A[2] = _______________ Ø In the year 1779.........A[3] = _______________ Ø In the year 1780.........A[4] = _______________ Once again, we need to look for a more general expression for A[k]. Re-express these new A[k] values in terms of the initial investment ($2), the interest rate (3%) and the number of times the bank has added interest to the account. Think about what you did to get your values in the preceding section to complete the more general statements below. Ø In the year 1776.........A[0] = 2 Ø In the year 1777.........A[1] = 2 * (1.03) Ø In the year 1778.........A[2] = ___________ Ø In the year 1779.........A[3] = ___________ Ø In the year 1780.........A[4] = ___________ Ø Do you see a general pattern emerging? Give a general formula to compute A as a function of k. Ø Place your formula for A[k] below. Ø A[k] = _______________ Ø Use your new function A[k] to compute the value of the account in the year 2001. Remember that k is the number of years since 1776, so in the year 2001, k = 2001-1776 = 225 Ø A[225] = ____________ Ø Does this value surprise you, i.e., was your conjecture far off? If so, why? Worksheet 2.3 - Compound Interest 4 5. What happens if the interest rate is doubled? Suppose your ancestor had only invested $1, but had obtained an interest rate of 6%. What do you guess your account would be worth in 2001? Conjecture: In order to compute the precise final balance if the initial investment is doubled, let's again follow the same path as in Section 2. We must record the value for A[0], then compute the values of A[1], A[2], A[3], A[4]…. These will be the dollar values of the account during the first few years of its existence. Fill in the appropriate values below Ø In the year 1776.........A[0] = ______________ Ø In the year 1777.........A[1] = ______________ Ø In the year 1778.........A[2] = ______________ Ø In the year 1779.........A[3] = ______________ Ø In the year 1780.........A[4] = ______________ Once again, we need to look for a more general expression for A[k]. Re-express these new A[k] values in terms of the initial investment ($1), the interest rate (6%) and the number of times the bank has added interest to the account. Think about what you did to get your values in the preceding section to complete the more general statements below. Ø In the year 1776.........A[0] = 1 Ø In the year 1777.........A[1] = 1 * (1.06) Ø In the year 1778.........A[2] = _______________ Ø In the year 1779.........A[3] = _______________ Ø In the year 1780.........A[4] = _______________ Do you see a general pattern emerging? Give a general formula to compute A as a function of k. Complete the following function definition statement. Ø A[k] = ___________ Use your new function A[k] to compute the value of the account in the year 2001. Again remember that k is the number of years since 1776, so in the year 2001, k = 2001-1776 = 225 Ø A[225] = _______________ Ø Does this value surprise you, i.e., was your conjecture far off? If so, why? 6. Why the difference between Sections 3 and 4? Ø Write a paragraph to explain how and why the answers to Parts 3 and 4 are so different. Ø Answer: Worksheet 2.3 - Compound Interest 5 7. What if the interest is compounded twice a year? Suppose your ancestor had only invested $1 at an interest rate of 3%, but that this interest was to be compounded twice a year, i.e., 1.5% interest on the balance of the account would be awarded each July 4 and January 4. What do you guess your account would be worth in 2001? Ø Conjecture: In order to determine this number precisely, it is convenient to slightly alter the meanings of k and A[k]. The new definitions will be as follows: k = the number of interest periods which have gone by (as of the specified date); and A[k] = the money in the account at the end of the k interest period (sometimes referred to as compounding period.) Ø In what ways do these definitions differ from the previous versions? In order to determine the value if the initial investment is compounded twice a year, let's follow a similar procedure as in Section 2. We must record the value for A[0], then compute the values for A[1], A[2], A[3], A[4], A[5], .... These will be the dollar values of the account during the first few interest periods of its existence. Fill in the appropriate values below. Ø On July 4, in the year 1776......A[0] = ______________ Ø On Jan 4,in the year 1777........A[1] = _____________ Ø On July 4, in the year 1777......A[2] = ______________ Ø On Jan 4, in the year 1778.......A[3] = ______________ Ø On July 4, in the year 1778.......A[4] = _____________ Ø On Jan 4, in the year 1779.......A[5] = ______________ Once again, we need to look for a more general expression for A[k]. Re-express these new A[k] values in terms of the initial investment ($1), the interest rate (1.5% every 6 months) and the number of times the bank has added interest to the account. Think about what you did to get your values in the preceding section to complete the more general statements below. Ø On July 4, in the year 1776...... A[0] = 1 Ø On Jan 4,in the year 1777........ A[1] = 1 * (1.015) Ø On July 4, in the year 1777...... A[2] = 1 * (1.015)* (1.015) Ø On Jan 4, in the year 1778....... A[3] = ____________ Ø On July 4, in the year 1778....... A[4] = ____________ Ø On Jan 4, in the year 1779....... A[5] = ____________ Ø Do you see the general pattern emerging? Complete the following statement. In the kth investment period, A[k] = ___________________ Worksheet 2.3 - Compound Interest 6 Now determine the k for the year 2001, i.e., the number of investment periods which will have occurred by July 4, 2001. Remember that k now represents the number of interest periods that have gone by. Ø k = ____________ Ø Redefine your function A[k] and compute the value of the account in the year 2001 Ø A[ ___ ] = ________________ Ø How does this value compare with the return you got on your money in part 2? 8. What if the interest is compounded three times a year? Suppose your ancestor had only invested $1 at an interest rate of 3%, but that this interest was to be compounded three times a year, i.e., 1% interest on the balance of the account would be awarded each July 4, November 4, and March 4. What do you guess your account would be worth in 2001? Conjecture: In order to determine the value if the initial investment is compounded three times a year, let's follow a similar procedure as in Section 2. We must record the value for A[0], then compute the values for A[1], A[2], A[3], A[4], A[5], .... These will be the dollar values of the account during the first few interest periods of its existence. Fill in the appropriate values below. Ø On July 4, in the year 1776.........A[0] = _______________ Ø On Nov 4, in the year 1776.........A[1] = _______________ Ø On Mar 4, in the year 1777.........A[2] = _______________ Ø On July 4, in the year 1777.........A[3] = _______________ Ø On Nov 4, in the year 1777.........A[4] = _______________ Ø On Mar 4, in the year 1778.........A[5] = _______________ Worksheet 2.3 - Compound Interest 7 Once again, we need to look for a more general expression for A[k]. Re-express these new A[k] values in terms of the initial investment ($1), the interest rate (1% every 4 months) and the number of times the bank has added interest to the account. Think about what you did to get your values in the preceding section to complete the more general statements below. Ø On July 4, in the year 1776.........A[0] = 1 Ø On Nov 4, in the year 1776.........A[1] =1 * (1.01) Ø On Mar 4, in the year 1777.........A[2] = _______________ Ø On July 4, in the year 1777.........A[3] = _______________ Ø On Nov 4, in the year 1777.........A[4] = _______________ Ø On Mar 4, in the year 1778.........A[5] = _______________ Do you see the general pattern emerging? Complete the following statement. Ø In the kth investment period, A[k] = _______________ Now determine k for the year 2001, i.e., the number of investment periods which will have occurred by July 4, 2001. Ø k = __________________ Redefine your function A[k] to compute the value of the account in the year 2001 Ø A[ _____ ] = _________________ Ø How does this value compare with the return you got for your money in parts 2 and 8? 9. But could it really happen? Yes! It really has happened! In 1790, Benjamin Franklin left 2,000 pounds (about $8,880) to be divided among Philadelphia, Boston, and their respective states. Part of the accumulated balance was to be paid out in 1890; the rest, in 1990. The 1890 Boston disbursement was marked by a 14 year legal battle for control of the money. In 1990, Boston and Massachusetts' share was estimated to be worth $4.5 million. Philadelphia's share was estimated to be worth $520,000. (Financial mismanagement, anyone?) ("Franklin's largesse has long reach," Boston Globe, April 17, 1990, pp. 15 and 17.) Worksheet 2.3 - Compound Interest 8