Math 1172 Project #2 Financial Applications of Geometric Series Due: Tuesday, October 11 Name(s): - - - - - - - - - - - - - - - - - - - - Description - - - - - - - - - - - - - - - - - - - Most people will have to deal with finances during their lives. Typical situations involve: Saving money Paying back student loans Understanding Credit Card Debt This assignment will explore these scenarios quantitatively. - - - - - - - - - - - - - - Purpose of the Assignment - - - - - - - - - - - - - - To apply knowledge of geometric sums to the above situations. To present the general methods behind modeling these situations (so they can be adapted to other similar situations should you need to do so!) - - - - - - - - - - - - - - - - - - - - - Directions - - - - - - - - - - - - - - - - - - - - This assignment is worth 15 pts. You may work in groups of up to 3 students. Each group will submit one copy of this assignment; group members should NOT submit individual assignments! Each group member's name should appear on the top of this page. Each member of the group will receive the same grade. If you need more space than what is provided, feel free to use scratch paper, but you must staple it to your assignment and clearly indicate to which problem any work belongs! 1 I. The Basics: Compound Interest Compound interest is an extremely important and useful concept! Should you be unfamiliar with it, please read the information posted here: http: //www.money-zine.com/investing/investing/understanding-compound-interest/ Using the ideas discussed in the above article, we can write a formula: \u0010 r \u0011nt F =P 1+ n F is the future value of the investment/loan, including interest. P is the principal investment amount (the initial deposit or loan amount). r is the annual interest rate, written as a decimal (ex: 6% =.06). n is the number of times that interest is compounded per year. t is the number of years the money is invested or borrowed. Problem 1: Suppose we invest $100 today. Calculate the future value 20 years from now under the following annual interest rates (i.e. n = 1) by using the formula above. a) 0.06% (a typical savings account rate) b) 1.2% (a typical high-yield money market rate) c) 3.1% (a typical 30-year US Treasury Bond Yield) d) 7.86% (the average annual growth rate of the S&P 500 between 1976-2016) 2 II. The Total Future Value of a Steady Stream of Payments Now, suppose we open an account and invest $100 at the beginning of every month, starting today, for 25 years. At the end of the 25 years, we make one final payment of $100. Problem 2: If we gain no interest on the monthly investments, how much will the account will be worth in 25 years? Remember to include the final payment! How should we go about calculating the value of the account at the end of the 25 years if we do invest? Let's first assume that the interest rate, r, is compounded monthly (so n = 12). In 25 years, the investment that we make today will be worth: \u0010 \u0010 r \u00111225 r \u0011300 100 1 + = 100 1 + 12 12 In 25 years, the investment that we make in 1 month from now will be worth: \u0010 \u0010 r \u0011299 r \u001112 239 12 = 100 1 + 100 1 + 12 12 (Note that 19 years and 11 months is 239/12 years). In 25 years, the investment that we make in 2 months from now will be worth: \u0010 r \u001112 238 r \u0011298 12 100 1 + = 100 1 + 12 12 \u0010 Problem 3: What will the future value of the payment that we make 47 months from now be worth in 25 years? Express your answer in terms of r. From the above, we can conclude that the amount of money we will have in 25 years will be the sum of the future values of the investments that we make every month! We will calculate this amount two different ways - by using Excel and by using finite geometric sums. Excel is a powerful tool for a wide variety of applications! If you are not familiar with using Excel, please see the instructions in the \"Projects\" Folder in Carmen. 3 Problem 4: Calculating the Total Future Value Using Excel For each interest rate r given in Problem 1, calculate the amount of money in the account by using an Excel spreadsheet to add up the future values of all of the monthly investments. If you are unfamiliar with Excel, please see the file in the project folder. When you are doing this, assume that the final payment of $100 will be made at the end of 25 years, so that the last payment will be 100 (1 + r/12)0 . Report your final answers below to 2 decimal places. You do NOT need to hand in the Excel sheet. Note that the first answer is provided. Once you have set up your spreadsheet, please use this result to verify that your spreadsheet is calculating the total future value correctly! At r = .06%, the account will be worth: $ 30326.88 . At r = 1.2%, the account will be worth: $ . At r = 3.1%, the account will be worth: $ . At r = 7.86%, the account will be worth: $ . It's possible that your answer may differ by a few decimal places. To ensure the most accurate response, make sure that you are not rounding until the end! Compare these results to the number you found in Problem 2 to see the effect of various interest rates! There is a way to do this using the geometric sum formula! First: Problem 5: To derive the formula: N X arn = a n=0 first notice that PN n=0 1 rN +1 1r arn is notation for a + ar + ar2 + + arN . Let Sn = PN n=0 arn . i Write out the expression Sn represents. On the line below it, write out the expression rSn represents by multiplying both sides of the equation for Sn by r. ii. Subtract both equations and solve for Sn . 4 Now, let's return to the cash flow described at the beginning of this section. Note that we can write the future value off ALL of the payments as: \u0010 \u0010 \u0010 \u0010 r \u0011300 r \u0011299 r \u0011298 r \u00110 + 100 1 + + 100 1 + + ... + 100 1 + F = 100 1 + 12 12 12 12 This is a geometric sum and we can use the formula in Problem 5 to evaluate it! Problem 6: A. Write the above expression for F in summation notation. Then, use the formula from Problem 5 (and a bit of algebra) to show that: \u0014 \u0015 r \u0011301 1200 \u0010 1+ 1 F = r 12 B. Using the rates in Problem 1 in the formula in Part A, show that the results here are consistent with the values generated in Excel from Problem 4. - At r = .06%, the account will be worth: $ . - At r = 1.2%, the account will be worth: $ . - At r = 3.1%, the account will be worth: $ . - At r = 7.86%, the account will be worth: $ 5 . III. The Annuity Formula We can of course generalize the result in the previous problem. In fact, if we invest a principal amount P at the beginning of each period and make a final investment of P at the end of the final period, we can write down a so-called \"annuity formula\": \u0014 \u0015 r \u0011nt+1 Pn \u0010 1+ 1 F = r n where: F is the future value of ALL of the investments/loans, including interest. P is the principal amount invested or paid at the beginning of each period. r is the annual interest rate, written as a decimal (ex: 6% =.06). n is the number of times that interest is compounded per year. t is the number of years the money is invested or borrowed. This formula is useful because, given a certain interest rate, it allows for one to figure out the total future value of a series of investments or the amount of money one should invest monthly to reach a certain goal. 6 Problem 7: Suppose you can afford to invest $1000 at the beginning of each month in hopes to save for retirement at the age of 65. As soon as you turn 65, you make one final investment of $1000. Using the annuity formula, it can be shown that the total future value of the account when you are 65. Assuming you start this process as soon as you turn 23 (so you invest for 42 years total): The account will be worth $1,037,402.58 under the annual rate of 3.1% The account will be worth $3,974,115.26 under the annual rate of 7.86% Make sure that you can calculate the values above. For both the annual rates r = 3.1% and r = 7.86%, compounded monthly: A. Calculate the total future value of the account when you are 65 assuming you start this process as soon as you turn 30 (so you invest for 35 years total). B. Calculate the total future value of the account when you are 65 assuming you start this process as soon as you turn 40 (so you invest for 25 years total). 7 Problem 8: Suppose you invest the same amount $P at the beginning of each month in a retirement account in hopes to save $700,000 for retirement at the age of 65. If you start this process as soon as you turn 23 (so you invest for 42 years total), the annuity formula can be used to show that the amount of money you will need to invest per month is: $674.76 under the annual rate of 3.1% $176.14 under the annual rate of 7.86% Make sure that you can calculate the values above. For both the annual rates r = 3.1% and r = 7.86%, compounded monthly, find the amount of money you will need to invest per month: A. if you start this process as soon as you turn 30 (so you invest for 35 years total). B. if you start this process as soon as you turn 40 (so you invest for 25 years total). 8 IV. Amortization Of course, investing (i.e. making money) is nice, but this same rationale can be applied to debt as well. The situation has a slightly added complication in that the debt owes grows in time, and so the payments made must cover not only the debt incurred at the present time, but must account for interest accrued along the way! This is the concept of amortization, a term used to describe the paying off of debt with a fixed repayment schedule in regular installments over a period of time. The following example walks you through a typical amortization calculation. Suppose Patrick has accumulated $10,000 in credit card debt. He decides to stop spending immediately, and pay P per month required until his debt is paid off. His credit card charges an annual interest rate of 18%, compounded monthly. We will answer the following questions: how long will it take him to pay off his debt and how much will he pay under different values of P ? To get started, we first figure out the amount owed after the first few months. To follow the line of reasoning here, it is highly recommended that you READ each line aloud and WRITE down each step after you say it. This will be hard to follow if you just try to read the lines below and not write anything down! While you are writing this down, try to get a sense for what it is you are computing! Month 1: The amount owed is $10000 and P is repaid. The remaining balance is thus 10000 P . This is the amount to which interest will be applied, so at the end of the first month (i.e. before the payment of P is made at the beginning of the second month, \u0012 \u0013 .18 (10000 P ) 1 + = 1.015(10000) 1.015P 12 is owed. Thus, the amount that Patrick owes after the payment is made at the start of month 2 is: 10000(1.015) [P + 1.015P ] Month 2: The amount owed at the end of the second month is 1.015(10000) [P + 1.015P ]. This is the amount to which the interest will be applied. After the second month (before the payment is made at the start of month 3), the amount owed is: [(1.015)10000 (P + 1.015P )] 1.015 = 10000(1.015)2 [P (1.015) + P (1.015)2 ]. Thus, the amount that Patrick owes after the payment is made at the start of month 3 is: 10000(1.015)2 [P + 1.015P + (1.015)2 P ] 9 Problem 9: By applying the same logic above: A. Repeat the logic above to calculate the amount owed after Patrick makes the payment after the start of Month 4. B. Hopefully, the pattern is clear! Call the amount owed after the payment at the start of Month n An . Based off of this pattern, write down a formula for An . Then, using the geometric sum formula developed in Problem 5, show that the sum can be expressed as: An = 10000(1.015)n1 10 P [(1.015)n 1] . .015 The result above can be interpreted as follows: After the payment at the beginning of Month n is made, the Debt of $10,000 have grown to: D = 10000(1.015)n1 . This is equivalent to applying the compound interest formula to the original debt if one notes that at the start of Month n, interest has been applied to the debt n 1 times (since interest is applied at the end of the previous month). After the payment at the beginning of Month n is made, the future value of the amount he will have Repaid will be: P R= [(1.015)n 1] . .015 This is equivalent to applying the annuity formula to calculate the total future value of his payments! When the future value of his debt is equal to the future value of his amount repaid, Patrick is out of debt! Problem 10: A. If P = $100, will Patrick ever get out of debt? Justify your response! 11 B. For both P = $500 answer both: i. How long will it take for Patrick to get out of debt? ii. How much will he have to pay his credit card company? To answer ii. note that if you get a fraction of a month, the amount owed at the beginning of the final month will be less than P ! You should calculate the amount owed at the end of the month before Patrick makes the payment at the start of Month n. His payment for Month n will just be the remaining balance! Hint: If you perform the calculations for P = 200, you should find it takes 91 months to get out of debt, and that the total amount repaid is $18,039.72. This is NOT an easy calculation, so ask if you have questions! 12 Paying Off Student Loans Suppose that after graduation that Sarah must pay back $80,000 in student loans and that she has 15 years to do so. She has a direct subsidized undergraduate loan with an interest rate of 4.29%, compounded monthly and this interest starts to accrue the month after she graduates. The terms of her loan are such that she will make a payment at the beginning of each month for 15 years, starting with the month she graduates. Thus, she will make a total of 15 12 = 180 payments. Problem 11: The following will guide you through calculating what Sarah will pay each month. Note that this can be treated analogously to the Credit Card Debt example! Make sure you adapt the formula used there! A. Show that the future value of the Debt, D, after 15 years (n = 180) is $151,534.07. B. Suppose Sarah pays back P at the start of each month. Find the total amount Repaid, R, after 15 years (n = 180) in terms of P . C. Sarah will be out of debt when D = R. Since you know D = R when n = 180, equate the above results to find the value for P that Sarah must repay at the start of each month. 13 Note that this example can be used as a template to pay off any type of loan; it can be modified to calculate car payments or mortgage payments! In fact, the repayment scheme in general can be modified to account for paying over the amount P required each term (though you will not be asked to do so here!). Hopefully, this assignment has convinced you that there is indeed a variety of uses for geometric sums! Though the above examples give only a glimpse into how geometric series can and do arise in personal finances, they provide solid examples that the geometric sum formula is much more powerful than one may think at a first glance! 14