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Chapter 10

Chapter 10. The Mathematics of Finance. Outline. 10.1 Interest 10.2 Annuities 10.3 Amortization of Loans 10.4 Personal Financial Decisions. 10.1 Interest. Definitions for Savings Account Common Compounding Periods New from Previous Balance Present and Future Value Simple Interest

lillian-schroeder
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Chapter 10

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  1. Chapter 10 The Mathematics of Finance Goldstein/Schnieder/Lay: Finite Math & Its Applications

  2. Outline • 10.1 Interest • 10.2 Annuities • 10.3 Amortization of Loans • 10.4 Personal Financial Decisions Goldstein/Schnieder/Lay: Finite Math & Its Applications

  3. 10.1 Interest • Definitions for Savings Account • Common Compounding Periods • New from Previous Balance • Present and Future Value • Simple Interest • Effective Rate of Interest • Calculator Solutions Goldstein/Schnieder/Lay: Finite Math & Its Applications

  4. Definitions for Savings Account • Interest is the fee a bank pays for the use of money deposited into a savings account. • The amount deposited is called the principal. • The amount to which the principal grows (after the addition of interest) is called the compound amount or balance. • If interest is compounded m times per year and the annual interest rate is r, then the interest rate per period is i = r/m. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  5. Example Definitions for Savings Account • For the passbook above, determine the principal, compound amount after 1 year, compound interest rate and annual interest rate. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  6. Example Definitions - Savings Account (2) • The principal is $100.00. • The compound amount after 1 year is $104.06. • The compound interest rate is 1% since the interest earned in the first period, $1.00, is 1% of the principal. • Interest is compounded 4 times per year so the annual interest rate is 4·1% = 4%. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  7. Common Compounding Periods Goldstein/Schnieder/Lay: Finite Math & Its Applications

  8. New from Previous Balance • For a savings account in which the interest rate per period is i, the interest earned during a period is i times the previous balance. • The new balance, Bnew is computed by adding the interest earned during the period to the previous balance, Bprevious. • Bnew = Bprevious + i · Bprevious • Bnew = (1 + i)Bprevious Goldstein/Schnieder/Lay: Finite Math & Its Applications

  9. Example New from Previous Balance • Compute the interest and the balance for the first two interest periods for a deposit of $1000 at 4% compounded semiannually. • For semiannually m = 2 so i = (4/2)% = 2% = .02. • First period: interest = .02(1000) = $20 • B1 = 1000 + 20 = $1020 • Second period: interest = .02(1020) = $20.40 • B2 = 1020 + 20.40 = $1040.40 Goldstein/Schnieder/Lay: Finite Math & Its Applications

  10. Present and Future Value • Let i be the interest per period, P the principal and F the balance after n periods, then • F is also referred to as the future value and P as the present value. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  11. Example Present and Future Value • If an account pays 6% compound quarterly, • a) find the amount in the account after 5 years if $1000 is initially deposited; • b) find the amount that must be initially deposited if $3000 is needed in 5 years. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  12. Example Present and Future Value (2) • Period interest is i = .06/4 = .015. • Number of periods is n = 4*5 = 20. • a) P = $1000 so F = 1000(1 + .015)20 = $1346.86. • b) F = $3000 so Goldstein/Schnieder/Lay: Finite Math & Its Applications

  13. Simple Interest • Simple interest is earned only on the principal and is not compounded. • If r is the annual percentage rate and n is the number of years, then • Interest = nrP, and • F = P + nrP = (1 + nr)P. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  14. Example Simple Interest • Calculate the amount after 4 years if $1000 is invested at 5% simple interest. • F = (1 + 4(.05))1000 = $1200 Goldstein/Schnieder/Lay: Finite Math & Its Applications

  15. Effective Rate of Interest • The effective rate of interest is the simple interest rate that yields the same amount after one year as the annual rate of interest. • If r is the annual interest rate compounded m times a year, then i = r/m and • reff = (1 + i)m – 1. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  16. Example Effective Rate of Interest • Calculate the effective rate of interest for a savings account paying 3.65% compounded quarterly. • reff = (1 + .0365/4)4 - 1 = .037 • So the effective rate is 3.7%. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  17. Calculator Solutions • Use a calculator to determine when the balance in a savings account in which $100 is deposited at 4% compounded quarterly reaches $130. • For a TI-83 set • Y1 = (1 + .04/4)^X*100 and • Y2 = 130. • Find the intersection of the two graphs. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  18. Calculator Solutions (2) • The intersection is at X = 26.367391, so in 27 quarters the balance will exceed $130. Graph of Y1 and Y2 with intersection Table of Y1 Goldstein/Schnieder/Lay: Finite Math & Its Applications

  19. Summary Section 10.1 - Part 1 • Money deposited into a savings account earns interest at regular time periods. Interest paid on the initial deposit only is called simple interest. Interest paid on the current balance (that is, on the initial deposit and the accumulated interest) is called compound interest. • Successive balances of a savings account with compound interest can be calculated with Bnew = (1 + i)Bprevious. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  20. Summary Section 10.1 - Part 2 • P - principal, the initial amount of money deposited into a savings account. P also represents the present value of a sum of money to be received in the future; that is, the amount of money needed to generate the future money. • r - annual rate of interest, interest rate stated by the bank and used to calculate the interest rate per period. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  21. Summary Section 10.1 - Part 3 • m - number of (compound) interest periods per year, most commonly 1, 4, or 12. • i - compound interest rate per period, calculated as r/m. • n - number of interest periods. • F - future value, compound amount, or balance, value in a savings account. F = (1 + i)nP with compound interest, and F = (1 + nr)P with simple interest. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  22. Summary Section 10.1 - Part 4 • reff - effective rate of interest, the simple interest rate that yields the same amount after one year as the annual rate of interest. reff = (1 + i)m – 1 Goldstein/Schnieder/Lay: Finite Math & Its Applications

  23. 10.2 Annuities • Definitions of Annuity • Future Value • Rent for a Future Value • Present Value and Rent • Storing and Goldstein/Schnieder/Lay: Finite Math & Its Applications

  24. Definitions of Annuities An annuity is a sequence of equal payments made at regular intervals of time. The payments are called rent. The amount in an increasingannuity gets larger with each payment and the final value is called the future value of the annuity. The amount in a decreasingannuity gets smaller with each payment and the amount at the beginning is called the present value of the annuity. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  25. Example Definitions of Annuities • Parents decide to deposit $100 at the end of each month into a savings account for the college education of their child. After 216 payments, the account will contain $38,735.32. • You have just sold your house and deposit your profit of $258,627.80 into an account so you can withdraw $5000 at the end of each month for 5 years at which time the balance will be $0. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  26. Example Definitions of Annuities (2) • The first example is of an increasing annuity with rent = $100 and future value = $38,735.32. • The second example is of a decreasing annuity with rent = $5000 and present value = $258,627.80. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  27. Future Value • Suppose that an increasing annuity consists of n payments of $R each, deposited at the ends of consecutive interest periods into an account with interest compounded at a rate i per period. Then the future value F of the annuity is Goldstein/Schnieder/Lay: Finite Math & Its Applications

  28. Example Future Value • Calculate the future value of an increasing annuity of $100 per month for 2 years at 6% interest compounded monthly. • R = 100, i = .06/12 = .005 and n = 2(12) = 24. • To calculate on a TI-83 calculator, key in • So = 25.43195524. • F = (25.43195524)(100) = $2,543.20. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  29. Rent for a Future Value • Suppose that an increasing annuity of n payments has future value F and has interest compounded at the rate i per period. Then the rent R is Goldstein/Schnieder/Lay: Finite Math & Its Applications

  30. Example Rent for a Future Value • Ms. Adams would like to buy a $30,000 airplane when she retires in 8 years. How much should she deposit at the end of each half-year into an account paying 4% interest compounded semiannually so that she will have enough money to purchase the airplane? Goldstein/Schnieder/Lay: Finite Math & Its Applications

  31. Example Rent for a Future Value (2) • F = 30,000, i = .04/2 = .02 and n = 8(2) = 16. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  32. Present Value and Rent • The present value P and the rent R of a decreasing annuity of n payments with interest compounded at a rate i interest per period are related by the formulas Goldstein/Schnieder/Lay: Finite Math & Its Applications

  33. Example Present Value and Rent • a) How much money must you deposit now at 6% interest compounded quarterly in order to be able to withdraw $3000 at the end of each quarter-year for 2 years? • b) How much money could you withdraw each quarter-year for 2 years if you deposited $24,000 into the same account? Goldstein/Schnieder/Lay: Finite Math & Its Applications

  34. Example Present Value and Rent • a) R = 3000, i = .06/4 = .015 and n = 4(2) = 8. • b) P = 24000, i = .06/4 = .015 and n = 4(2) = 8. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  35. Storing and • As a time-saving device, the formulas for • can be assigned to the Y= editor on the TI-83 calculator. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  36. Example Calculating Number of Periods • Use a graphing calculator to determine when the balance in an account in which $100 is deposited monthly at 6% interest compounded monthly will exceed $10,000. • Assuming Y4 contains the formula for • store .005 into I on the home screen. • In the Y= menu, define Y1 = 100Y4. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  37. Example Calculating Number of Periods (2) • Scroll down the table for Y1 until Y1 exceeds 10000. This occurs when X = 82. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  38. Summary Section 10.2 - Part 1 • An increasing (decreasing) annuity is a sequence of equal deposits (withdrawals) made at the ends of regular time intervals. • F - future value, compound amount, or balance, value in an annuity at some point in the future. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  39. Summary Section 10.2 - Part 2 • R - rent, periodic deposit into or withdrawal from an annuity. • - s sub n angle i, future value of an increasing annuity of n $1 payments at compound interest rate i per period. For an increasing annuity, Goldstein/Schnieder/Lay: Finite Math & Its Applications

  40. Summary Section 10.2 - Part 3 • - a sub n angle i, present value of a decreasing annuity of n $1 payments at compound interest rate i per period. For a decreasing annuity, Goldstein/Schnieder/Lay: Finite Math & Its Applications

  41. Summary Section 10.2 - Part 4 • Successive balances of an increasing annuity can be calculated with Bnew = (1 + i)Bprevious + R. • Successive balances of a decreasing annuity can be calculated with Bnew = (1 + i)Bprevious - R. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  42. 10.3 Amortization of Loans • Amortization and Mortgage • Repayment Process • Unpaid Balance I • Unpaid Balance II • Balloon Payment • Calculator Application Goldstein/Schnieder/Lay: Finite Math & Its Applications

  43. Amortization and Mortgage • Loans under consideration will be repaid in a sequence of equal payments at regular time intervals, with the payment intervals coinciding with the interest periods. The process of paying off such a loan is called amortization. • A mortgage is a long-term loan used to purchase real estate. The real estate is used as collateral to guarantee the loan. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  44. Repayment Process • 1. Payments are made at the end of each interest period. • 2. The interest to be paid each interest period is the period interest rate, i, times the unpaid balance at the end of the previous interest period. • 3. The unpaid balance at the end of the interest period is the previous unpaid balance plus the interest owed for the current interest period minus the payment. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  45. Unpaid Balance I • For a loan amortized over n payments with payments R, the unpaid balance at the end of the kth payment is the present value of a decreasing annuity with the same i and R but with n - k payments. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  46. Example Amortization • On Dec. 31, 1990, a house was purchased with the buyer taking out a 30-year, $112,475 mortgage at 9% interest, compounded monthly. The mortgage payments are made at the end of each month. • a) Calculate the amount of the monthly payment. • b) Calculate the unpaid balance of the loan on Dec. 31, 2016, just after the 312th payment. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  47. Example Amortization - continued • c) How much interest will be paid during the month of January 2017? • d) How much of the principal will be paid off during the year 2016? • e) How much interest will be paid during the year 2016? Goldstein/Schnieder/Lay: Finite Math & Its Applications

  48. Example Amortization (a) • A mortgage is a decreasing annuity. P = 112475, i = .09/12 = .0075 and n = (30)(12) = 360. Goldstein/Schnieder/Lay: Finite Math & Its Applications

  49. Example Amortization (b) • There are 360 - 312 = 48 remaining payments. • Therefore, Goldstein/Schnieder/Lay: Finite Math & Its Applications

  50. Example Amortization (c) • The interest paid during January 2017 is i times the unpaid balance at the end of December 2016 which was calculated in (b). • Interest = .0075(36367.23) = $272.75. Goldstein/Schnieder/Lay: Finite Math & Its Applications

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