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CHAPTER 13 Compound Interest, Future Value, and Present Value

CHAPTER 13 Compound Interest, Future Value, and Present Value. Learning Outcomes. 13-1. Find the future value and compound interest by compounding manually. Find the future value and compound interest using a $1.00 future value table.

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CHAPTER 13 Compound Interest, Future Value, and Present Value

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  1. CHAPTER 13 Compound Interest, Future Value, and Present Value

  2. Learning Outcomes 13-1 • Find the future value and compound interest bycompounding manually. • Find the future value and compound interest using a $1.00 future value table. • Find the future value and compound interest using a formula or a calculator application (optional). • Find the effective interest rate. • Find the interest compounded daily using a table.

  3. Interest Section 13-1 Compound Interest and Future Value • In some loans, interest is computed once during the life of the loan, using the simple interest formula. • In other loans, interest is computed more than once during the life of the loan or investment. • The interest is added to the principal, the new amount becoming the principal for the next interest calculation. • This process is called compounding interest.

  4. Interest period The amount of time after which interest is calculated and added to the principal. Compound interest The total interest that has accumulated aftermore than one interest period. Key Terms… Section 13-1 Compound Interest and Future Value

  5. Future value, maturity value, compound amount The accumulated principal and interest after one or more interest periods. Period interest rate The rate for calculating interest for one interest period; the annual interest rate divided by the number of interest periods per year. Key Terms… Section 13-1 Compound Interest and Future Value

  6. Find the Future Value and Compound Interest by Compounding Manually 13-1-1 Section 13-1 Compound Interest and Future Value • Dividing the annual interest rate by the annual number of interest periods gives us the period interest rate. • For example, 12% annual interest rate divided by 2 interest periods yields a period interest rate of 6%. • Using I = P x R x T, we can calculate the interest per period. • Simplifying the formula to I = P x R, since T isone period.

  7. Period interest rate = Find the period interest rate HOW TO: Section 13-1 Compound Interest and Future Value

  8. A 12% annual interest rate with 4 interest periods per year. 3% An 18% annual rate with 12 interest periods per year. 1 ½ % An 8% annual rate with 4 interest periods per year. 2% Examples… Section 13-1 Compound Interest and Future Value

  9. Using the simple interest formula method: Find the future value HOW TO: Section 13-1 Compound Interest and Future Value STEP 1 Find the end of period principal—multiply the original principal by the sum of 1 and the period interest rate. STEP 2 For each remaining period in turn, find the next end of period principal: multiply the previous end of period principal by the sum of 1 and the period interest rate. STEP 3 Identify the last end-of-period principal as the future value.

  10. An Example… Section 13-1 Compound Interest and Future Value Find the future value of a loan of$800 at 13% for three years. The period interest rate is 13%since it is calculated annually. First end-of-year = $800 x 1.13 = $904 Second end-of-year =$904 x 1.13 = $1021.52 Third end-of-year = $1021.52 x 1.13 = $1,154.32 The FV of this loan is $1,154.32.

  11. First end-of-year = $800 x 1.13 = $904 Second end-of-year =$904 x 1.13 = $1021.52 Third end-of-year = $1021.52 x 1.13 = $1,154.32 The FV of this loan is $1,154.32. Find the compound interest HOW TO: Section 13-1 Compound Interest and Future Value Compound interest =future value – original principal In this example, thecompound interest is equal to: CI = $1,154.32 – $800 = $354.32 The compound interest is $354.32.

  12. Compare the compound interestamount to the simple interest CI = $354.32 Simple interest for the same loan would be: I = PRT I = $800 x 0.13 x 3 = $312.00 Simple interest would be $312.00 The difference between compound interest andsimple interest for this loan = $354.32 – $312 The difference is $43.32. HOW TO: Section 13-1 Compound Interest and Future Value

  13. Principal = $10,000 8% annual interest rate,compounded semi-annually Find the FV at the end of three years. Find the period interest rate: 8% ÷ 2 = 4% Determine number of periods: 3 x 2 = 6 Calculate each end-of-period principal. Period 1 = $10,000 x 1.04 = $10,400 Find the FV of an investment HOW TO: Section 13-1 Compound Interest and Future Value

  14. Find the FV of an investment HOW TO: Section 13-1 Compound Interest and Future Value Principal = $10,0008% annual interest rate,compounded semi-annually Find the FV at the end of three years. Find the period interest rate: 8% ÷ 2 = 4% Determine number of periods: 3 x 2 = 6 Calculate each end-of-period principal. Period 1 = $10,000 x 1.04 = $10,400 Period 2 = $10,400 x 1.04 = $10,816 Calculate through sixth end-of-period principal. Final end-of-principal amount = $12,653.19

  15. Find the future value and compound interestusing a $1.00 future value table. See page461 13-1-2 Section 13-1 Compound Interest and Future Value • It would be tedious and time-consuming to calculate a large number of periods with the previous method. • Use Table 13-1, which is the futurevalue or compound amount of $1.00. • Find the number of periodsand the rate per period to identify the value by which theprincipal is multiplied.

  16. Find the future value and compound interestusing a $1.00 future value table. HOW TO: Section 13-1 Compound Interest and Future Value Using Table 13-1, find the compound interest on$500 for six years compounded annually at 8%. Determine the number of periods: 6 Determine the interest rate per period: 8% Locate the value in the intersecting cell: 1.58687 Multiply the principal: $500 x 1.58687 = $793.44 The FV of the loan is $793.44. Compound interest = $793.44 – $500 =$293.44

  17. Using Table 13-1, find the future valueandcompound intereston $2,000 invested forfour years compounded semiannually at 8%. FV = $2,737.14 CI = $737.14 What would the simple interestbe for the same loan? $640 An Example… Section 13-1 Compound Interest and Future Value

  18. Find the Future Value and CompoundInterest Using a Formula (optional) 13-1-3 Section 13-1 Compound Interest and Future Value The formula for finding future value willrequire a calculator that has a power function. The future value formula is: FV = P(1 + R)N …where FV is the future value, P isthe principal, R is the period interestrate, and N is the number of periods.

  19. Find the future value and compound interest for a 3-year $5,000 investment that earns 6% compounded monthly. An Example… Section 13-1 Compound Interest and Future Value FV = P(1 + R)N FV = 5,000(1 + 0.005)36 FV = $5,983.40 CI = $5,983.40 – $5,000 = $983.40

  20. Scientific: BA II Plus: Same Example, using a calculator… Section 13-1 Compound Interest and Future Value

  21. TI-84 Highlight END Same Example, using a calculator… Section 13-1 Compound Interest and Future Value

  22. Find the effective interest rate 13-1-4 Section 13-1 Compound Interest and Future Value • Effective interest rate is also called the annual percentage yield or APY when identifying rateof earning on an investment. • It is called APR, annual percentage rate, when identifying the rate of interest on a loan.

  23. Effective rate The equivalent simple interest rate that is equivalent to a compound rate. Key Terms… Section 13-1 Compound Interest and Future Value

  24. Marcia borrowed $600 at 10%compounded semiannually. What is the effective interest rate? Using the manual compound interest method: First end-of-period principal = $600 x 1.05 = $630 Second end-of-principal = $630 x 1.05 = $661.50 Compound interest after first year = $61.50 Period rate interest = = 5% = 0.05 An Example… Section 13-1 Compound Interest and Future Value

  25. Annual effective interest rate = Multiplied by 100%: 0.1025 x 100% = 10.25% Using the table method (Table 13-1) The table value is 1.10250. Subtract 1.00 and multiply by 100%. The effective rate is 10.25%. Effective interest rate HOW TO: Section 13-1 Compound Interest and Future Value

  26. Find the Interest CompoundedDaily Using a Table See page469 13-1-5 Section 13-1 Compound Interest and Future Value • Table 13-2 gives compound interestfor $100 compounded daily (using 365 days as a year). • Exactly like using Table 11-2, which gives the simple intereston $100. Table 13-2 uses $100 as the principal amount;other tables may use $1, $10 or other amounts.

  27. Find the interest on $800 at 7.5% annually,compounded daily for 28 days. Find the corresponding value by intersectingthe number of days (28) and annualinterest rate (7.5%): value =0.576941 Multiply this value by 8 to get $4.62. The compounded interest is $4.62. See page469 An Example… Section 13-1 Compound Interest and Future Value Divide the principal by $100 as you areusing Table 13-2. ($800 ÷ $100 = 8)

  28. Find the interest on $1,000 for 30 dayscompounded at a 6% annual rate. Divide $1,000 ÷ 100 = 10Locate the cell where 30 days and 6%intersect to determine the value: 0.494328 Multiply by 10. The interest is$4.94. See page466 An Example… Section 13-1 Compound Interest and Future Value

  29. Learning Outcomes 13-2 • Find the present value based on annual compounding for one year. • Find the present value using a $1.00 present value table. • Find the present value using a formula or a calculator application (optional).

  30. Find the Present Value Based onAnnual Compounding for One Year 13-2-1 Section 13-2 Present Value • Using the concepts of compound interest, you can determine amounts needed now to cover expenses in the future. • The amount of money you set aside now is called present value.

  31. The simplest case is annualcompounding interest for one year. The number of interest periods is 1 and theperiod interest rate is the annual interest rate. Principal (present value) = * denotes decimal equivalent Present value HOW TO: Section 13-2 Present Value

  32. Find the amount of money that The 7th Inning needs to set aside today to ensure that $10,000 will be available to buy a new large screen plasma television in one year if the annual interest rate is 4% compounded annually. An investment of $9,615.38 at 4%would have a value of $10,000 in one year. PV = An Example… Section 13-2 Present Value

  33. Calculate the amount of money needed now to purchase a laptop computer and accessories valued at $2,000 in a year if you invest the money at 6%. $1,886.79 John wants to replace a tool valued at $150 in a year. How much money will he have to put into an account that pays 3% annual interest? $145.63 Examples… Section 13-2 Present Value

  34. See page475 Find Present Value Using a $1.00 PV Table 13-2-2 Section 13-2 Present Value • Using a present value table is the most efficient way to calculate the money needed now for a future expense or investment. Table 13-3 shows the present value of $1.00at different interest rates for different periods.

  35. Period interest rate = Use table 13-3 HOW TO: Section 13-2 Present Value STEP 1 Find the number of interest periods—multiply the time period in years by number of interest periods per year. Interest periods =number of years x number of interest periods per year STEP 2 Find the interest rate—divide the annual interest rate by the number of interest periods per year.

  36. Use table 13-3 HOW TO: Section 13-2 Present Value STEP 3 Select the periods row corresponding to the number of interest periods. STEP 4 Select the rate per period column corresponding to the period interest rate. STEP 5 Locate the value in the cell where the periods row intersects the rate-per-period column. STEP 6 Multiply the future value by value from step 5.

  37. The 7th Inning needs $35,000 in 4 yearsto buy new framing equipment. How much should be invested at 4%interest compounded annually? 4 periods at 4% shows a value of 0.85480 Multiply this value by $35,000 The result is $29,918. They must invest $29,918 at 4% compoundedannually for four years to have $35,000. An Example… Section 13-2 Present Value

  38. How much money would you have to invest for 5 years at 6% paid semi-annually to make a down payment of $20,000 on a house? $14,881.80 How much money would you have to invest for 3 years at 10% paid semi-annually to purchase an automobile that costs $20,000? $14,924.40 Examples… Section 13-2 Present Value

  39. Find the Present Value Using a Formula (optional) PV = 13-2-3 Section 13-2 Present Value The present value formula: …where PV is the present value, FVis the future value, Ris the period interest rate, and N isthe number of periods.

  40. PV = PV = = $6,846.78 Period int. rate = = 0.0043333333 Examples… Section 13-2 Present Value Find the present value required at 5.2% compoundedmonthly to total $8,000 in three years.

  41. The same example using calculator keystrokes Scientific: BA II Plus: Section 13-2 Present Value

  42. The same example using calculator keystrokes TI-84: HighlightEND Section 13-2 Present Value

  43. Exercises Set A

  44. EXERCISE SET A 2. Use Table 13-1 or the appropriate formula for the Exercise. Number of periods = 4(4) = 16 Period rate = 4%/4 = 1% Table value = 1.17258 Compound amount = 5,000(1.17258) = 5,862.90 Compound interest = 5,862.90 - 5,000 = 862.90

  45. EXERCISE SET A 4. Use Table 13-1 or the appropriate formula for the Exercise. Number of periods = 4(2) = 8 Period rate = 1%/2 = 0.5% Table value = 1.04071 Compound amount = 8,000(1.04071) = 8,325.68 Compound interest = 8,325.68 - 8,000 = 325.68

  46. EXERCISE SET A 6. Find the amount that should be set aside today to yield the desired future amount. Use Table 13-3 or the present value formula. Number of periods = 6(4) = 24 periods Period rate = 6%/4 = 1.5% Table value = 0.69954 Present value = 8,000(0.69954) = 5,596.32

  47. EXERCISE SET A 8. Find the amount that should be set aside today to yield the desired future amount. Use Table 13-3 or the present value formula. Number of periods = 20(1) = 20 Period rate = 3%/1 = 3% Table value = 0.55368 Present value = 14,700(0.55368) = 8,139.10

  48. EXERCISE SET A 10. Manually calculate the compound interest on a 13% loan of $1,600 for three years if the interest is compounded annually. 1,600(1.13) = 1,808 1,808(1.13) = 2,043.04 2,043.04(1.13) = 2,308.64 compound amount 2,308.64 - 1,600 = 708.64 interest

  49. EXERCISE SET A 12. Use Table 13-1 or the appropriate formula to find the interest on a certificate of deposit (CD) of $10,000 for five years at 4% compounded semiannually. 5 years(2 periods per year) = 10 periods 4%  2 periods per year = 2% per period Table value = 1.21899 10,000(1.21899) = 12,189.90 compound amount 12,189.90 - 10,000 = 2,189.90

  50. EXERCISE SET A 14. Find the compound interest on a loan of $5,000 for two years if the interest is compounded quarterly at 12%. 2(4) = 8 periods 12%  4 = 3% per period Table value = 1.26677 5,000(1.26677) = 6,333.85 compound amount Compound interest = 6,333.85 - 5,000 = 1,333.85

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