Compound Interest and Present Value ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

PERFORMANCE OBJECTIVES Section I Compound Interest—The Time Value of Money 11-1: Manually calculating compound amount (future value) and compound interests 11-2: Computing compound amount (future value) and compound interest by using compound interest tables 11-3: Creating compound interest table factors for periods beyond the table 11-4: Calculating annual percentage yield (APY) or effective interest rate 11-5: Calculating compound amount (future value) by using the compound interest formula ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

PERFORMANCE OBJECTIVES continued Section II Present Value 11-6: Calculating the present value of a future amount by using present value tables 11-7: Creating present value table factors for periods beyond the table 11-8: Calculating present value of a future amount by using the present value formula ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Compound Interest—The Time Value of Money Interest that is applied a number of times during the term of a loan or investment. Interest paid on principal and previously earned interest. time value of money The idea that money “now,” or in the present, is more desirable than the same amount of money in the future, because it can be invested and earn interest as time goes by. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Compound Interest—The Time Value of Money continued compound amount, or future value (FV) The total amount of principal and accumulated interest at the end of a loan or investment. present amount, or present value (PV) An amount of money that must be deposited today, at compound interest, to provide a specified lump sum of money in the future. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

EXHIBIT 11-1 The Time Value of Money ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

EXHIBIT 11-2 The Time Value of Money ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Manually Calculating Compound Interest Using the simple interest formula, I = PRT, calculate the interest earned per period. Each time interest is calculated, it is then added to the principal to determine the new principal. Compound interest = Compound amount – Principal ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Calculating Compound Interest Example What is the compound amount and the compound interest of $10,000 at 12% interest, compounded semiannually for 3 years? Original principal = 10,000.00 Interest period 1 = PRT = 10,000.00 × .12 × .5 = + 600.00 Principal period 2 = 10,600.00 Interest period 2 = PRT = 10,600.00 × .12 × .5 = + 636.00 Principal period 3 = 11,236.00 Interest period 3 = PRT = 11,236.00 × .12 × .5 = + 674.16 Principal period 4 = 11,910.16 Interest period 4 = PRT = 11,910.16 × .12 × .5 = + 714.61 Principal period 5 = 12,624.77 Interest period 5 = PRT = 12,624.77 × .12 × .5 = + 757.49 Principal period 6 = 13,382.26 Interest period 6 = PRT = 13,382.26 × .12 × .5 = + 802.94 Compound amount = 14,185.20 Original principal = -10,000.00 Compound interest = 4,185.20 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Compound amount (future value) Compounding periods Computing Compound Amount (Future Value) and Compound Interest By Using Compound Interest Tables Compound amount (future value) Table factor × Principal Compounding periods Years × Periods per year Interest rate per period Nominal rate Periods per year ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

TABLE 11-1 Compound Interest Table (Future Value of $1 at Compound Interest) ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Compounding Periods per Year EXHIBIT 11-3 Compounding Periods per Year ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

For using compound interest tables Step 1 Scan across the top row to find the interest rate per period. Step 2 Look down that column to the row corresponding to the number of periods. Step 3 The table factor at the intersection of the rate-per- period column and the number-of-periods row is the future value of $1 at compound interest. Multiply the table factor by the principal to determine the compound amount. Compound amount = Table factor × Principal ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Using Compound Interest Tables Example Compounding periods = years  periods per year Compounding periods = Interest rate per period = nominal rate ÷ periods per year Interest rate period = Compound amount (FV) = table factor  principal Compound amount (FV) = Compound interest = 3 × 2 = 6 12% ÷ 2 = 6% 1.41852  10,000 = 14,185.20 14,185.20 – 10,000 = 4,185.20 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

For creating new compound interest table factors Step 1 For the stated interest rate per period, find the two table factors that represent half, or values as close as possible to half, of the periods required. Step 2 Multiply the two table factors from Step 1 to form the new factor. Step 3 Round the new factor to five decimal places. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Creating Compound Interest Table Factors for Periods Beyond the Table Example Calculate a new table factor and find the compound amount of $3,500 invested at 16% interest compounded quarterly, for 7 years. Interest rate per period = 16% ÷ 4 = 4% 28 Compounding periods = 7  4 = 4%, 14 periods = 1.73168 4%, 14 periods =  1.73168 28 periods 2.9987156 Rounded new table value = 2.99872  3,500 Compounded amount = $10,495.52 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Annual Percentage Yield (APY) or Effective Interest Rate annual or nominal rate The advertised or stated interest rate of an investment or loan. The rate used to calculate the compound interest. annual percentage yield (APY) or effective rate The real or true rate of return on an investment. It is the total compound interest earned in 1 year divided by the principal. The more compounding periods per year, the higher the APY. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Compound Interest Earned on $100 at 12% EXHIBIT 11-4 Compound Interest Earned on $100 at 12% ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Calculating Annual Percentage Yield (APY) or Effective Interest Rate What is the compound amount, compound interest, and annual percentage yield of $7,000 invested for 1 year, at 6% interest, compounded quarterly. Interest rate per period = Compound periods = Compound amount = Compound interest = 6% ÷ 4 = 1.5% 1  4 = 4 1.06136  7,000 = $7,429.52 7,492.52 – 7,000 = $429.52 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Calculating Compound Amount (Future Value) by Using the Compound Interest Formula ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

For using present value tables Step 1 Add the 1 and the interest rate per period, i. Step 2 Raise the sum from Step 1 to the nth (number of compounding periods) power by using the yx key on your calculator. Step 3 Multiply the principal, P, by the answer from Step 2. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

For using present value tables Step 1 Scan across the top row to find the interest rate per period. Step 2 Look down that column to the row corresponding to the number of periods. Step 3 The table factor found at the intersection of the rate-per-period column and the number-of-periods row is the present value of $1 at compound interest. Multiply the table factor by the compound amount to determine the present value. Present value = Table factor × Compound amount (future value) ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Present Value to Future Value EXHIBIT 11-5 Present Value to Future Value ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

TABLE 11-2 Present Value Table (Present Value of $1 at Compound Interest) ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Calculating Present Value Use Table 11-2 to find how much must be invested now at 8% interest compounded quarterly to have $3,000,000, 3 years from now. Interest rate per period = Compounding periods = Present value = 8% ÷ 4 = 2% 3  4 = 12 .78849  3,000,000 = $2,365,470 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

For Creating New Table Factors Step 1 For the stated interest rate per period, find the two table factors that represent half, or values as close as possible to half, of the periods required. Step 2 Multiply the two table factors from Step 1 to form the new factor. Step 3 Round the new factor to five decimal places. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Creating Present Value Table Factors for Periods Beyond the Table Example Calculate a new table factor and find the present value of $8,500, if the interest rate is 6% compounded quarterly, for 10 years. Interest rate per period = 6% ÷ 4 = 1.5% Compound periods = 10  4 = 40 1.5%, 20 periods = .74247 1.5%, 20 periods =  .74247 1.5%, 40 periods = .5512617 New table factor (rounded)= .55126  8,500 Present value of $8,500.00 = $4,685.71 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Calculating the Present Value of a Future Amount by Using the Present Value Formula ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

For SOLVING THE PRESENT VALUE FORMULA Step 1 Add the 1 and the interest rate per period, i. Step 2 Raise the sum from Step 1 to the nth power by using the yx key on your calculator. Step 3 Divide the compound amount, A, by the answer from step 2. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter Review Problem 1 A firm is planning to build five homes, each costing $125,000, in 2 ½ years. The bank pays 6% interest compounded semiannually. How much should the firm invest now to have sufficient funds to build the homes in the future? Amount needed = Interest rate per period = Compound periods = Present value = 125,000  5 = $625,000 6% ÷ 2 = 3% 2.5  2 = 5 .86261  625,000 = $539,131.25 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter Review Problem 2 As a savings plan for college, a family deposited $10,000 in an account paying 8% compounded annually when their son was born. How much will the account be worth when their son is 18 years old? Interest rate per period = Compound periods = Present value = 8% ÷ 1 = 8% 18  1 = 18 3.99602  10,000 = $39,960.20 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter Review Problem 3 A firm deposited $500,000 in an account earning 12% compounded monthly to pay for the future construction of a warehouse. How much will be available for the project in 2 ½ years? Interest rate per period = 12% ÷ 12 = 1% 30 Compounding periods = 2.5  12 = 1%, 15 periods = 1.16097 1%, 15 periods =  1.16097 30 periods 1.34785 Rounded new table value = 1.34785 Amount deposited =  500,000 Compounded amount = $673,925.65 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter Review Problem 4 A bank is offering a 6-year certificate of deposit (CD) at 4% interest, compounded quarterly. What is the annual percentage yield for an investment of $8,000? Interest rate per period = Compound periods = Compound amount = Compound interest = 4% ÷ 4 = 1% 1  4 = 4 1.04060  8,000 = $8,324.80 8,324.80 – 8,000 = $324.80 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.