Annuities ©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 Future Value of an Annuity: Ordinary and Annuity Due 12-1: Calculating the future value of an ordinary annuity by using tables 12-2: Calculating the future value of an annuity due by using tables 12-3: Calculating the future value of an ordinary annuity and an annuity due by formula Section II Present Value of an Annuity: Ordinary and Annuity Due 12-4: Calculating the present value of an ordinary annuity by using tables 12-5: Calculating the present value of an annuity due by using tables 12-6: Calculating the present value of an ordinary annuity and an annuity due by 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 III Sinking Funds and Amortization 12-7: Calculating the amount of a sinking fund payment by table 12-8: Calculating the amount of an amortization payment by table 12-9: Calculating sinking fund payments by formula 12-10: Calculating amortization payments by 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.

Types of Annuities annuity simple annuity complex annuity Payment or receipt of equal amounts of money per period for a specified amount of time. simple annuity Annuity in which the number of compounding periods per year coincides with the number of annuity payments per year. complex annuity Annuity in which the annuity payments and compounding periods do not coincide. annuities certain Annuities that have a specified number of time periods. contingent annuities Annuities based on an uncertain time period, such as the life of a person. ©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.

Timeline Illustrating Present and Future Value of an Annuity EXHIBIT 12-1 Timeline Illustrating Present and Future Value of an Annuity ©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.

Annuity Payment Types ordinary annuity annuity due Annuity that is paid or received at the end of each time period. annuity due Annuity that is paid or received at the beginning of each time period. future value of an annuity The total amount of the annuity payments and the accumulated interest on those payments. Also known as the amount of an annuity. ©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 the Future Value of an Ordinary Annuity Example A bank is paying 8% interest compounded annually. Find the future value of $1,000 deposited at the end of every year for 3 years. Beginning of period 1 = 0 First annuity payment = +1,000.00 End of period 1 = 1,000.00 Begin period 2 + second payment = 2,000.00 I = PRT = 1,000 × .08 × 1 = + 80.00 End of period 2 = 2,080.00 Begin period 3 + third payment = 3,080.00 I = PRT = 2,080 × .08 × 1 = + 166.40 Future value of the ordinary annuity = 3,246.40 ©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 calculating future value (amount) of an ordinary annuity Step 1 Calculate the interest rate per period for the annuity (nominal rate ÷ periods per year). Step 2 Determine the number of periods of the annuity (years × periods per year). Step 3 From Table 12-1, locate the ordinary annuity table factor at the intersection of the rate-per-period column and the number-of-periods row. Step 4 Calculate the future value of the ordinary annuity. Future value (ordinary annuity) = Ordinary annuity table factor × Annuity payment ©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.

Future Value (Amount) of an Ordinary Annuity of $1 TABLE 12-1 Future Value (Amount) of an Ordinary Annuity of $1 ©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 Future Value of an Ordinary Annuity Example A bank is paying 8% interest compounded annually. Find the future value of $1,000 deposited at the end of every year for 3 years. Periods = 3 × 1 = 3 Interest rate per period = 8% ÷ 1 = 8% Future value = Table factor × Annuity payment Future value = 3.24640 × 1,000 = $3,246.40 ©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 the Future Value of an Annuity Due A bank is paying 6% interest compounded annually. Calculate the future value of $1,000, deposited at the beginning of each year for three years. Beginning of period 1 = 1,000.00 I = PRT = 1,000 × .06 × 1 = + 60.00 End of period 1 = 1,060.00 Beginning of period 2 (includes 2nd payment) = 2,060.00 I = PRT = 2,060 × .06 × 1 = 123.60) = + 123.60 End of period 2 = 2,183.60 Beginning of period 3 (includes 3rd payment) = 3,183.60 I = PRT = 3,183.60 × .06 × 1 = + 191.02 End of period 3 = 3,374.62 ©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 calculating future value (amount) of an annuity due Step 1 Calculate the number of periods of the annuity (years × periods per year) and add one period to the total. Step 2 Calculate the interest rate per period (nominal rate ÷ periods per year). Step 3 From Table 12-1, locate the table factor at the intersection of the rate-per-period column and the number-of-periods row. Step 4 Subtract 1.00000 from the ordinary annuity table factor to get the annuity due table factor. Step 5 Calculate the future value of the annuity due. Future value (annuity due) = Annuity due table factor × Annuity payment ©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 Future Value (Amount) of an Annuity Due Example A bank is paying 6% interest compounded annually. Calculate the future value of $1,000, deposited at the beginning of each year for 3 years. Interest rate per period = 6% ÷ 1 = 6% Periods: 3 × 1 = 3 + 1 = 4 Table factor = 4.37462 – 1.0000 = 3.37462 Future value = 3.37462 × 1,000 = 3,374.62 ©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 Future Value of an Ordinary and Annuity Due by 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 calculating present value of an ordinary annuity Step 1 Calculate the interest rate per period for the annuity (nominal rate ÷ periods per year). Step 2 Determine the number of periods of the annuity (years × periods per year). Step 3 From Table 12-2, locate the present value table factor at the intersection of the rate-per-period column and the number-of-periods row. Step 4 Calculate the present value of the ordinary annuity. Present value (ordinary annuity) = Ordinary annuity table factor × Annuity payment ©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 (Amount) of an Ordinary Annuity of $1 TABLE 12-2 Present Value (Amount) of an Ordinary Annuity of $1 ©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 an Ordinary Annuity A theater wants $20,000 available at the end of each 6-month theater season for renovations and new stage and lighting equipment. How much must be deposited now, at 8% compounded semiannually, to yield this annuity payment for the next 6 years? Interest per period = 8% ÷ 2 = 4% Periods = 6 × 2 = 12 Present value = 9.38507 × 20,000 = $187,701.40 ©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 calculating Present value of an annuity due Step 1 Calculate the number of periods of the annuity (years × periods per year) and subtract one period from the total. Step 2 Calculate the interest rate per period (nominal rate ÷ periods per year). Step 3 From Table 12-2, locate the table factor at the intersection of the rate-per-period column and the number-of-periods row. Step 4 Add 1.00000 from the ordinary annuity table factor to get the annuity due table factor. Step 5 Calculate the future value of the annuity due. Present value (annuity due) = Annuity due table factor × Annuity payment ©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 an Annuity Due Example Based on sales and revenue expense forecasts, it is estimated that $10,000 must be sent to the IRS for income tax purposes at the beginning of each 3-month period for the next 3 years. How much must be deposited now, at 6% compounded quarterly, to yield the annuity payment needed? Interest per period = 6% ÷ 4 = 1.5% Periods = 3 × 4 = 12 – 1 = 11 Table factor = 10.07112 + 1.0000 = 11.07112 Present value = 11.07112 × 10,000 = $110,711.20 ©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.

Calculating the Present Value of an Ordinary Annuity and an Annuity Due by 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 calculating the amount of a sinking fund payment Step 1 Using the appropriate rate per period and number of periods of the sinking fund, find the future value table factor from Table 12-1. Step 2 Calculate the amount of the sinking fund payment. Sinking fund payment = Future value of the sinking fund Future value table factor ©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 Amount of a Sinking Fund Example Steve wants to accumulate $8,000 in 5 years. If his bank is paying 12% interest compounded quarterly, how much must he deposit at the end of each 3-month period to reach his desired goal? Interest per period: 12% ÷ 4 = 3% Periods = 5 × 4 = 20 Sinking fund payment = Future value of the sinking fund Future value table factor Sinking fund payment = 8,000 = $297.73 26.87037 ©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 Sinking Fund Payments by 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 calculating the amount of an amortization payment Step 1 Using the appropriate rate per period and number of periods of the amortization, find the present value table factor from Table 12-2. Step 2 Calculate the amount of the amortization payment. Amortization payment = Original amount of obligation Present value table factor ©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 Amount of an Amortization Payment Example A fisherman purchases a new fishing boat for $130,000. He made a $20,000 down payment and financed the balance at his bank for 7 years. What amortization payments are required every 3 months, at 16% interest, to pay off the boat loan? Interest per period: 16% ÷ 4 = 4% Periods = 7 × 4 = 28 Sinking fund payment = Original amount of obligation Present value table factor Sinking fund payment = 110,000 = $6,601.43 16.66306 ©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 Amortization Payments by 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.

Chapter Review Problem 1 Jill has saved $200,000 and she wants to amortize (liquidate) that amount in a retirement fund so that she will receive equal annual payments over the next 25 years. At the end of the 25 years, there will be no funds left in the account. If the fund earns 12% interest, how much will Karen receive each year? Interest per period: 16% ÷ 4 = 4% Periods = 7 × 4 = 28 Payment = 200,000 = $25,499.99 7.84314 ©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 Calculate the amount of the periodic payment needed to amount to $50,000 in 8 years compounded semi-annually at 10% interest. Interest per period: 10% ÷ 2 = 5% Periods = 8 × 2 = 16 Sinking fund payment = Future value of the sinking fund Future value table factor Sinking fund payment = 50,000 = $2,113.50 23.65749 ©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 bank is paying 6% interest compounded monthly. Find the future value of $100 deposited at the end of each month be worth after 2 years. Periods = 2 × 12 = 24 Interest rate per period = 6% ÷ 12 = .5% Future value = 25.43196 × 100 = $2,543.20 ©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.