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Chapter Twenty ANNUITIES AND SINKING FUNDS Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

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1. Differentiate between contingent annuities and annuities certain. 2. Calculate the future value of an ordinary annuity and an annuity due manually and by formula. LU20-1: Annuities: Ordinary Annuity and Annuity Due (Find Future Value) LEARNING UNIT OBJECTIVES LU 20-2: Present Value of an Ordinary Annuity (Find Present Value) 1. Calculate the present value of an ordinary annuity by formula. 2. Compare the calculation of the present value of one lump sum versus the present value of an ordinary annuity. 20-2 LU 20-3: Sinking Funds (Find Periodic Payments) 1. Calculate the payment made at the end of each period by formula.

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COMPOUNDING INTEREST (FUTURE VALUE) Term of the annuity – the time from the beginning of the first payment period to the end of the last payment period Future value of annuity – the future dollar amount of a series of payments plus interest Present value of an annuity – the amount of money needed to invest today in order to receive a stream of payments for a given number of years in the future Annuity – a series of payments 20-3

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End of period $1.00 $2.0800 $3.2464 FUTURE VALUE OF AN ANNUITY OF $1 AT 8% (FIGURE 20.1) 20-4

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CLASSIFICATION OF ANNUITIES Contingent annuities – have no fixed number of payments but depend on an uncertain event Life Insurance payments Annuities certain – have a specific stated number of payments Mortgage payments 20-5

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CLASSIFICATION OF ANNUITIES Ordinary annuity – regular deposits (payments) made at the end of the period Annuity due – regular deposits (payments) made at the beginning of the period Jan. 31MonthlyJan. 1 June 30QuarterlyApril 1 Dec. 31SemiannuallyJuly 1 Dec. 31AnnuallyJan. 1 20-6 Example: salaries, stock dividends Example: rent, life insurance premiums

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Step 1. For period 1, no interest calculation is necessary, since money is invested at the end of the period. Step 2. For period 2, calculate interest on the balance and add the interest to the previous balance. Step 3. Add the additional investment at the end of period 2 to the new balance. CALCULATING FUTURE VALUE OF AN ORDINARY ANNUITY MANUALLY 20-7 Step 4. Repeat Steps 2 and 3 until the end of the desired period is reached.

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CALCULATING FUTURE VALUE OF AN ORDINARY ANNUITY MANUALLY Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%. 20-8

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Step 1. Calculate the number of periods, n, and rate per period, i. Step 2. Determine the payment, PMT, given in the word problem. Step 3. Plug these values into the Future Value of an Ordinary Annuity Formula: CALCULATING FUTURE VALUE OF AN ORDINARY ANNUITY BY FORMULA 20-9 FV = PMT EXAMPLE: Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%. CALCULATOR: ((1 +.08) y X 3 – 1)) ÷.08 x 3,000 = 9,739.20

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CALCULATING FUTURE VALUE OF ORDINARY ANNUITIES BY FINANCIAL CALCULATOR 20-10 EXAMPLE: Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%. Input 3 and then press N. Input 8 and then press I/Y. Input 0, and then press PV. Input 3,000 +/-, and then press PMT. Press CPT FV = 9,739.20 Remember to clear the TVM each time you work with new data: 2ND CLR TVM

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CALCULATING FUTURE VALUE OF AN ANNUITY DUE MANUALLY Step 1. Calculate the interest on the balance for the period and add it to the previous balance. Step 2. Add additional investment at the beginning of the period to the new balance. Step 3. Repeat Steps 1 and 2 until the end of the desired period is reached. 20-11

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CALCULATING FUTURE VALUE OF AN ANNUITY DUE MANUALLY Find the value of an investment after 3 years for a $3,000 annuity due at 8%. 20-12

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CALCULATING FUTURE VALUE OF AN ANNUITY DUE BY FORMULA 20-13 Step 1. Calculate the number of periods, n, and rate per period, i. Step 2. Determine the payment, PMT, given in the word problem. Step 3. Plug these values into the Future Value of an Annuity Due Formula and solve: FV due = PMT (1 +i) EXAMPLE: Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%. CALCULATOR: ((1 +.08) y X 3 – 1) ÷.08 x 3,000 = 9,739.20 STO 1 1 +.08 x RCL 1 = 10,518.34

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CALCULATING FUTURE VALUE OF ANNUITY DUE BY FINANCIAL CALCULATOR 20-14 EXAMPLE: Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%. Input 3 and then press N. Input 8 and then press I/Y. Input 0, and then press PV. Input 3,000 +/-, and then press PMT. Press CPT FV = 9,739.20 Remember to clear the TVM each time you work with new data: 2ND CLR TVM Press 2ND BGN, 2ND SET, 2ND QUIT, CPT FV $10,518.34

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DIFFERENT NUMBER OF PERIODS AND RATES 20-15 EXAMPLE: ORDINARY ANNUITY Find the value of a $3,000 investment for 3 years made quarterly at 8%. CALCULATOR: ((1 +.02) y X 12 – 1) ÷.02 x 3,000 = 40,236.27 Remember to clear the TVM each time you work with new data: 2ND CLR TVM FINANCIAL CALCULATOR: Input 12 and then press N. Input 2 and then press I/Y. Input 0, and then press PV. Input 3,000 +/-, and then press PMT. Press CPT FV = 40,236.27

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DIFFERENT NUMBER OF PERIODS AND RATES EXAMPLE: ANNUITY DUE Find the value of a $3,000 investment for 3 years made quarterly at 8%. CALCULATOR: ((1 +.02) y X 12 – 1) ÷.02 x 3,000 = 40,236.27 STO 1 1 +.02 x RCL 1 = 41,040.99 FINANCIAL CALCULATOR: Remember to clear the TVM each time you work with new data: 2ND CLR TVM Input 12 and then press N. Input 2 and then press I/Y. Input 0, and then press PV. Input 3,000 +/-, and then press PMT. Press CPT FV = 40,236.27 Press 2ND BGN, 2ND SET, 2ND QUIT, CPT FV 20-16 $41,040.99

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Number of periods $.9259 $1.7833 $2.5771 PRESENT VALUE OF AN ANNUITY OF $1 AT 8% (FIGURE 20.2) 20-17

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CALCULATING PRESENT VALUE OF AN ORDINARY ANNUITY BY FORMULA Step 1. Calculate the number of periods, n, and rate per period, i. Step 2.Determine the payment, PMT, given in the word problem. Step 3.Plug these values into the Present Value of an Ordinary Annuity Formula. 20-18 EXAMPLE: John Fitch wants to receive an $8,000 annuity in 3 years. Interest on the annuity is 8% annually. John will make withdrawals at the end of each year. How much must John invest today to receive a stream of payments for 3 years? CALCULATOR: (1 +.08) y X 3 = STO 1 1 ÷ RCL 1 = STO 1 (1 - RCL 1) ÷.08 X 8,000 = 20,616.78 PV oa = PMT

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PRESENT VALUE OF AN ANNUITY John Fitch wants to receive a $8,000 annuity in 3 years. Interest on the annuity is 8% semiannually. John will make withdrawals at the end of each year. How much must John invest today to receive a stream of payments for 3 years. Interest ==> Payment ==> End of Year 3 ==> Interest ==> Payment ==> 20-19

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CALCULATING PRESENT VALUE OF AN ORDINARY ANNUITY BY FINANCIAL CALCULATOR 20-20 EXAMPLE: John Fitch wants to receive an $8,000 annuity in 3 years. Interest on the annuity is 8% annually. John will make withdrawals at the end of each year. How much must John invest today to receive a stream of payments for 3 years? Remember to clear the TVM each time you work with new data: 2ND CLR TVM FINANCIAL CALCULATOR: Input 3 and then press N. Input 8 and then press I/Y. Input 0, and then press FV. Input 8,000 +/-, and then press PMT. Press CPT PV = 20,616.78

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LUMP SUMS VERSUS ANNUITIES John Sands made deposits of $200 semiannually to Floor Bank, which pays 8% interest compounded semiannually. After 5 years, John makes no more deposits. What will be the balance in the account 6 years after the last deposit? 20-21 FINANCIAL CALCULATOR: Input 10 and then press N. Input 4 and then press I/Y. Input 0, and then press PV. Input 200 +/-, and then press PMT. Press CPT FV = 2,401.22 Calculate the first 5 years: For John, the stream of payments grows to $2,401.22. Then this lump sum grows for 6 years to $3,844.43. Input 12 and then press N. Input 4 and then press I/Y. Input 2,401.22, and then press PV. Input 0 and then press PMT. Press CPT FV = 3,844.43

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SINKING FUNDS (FIND PERIODIC PAYMENTS) 20-22 Sinking fund –a financial arrangement that sets aside regular periodic payments of a particular amount of money. Compound interest accumulates on these payments to a specific sum at a predetermined future date. Corporations use sinking funds to: discharge bonded indebtedness replace worn-out equipment purchase plant expansion, etc.

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CALCULATING SINKING FUND PAYMENTS BY FORMULA To retire a bond issue, Moore Company needs $60,000 18 years from today. The interest rate is 10% compounded annually. What payment must Moore make at the end of each year? Use Table 13.3. 20-23 Sinking Fund Payment = CALCULATOR: 60,000 X.10 = STO 1 (1 +.10) y x 18 = ─1 = STO 2 RCL 1 ÷ RCL 2 = 1,315.81

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CALCULATING SINKING FUND PAYMENTS BY FINANCIAL CALCULATOR To retire a bond issue, Moore Company needs $60,000 in 18 years from today. The interest rate is 10% compounded annually. What payment must Moore make at the end of each year? Input 18 and then press N. Input 10 and then press I/Y. Input 0, and then press PV. Input 60,000 and then press FV Press CPT PMT = 1,315.81 20-24 Remember to clear the TVM each time you work with new data: 2ND CLR TVM If Moore Company pays $1,315.81 at the end of each period for 18 years, then $60,000 will be available to pay off the bond issue at maturity.

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