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1 Chapter 13 Annuities and Sinking Funds McGrawHill/Irwin Copyright 2011 by the McGrawHill Companies, Inc. All rights reserved.
2 #13 LU13.1 Annuities and Sinking Funds Learning Unit Objectives Annuities: Ordinary Annuity and Annuity Due (Find Future Value) 1. Differentiate between contingent annuities and annuities certain 2. Calculate the future value of an ordinary annuity and an annuity due manually and by table lookup 132
3 #13 Annuities and Sinking Funds LU13.2 Learning Unit Objectives Present Value of an Ordinary Annuity (Find Present Value) 1. Calculate the present value of an ordinary annuity by table lookup and manually check the calculation 2. Compare the calculation of the present value of one lump sum versus the present value of an ordinary annuity 133
4 #13 Annuities and Sinking Funds LU13.3 Learning Unit Objectives Sinking Funds (Find Periodic Payments 1. Calculate the payment made at the end of each period by table lookup 2. Check table lookup by using ordinary annuity table 134
5 Compounding Interest (Future Value) Annuity  A series of payments 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 135
6 Figure 13.1 Future value of an annuity of $1 at 8% $3.50 $ $3.00 $2.50 $2.08 $2.00 $1.50 $1.00 $1.00 $0.50 $ End of period 136
7 Classification of Annuities Contingent Annuities  have no fixed number of payments but depend on an uncertain event Annuities certain  have a specific stated number of payments Life Insurance payments Mortgage payments 137
8 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. 31 Monthly Jan. 1 June 30 Quarterly April 1 Dec. 31 Semiannually July 1 Dec. 31 Annually Jan
9 Calculating Future Value of an Ordinary Annuity Manually Step 4. Repeat steps 2 and 3 until the end of the desired period is reached. Step 3. Add the additional investment at the end of period 2 to the new balance. Step 2. For period 2, calculate interest on the balance and add the interest to the previous balance. Step 1. For period 1, no interest calculation is necessary, since money is invested at the end of period 139
10 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% Manual Calculation $ 3, End of Yr , , , End of Yr , , , End of Yr
11 Calculating Future Value of an Ordinary Annuity by Table Lookup Step 3. Multiply the payment each period by the table factor. This gives the future value of the annuity. Future value of = Annuity pymt. x Ordinary annuity ordinary annuity each period table factor Step 2. Lookup the periods and rate in an ordinary annuity table. The intersection gives the table factor for the future value of $1 Step 1. Calculate the number of periods and rate per period 1311
12 Table 13.1 Ordinary annuity table: Compound sum of an annuity of $1 Ordinary annuity table: Compound sum of an annuity of $1 (Partial) Period 2% 3% 4% 5% 6% 7% 8% 9% 10%
13 Future Value of an Ordinary Annuity Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8% N = 3 x 1 = 3 R = 8%/1 = 8% x $3,000 $9,
14 Calculating Future Value of an Annuity Due Manually Step 3. Repeat steps 1 and 2 until the end of the desired period is reached. Step 2. Add additional investment at the beginning of the period to the new balance. Step 1. Calculate the interest on the balance for the period and add it to the previous balance 1314
15 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% Manual Calculation $ 3, Beginning Yr , , Beginning Yr 2 6, , , Beginning Yr 3 9, , End of Yr
16 Calculating Future Value of an Annuity Due by Table Lookup Step 4. Subtract 1 payment from Step 3. Step 3. Multiply the payment each period by the table factor. Step 2. Look up the periods and rate in an ordinary annuity table. The intersection gives the table factor for the future value of $1 Step 1. Calculate the number of periods and rate per period. Add one extra period
17 Future Value of an Annuity Due Find the value of an investment after 3 years for a $3,000 annuity due at 8% N = 3 x 1 = = 4 R = 8%/1 = 8% x $3,000 $13, $3,000 $10,
18 Figure Present value of an annuity of $1 at 8% $3.50 $3.00 $ $2.50 $2.00 $1.78 $1.50 $1.00 $.93 $0.50 $ End of period 1318
19 Calculating Present Value of an Ordinary Annuity by Table Lookup Step 3. Multiply the withdrawal for each period by the table factor. This gives the present value of an ordinary annuity Present value of = Annuity x Present value of ordinary annuity pymt. Pymt. ordinary annuity table Step 2. Look up the periods and rate in an ordinary annuity table. The intersection gives the table factor for the present value of $1 Step 1. Calculate the number of periods and rate per period 1319
20 Table Present Value of an Annuity of $1 Present value of an annuity of $1 (Partial) Period 2% 3% 4% 5% 6% 7% 8% 9% 10%
21 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. N = 3 x 1 = 3 R = 8%/1 = 8% x $8,000 $20, Interest ==> Payment ==> Interest ==> Payment ==> Interest ==> Payment ==> End of Year 3 ==> Manual Calculation $ 20, , , (8,000.00) 14, , , (8,000.00) 7, , (8,000.00)
22 Lump Sums versus Annuities John Sands made deposits of $200 to Floor Bank, which pays 8% interest compounded annually. After 5 years, John makes no more deposits. What will be the balance in the account 6 years after the last deposit? N = 5 x 2 = 10 R = 8%/2 = 4% x $200 $2, Step 1 Future value of an annuity Future value of a lump sum N = 6 x 2 = 12 R = 8%/2 = 4% x $2, $3, Step 2
23 1323 Lump Sums versus Annuities Mel Rich decided to retire in 8 years to New Mexico. What amount must Mel invest today so he will be able to withdraw $40,000 at the end of each year 25 years after he retires? Assume Mel can invest money at 5% interest compounded annually. N = 25 x 1 = 25 R = 5%/1 = 5% x $40,000 $563,756 Step 1 Present value of an annuity Present value of a lump sum N = 8 x 1 = 8 R = 5%/1 = 5%.6768 x $563,756 $381, Step 2
24 Sinking Funds (Find Periodic Payments) Bonds Bonds Sinking Fund = Future x Sinking Fund Payment Value Table Factor 1324
25 Table Sinking Fund Table Based on $1 Period 2% 3% 4% 5% 6% 8% 10%
26 Sinking Fund 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? Use Table Check $1,314 x ,917.35* N = 18 x 1 = 18 R = 10%/1 = 10% x $60,000 $1,314 N = 18, R= 10% Future Value of an annuity table * Off due to rounding 1326
27 Problem 1313: Solution: 18 periods + 1 = 19, 5% X $2,000 $61, $ 2, $59,
28 Problem 1317: Solution: 20 periods, 12% (Table 13.1) $12,500 x = $900,
29 Problem 1318: Solution: 10 periods, 11% (Table 13.2) $15,000 x = $88,
30 Problem 1323: Solution: 16 periods, 8% = 2% 4 $900 x = $12, OR $900 x = $16, x.7284 (Table 12.3) $12, % 16 periods 1330
31 Problem 1325: Solution: 20 periods, 2% (Table 13.3).0412 x $88,000 = $3, quarterly payment 1331
32 Problem 1326: Solution: Morton: 5 periods, 8% x $35,000 = $139, $40,000 = $179, Flynn: 5 periods, 8% x $38,000 = $151, $25,000 = $176, Morton offered a better
33 Problem 1327: Solution: PV annuity table: 15 periods, 8% x $28,000 = $239,666 PV table: 10 years, 8%.4632 x $239,666 = $111,
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