ANNUITIES AND SINKING
FUNDS
McGraw-Hill/Irwin
Chapter Thir teen
Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.
LEARNING UNIT OBJECTIVES
LU13-1: 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.
LU 13-2: 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.
LU 13-3: 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.
13-2
ANNUITIES
Annuities have many uses in addition to lottery payoffs. Some of these uses are insurance companies'
pension installments, Social Security payments, home mortgages, businesses paying off notes, bond
interest, and savings for a vacation trip or college education.
Example 1
What happens when you have the winning lottery ticket? You take it to the lottery headquarters. When
you turn in the ticket, do you immediately receive a check for $1 million? No. Lottery payoffs are not
usually made in lump sums. Lottery winners receive a series of payments over a period of time—
usually years. This stream of payments is an annuity. By paying the winners an annuity, lotteries do
not actually spend $1 million. The lottery deposits a sum of money in a financial institution. The
continual growth of this sum through compound interest provides the lottery winner with a series of
payments.
Example 2
Many parents of small children are concerned about being able to afford to pay for their children's
college educations. Some parents start an annuity by depositing a series of payments in a financial
institution (usually of equal amounts over a period of time) from the time when the child is in diapers.
The interest on these deposits is compounded until the child is 18, when the parents withdraw the sum
of all deposits plus the interest that accumulates for college expenses.
We begin the chapter by explaining the difference between calculating the future value of an ordinary
annuity and an annuity due. Then you learn how to find the present value of an ordinary annuity. The
chapter ends with a discussion of sinking funds.
13-3
COMPOUNDING INTEREST (FUTURE VALUE)
Annuity –
Term of the annuity –
A series of payments
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
13-4
FUTURE VALUE OF AN ANNUIT Y
• Annuity is stream of equal payments made at periodic times.
•
•
•
•
•
The future value of an annuity is the future dollar amount of a series of payments plus interest. The term of the
annuity is the time from the beginning of the first payment period to the end of the last payment period.
The concept of the future value of an annuity is illustrated in the following figure
At end of period 1: $1 is invested.
At end of period 2: An additional $1 is invested. The $1 from period 1 earns interest and is now worth $1.08.
The $1 invested at the end of period 2, does not earn any interest, because it was invested at the end of the
period. The $2.00 is now worth $2.08
At end of period 3: An additional $1 is invested. The $3.00 is now worth $3.25. Remember that the last dollar
invested earns no interest.
13-5
CALCULATING FUTURE VALUE OF AN
ORDINARY ANNUIT Y MANUALLY
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.
Step 4. Repeat Steps 2 and 3 until the end of the desired period is
reached.
13-6
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
$ 3,000.00 End of Yr 1
240.00 plus interest
3,240.00
3,000.00 Yr. 2 Investment
6,240.00 End of Yr 2
499.20 plus interest
6,739.20
3,000.00 Yr. 3 Investment
$ 9,739.20 End of Yr 3
13-7
CALCULATING FUTURE VALUE OF AN
ORDINARY ANNUITY BY TABLE LOOKUP
Step 1. Calculate the number of periods and rate per period.
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 3. Multiply the payment each period by the table factor. This
gives the future value of the annuity.
Future value of =
ordinary annuity
Annuity payment
each period
x
Ordinary annuity
table factor
13-8
ORDINARY ANNUITY TABLE: COMPOUND
SUM OF AN ANNUITY OF $1 (TABLE 13.1)
Ordinary annuity table: Compound sum of an annuity of $1 (partial)
Period
2%
3%
4%
5%
6%
7%
8%
9%
10%
1
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
2
2.0200
2.0300
2.0400
2.0500
2.0600
2.0700
2.0800
2.0900
2.1000
3
3.0604
3.0909
3.1216
3.1525
3.1836
3.2149
3.2464
1.0000
3.3100
4
4.1216
4.1836
4.2465
4.3101
4.3746
4.4399
4.5061
4.5731
4.6410
5
5.2040
5.3091
5.4163
5.5256
5.6371
5.7507
5.8666
5.9847
6.1051
6
6.3081
6.4684
6.6330
6.8019
6.9753
7.1533
7.3359
7.5233
7.7156
7
7.4343
7.6625
7.8983
8.1420
8.3938
8.6540
8.9228
9.2004
9.4872
8
8.5829
8.8923
9.2142
9.5491
9.8975
10.2598
10.6366
11.0285
11.4359
9
9.7546
10.1591
10.5828
11.0265
11.4913
11.9780
12.4876
13.0210
13.5795
10
10.9497
11.4639
12.0061
12.5779
13.1808
13.8164
14.4866
15.1929
15.9374
11
12.1687
12.8078
13.4863
14.2068
14.9716
15.7836
16.6455
17.5603
18.5312
12
13.4120
14.1920
15.0258
15.9171
16.8699
17.8884
18.9771
20.1407
21.3843
13
14.6803
15.6178
16.6268
17.7129
18.8821
20.1406
21.4953
22.9534
24.5227
14
15.9739
17.0863
18.2919
19.5986
21.0150
22.5505
24.2149
26.0192
27.9750
15
17.2934
18.5989
20.0236
21.5785
23.2759
25.1290
27.1521
29.3609
31.7725
13-9
FUTURE VALUE OF
AN ORDINARY ANNUITY
Find the value of an investment after 3 years for a $3,000 ordinary
annuity at 8%.
Periods (N) = 3 x 1 = 3
Rate (R) = 8%/1 = 8%
3.2464 (table factor) x $3,000 = $9,739.20
13-10
CLASSIFICATION OF ANNUITIES
Annuities are classified into two major groups: contingent
annuities and annuities certain
Contingent annuities –
Annuities certain –
have no fixed number of payments
but depend on an uncertain event
have a specific stated number of
payments
Life Insurance payments
Mortgage payments
13-11
CLASSIFICATION OF ANNUITIES
we can divide each of the major annuity groups (Contingent annuities and Annuities
certain) into Ordinary annuity and Annuity due:
Ordinary annuity –
Annuity due –
regular deposits/payments
made at the end of the period
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. 1
13-12
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.
13-13
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
$ 3,000.00 Beginning Yr 1
240.00 Yr 1 Interest
3,240.00
3,000.00 Beginning Yr 2
6,240.00
499.20 Yr 2 Interest
6,739.20
3,000.00 Beginning Yr 3
9,739.20
779.14 Yr 3 Interest
10,518.34 End of Yr. 3
13-14
CALCULATING FUTURE VALUE OF AN
ANNUITY DUE BY TABLE LOOKUP
Step 1. Calculate the number of periods and rate per period. Add
one extra period.
Step 2. Look up in an ordinary annuity table the periods and rate.
The intersection gives the table factor for the future value of
$1.
Step 3. Multiply the payment each period by the table factor.
Step 4. Subtract 1 payment from Step 3.
13-15
FUTURE VALUE OF AN ANNUITY DUE
Find the value of an investment after 3 years for a $3,000
annuity due at 8%.
Periods (N) = 3 x 1 = 3 + 1 = 4
Rate (R) = 8%/1 = 8%
4.5061 (table factor) x $3,000 = $13,518.30
$13,518.30 -- $3,000 = $10,518.30
13-16
PRACTICE QUIZ
1-Using Table 13.1, (a) find the value of an investment after 4 years on an ordinary annuity of $4,000
made semiannually at 10%; and (b) recalculate, assuming an annuity due.
2-Wally Beaver won a lottery and will receive a check for $4,000 at the beginning of each 6 months for
the next 5 years. If Wally deposits each check into an account that pays 6%, how much will he have at
the end of the 5 years?
For step by step solution watch the video for LU 13-1 ( Go to: McGraw-Hill’s Connect; Assignment # 5; Question
1; Click the eBook & resources options drop down menu; scroll down to LU13-1 and click
13-17
PRESENT VALUE OF AN ANNUITY
• Let's assume that we want to know how much money we need to invest today to receive a stream of payments for
a given number of years in the future. This is called the present value of an ordinary annuity.
• In the following figure you can see that if you wanted to withdraw $1at the end of one period, you would
have to invest 93 cents today. If at the end of each period for three periods you wanted to withdraw $1, you
Would have to put $2.58 in the bank today at 8% interest. (Note that we go from the future back to the
present.)
$3.50
$2.5771
$3.00
$2.50
$1.7833
$2.00
$1.50
$.9259
$1.00
$0.50
$0.00
1
2
Number of periods
3
13-18
CALCULATING PRESENT VALUE OF AN
ORDINARY ANNUITY BY TABLE LOOKUP
Step 1. Calculate the number of periods and rate per period.
Step 2. Look up the periods and rate in the present value of an
annuity table. The intersection gives the table factor for
the present value of $1.
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
ordinary annuity payment = payment
x
Present value of
ordinary annuity table
13-19
PRESENT VALUE OF AN ANNUITY OF $1
(TABLE 13.2)
Present value of an annuity of $1 (partial)
Period
2%
3%
4%
5%
6%
7%
8%
9%
10%
1
0.9804
0.9709
0.9615
0.9524
0.9434
0.9346
0.9259
0.9174
0.9091
2
1.9416
1.9135
1.8861
1.8594
1.8334
1.8080
1.7833
1.7591
1.7355
3
2.8839
2.8286
2.7751
2.7232
2.6730
2.6243
2.5771
2.5313
2.4869
4
3.8077
3.7171
3.6299
3.5459
3.4651
3.3872
3.3121
3.2397
3.1699
5
4.7134
4.5797
4.4518
4.3295
4.2124
4.1002
3.9927
3.8897
3.7908
6
5.6014
5.4172
5.2421
5.0757
4.9173
4.7665
4.6229
4.4859
4.3553
7
6.4720
6.2303
6.0021
5.7864
5.5824
5.3893
5.2064
5.0330
4.8684
8
7.3255
7.0197
6.7327
6.4632
6.2098
5.9713
5.7466
5.5348
5.3349
9
8.1622
7.7861
7.4353
7.1078
6.8017
6.5152
6.2469
5.9952
5.7590
10
8.9826
8.5302
8.1109
7.7217
7.3601
7.0236
6.7101
6.4177
6.1446
11
9.7868
9.2526
8.7605
8.3064
7.8869
7.4987
7.1390
6.8052
6.4951
12
10.5753
9.9540
9.3851
8.8632
8.3838
7.9427
7.5361
7.1607
6.8137
13
11.3483
10.6350
9.9856
9.3936
8.8527
8.3576
7.9038
7.4869
7.1034
14
12.1062
11.2961
10.5631
9.8986
9.2950
8.7455
8.2442
7.7862
7.3667
15
12.8492
11.9379
11.1184
10.3796
9.7122
9.1079
8.5595
8.0607
7.6061
13-20
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 ==>
Interest ==>
N = 3 x 1 = 3 periods
R = 8%/1 = 8%
2.5771 (table factor) x $8,000 =
$20,616.80
Payment ==>
Interest ==>
Payment ==>
End of Year 3 ==>
Manual Calculation
$
20,616.80
1,649.34
22,266.14
(8,000.00)
14,266.14
1,141.29
15,407.43
(8,000.00)
7,407.43
592.59
8,000.02
(8,000.00)
0.02
13-21
LUMP SUMS VERSUS ANNUITIES
John Sands made deposits of $200 to Floor Bank, which pays 8% interest
compounded semi annually. After 5 years, John makes no more deposits. What will
be the balance in the account 6 years after the last deposit?
Step 1. Future value of an annuity
N = 5 x 2 = 10 periods
R = 8%/2 = 4%
12.0061 (table factor) x $200 =
$2,401.22
Step 2. Future value of a lump sum
N = 6 x 2 = 12 periods
R = 8%/2 = 4%
1.6010 (table factor) x $2,401.22 =
$3,844.35
13-22
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.
Step 1. Present value of an annuity
N = 25 x 1 = 25 periods
R = 5%/1 = 5%
Step 2. Present value of a lump sum
N = 8 x 1 = 8 periods
R = 5%/1 = 5%
.6768 x $563,756 = $381,550.06
14.0939 x $40,000 = $563,756
13-23
PRACTICE QUIZ
1-What must you invest today to receive an $18,000 annuity for 5 years semiannually at a 10% annual rate? All
withdrawals will be made at the end of each period.
2-Rase High School wants to set up a scholarship fund to provide five $2,000 scholarships for the next 10 years. If
money can be invested at an annual rate of 9%, how much should the scholarship committee invest today?
3-Joe Wood decided to retire in 5 years in Arizona. What amount should Joe invest today so he can withdraw $60,000
at the end of each year for 30 years after he retires? Assume Joe can invest money at 6% compounded annually.
For step by step solution watch the video for LU 13-2 ( Go to: McGraw-Hill’s Connect;
Assignment # 5; Question 2; Click the eBook & resources options drop down menu; scroll
13-24
down to LU13-2 and click
SINKING FUNDS
(FIND PERIODIC PAYMENTS)
• A sinking fund is 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, to
replace worn-out equipment, to purchase plant expansion, and so on.
• A sinking fund is a different type of an annuity. In a sinking fund, you
determine the amount of periodic payments you need to achieve a given
financial goal. In the annuity, you know the amount of each payment and
must determine its future value.
Let's work with the following formula:
Sinking fund = Future x Sinking fund
payment
value
table factor
13-25
SINKING FUND TABLE BASED ON $1
(Table 12.3)
Period
2%
3%
4%
5%
6%
8%
10%
1
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
2
0.4951
0.4926
0.4902
0.4878
0.4854
0.4808
0.4762
3
0.3268
0.3235
0.3203
0.3172
0.3141
0.3080
0.3021
4
0.2426
0.2390
0.2355
0.2320
0.2286
0.2219
0.2155
5
0.1922
0.1884
0.1846
0.1810
0.1774
0.1705
0.1638
6
0.1585
0.1546
0.1508
0.1470
0.1434
0.1363
0.1296
7
0.1345
0.1305
0.1266
0.1228
0.1191
0.1121
0.1054
8
0.1165
0.1125
0.1085
0.1047
0.1010
0.0940
0.0874
9
0.1025
0.0984
0.0945
0.0907
0.0870
0.0801
0.0736
10
0.0913
0.0872
0.0833
0.0795
0.0759
0.0690
0.0627
11
0.0822
0.0781
0.0741
0.0704
0.0668
0.0601
0.0540
12
0.0746
0.0705
0.0666
0.0628
0.0593
0.0527
0.0468
13
0.0681
0.0640
0.0601
0.0565
0.0530
0.0465
0.0408
14
0.0626
0.0585
0.0547
0.0510
0.0476
0.0413
0.0357
15
0.0578
0.0538
0.0499
0.0463
0.0430
0.0368
0.0315
16
0.0537
0.0496
0.0458
0.0423
0.0390
0.0330
0.0278
17
0.0500
0.0460
0.0422
0.0387
0.0354
0.0296
0.0247
18
0.0467
0.0427
0.0390
0.0355
0.0324
0.0267
0.0219
13-26
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 13.3.
N = 18 x 1 = 18 periods
R = 10%/1 = 10%
0.0219 x $60,000 = $1,314
Check
Future Value of an annuity table
N = 18, R= 10%
$1,314 x 45.5992 = $59,917.35*
* Off due to rounding
13-27
PRACTICE QUIZ
Today, Arrow Company issued bonds that will mature to a value of $90,000 in
10 years. Arrow's controller is planning to set up a sinking fund. Interest rates
are 12% compounded semiannually. What will Arrow Company have to set
aside to meet its obligation in 10 years? Check your answer. Your answer will
be off due to the rounding of Table 13.3.
For step by step solution watch the video for LU 13-3 ( Go to: McGraw-Hill’s
Connect; Assignment # 5; Question 3; Click the eBook & resources options drop
down menu; scroll down to LU13-3 and click
13-28
PROBLEM 13-13
To help you reach financial security upon retirement, you should invest 20% of
your income annually. If you automatically transferred $3,000 at the end of each
year to a retirement account earning 4% interest compounded annually, how
much would you have after 25 years? 30 years? LU 13-1(2)
Solution:
Periods = 25 years x 1 = 25 periods
Interest rate per period = 4%/1 = 4%
$3,000 x 41.6459 = $124,937.70 after 25 years
Periods = 30 years x 1 = 30 periods
Interest rate per period = 4%/1 = 4%
$3,000 x 56.0849 = $168,254.70 after 30 years
13-29
PROBLEM 13-17
Josef Company borrowed money that must be repaid in 20 years. The company
wants to make sure the loan will be repaid at the end of year 20, so it invests
$12,500 at the end of each year at 12% interest compounded annually. What
was the amount of the original loan? LU 13-1(2)
Solution:
20 periods, 12% (Table 13.1)
$12,500 X 72.0524 = $900,655
13-30
PROBLEM 13-18
Bankrate.com reported on a shocking statistic: only 54% of workers participate in
their company’s retirement plan. This means that 46% do not. With such an uncertain
future for Social Security, this can leave almost 1 in 2 individuals without proper
income during retirement. Jill Collins, 20, decided she needs to have $250,000 in her
retirement account upon retiring at 60. How much does she need to invest each year
at 5% compounded annually to meet her goal? Tip: She is setting up a sinking fund.
LU 13-3(1)
Solution:
Periods = 40 years X 1 = 40 periods
Interest rate per period = 5%/1 = 5%
$250,000 X .0083 = $2,075 each year
13-31
PROBLEM 13-23
On Joe Martin’s graduation from college, Joe’s uncle promised him a gift of
$12,000 in cash or $900 every quarter for the next 4 years after graduation. If
money could be invested at 8% compounded quarterly, which offer is better for
Joe? LU 13-1(2), LU 13-2(1)
Solution:
8 periods 8%/ 4 = 2%
$900 x 13.5777 = $12,219.93
(Table 13.2)
or $900 x 18.6392 = $16,775.28
(Table 13.1)
x
.7284 (Table 12.3)
$12,219.11
2%, 16 periods
13-32
PROBLEM 13-25
A local Dunkin’ Donuts franchise must buy a new piece of equipment in 5 years
that will cost $88,000. The company is setting up a sinking fund to finance the
purchase. What will the quarterly deposit be if the fund earns 8% interest?
LU 13-3(1)
Solution:
20 periods, 2% (Table 13.3)
.0412 X $88,000 = $3,625.60 quarterly payment
13-33
PROBLEM 13-26
Mike Macaro is selling a piece of land. Two offers are on the table. Morton
Company offered a $40,000 down payment and $35,000 a year for the next 5
years. Flynn Company offered $25,000 down and $38,000 a year for the next 5
years. If money can be invested at 8% compounded annually, which offer is better
for Mike? LU 13-1(2)
Solution:
Morton: 5 periods, 8% (Table 13.2)
3.9927 X $35,000 = $139,744.50 + $40,000 = $179,744.50
Flynn: 5 periods, 8% (Table 13.2))
3.9927 X $38,000 = $151,722.60 + $25,000 = $176,722.60
Morton’s offer is the better deal.
13-34