Download Present Value of an Ordinary Annuity

ANNUITIES AND
SINKING FUNDS
McGraw-Hill/Irwin
Chapter Twenty
Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.
LEARNING UNIT OBJECTIVES
LU20-1: 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 formula.
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.
LU 20-3: Sinking Funds (Find Periodic Payments)
1.
Calculate the payment made at the end of each period by
formula.
20-2
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
20-3
FUTURE VALUE OF AN ANNUITY OF $1
AT 8% (FIGURE 20.1)
$3.2464
$3.50
$3.00
$2.0800
$2.50
$2.00
$1.50
$1.00
$1.00
$0.50
$0.00
1
2
3
End of period
20-4
CLASSIFICATION OF ANNUITIES
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
20-5
CLASSIFICATION OF ANNUITIES
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
Example: salaries, stock
dividends
Example: rent, life insurance
premiums
20-6
CALCULATING FUTURE VALUE OF AN
ORDINARY ANNUITY 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.
20-7
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
20-8
CALCULATING FUTURE 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 Future Value of an Ordinary Annuity
Formula:
FV = PMT
EXAMPLE:
Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%.
CALCULATOR:
((1 + .08) yX 3 – 1)) ÷ .08 x 3,000 = 9,739.20
20-9
CALCULATING FUTURE VALUE OF ORDINARY
ANNUITIES BY FINANCIAL CALCULATOR
EXAMPLE:
Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%.
Remember to clear the TVM
each time you work with new
data: 2ND CLR TVM
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
20-10
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
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
20-12
CALCULATING FUTURE VALUE OF AN
ANNUITY DUE 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 Future Value of an Annuity Due
Formula and solve:
FVdue = PMT
(1 +i)
EXAMPLE:
Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%.
CALCULATOR:
((1 + .08) yX 3 – 1) ÷ .08 x 3,000 = 9,739.20 STO 1 1 + .08 x RCL 1 = 10,518.34
20-13
CALCULATING FUTURE VALUE OF ANNUIT Y
DUE BY FINANCIAL CALCULATOR
EXAMPLE:
Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%.
Remember to clear the TVM
each time you work with new
data: 2ND CLR TVM
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
Press 2ND BGN, 2ND SET, 2ND QUIT, CPT FV
$10,518.34
20-14
DIFFERENT NUMBER OF PERIODS
AND RATES
EXAMPLE: ORDINARY ANNUITY
Find the value of a $3,000 investment for 3 years made quarterly at 8%.
CALCULATOR:
((1 + .02) yX 12 – 1) ÷ .02 x 3,000 = 40,236.27
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
Remember to clear the TVM
each time you work with new
data: 2ND CLR TVM
20-15
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) yX 12 – 1) ÷ .02 x 3,000 = 40,236.27 STO 1 1 + .02 x RCL 1 = 41,040.99
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
Remember to clear the TVM
each time you work with new
data: 2ND CLR TVM
Press 2ND BGN, 2ND SET, 2ND QUIT, CPT FV
$41,040.99
20-16
PRESENT VALUE OF AN ANNUITY OF $1
AT 8% (FIGURE 20.2)
$3.50
$2.5771
$3.00
$2.50
$1.7833
$2.00
$1.50
$.9259
$1.00
$0.50
$0.00
1
2
3
Number of periods
20-17
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.
PVoa = PMT
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) yX 3 = STO 1 1 ÷ RCL 1 = STO 1 (1 - RCL 1) ÷ .08
20-18
X 8,000 = 20,616.78
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 ==>
Payment ==>
Interest ==>
Payment ==>
End of Year 3 ==>
Manual Calculation
$
20,616.78
1,649.34
22,266.12
(8,000.00)
14,266.12
1,141.29
15,407.41
(8,000.00)
7,407.41
592.59
8,000.00
(8,000.00)
-
20-19
CALCULATING PRESENT VALUE OF AN
ORDINARY ANNUIT Y BY
FINANCIAL CALCULATOR
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?
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
Remember to clear the TVM
each time you work with new
data: 2ND CLR TVM
20-20
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?
FINANCIAL CALCULATOR:
Calculate the first 5 years:
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
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
20-21
SINKING FUNDS
(FIND PERIODIC PAYMENTS)
 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.
20-22
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.
Sinking Fund Payment =
CALCULATOR:
60,000 X .10 = STO 1 (1 + .10) yx 18 = ─1 = STO 2 RCL 1 ÷
RCL 2 = 1,315.81
20-23
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?
Remember to clear the TVM
each time you work with new
data: 2ND CLR TVM
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
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.
20-24