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

Chapter Twenty

ANNUITIES AND SINKING FUNDS

Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

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.

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

$0.00

$0.50

$1.00

$1.50

$2.00

$2.50

$3.00

$3.50

1 2 3End of period

$1.00

$2.0800

$3.2464

FUTURE VALUE OF AN ANNUITY OF $1

AT 8% (FIGURE 20.1)

20-4

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

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. 1

20-6

Example: salaries, stock dividends

Example: rent, life insurance premiums

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.

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 Calculation3,000.00$ End of Yr 1

240.00 plus interest3,240.00 3,000.00 Yr. 2 Investment6,240.00 End of Yr 2

499.20 plus interest6,739.20 3,000.00 Yr. 3 Investment9,739.20$ End of Yr 3

20-8

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) yX 3 – 1)) ÷ .08 x 3,000 = 9,739.20

CALCULATING FUTURE VALUE OF ORDINARY ANNUITIES BY FINANCIAL

CALCULATOR

20-10


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

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 Calculation3,000.00$ Beginning Yr 1

240.00 Yr 1 Interest3,240.00 3,000.00 Beginning Yr 26,240.00

499.20 Yr 2 Interest6,739.20 3,000.00 Beginning Yr 39,739.20

779.14 Yr 3 Interest10,518.34 End of Yr. 3

20-12

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:

FVdue = PMT

(1 +i)


CALCULATOR:((1 + .08) yX 3 – 1) ÷ .08 x 3,000 = 9,739.20 STO 1 1 + .08 x RCL 1 = 10,518.34

CALCULATING FUTURE VALUE OF ANNUITY DUE BY FINANCIAL

CALCULATOR

20-14




Press 2ND BGN, 2ND SET, 2ND QUIT, CPT FV

$10,518.34

DIFFERENT NUMBER OF PERIODS AND RATES

20-15

EXAMPLE: ORDINARY ANNUITYFind 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

DIFFERENT NUMBER OF PERIODS AND RATES

EXAMPLE: ANNUITY DUEFind 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: Remember to clear the

TVM each time you work with new data: 2ND CLR TVM


Press 2ND BGN, 2ND SET, 2ND QUIT, CPT FV

20-16

$41,040.99

$0.00

$0.50

$1.00

$1.50

$2.00

$2.50

$3.00

$3.50

1 2 3

Number of periods

$.9259

$1.7833

$2.5771

PRESENT VALUE OF AN ANNUITY OF $1 AT 8% (FIGURE 20.2)

20-17

CALCULATING PRESENT VALUE OF AN ORDINARY ANNUITY BY

FORMULAStep 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) yX 3 = STO 1 1 ÷ RCL 1 = STO 1 (1 - RCL 1) ÷ .08 X 8,000 = 20,616.78

PVoa = PMT

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.

Manual Calculation20,616.78$ 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) -

Interest ==>

Payment ==>

End of Year 3 ==>

Interest ==>

Interest ==>

Payment ==>

Payment ==>

20-19

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?


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

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

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.

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) yx 18 = ─1 = STO 2 RCL 1 ÷ RCL 2 = 1,315.81

CALCULATING SINKING FUND PAYMENTS BY FINANCIAL

CALCULATORTo 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 FVPress CPT PMT = 1,315.81

20-24


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.

Documents

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