annuities and sinking chapter thirteen...

Chapter Thirteen

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 table lookup.

LU13-1: Annuities: Ordinary Annuity and Annuity Due (Find Future Value)

LEARNING UNIT OBJECTIVES

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.

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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 1What 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 2Many 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.

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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

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FUTURE VALUE OF AN ANNUITY

• 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.

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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

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%.

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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.

CALCULATING FUTURE VALUE OF AN ORDINARY ANNUITY BY TABLE LOOKUP

Future value of = Annuity payment x Ordinary annuity

ordinary annuity each period table factor

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ORDINARY ANNUITY TABLE: COMPOUND SUM OF AN ANNUITY OF $1 (TABLE 13.1)

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Periods (N) = 3 x 1 = 3

FUTURE VALUE OF AN ORDINARY ANNUITY

Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%.

Rate (R) = 8%/1 = 8%3.2464 (table factor) x $3,000 = $9,739.20

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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

Annuities are classified into two major groups: contingent annuities and annuities certain

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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. 31 Monthly Jan. 1

June 30 Quarterly April 1

Dec. 31 Semiannually July 1

Dec. 31 Annually Jan. 1

we can divide each of the major annuity groups (Contingent annuities and Annuities certain) into Ordinary annuity and Annuity due:

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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.

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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%.

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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.

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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

4.5061 (table factor) x $3,000 = $13,518.30

Rate (R) = 8%/1 = 8%

$10,518.30$13,518.30 -- $3,000 =

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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

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• Let's assume that we want to know how much money we need to invest today to receive a stream of payments fora 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, youWould have to put $2.58 in the bank today at 8% interest. (Note that we go from the future back to the present.)

Number of periods

$.9259

$1.7833

$2.5771

PRESENT VALUE OF AN ANNUITY

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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 Present value ofordinary annuity payment payment ordinary annuity table

= x

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PRESENT VALUE OF AN ANNUITY OF $1(TABLE 13.2)

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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.

N = 3 x 1 = 3 periods

Interest ==>

Payment ==>

End of Year 3 ==>

Interest ==>

Interest ==>

Payment ==>

Payment ==>

R = 8%/1 = 8%

2.5771 (table factor) x $8,000 =

$20,616.80

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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.

Step 2.R = 8%/2 = 4%

12.0061 (table factor) x $200 =$2,401.22

Future value of an annuity

Future value of a lump sum

R = 8%/2 = 4%

1.6010 (table factor) x $2,401.22 =

$3,844.35

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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 periods R = 5%/1 = 5%

Step 1. Present value of an annuity Step 2. Present value of a lump sum

R = 5%/1 = 5%

14.0939 x $40,000 = $563,756


.6768 x $563,756 = $381,550.06

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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.


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SINKING FUNDS (FIND PERIODIC PAYMENTS)

Sinking fund = Future x Sinking fund

payment value table factor

• 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:

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SINKING FUND TABLE BASED ON $1 (Table 12.3)

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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

Check

Future Value of an annuity table

N = 18, R= 10%

* Off due to rounding

R = 10%/1 = 10%

0.0219 x $60,000 = $1,314

$1,314 x 45.5992 = $59,917.35*

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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.


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PROBLEM 13-13

Solution:

Periods = 25 years x 1 = 25 periods

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)

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

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PROBLEM 13-17

Solution:

20 periods, 12% (Table 13.1)

$12,500 X 72.0524 = $900,655

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)

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PROBLEM 13-18

Solution:

Periods = 40 years X 1 = 40 periods

$250,000 X .0083 = $2,075 each year

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)

Interest rate per period = 5%/1 = 5%

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PROBLEM 13-23

Solution:

8 periods 8%/ 4 = 2%

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)

or $900 x 18.6392 = $16,775.28 (Table 13.1) x .7284 (Table 12.3) $12,219.11 2%, 16 periods

$900 x 13.5777 = $12,219.93(Table

13.2)

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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

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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 =

Flynn: 5 periods, 8% (Table 13.2)) 3.9927 X $38,000 =

Morton’s offer is the better deal.

$139,744.50 + $40,000 =$179,744.50

$151,722.60 + $25,000 =$176,722.60

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