Tan S.T. Finite Mathematics for the Managerial, Life, and Social Sciences

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b. Compute the interest earned on each account.

c. Show that even if the interest on Colby’s investment were subjected to a tax of

28% upon withdrawal of his investment at the end of 20 years, the net accumu-

lated amount of his investment would still be greater than that of Clark’s.

Solution

a. Because Clark is in the 28% marginal tax bracket, the net yield for his invest-

ment is (0.72)(8), or 5.76%, per year.

Using Formula (9) with R  1000, r  0.0576, m  1, and t  20, so that

i  0.0576 and n  mt  20, we see that Clark’s investment will be worth

or $35,850.49 at his retirement.

Colby has a tax-sheltered investment with an effective yield of 8% per

year. Using Formula (9) with R  1000, r  0.08, m  1, and t  20, so that

i  0.08 and n  mt  20, we see that Colby’s investment will be worth

or $45,761.96 at his retirement.

b. Each man will have paid 20(1000), or $20,000, into his account. Therefore,

the total interest earned in Clark’s account will be (35,850.49  20,000), or

$15,850.49, whereas the total interest earned in Colby’s account will be

(45,761.96  20,000), or $25,761.96.

c. From part (b) we see that the total interest earned in Colby’s account will be

$25,761.96. If it were taxed at 28%, he would still end up with (0.72)(25,761.96),

or $18,548.61. This is larger than the total interest of $15,850.49 earned by

Clark.

⬇ 45,761.96

S 

1000311  0.082

 1 4

0.08

⬇ 35,850.49

S 

1000311  0.05762

 1 4

0.0576

282 5 MATHEMATICS OF FINANCE

5.2 Self-Check Exercises

5.2 Concept Questions

1. Phyliss opened an IRA on January 31, 1993, with a contri-

bution of $2000. She plans to make a contribution of $2000

thereafter on January 31 of each year until her retirement

in the year 2012 (20 payments). If the account earns inter-

est at the rate of 8%/year compounded yearly, how much

will Phyliss have in her account when she retires?

2. Denver Wildcatting Company has an immediate need for a

loan. In an agreement worked out with its banker, Denver

assigns its royalty income of $4800/month for the next 3 yr

from certain oil properties to the bank, with the first pay-

ment due at the end of the first month. If the bank charges

interest at the rate of 9%/year compounded monthly, what

is the amount of the loan negotiated between the parties?

Solutions to Self-Check Exercises 5.2 can be found on

page 284.

1. In an ordinary annuity, is the term fixed or variable? Are

the periodic payments all of the same size, or do they vary

in size? Are the payments made at the beginning or the end

of the payment period? Do the payment periods coincide

with the interest conversion periods?

2. What is the difference between an ordinary annuity and an

annuity due?

3. What is the future value of an annuity? Give an example.

4. What is the present value of an annuity? Give an example.

87533_05_ch5_p257-312 1/30/08 9:55 AM Page 282

5.2 ANNUITIES 283

In Exercises 1–8, find the amount (future value) of each

ordinary annuity.

1. $1000/year for 10 yr at 10%/year compounded annually

2. $1500/semiannual period for 8 yr at 9%/year compounded

semiannually

3. $1800/quarter for 6 yr at 8%/year compounded quarterly

4. $500/semiannual period for 12 yr at 11%/year com-

pounded semiannually

5. $600/quarter for 9 yr at 12%/year compounded quarterly

6. $150/month for 15 yr at 10%/year compounded monthly

7. $200/month for yr at 9%/year compounded monthly

8. $100/week for yr at 7.5%/year compounded weekly

In Exercises 9–14, find the present value of each ordinary

annuity.

9. $5000/year for 8 yr at 8%/year compounded annually

10. $1200/semiannual period for 6 yr at 10%/year compounded

semiannually

11. $4000/year for 5 yr at 9%/year compounded yearly

12. $3000/semiannual period for 6 yr at 11%/year compounded

semiannually

13. $800/quarter for 7 yr at 12%/year compounded quarterly

14. $150/month for 10 yr at 8%/year compounded monthly

15. IRA

If a merchant deposits $1500 at the end of each tax

year in an IRA paying interest at the rate of 8%/year com-

pounded annually, how much will she have in her account

at the end of 25 yr?

16. S

AVINGS

CCOUNTS

If Jackson deposits $100 at the end of

each month in a savings account earning interest at the rate

of 8%/year compounded monthly, how much will he have

on deposit in his savings account at the end of 6 yr, assum-

ing that he makes no withdrawals during that period?

17. S

AVINGS

CCOUNTS

Linda has joined a “Christmas Fund

Club” at her bank. At the end of every month, December

through October inclusive, she will make a deposit of $40

in her fund. If the money earns interest at the rate of

7%/year compounded monthly, how much will she have in

her account on December 1 of the following year?

18. K

EOGH

CCOUNTS

Robin, who is self-employed, con-

tributes $5000/year into a Keogh account. How much will

he have in the account after 25 yr if the account earns inter-

est at the rate of 8.5%/year compounded yearly?

19. R

ETIREMENT

LANNING

As a fringe benefit for the past

12 yr, Colin’s employer has contributed $100 at the end

of each month into an employee retirement account for

Colin that pays interest at the rate of 7%/year compounded

monthly. Colin has also contributed $2000 at the end of

each of the last 8 yr into an IRA that pays interest at the

rate of 9%/year compounded yearly. How much does Colin

have in his retirement fund at this time?

20. S

AVINGS

CCOUNTS

The Pirerras are planning to go to

Europe 3 yr from now and have agreed to set aside

$150/month for their trip. If they deposit this money at the

end of each month into a savings account paying interest at

the rate of 8%/year compounded monthly, how much money

will be in their travel fund at the end of the third year?

21. I

NVESTMENT

NALYSIS

Karen has been depositing $150 at

the end of each month in a tax-free retirement account

since she was 25. Matt, who is the same age as Karen,

started depositing $250 at the end of each month in a tax-

free retirement account when he was 35. Assuming that

both accounts have been and will be earning interest at the

rate of 5%/year compounded monthly, who will end up

with the larger retirement account at the age of 65?

22. I

NVESTMENT

NALYSIS

Luis has $150,000 in his retirement

account at his present company. Because he is assuming a

position with another company, Luis is planning to “roll

over” his assets to a new account. Luis also plans to put

$3000/quarter into the new account until his retirement

20 yr from now. If the account earns interest at the rate of

8%/year compounded quarterly, how much will Luis have

in his account at the time of his retirement?

Hint: Use the compound interest formula and the annuity formula.

23. A

UTO

EASING

The Betzes have leased an auto for 2 yr at

$450/month. If money is worth 9%/year compounded

monthly, what is the equivalent cash payment (present

value) of this annuity?

24. A

UTO

INANCING

Lupé made a down payment of $4000

toward the purchase of a new car. To pay the balance of the

purchase price, she has secured a loan from her bank at the

rate of 12%/year compounded monthly. Under the terms of

her finance agreement, she is required to make payments of

$420/month for 36 mo. What is the cash price of the car?

25. I

NSTALLMENT

LANS

Mike’s Sporting Goods sells elliptical

trainers under two payment plans: cash or installment.

Under the installment plan, the customer pays $22/month

over 3 yr with interest charged on the balance at a rate of

18%/year compounded monthly. Find the cash price for an

elliptical trainer if it is equivalent to the price paid by a cus-

tomer using the installment plan.

26. L

OTTERY

AYOUTS

A state lottery commission pays the

winner of the “Million Dollar” lottery 20 installments of

$50,000/year. The commission makes the first payment of

$50,000 immediately and the other n  19 payments at the

end of each of the next 19 yr. Determine how much money

the commission should have in the bank initially to guar-

antee the payments, assuming that the balance on deposit

with the bank earns interest at the rate of 8%/year com-

pounded yearly.

Hint: Find the present value of an annuity.

5.2 Exercises

87533_05_ch5_p257-312 1/30/08 9:55 AM Page 283

27. P

URCHASING A

OME

The Johnsons have accumulated a

nest egg of $40,000 that they intend to use as a down pay-

ment toward the purchase of a new house. Because their

present gross income has placed them in a relatively high

tax bracket, they have decided to invest a minimum of

$2400/month in monthly payments (to take advantage of

the tax deduction) toward the purchase of their house.

However, because of other financial obligations, their

monthly payments should not exceed $3000. If local mort-

gage rates are 7.5%/year compounded monthly for a con-

ventional 30-yr mortgage, what is the price range of houses

that they should consider?

28. P

URCHASING A

OME

Refer to Exercise 27. If local mort-

gage rates were increased to 8%, how would this affect the

price range of houses that the Johnsons should consider?

29. P

URCHASING A

OME

Refer to Exercise 27. If the Johnsons

decide to secure a 15-yr mortgage instead of a 30-yr mort-

gage, what is the price range of houses they should con-

sider when the local mortgage rate for this type of loan is

7%?

30. S

AVINGS

LAN

Lauren plans to deposit $5000 into a bank

account at the beginning of next month and $200/month

into the same account at the end of that month and at the

end of each subsequent month for the next 5 yr. If her bank

pays interest at the rate of 6%/year compounded monthly,

how much will Lauren have in her account at the end of

5 yr? (Assume she makes no withdrawals during the 5-yr

period.)

31. F

INANCIAL

LANNING

Joe plans to deposit $200 at the end

of each month into a bank account for a period of 2 yr, after

which he plans to deposit $300 at the end of each month

into the same account for another 3 yr. If the bank pays

interest at the rate of 6%/year compounded monthly, how

much will Joe have in his account by the end of 5 yr?

(Assume no withdrawals are made during the 5-yr period.)

32. I

NVESTMENT

NALYSIS

From age 25 to age 40, Jessica de-

posited $200 at the end of each month into a tax-free retire-

ment account. She made no withdrawals or further contri-

butions until age 65. Alex made deposits of $300 into his

tax-free retirement account from age 40 to age 65. If both

accounts earned interest at the rate of 5%/year com-

pounded monthly, who ends up with a bigger nest egg

upon reaching the age of 65?

Hint: Use both the annuity formula and the compound interest

formula.

In Exercises 33 and 34, determine whether the statement

is true or false. If it is true, explain why it is true. If it is

false, give an example to show why it is false.

33. The future value of an annuity can be found by adding

together all the payments that are paid into the account.

34. If the future value of an annuity consisting of n payments

of R dollars each—paid at the end of each investment

period into an account that earns interest at the rate of i per

period—is S dollars, then

R 

11  i 2

 1

284 5 MATHEMATICS OF FINANCE

5.2 Solutions to Self-Check Exercises

1. The amount Phyliss will have in her account when she

retires may be found by using Formula (9) with R  2000,

r  0.08, m  1, and t  20, so that i  r  0.08 and n 

mt  20. Thus,

or $91,523.93.

⬇ 91,523.93

S 

2000311.082

 1 4

0.08

2. We want to find the present value of an ordinary annuity

consisting of 36 monthly payments of $4800 each and

earning interest at 9%/year compounded monthly. Using

Formula (11) with R  4800, m  12, and t  3, so that

i  0.0075 and n  (12)(3)  36, we find

or $150,944.67, the amount of the loan negotiated.

P 

480031  11.00752

36

0.0075

⬇ 150,944.67

0.09

USING

TECHNOLOGY

Finding the Amount of an Annuity

Graphing Utility

As mentioned in Using Technology, Section 5.1, the TI-83/84 can facilitate the solu-

tion of problems in finance. We continue to exploit its versatility in this section.

EXAMPLE 1

Finding the Future Value of an Annuity Find the amount of an

ordinary annuity of 36 quarterly payments of $220 each that earn interest at the rate

of 10% per year compounded quarterly.

87533_05_ch5_p257-312 1/30/08 9:55 AM Page 284

Solution

We use the TI-83/84

TVM SOLVER

with the following inputs:

N  36

I%  10

PV  0

PMT 220

Recall that a payment is an outflow.

FV  0

P/Y  4

The number of payments each year

C/Y  4 The number of conversion periods each year

PMT:END BEGIN

The result is displayed in Figure T1. We deduce that the desired amount is $12,606.31.

EXAMPLE 2

Finding the Present Value of an Annuity Find the present value

of an ordinary annuity consisting of 48 monthly payments of $300 each and earning

interest at the rate of 9% per year compounded monthly.

Solution

We use the TI-83/84

TVM SOLVER

with the following inputs:

N  48

I%  9

PV  0

PMT 300

A payment is an outflow.

FV  0

P/Y  12

The number of payments each year

C/Y  12 The number of conversion periods each year

PMT:END BEGIN

The output is displayed in Figure T2. We see that the required present value of the

annuity is $12,055.43.

Excel

Now we show how Excel can be used to solve financial problems involving annuities.

EXAMPLE 3

Finding the Future Value of an Annuity Find the amount of an

ordinary annuity of 36 quarterly payments of $220 each that earn interest at the rate

of 10% per year compounded quarterly.

N = 48

I% = 9

PV = 12055.43457

PMT = −300

FV = 0

P/Y = 12

C/Y = 12

PMT : END BEGIN

N = 36

I% = 10

PV = 0

PMT = −220

FV = 12606.31078

P/Y = 4

C/Y = 4

PMT : END BEGIN

5.2 ANNUITIES 285

FIGURE T1

The TI-83/84 screen showing the future

value (FV) of an annuity

FIGURE T2

The TI-83/84 screen showing the present

value (PV) of an ordinary annuity

(continued)

87533_05_ch5_p257-312 1/30/08 9:55 AM Page 285

Solution

Here we are computing the future value of a series of equal payments, so

we use the FV (future value) function. As before, we access the Insert Function dia-

log box to obtain the Function Arguments dialog box. After making each of the

required entries, we obtain the dialog box shown in Figure T3.

Note that a 0 is entered in the Type edit box because payments are made at the end

of each payment period. Once again, the answer is negative because cash is paid out.

We deduce that the desired amount is $12,606.31.

EXAMPLE 4

Finding the Present Value of an Annuity Find the present value

of an ordinary annuity consisting of 48 monthly payments of $300 each and earning

interest at the rate of 9% per year compounded monthly.

Solution

Here we use the PV function to compute the present value of an annuity.

Accessing the PV (present value) function from the Insert Function dialog box and

making the required entries, we obtain the PV dialog box shown in Figure T4. We

see that the required present value of the annuity is $12,055.43.

286 5 MATHEMATICS OF FINANCE

Note: Boldfaced words/characters enclosed in a box (for example, ) indicate that an action (click, select,

or press) is required. Words/characters printed blue (for example, Chart sub-t

ype:) indicate words/characters that appear on the

screen.

Enter

FIGURE T3

Excel’s dialog box for the future value

(FV) of an annuity

FIGURE T4

Excel’s dialog box for computing the

present value (PV) of an annuity

87533_05_ch5_p257-312 1/30/08 9:55 AM Page 286

1. Find the amount of an ordinary annuity of 20 payments of

$2500/quarter at %/year compounded quarterly.

2. Find the amount of an ordinary annuity of 24 payments of

$1790/quarter at %/year compounded quarterly.

3. Find the amount of an ordinary annuity of $120/month for

5 yr at %/year compounded monthly.

4. Find the amount of an ordinary annuity of $225/month for

6 yr at %/year compounded monthly.

5. Find the present value of an ordinary annuity of $4500/

semiannual period for 5 yr earning interest at 9%/year

compounded semiannually.

5.3 AMORTIZATION AND SINKING FUNDS 287

6. Find the present value of an ordinary annuity of $2100/

quarter for 7 yr earning interest at %/year compounded

quarterly.

7. Find the present value of an ordinary annuity of $245/month

for 6 yr earning interest at %/year compounded monthly.

8. Find the present value of an ordinary annuity of $185/month

for 12 yr earning interest at %/year compounded monthly.6

TECHNOLOGY EXERCISES

5.3 Amortization and Sinking Funds

Amortization of Loans

The annuity formulas derived in Section 5.2 may be used to answer questions involv-

ing the amortization of certain types of installment loans. For example, in a typical

housing loan the mortgagor makes periodic payments toward reducing his indebted-

ness to the lender, who charges interest at a fixed rate on the unpaid portion of the

debt. In practice, the borrower is required to repay the lender in periodic installments,

usually of the same size and over a fixed term, so that the loan (principal plus interest

charges) is amortized at the end of the term.

By thinking of the monthly loan repayments R as the payments in an annuity, we

see that the original amount of the loan is given by P, the present value of the annu-

ity. From Equation (11), Section 5.2, we have

(12)

A question a financier might ask is: How much should the monthly installment be so

that a loan will be amortized at the end of the term of the loan? To answer this ques-

tion, we simply solve (12) for R in terms of P, obtaining

R 

1  11  i 2

n



P  R c

1  11  i2

n

d Ra

Amortization Formula

The periodic payment R on a loan of P dollars to be amortized over n periods

with interest charged at the rate of i per period is

(13)R 

1  11  i 2

n

87533_05_ch5_p257-312 1/30/08 9:55 AM Page 287

APPLIED EXAMPLE 1

Amortization Schedule A sum of $50,000 is

to be repaid over a 5-year period through equal installments made at the

end of each year. If an interest rate of 8% per year is charged on the unpaid bal-

ance and interest calculations are made at the end of each year, determine the size

of each installment so that the loan (principal plus interest charges) is amortized

at the end of 5 years. Verify the result by displaying the amortization schedule.

Solution

Substituting P  50,000, i  r  0.08 (here, m  1), and n  5 into

Formula (13), we obtain

giving the required yearly installment as $12,522.82.

The amortization schedule is presented in Table 4. The outstanding principal

at the end of 5 years is, of course, zero. (The figure of $.01 in Table 4 is the

result of round-off errors.) Observe that initially the larger portion of the repay-

ment goes toward payment of interest charges, but as time goes by more and

more of the payment goes toward repayment of the principal.

Financing a Home

APPLIED EXAMPLE 2

Home Mortgage Payments The Blakelys bor-

rowed $120,000 from a bank to help finance the purchase of a house. The

bank charges interest at a rate of 9% per year on the unpaid balance, with interest

computations made at the end of each month. The Blakelys have agreed to repay

the loan in equal monthly installments over 30 years. How much should each

payment be if the loan is to be amortized at the end of the term?

Solution

Here, P  120,000, , and n  (30)(12)  360.

Using Formula (13) we find that the size of each monthly installment required is

given by

or $965.55.

⬇ 965.55

R 

1120,000210.0075 2

1  11.00752

360

i 



0.09

 0.0075

R 

150,000210.08 2

1  11.082

5

⬇ 12,522.82

288 5 MATHEMATICS OF FINANCE

TABLE 4

An Amortization Schedule

Payment

End of Interest Repayment Toward Outstanding

Period Charged Made Principal Principal

0 — — — $50,000.00

1 $4,000.00 $12,522.82 $ 8,522.82 41,477.18

2 3,318.17 12,522.82 9,204.65 32,272.53

3 2,581.80 12,522.82 9,941.02 22,331.51

4 1,786.52 12,522.82 10,736.30 11,595.21

5 927.62 12,522.82 11,595.20 0.01

87533_05_ch5_p257-312 1/30/08 9:55 AM Page 288

APPLIED EXAMPLE 3

Home Equity Teresa and Raul purchased a

house 10 years ago for $200,000. They made a down payment of 20% of

the purchase price and secured a 30-year conventional home mortgage at 9% per

year on the unpaid balance. The house is now worth $380,000. How much

equity do Teresa and Raul have in their house now (after making 120 monthly

payments)?

Solution

Since the down payment was 20%, we know that they secured a

loan of 80% of $200,000, or $160,000. Furthermore, using Formula (13) with

P  160,000, and n  (30)(12)  360, we determine that

their monthly installment is

or $1287.40.

After 120 monthly payments have been made, the outstanding principal

is given by the sum of the present values of the remaining installments (that

is, 360  120  240 installments). But this sum is just the present value of

an annuity with n  240, R  1287.40, and i  0.0075. Using Formula (11),

we find

or approximately $143,088. Therefore, Teresa and Raul have an equity of

380,000  143,088, that is, $236,912.

⬇ 143,088.01

P  1287.40 c

1  11  0.00752

240

0.0075

⬇ 1287.40

R 

1160,000210.0075 2

1  11.00752

360

i 



0.09

 0.0075

5.3 AMORTIZATION AND SINKING FUNDS 289

Explore & Discuss and Exploring with Technology

1. Consider the amortization Formula (13):

Suppose you know the values of R, P, and n and you wish to determine i. Explain why

you can accomplish this task by finding the point of intersection of the graphs of the

functions

2. Thalia knows that her monthly repayment on her 30-year conventional home loan of

$150,000 is $1100.65 per month. Help Thalia determine the interest rate for her loan by

verifying or executing the following steps:

a. Plot the graphs of

using the viewing window [0, 0.01]  [0, 1200].

b. Use the

ISECT

(intersection) function of the graphing utility to find the point of inter-

section of the graphs of part (a). Explain why this gives the value of i.

c. Compute r from the relationship r  12i.

 1100.65

and



150,000x

1  11  x 2

360

 R

and



1  11  i 2

n

R 

1  11  i 2

n

87533_05_ch5_p257-312 1/30/08 9:55 AM Page 289

APPLIED EXAMPLE 4

Home Affordability The Jacksons have deter-

mined that, after making a down payment, they could afford at most $2000

for a monthly house payment. The bank charges interest at the rate of 7.2% per

year on the unpaid balance, with interest computations made at the end of each

month. If the loan is to be amortized in equal monthly installments over 30 years,

what is the maximum amount that the Jacksons can borrow from the bank?

Solution

Here, , n  (30)(12)  360, and R  2000; we

are required to find P. From Equation (12), we have

Substituting the numerical values for R, n, and i into this expression for P, we

obtain

Therefore, the Jacksons can borrow at most $294,643.

An adjustable-rate mortgage (ARM) is a home loan in which the interest rate is

changed periodically based on a financial index. For example, a 5/1 ARM is one that

P 

200031  11.0062

360

0.006

⬇ 294,643

P 

R31  11  i2

n

i 



0.072

 0.006

290 5 MATHEMATICS OF FINANCE

Explore & Discuss and Exploring with Technology

1. Suppose you secure a home mortgage loan of $P with an interest rate of r per year to be

amortized over t years through monthly installments of $R. Show that, after N install-

ments, your outstanding principal is given by

(0  N  n)

Hint: . To see this, study Example 3, page 289. Replace R using

Formula (13).

2. Refer to Example 3, page 289. Using the result of part 1, show that Teresa and Raul’s

outstanding balance after making N payments is

(0  N  360)

3. Using a graphing utility, plot the graph of

using the viewing window [0, 360]  [0, 160,000].

4. Referring to the graph in part 3, observe that the outstanding principal drops off slowly

in the early years and accelerates quickly to zero toward the end of the loan. Can you

explain why?

5. How long does it take Teresa and Raul to repay half of the loan of $160,000?

Hint: See the previous Explore & Discuss and Exploring with Technology box.

E1x2

160,00011.0075

360

 1.0075

1.0075

360

 1

E1N2

160,00011.0075

360

 1.0075

1.0075

360

 1

B1N2 R c

1  11  i2

nN

B1N2 P c

11  i 2

 11  i2

11  i 2

 1

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5.3 AMORTIZATION AND SINKING FUNDS 291

has an initial rate for the first 5 years, and thereafter is adjusted every year for the

remaining term of the loan.

APPLIED EXAMPLE 5

Adjustable Rate Mortgages Five years ago,

the Campbells secured a 5/1 ARM to help finance the purchase of their

home. The amount of the original loan was $350,000 for a term of 30 years, with

interest at the rate of 5.76% per year, compounded monthly. The Campbells’ mort-

gage is due to reset next month and the new interest rate will be 6.96% per year,

compounded monthly.

a. What was the Campbells’ monthly mortgage payment for the first 5 years?

b. What will the Campbells’ new monthly mortgage payment be (after the reset)?

By how much will the monthly payment increase?

Solution

a. First, we find the Campbells’ monthly payment on the original loan amount.

Using Formula (13) with P  350,000, , and n  mt 

(12)(30)  360, we find that the monthly payment was

or $2044.73 for the first 5 years.

b. In order to find the amount of the Campbells’ new mortgage payment, we first

need to find their outstanding principal. This is given by the present value of

their remaining mortgage payments. Using Formula (11), with R  2044.729,

, and n  mt  360  5(12)  300, we find that their outstand-

ing principal is

or $324,709.19.

Next, we compute the amount of their new mortgage payment for the

remaining term (300 months). Using Formula (13) with P  324,709.194,

, and n  mt  300, we find that the monthly payment is

or $2286.70—an increase of $241.97.

Sinking Funds

Sinking funds are another important application of the annuity formulas. Simply

stated, a sinking fund is an account that is set up for a specific purpose at some future

date. For example, an individual might establish a sinking fund for the purpose of dis-

charging a debt at a future date. A corporation might establish a sinking fund in order

to accumulate sufficient capital to replace equipment that is expected to be obsolete at

some future date.

By thinking of the amount to be accumulated by a specific date in the future as

the future value of an annuity [Equation (9), Section 5.2], we can answer questions

about a large class of sinking fund problems.

R 

324,709.1941

0.0696

1  11 

0.0696

300

⬇ 2286.698

i 



0.0696

P  2044.729 c

1  11 

0.0576

300

0.0576

d⬇ 324,709.194

i 



0.0576

R 

350,0001

0.0576

1  11 

0.0576

360

⬇ 2044.729

i 



0.0576

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