Problems

1. M purchased a small lot in a subdivision, paying P200, 000 down and promising to pay P15, 000
every 3 months for the next 10 years. The seller figured interest at 12% compounded quarterly.
a).What was the cash price of the lot?

b.) If M missed the first 12 payments, what must he pay at the time the 13th is due to bring
him up to date?

c.) After making 8 payments, M wished to discharge his remaining

indebtedness by a single payment at the time when the 9th regular payment was due, what must he
pay in addition to the regular payment then due?

d.)If M missed the first 10 payments, what must he pay when the 11th
payment is due to discharge his entire indebtedness?

2. A man approaches the ABC Loan Agency for P100, 000 to be paid in 24
monthly installments. The agency advertises an interest rate of 1.5% per month. They proceed to
calculate the amount of his monthly payment in the following manner
Amount requested P100,000
Credit investigation 500
 Credit risk insurance 1,000

Total P101,500

Interest: (P101,500) (24) (0.015) = P36,540

Total owned: P101,500 + P36,540 = P138,040

Payment = P138,040/24 = P5,751.67

What is the effective rate of interest of the loan?

3. A new office building was constructed 5 years ago by a consulting engineering firm. At that time
the firm obtained the bank loan for P 10,000,000 with a 20% annual interest rate, compounded quarterly.
The terms of the loan called for equal quarterly payments for a 10-year period with the right of prepayment
any time without penalty.

Due to internal changes in the firm, it is now proposed to refinance the loan through an insurance
company. The new loan is planned for a 20- year term with an interest rate of 24% per annum, compounded
quarterly. The insurance company has a onetime service charge 5% of the balance. This new loan also calls
for equal quarterly payments.

a.) What is the balance due on the original mortgage (principal) if all
payments have been made through a full five years?

b.) What will be the difference between the equal quarterly payments in
the existing arrangement and the revised proposal?
 4. An asphalt road requires no upkeep until the end of 2 years when P60, 000 will be needed for
repairs. After this P90, 000 will be needed for repairs at the end of each year for the next 5 years, then P120,
000 at the end of each year for the next 5 years.
If money is worth 14% compounded annually, what was the equivalent uniform annual cost for the
12-year period?

5. A man wishes to provide a fund for his retirement such that from his 60 th to 70th birthdays he
will be able to withdraw equal sums of P18, 000 for his yearly expenses. He invests equal amount for his
41st to 59th birthdays in a fund earning 10% compounded annually. How much should each of these amounts
be?
 6.Juan Reyes inherited P280,000 on his 30th birthday. This he placed in his own person fund which
earns interest 12% compounded annually. He withdraws P450,000 on his 35th birthday to make the initial
payment on a house. He drew his annual uniform payment for the person fund on his 65 th birthday and
planned to make 20 such withdraws so as to just exhaust the fund. He died at the age of 71, however, just
before withdrawing the amount due on that date. How much is left in the fund to be passed on the heirs at
the time of Juan’s death?

7. Determine the present worth and the accumulated amount of an annuity consisting of 6 payments
of P120, 000 each, the payment are made at the beginning of each year. Money is worth 15% compounded
annually.

8. Calculate the capitalized cost of a project that has an initial cost of P3, 000,000 and an additional
cost of P100, 000 at the end of every 10 yrs. The annual operating costs will be P100, 000 at the end of
every year for the first 4 years and P160, 000 thereafter. In addition, there is expected to be recurring major
rework cost of P300, 000 every 13 yrs. Assume i =15%.
 9. The will of a wealthy philanthropist left P5, 000,000 to establish a perpetual charitable
foundation. The foundation trustees decided to spend P1, 200,000 to provide facilities immediately and to
provide P100, 000 of capital replacement at the end of each 5 year period. If the invested funds earned 12%
per annum, what would be the year end amount available in perpetuity from the endowment for charitable
purposes?

10.The surface area of a certain plant requires painting is 8,000 sq. ft. Two kinds of paint are
available whose brands are A and B. Paint A cost P 1.40 per sq. ft. but needs renewal at the end of 4 yrs.,
while paint B cost P 1.80 per sq. ft. If money is worth 12% effective, how often should paint B be renewed
so that it will be economical as point A?
 11.A contract has been signed to lease a building at P20,000 per year with an annual increase of
P1,500 for 8 years. Payments are to be made at the end of each year, starting one year from now. The
prevailing interest rate is 7%. What lump sum paid today would be equivalent to the 8-year lease-payment
plan?

12. A mechanical engineer designed a mechanical device costing P20,000. Maintenance will cost
P1,000 each year. The device will generate revenues of P5,000 each year for 5 years, after which the salvage
value is expected to be P7,000. Draw the cash flow diagram.
 13. A father wishes to provide P4,000 for his son on his 21th birthday. How much should he deposit
every 6 months in a savings bank which pays 3% compounded semi annually, if the first deposit is made
1
when the sons is 3 2
years old?

14. If money is worth 12% compounded quarterly and n = 5, compute the present value of the
following.

a.) P1.00 received monthly in year n

b.) P1.00 received monthly for n years

15. A Filipino contractor in the Middle east plans to purchase a new office
Building costing P1,000,000. He agrees to pay P150,000 annually for 20 years. At the end of this
time he estimate that he can resell the building for at least P600,000. Instead of buying the building he can
lease it for P140,000 a year. All payments are to be made at the beginning of each year and the rate of
interest is 10%. Should the contractor purchase or lease the building?
 16. XYZ Inc. plans to construct an additional building at the end of 10 years at an estimated cost
of P5,000,000. To accumulate this amount it will deposit equal year end amounts in a fund earning 13%.
However at the end of the 5th year it decrease to have a longer building estimated to cost P8,000,000. What
should be the annual deposit for the last 5 years?

17. A computer costs P105,000. It can be leased for an annual rent of P30,000 payable at the
beginning of each year throughout its economic life of 5 years. No salvage value if expected at the end of
5 years. Cost of capital is 10% and the applicable in rate is 40%. Should the computer purchased or lease?
Solution:

18. A refrigerator equipment fabricator sold an ice drop making machine with condition that in
case the machine will not be able to produce the guaranteed capacity, he will deduct from the agreed price
of the machine the loss of revenue that the buyer will incur during the economic life of the machine which
is 5 years, plus 12% cost of money. One month after the machine was delivered the buyer complained to
the fabricator that he was not getting the guaranteed production. After a series of test it was determined that
the buyer would be costing P4,000 worth of unrealized sales per year. Assuming no increase in operation
cost if guaranteed production is attained, how much must the buyer deduct from the agreed price of the
machine?
 19. A company manufacturing acids, upon inspection of the plant, found out that it is badly
corroded from the acid fumes and would need to be replaced, to try to get some more life out of the roofing,
the company consulted a roofing coating contractor who presented the company with two options. The first
option is a coating that will cost P20,000 which would extend the life of the roofing for 3 years from the
date of application, and the second option will cost P30,000 and which would extend the life of the roofing
for 5 years from the date of application. At what rate of return are the investment equal?

20. A fully secured loan of P30,000 was to be amortized by 10 equal payment semi-annual, the first
payment to be made 6 months after the loan is approved. After the 6th payment was mae the debtor was in
a position to settle the entire balance remaining by a single payment on the date. If the interest on the loan
is 12% compounded semi-annually, what be the amount of single payment?
 21. A machine has just been purchased for P1.2 million. It is anticipated that the machine will be
held 5 years, that it will have a salvage value of P80,000 based on current prices, and that the annual rate
of inflation during the next 5 years will be 7%. The machine will be replaced by a duplicate and the firm
will accumulate the necessary capital by making equal year end deposits in a reserve fund that earns 6%
per annum. Determine the amount of the annual deposit.

22. A man agreed to pay the loan he is borrowing from a bank in six equal year end payments of
P71,477.70. Interest is 18% per annum, compounded annually, and is included in the yearly amount he will
be paying the bank. How much money is the man borrowing from the bank?

23. A man wishes to purchase a set of appliances with a total cost of P20,000. He made a down
payment of P5,000 and the balance payable in 24 equal monthly installments. If financing charge is 12%
each year computed on the total balance to be paid by installment and interest is 12%,

a.) how much would each installment payment be?

 b.)what will be the actual cost of money?

24. A new company developed a program in which employees will be allowed to purchase shares
of stock of the company at the end of its fifth year of operation, when the company is thought to have gained
stability, at par value of P100 per share. Believing in the good potential of the company, an employee
decided to save in a bank the amount of P8,000 at the end of every year which will earn 9% interest,
compounded annually. How many shares will he able to purchase at the end of the fifth year?

25. If P10,000 is deposited each year for 9 years, how much annuity can a person get annually from
a bank every year for 8 years starting one year after the ninth deposit is made? Money is worth 14%.

26. A company has approved a car plan for its six senior officers in which the company will
shoulder 25% of the cost and the difference payable by each officer to a financing company in 48 equal
end-of-the-month installments at an interest rate of 1.5% per month. If the cost of each car is P350,000
determine the following:
 27. A machinery supplier is offering a certain machine on a 10% down payment and the balance
payable in equal year end payments without interest for 2 years. Under this arrangement the price is pegged
at P250,000. However, for cash purchase the machine would only cost P195,000. What is the equivalent
interest rate that is charged on the two year payment plan if interest is compounded quarterly?

28. Determine the amount needed now to purchase a machine for P100,000, provide an annual fund
of P15,000 for operation and maintenance, and to replace it at the same cost at the end of every ten-year
1
period. Money is worth 7 % compounded annually.
2

29. A new storage tank can be purchased and installed for P50,000 and is expected to last for 10
years. However, an existing tank of equivalent capacity may be continued to be used provided it is repaired.
If this tank were repaired its useful life is estimated to be done again. Assume that future costs will remain
the same and that the tanks will have no scrap value. Money is worth 10% compounded annually. How
much can be spent for repairing the existing tank if the capitalized cost of the two tanks are the same?
 30. A chemical reactor , which will contain corrosive liquids, has been designed. The reactor may
be made of mild steel or stainless steel. The initial cost, including installation, of a mild steel reactor is
expected to be P72,000 and would last 5 years. Stainless steel is more resistant to be corrosive action of the
liquids, but is more expensive and will cost P120,000. Assuming that both types of reactors will have no
salvage or scrap value at the end of their useful lives and could be replaced at the same costs as before, and
if money is worth 12% compounded annually, what should the useful life of the stainless reactor be so that
the capitalized costs are equal?

31. Compute the values of P and F

320 400
240
160
80
0
1 2 3 4 5 6
i = 10%
P
F
32. Compute the values of P and F
P450
P370
P290
P210
P130
P50 P50

0 1 2 3 4 5 6
i = 10%
P F

32. Compute the values of X and P

6x
5x
4x
3x
2x

X
0 1 2 3 4 5 6
i = 10%
P F = P10,000
 33. For the gradient series, compute the values of P and F, and the value of A for the equivalent
uniform series.

P1,000
P900
P800 P700
P600 P500
0 1 2 3 4 5 6
i = 10%
P F

34. Compute the value X so that the two cash flow will be equivalent. Hint the present or
accumulated values for the cash flow should be equal.

P2,800 5x

P2,100 4x

P1,400 3x

P700 2x

0 x

1 2 3 4 5 = 1 2 3 4 5
i = 12% i = 12%
 35. Suppose an engineer receives an initial annual salary of P24,000 increasing at the rate of P1,200
a year. If money is worth 12%, determine his equivalent annual salary for a period of 10 years.

36. The year end operating and maintenance costs of a certain machine are estimated to be P4,800
the first year and to increase by P800.

37. An inventor was offered the following alternatives as royalties for his invention:
a.) A single lump sum payment of P350,000 for the right to manufacture the invention for the next
5 years
b.) An initial payment of P100,000 and year end payments as follows:

End of Year Royalty Payments

 1 P 0
2 50,000
3 100,000
4 150,000
5 200,000

38. For the following arithmetic gradient series, with interest at 10% find:
a.) the present worth
b.) the value at the end of year 6
c.) the value at the end of year 10, assuming no further deposits after the
sixth year
d.). the equivalent uniform series for 6 years.

End of year Deposits

1 P 0
2 800
3 1,600
4 2,400
5 3,200
6 4,000
 39. A machine in a factory has the following end-of-year maintenance costs

Year Maintenance Costs

1 P 720
2 800
3 880
4 960
5 1,040
6 1,120
7 1,200

If Interest is 14.8% compounded annually, determine the present worth of this series.

Solution

40. A company owes P 80,000, which includes the interest, to be paid one year from now. To
provide for the repayment of this debt, the company deposits P2, 000 at the beginning of the first month,
P4, 000 at the beginning of the second moth, increasing the deposits in each succeeding month by P2, 000
until the beginning of the eighth month. These amounts earn interest at the rate of 9% compounded monthly.
Will the total amount in the fund at the end of 12 months be sufficient to repay the debt? If not, how more
is needed?
41. For the cash flow, find P and F if i = 18% per year

P881.17
P786.76
P702.46
P627.20
P560
P500

0 1 2 3 4 5 6

P r = 12% , i = 18% F

42. For the cash flow , find P and F if i=18% per year.

P966.31
P878.46
P798.60
P726

P660
P600

0 1 2 3 45 6 7 8
r = 10% , i = 18%
P
F
 43. For the cash flow, find P and if i= 15%.

P5,000
P4,000

P3,200
P2,560
P2,048
P1,638.40

0 1 2 3 4 5 6
i = - 20% , r = 15%
P F

44. Find the value of x in the arithmetic gradient ,so that the two cash flows will be equivalent if i
= 15% per year.

P1,464.10 5x

P1,321 4x
P1,200 3x
 P1,100 2x
P1,000 x

1 2 3 4 5 = 1 2 3 4 5
r=10% , i=15% r=10% , i=15%

45. A man makes a series of ten annual deposits starting at P2, 000 at the end of the first year and
increasing the account deposited by 10% every year thereafter. Find the total account on deposit in (a) 8%
(b) 10% (c) 12%.

46. A parcel of land in a downtown area, suitable for a making lot, can be leased for a period of 10
years. Initial development costs for clearing, paving and constructing a small office shed on the lot is
estimated to be P150,000. If the net annual revenue for the first year is P22,000 and increases by 10% per
year thereafter until the tenth year, and the improvement revert to the owner of the land at the end of 10
years, what is the rate of return on the investment?
 47. A man invested P50, 000 now for the college education of his 2-year old son. If the fund earns
14% effective, how much will the son get each year starting from his 17𝑡ℎ to the 22𝑡ℎ birthday?

48. Find the uniform annual amount that is equivalent to a uniform gradient series in which the first
year’s payment is P500, the second year’s payment is P600, third year’s payment is P700 and so on and
there is a total of 20 payments. The annual interest is 8%.

49. You are purchasing a P900,000 used automobile, which is to be paid in 36 monthly installment
of P28,872.00. What nominal interest rate are you paying on this financing arrangement?
 50. What is the future worth of the following series of payments?
(a) $3,000 at the end of each six-month period for 10 years at 6% compounded semiannually.

(b) $4,000 at the end of each quarter for six years at 8% compounded quarterly.

(c) $7,000 at the end of each month for 13 years at 9% compounded monthly.

51.What equal series of payments must be paid into a sinking fund in order to accumulate the
following amounts?
(a) $12,000 in 10 years at 6% compounded semi annually when payments are semi annually

(b) $7,000 in 15 years at 9% compounded quarterly when payments are quarterly

(c) $34,000 in 5 years at 7.55% compounded monthly when payments are month.
 52. What is the present worth of the following series of payments?
(a) $500 at the end of each six-month period for 10 years at 8% compounded semiannually

(b) $2,000 at the end of each quarter for five years at 8% compounded quarterly

(c) $3,000 at the end of each month for eight years at 9% compounded monthly

53. Find the effective interest rate per payments period for an interest rate at 9% compounded
monthly if the payment period is
(a) Monthly
(b) Quarterly

(c) Semiannual

(d) Annually

54.What will be the amount accumulated by each of the following present investments?
(a) $4,638 in 10 years at 6% compounded semiannually

(b) $6,500 in 15 years at 8% compounded quarterly

(c) $28,300 in seven years at 9% compounded monthly

 55.A man is planning to retire in 25 years. He wishes to deposit a regular amount every three
months until he retires so that, beginning one year following his retirement, he will receive annual payments
of $32,000 or the next 10 years. How much he deposit if the interest rate is 8% compounded quarterly?

56. A building is priced at $125,000. If the buyer makes a down payment of $25,000 and a payment
of $1,000 every month thereafter, how many months will it take for the buyer to completely pay for the
building? Interest is charged at a rate of 9% compounded monthly

57.A couple is planning to finance its three-year old son’s college education. The couple can deposit
money at 6% compounded quarterly. What quarterly deposit must be made from the son’s 3rd birthday to
his lash birthday in order to provide $50,000 on each birthday from the 18th to the 21st? (Note that the last
deposit is made on the date of the first withdrawal)

58.Sam Santos is planning to retire in 15 years. He can deposit money at 8% compounded quarterly.
Which deposit must he make at the end of each quarter until he retires so that he can make a withdrawal of
$25,000 semiannually over the five years after his retirement? Assume that his first withdrawal occurs at
the end of six months after his retirement.

59.Emily Lacy received $500,000 from an insurance company after her husband’s death. She wants
to deposit this amount in a savings account that earns interest at a rate of 6% compounded monthly. Then
she would like to make 60 equal withdrawals over five years such that, when she makes the last withdrawal
the savings account will have a balance of zero. How much should she withdraw each month?

60.Bob Pearson borrowed $20,000 from a bank at an interest rate of 12% compounded month. This
loan is to be repaid in 36 equal monthly installments over three years. Immediately after his 20 th payment,
Bob desires to pay the remainder of the load in a single payment. Compute the total amount he must pay at
the time.
 61. You are buying a home for $190,000. If you make a down payment of $40,000 and take out a
mortgage on the rest at 8.5% compounded monthly, what will be your monthly payment if the mortgage is
to be paid off in 15 years?

Annuity with Continuous Compounding

62.What is the present equivalent of the following continuous fund flow situation?
a) P1,000,000 per year for 4 years at 10% compounded continuously
b) P500 per quarter for 6.75 years at 20% compounded continuously

63.A series of equal quarterly payments of $5.0(1 for 12 years is equivalent to what present amount
at an interest rate of 9% compounds
(a) Quarterly?
(b) Monthly?
(c) Continuously?

64.What is the future worth of an equal-payment series of $5,000 per year for five years if the
interest rate is 8% compounded continuously?
 65. Suppose that $1,000 is placed in the bank account at the end of each quarter over the 20 years.
What is the account’s future worth at the end of 20 years when interest rate is 8 compounded.
(a) Quarterly?
(b) Monthly?
(c) Continuously?

66.If the interest rate is 7.5% compounded continuously, what is the required quarterly payment to
repay a loan of $10,000 in four years?

67.What is the required payment to repay a loan of $20,000 in five years if the interest rate is 8%
compounded continuously?

68.A series of equal quarterly payments of $1,000 extends over a period of five years. What is the
present worth of this quarterly-payment series at 9.75% interest compounded continuously
 69.A series of equal quarterly payments of $2,000 for 15 years is equivalent to what future lump
sum amount at the end of 10 years at an interest rate of 8% compounded continuously?

You have just invested a one-time amount of P5,000 in a stock-based mutual fund. This fund should
earn (on average) 9% per year over a long period of time. How much will your investment be worth in 35
years?

A 12-cylinder heavy-duty diesel engine will have a guaranteed residual value of P1,000 in five
years. Today (year 0) the equivalent worth of this engine is how much if the interest rate is 9% per year?

You just inherited P10,000. While you plan to squander some of it away, how much should you
deposit in an account earning 5% interest per year if you’d like to have P10,000 in the account in 10 years?

How long does it take (to the nearest whole year) for P1,000 to quadruple in value when the interest
rate is 15% per year?
 An enterprising student invests P1,000 at an annual interest rate that will grow the original
investment to P2,000 in 4 years. In 4 more years, the amount will grow to P4,000, and this pattern of
doubling every 4 years repeats over a total time span of 36 years. What is the annual interest rate the student
is earning?

If a certain machine undergoes a major overhaul now, its output can be increased by 20%, which
translates into additional cash flow of P20,000 at the end of each year for 5 years. If i = 15% per year, how
much can we afford to invest to overhaul this machine?

One of life’s great lessons is to start early and save all the money you can! If you save P2 today
and P2 each and every day thereafter until you are 60 years old (say P730 per year for 35 years), how much
money will you accumulate if the annual interest rate is 7%?
 You can buy a machine for P100,000 that will produce a net income, after operating expenses, of
P10,000 per year. If you plan to keep the machine for 4 years, what must the market (resale) value be at the
end of 4 years to justify the investment? You must make a 15% annual return on your investment.

A certain college graduate, Sallie Evans, has P24,000 in student-loan debt at the end of her college
career. The interest rate on this debt is 0.75% per month. If monthly payments on this loan are P432.61,
how many months will it take for Sallie to repay the entire loan?

A 45-year-old person wants to accumulate P750,000 by age 70. How much will she need to save
each month, starting one month from now, if the interest rate is 0.5% per month?
 Jason makes six EOY deposits of P2,000 each in a savings account paying 5% compounded
annually. If the accumulated account balance is withdrawn 4 years after the last deposit, how much money
will be withdrawn?

A friend of yours just bought a new sports car with a P5,000 down payment, and her P30,000 car
loan is financed at an interest rate of 0.75% per month for 48 months. After two years, the value of her
vehicle in the used-car marketplace is P15,000. How much does your friend still owe on the car loan
immediately after she makes her 24th payment?

A certain fluidized-bed combustion vessel has an investment cost of P100,000, a life of 10 years,
and negligible market (resale) value. Annual costs of materials, maintenance, and electric power for the
vessel are expected to total P10,000. A major relining of the combustion vessel will occur during the fifth
year at a cost of P30,000. If the interest rate is 15% per year, what is the lump-sum equivalent cost of this
project at the present time?

Enrico Suarez has decided to purchase a house instead of renting an apartment. He can afford a
monthly payment of P800, and he has saved P6,000 to use as a down payment on a house. If the mortgage
is 4% nominal interest (compounded monthly) on a 30-year loan, how much can Enrico afford to spend on
a house?
 Determine the current amount of money that must be invested at 12% nominal interest,
compounded monthly, to provide an annuity of P10,000 (per year) for six years, starting 12 years from now.
The interest rate remains constant over this entire period of time.

To pay off P50,000,000 worth of new construction bonds when they come due in 20 years, a water
municipality must deposit money into a sinking fund. Payments to the fund will be made quarterly, starting
three months from now. If the interest rate for the sinking fund is 8% compounded quarterly, how much
will each deposit be?