(Business BUSMATH Math)
BUSMATH (Business Math)
Kristine Joy E. Carpio
Department of Mathematics
De La Salle University Manila
Term 2 2009-2010
BUSMATH (Business Math)
Outline
Annuities and Perpetuities
Ordinary Annuity
Annuity Due
Deferred Annuity
BUSMATH (Business Math)
Course Description
This is a 3-unit course in business mathematics which covers
pre-calculus algebra, theories of investment mathematics, and
their applications to commerce and economics. The course also
introduces students to the rudiments of nancial mathematics.
Discussions on logarithms, sequences, binomial theorem, simple
and compound interest are likewise included.
BUSMATH (Business Math)
Annuities and Perpetuities
Annuity
Denition
An annuity is a sequence of periodic payments made at equal
intervals of time and most of the time in equal amounts.
The time between two consecutive payments of annuity is called
payment interval and is equivalent to one period. The total
number of periods is the same as the total number of payments.
Payments maybe made monthly, quarterly, semi-annually,
annually or at other periods. The term of each annuity starts
immediately and ends on the last payment interval. The size of
each payment is termed as periodic rent or periodic payment,
and is denoted by R.
BUSMATH (Business Math)
Annuities and Perpetuities
Examples of annuities are
a. monthly payments of rent
b. weekly wages
c. annual premiums on a life insurance policy
d. periodic pensions
e. periodic payments on installment purchases
f. semi-annual interest payments on a bond
BUSMATH (Business Math)
Annuities and Perpetuities
Types of Annuities
a. Annuity Certain
Denition
An annuity certain is an annuity in which payment begin and
end at xed times.
b. Contingent Annuity
Denition
A contigent annuity is a sequence of periodic payments in which
the payments extend over an indeterminate length of time.
BUSMATH (Business Math)
Annuities and Perpetuities
Types of Annuities Certain
a. Ordinary Annuity
Denition
An ordinary annuity is a sequence od periodic payments made
at the end of each period.
b. Annuity Due
Denition
An annuity due is a sequence of periodic payments made at the
start of each period.
c. Deferred Annuity
Denition
A deferred annuity is a sequence of periodic payments in which
the rst payment is not made at the beginning nor at the end of
the rst period but at some later date.
BUSMATH (Business Math)
Ordinary Annuity
Present Value
Denition
The present value of an ordinary annuity, denoted by A, is the
sum of the discounted payments at the beginning of the term.
This is the value at the beginning of the term; this is the value
before the rst payment.
The sum of the periodic payments discounted to the present
time is
A = R(1 + i )1 + R(1 + i )2 + + R(1 + i )n
=R
1 (1 + i )n
i
where R is the periodic payment of the annuity, i is the interest
rate per period and n is the total number of payments or period.
BUSMATH (Business Math)
Ordinary Annuity
Cash Value or Cash Price
Denition
The cash value or cash price of a contract is equal to the down
payment (if there is any) plus the discounted value of all future
payments, that is,
Cash Price = Down Payment + A
BUSMATH (Business Math)
Ordinary Annuity
Exercises
1. The purchaser of a piece of property pays P250,000 cash
and the balance in 20 annual payments of P100,000 each.
If money is worth 8%, what is the propertys cash value?
2. An air conditioner is for sale at P16,000 in cash or on terms
P2,500 down and P1,200 each month for the next twelve
months. If you were the buyer, which purchase plan would
you prefer? Money is worth 15% compounded monthly.
3. A philantropist wishes to donate an amount to an
orphanage. The amount will provide P15,000 a year to
support a child for 8 years. If money is invested at 8%
compounded annually, how much should he donate?
4. A furniture set is purchased with a downpayment of P7,500
and the balance at P1,000 a month for two years. What is
the cash price if the interest rate is 18% converted
monthly?
BUSMATH (Business Math)
Ordinary Annuity
Amount
Denition
The sum of all the periodic payments made at the end of each
term plus all accumulated interests is called the amount of
ordinary annuity, denoted by S . This is the value at the end of
the term; this is the value on the last payment date.
The sum of the periodic payments made at the end of each
term with accumulated interest is
S
= R + R(1 + i ) + R(1 + i )2 + + R(1 + i )n 1
(1 + i )n 1
=R
i
BUSMATH (Business Math)
Ordinary Annuity
The present value A and amount S are related by the equations
A = S (1 + i )n
S
= A(1 + i )n
This means that A is the present value of S due in n periods,
and S is the amount of A for n periods.
BUSMATH (Business Math)
Ordinary Annuity
Amount of an Annuity at Any Time
The amount of an annuity can be computed at any time before
the end of the term. This is denoted by Sk and is given by the
equation
(1 + i )k 1
Sk = R
,
i
where k is the number of deposits or payments made.
BUSMATH (Business Math)
Ordinary Annuity
Exercises
1. A housewife deposits P10,000 every three months for seven
years in a savings account that pays 4% compounded
quarterly. How much would she have in her account at the
end of seven years, assuming no withdrawals were made?
2. Every 6 months for 5 years, a father deposits P30,000 in a
trust company for his daughters education. If money earns
at least 16% compounded semi-annually, how will be in the
fund after the seventh deposit? After the last deposit?
3. P450,000 is the amount of a 4 year ordinary annuity. What
is the present value of the annuity if money is worth 8%
(m = 4)?
4. A student invests P5,000 every six months at 13%
compounded semi-annually. Find his savings in twelve
years.
BUSMATH (Business Math)
Ordinary Annuity
Debtors Remaining Liability
Denition
The debtors remaining liability is the sum of the present value
of all unpaid periodic payments.
The remaining liability of the debtor after any k th payment
maybe computed by
RLk = R
1 (1 + i )(n k )
i
The remaining liability of the debtor before any k th payment
maybe computed by
k RL
= R + RLk
BUSMATH (Business Math)
Ordinary Annuity
Finding the Unknown Periodic Payment
The periodic payment given the present value A is
R=
Ai
1 (1 + i )n
The periodic payment given the amount S is
R=
Si
(1 + i )n 1
BUSMATH (Business Math)
Ordinary Annuity
Exercises
1. A loan of P500,000 with interest at 15% compounded
semi-annually is said to be repaid by twenty equal
payments (principal and interest included) made at the end
of each six months. Find the size of each payment.
2. The Citizen Cottage Industry has a high speed machine
that will be retired after 5 years. How much must be set
aside each year in a fund investment at 8% to buy a new
machine costing P250,000 to replace the old one?
3. In purchasing a television worth P30,000, a man pays
P8,000 cash and agrees to make twenty monthly payments.
If interest is 18% (m=12), nd the monthly payment.
4. A student received P500,000 as a graduation gift from his
mother. If he deposited it in a bank paying 12%
compounded quarterly how much would he get every three
months over the next ve years.
BUSMATH (Business Math)
Ordinary Annuity
Finding the Unknown Time
Solving for the number of full payments n given the present
value A gives
Ai
log 1
R
n=
.
log(1 + i )
Solving for the number of full payments n given the amount S
gives
Si
log 1 +
R
n=
.
log(1 + i )
If the number of years t is required divide n by m since n = mt .
BUSMATH (Business Math)
Ordinary Annuity
Exercises
1. A man borrows P50,000 with interest 20% compounded
quarterly. He will discharge the debt by paying P4,000
quarterly.
a. Find the number of regular payments.
b. Find the nal payment after the last full payment.
2. A fund of P120,000 is to be created by depositing P6,000
monthly. If the fund earns 8%,
a. how many deposits of P6,000 will be needed?
b. how much would have been the nal deposit if it were to fall
on the last full deposit date?
c. how much would have been the nal deposit if it is made
one month after the last full deposit?
BUSMATH (Business Math)
Ordinary Annuity
Exercises
3. A mother deposits P12,500 each quarter in a savings bank
paying 10% compounded quarterly. If she desires to
accumulate P900,000, how many full payments must she
make? How much will the nal payment (if many) be if it is
made three months after the last full payment of P12,500.
4. A building costs P25,000,000. It is sold for P5,000,000
downpayment and yearly payments of P4,000,000. If
money is worth 7% eective, nd how many full payments
there will be and how much the nal payment will be it
is if made one year after the last payment of P4,000,000.
5. A fund on P150,000 is invested at 9% compounded
semi-annually. Principal plus interest will provide for
semi-annual withdrawals of P25,000 each. How many such
withdrawals can be made? How much will the nal
withdrawal be?
BUSMATH (Business Math)
Ordinary Annuity
Solving for the Rate j
When the interest rate is unknown, the problem can be solved
using formulas that were derived by series expansion. These
formulas are
n 2 1 i 2 + 6(n + 1)i + 12 1
nR
A
=0
n 2 1 i 2 6(n 1)i + 12 1
nR
S
= 0.
and
The interest rate i is then solved by the quadratic formula.
BUSMATH (Business Math)
Ordinary Annuity
Exercises
1. A used car maybe bought for P760,000 cash or on credit
with a downpayment of P160,000 plus P30,000 monthly for
two years. At what rate compounded monthly is interest
charged?
2. At what rate converted annually will yearly deposits of
P25,000 each amount to P600,000 after the 15th deposit?
3. A man invests P10,000 every three months. If he has
P390,000 in seven years, at what nominal rate compounded
quarterly did his investment earn?
4. At what nominal rate compounded semi-annually is
P500,000 the present value of a P100,000 ordnary annuity
payable semi-annually for three years?
BUSMATH (Business Math)
Annuity Due
Annuity Due
Denition
An annuity due is an annuity paid at the beginning of the
payment interval.
The present value of an annuity due, denoted by A, is its value
on the rst payment date. It is the present value of an ordinary
annuity with (n 1) payments plus the rst payment. This
leads to the formula:
1 (1 + i )(n 1)
A=R+R
.
i
If R is factored from the given equation, the following is
obtained
1 (1 + i )n
A=R
.
i
BUSMATH (Business Math)
Annuity Due
The amount of an annuity due, denoted by S , is its value one
period after the payment. It is the amount of an ordinary
annuity with (n + 1) payments minus one payment. This leads
to the formula:
(1 + i )(n +1) 1
R.
S =R
i
If R is factored from the given equation, the following is
obtained
(1 + i )n 1
S =R
(1 + i ) = S (1 + i ).
i
BUSMATH (Business Math)
Annuity Due
The relationship between the present value and amount of an
annuity due is the same as that of the present value and
annuity due of an ordinary annuity, that is,
A = S (1 + i )n
and
S = A(1 + i )n
BUSMATH (Business Math)
Annuity Due
Exercises
1. Jerome deposits P50,000 at the beginning of each year at
6.5% compounded annually. How much money does he
have in eight years?
2. A P65,000 debt bears interest at 18% compounded
quarterly. It is to be repaid in installments at the
beginning of every three months for ve years. Find the
quarterly payment.
3. A family rents a house for P20,000 payable at the
beginning of each month. Find the cash equivalent of six
years rent if money is worth 18% compounded monthly.
4. A camera can be purchased for 6 monthly payments of
P2,500 each. the rst payment is due on the day of the
purchase. Find the equivalent cash price if money is worth
18% (m = 12).
BUSMATH (Business Math)
Annuity Due
Finding the Periodic Payment of Annuity Due
The periodic payment in an annuity due can be obtained by the
following formulas
R=
Ai
[1 (1 + i )n ] (1 + i )
and
R=
[(1 +
i )n
Si
.
1] (1 + i )
BUSMATH (Business Math)
Annuity Due
Exercises
1. How much must be deposited at the beginning of every
three months in a fund giving 15% compounded quarterly
in order to have P300,000 in ve years.
2. Five years from now, a man will need P100,000. How much
must he invest in a fund at the beginning of each year
starting now to accumulate this sum if the fund pays 8%
compounded annually?
3. Mr. De Jesus borrows P200,000 and agrees to discharge his
liability by making equal payments at the beginning of
each six months for three years. If money is worth
{4.5%, m = 2}, nd his semi-annual payment.
BUSMATH (Business Math)
Annuity Due
Exercises
4. An amount of P350,000 is needed by Mr. Solis for his
business expansion at the end of two years. To accumulate
this amount, Mr. Solis will deposit equal amounts at the
beginning of each month for two years. How much must be
deposited monthly if money is worth 5% converted
monthly?
5. A debtor, who receives a loan of $30,000, agrees to deposit
equal amount of sums at the beginning of each 3 months
for 10 years with a trust company, in order to pay the
principal at the end of 10 years. If the fund earns 3%
compounded quarterly, nd the periodic deposit.
BUSMATH (Business Math)
Deferred Annuity
Deferred Annuity
Denition
A deferred annuity is an annuity where the rst payment does
not coincide with the rst interest period. It is put o to a later
date.
The value of the deferred annuity at the end of the term in the
amount or S . This is the same as the amount of the ordinary
annuity. The value of the deferred annuity at the beginning of
its term is the present value or A. If A is discounted to the
present time, the present value of the deferred annuity is
obtained. Thus,
A = A(1 + i )d ,
where A is the present value of the deferred annuity and d is
the number of deferment periods. Denote the number of years
payment is deferred by td and we have d = mtd .
BUSMATH (Business Math)
Deferred Annuity
Exercises
1. Find the present value of a P50,000 annuity payable eahc
three months for ten years but deferred for ve years.
Money is worth 12% compounded quarterly.
2. On the birth of a child, the father wants to deposit a sum
in a savings account. He intends to privide his daughter
P50,000 every month for four years, starting on her 18th
birthday. If the savings bank pays 12% compounded
monthly, how much ahould the father deposit?
3. A house and lot can be bought for P500,000 downpayment
and 25 quarterly payments of P80,000 each. The rst
installment is due at the end of ve years and three
months. If money is worth 12% compounded quarterly,
how much is the cash value of the house and lot?
BUSMATH (Business Math)
Deferred Annuity
4. Find the present value of annual payments of P50,000 each,
the rst of which is due in four years and the last in twelve
years. Money is worth 8% eective.
5. An investment in an oil eld will yield no operating prot
until the end of 4 years, when the investor will receive
$20,000. After that, he will receive $20,000 at the end of
each years for 15 more years. Find the present value of this
income if money is worth 6%.
6. A purchaser of a farm pays $80,000 cash and also agrees to
pay a sequence of 12 semi-annual payments of $7,500 each,
the rst due at the end of 2.5 years. If money is worth 7%
compounded semi-annually, nd the cash value of the farm.
BUSMATH (Business Math)
Deferred Annuity
Finding the Periodic Payment under Deferred Annuity
Finding the periodic payment for deferred annuty when the
amount given is the same as in the ordinary annuity. The
problem lies when the present value is given. Under this
condition, the periodic payment can be obtained by the formula
R=
Ai
.
[1 (1 + i )n ] (1 + i )d
BUSMATH (Business Math)
Deferred Annuity
Exercises
1. On July 19, 2001, Jeerson deposited P2,000,000 in a
savings account paying 9% compounded semi-annually.
Jeerson planned to make ten semi-annual withdrawals
starting July 19, 2003. Find the amount of each
withdrawal.
2. A man borrowed P300,000 from a nance company that
charges interest at 16% compounded semi-annually. He
promised to pay o the loan in twelve semi-annual
payments. The rst payment is to be made at the end of
two years. Find his semi-annual payment.
BUSMATH (Business Math)
Deferred Annuity
3. In buying a piece of property worth P2.5M cash, the buyer
pays P1.5M as down payment and agrees to pay the
balance, including interest at 15.5% compounded
semi-annually, by a sequence of ten equal semiannual
payements, the rst due at the end of four years. Find the
semi-annual payment.
4. The owner of a handicraft business borrows P500,000 with
interest at 24% compounded monthly. He agrees to
discharge the loan by a sequence of equal payments at the
end of each month for three years, with the rst payment
at the end of one year. Find the periodic payment.
5. A house costs $250,000 cash. A purchaser will pay $50,000
cash and a sequence of 8 equal payments, the rst due at
the end of 3 years. If money is worth 7%, nd the annual
payment.
BUSMATH (Business Math)
Deferred Annuity
References
W. Cordova, C.M. Gotauco, F.F. Ledesma and M.C.R.
Tabuloc. Mathematics of Finance. Mobius Strip
Corporation, 2006.
P.B. Gabriel, and A.C. Ong. Fundamentals of Investment
Mathematics. Island Publishing House, Inc., Revised
Edition, 1994.
W.L. Hart. Mathematics of Investment. D. C. Heath and
Company, 1980.