Annuities and gradient

ANNUITY
 consists of a series of equal payments made
at equal intervals of time

OCCURRENCES OF ANNUITY
 Payment of debt by a series of equal
payments at equal intervals of time.
 Accumulation of a certain amount by setting
equal amounts periodically
 Substitution of a series of equal amounts
periodically in lieu of a lump sum at
retirement of an individual.

TYPES OF ANNUITIES
 Ordinary annuity
 Deferred annuity
 Annuity due
 Perpetuity

ORDINARY ANNUITY
 One where equal payments are made at the
end of each payment period starting form the
first period

4 ESSENTIAL ELEMENTS OF ORDINARY ANNUITY
 The amount of payments are equal
 The payments made at equal intervals of
time
 The first payment is made at the end of the
first period and all payments thereafter are
made at the end of the corresponding period
 Compound interest is paid on all amounts in
the annuity

 1 2 n-2 n-1 n
 $1 $1 $1 $1 $1
(P/A, i%,n) (F/A, i%,n)

EXAMPLE FOR ORDINARY ANNUITY
 A steam boiler is purchased on the basis of
guaranteed performance. However, initial
tests indicate that the only operating cost will
be P 400 more per year than guaranteed. If
the expected life is 25 years and money is
worth 10%, what deduction from the
purchase price would compensate the buyer
for the additional operating cost?

EXAMPLE ON ORDINARY ANNUITY
 How much money would you have to deposit
for five consecutive years starting one year
from now if you want to be able to withdraw
P50,000 ten years from now? Assume
interest is 14% compounded annually.

DEFERRED ANNUITY
 One where the payment of the first amount is
deferred a certain number of periods after the
first.
 It will be noted that the first payment is made
at a period later than the first. After the first
payment is made, all succeeding payments
are paid at the end of the periods extending
to the end of the annuity.

 1 2 k k+1 k+2
k+n
 A 0 1 2 n-1 n
k/(P/A,i%,n
)
(P/A,i%,n)
Deferment, k
periods
Ordinary Annuity, n periods
Deferred Annuity, (k+n) periods

EXAMPLE ON DEFERRED PAYMENT
 A lathe for a machine shop costs P60,000 if
paid in cash. On the instalment plan, a
purchaser should pay P20,000 down
payment and 10 quarterly instalments, the
first due at the end of the of the first year
purchase. If the money is worth 15%
compounded quarterly, determine the
quarterly instalment.

EXAMPLE ON DEFERRED ANNUITY
 A man invests P10,000 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 18th to the
22nd birthday?

ANNUITY DUE
 One where the payments are made at the
start of each period, beginning from first
period.

EXAMPLE ON ANNUITY DUE
 A farmer bought a tractor costing P25,000
payable in 10 semi-annual payments, each
instalment payable at the beginning of each
period. If the rate of interest is 26%
compounded semi-annually, determine the
amount of each instalment.

PERPETUITY
 An annuity where the payment periods
extend FOREVER or in which the periodic
payments continue indefinitely

CAPITALIZED COST
 The capitalized cost of any structure or
property is the sum of its first cost and the
present worth of all cost for replacement,
operation, and maintenance for a long period
of time or forever.

EXAMPLE ON PERPETUITY
 If money is worth 8% compounded quarterly,
compare the present values of the following:
 Annuity of P1000 payable quarterly for 50 years
 Annuity of P1000 payable quarterly for 100 years
 A perpetuity of P1000 payable quarterly

EXAMPLES ON CAPITALIZED COST AND
PERPETUITY
 The capitalized cost of a piece of equipment
was found to be P142,000. the rate of
interest used in the computations was 12%,
with a salvage value of P10,000 at the end of
a service life of 8 years. Assuming that the
cost of perpetual replacement remains
constant, determine the original cost of the
equipment.

EXAMPLES ON CAPITALIZED COST AND
PERPETUITY
 Compare the capitalized costs of the following
road pavements:
 An asphalt pavement costing P 100,000 which
would last for 5 years with negligible repairs. At
the end of 5 years, P 5,000 would spent to
remove the old surface before P 100,000 is
spent again for a new surface.
 A thick concrete pavement costing P 250,000
which would last indefinitely, with a cost of P
20,000 for minor repairs at the end of every 3
years. Money is worth 8% compounded
annually.

EXAMPLE ON CAPITALIZED COST AND
PERPETUITY
 It costs P 50,000 at the end of each year to
maintain a section of Kennon Road in Baguio
City. If money is worth 10% how much would
it pay to spend immediately to reduce the
annual cost to P10,000?

GRADIENT
 A series of disbursement or receipts that
increase or decrease in each succeeding
periods.

FORMULAS FOR UNIFORM ARITHMETIC
GRADIENT
(A/G,i%,n) = 1/i – n/[(1+i)^n - 1)]
(P/G,i%,n) = 1/i [{1-(1+i)^-n /i} – {n/(1+i)^n}]
(F/G,i%,n) = 1/i [{(1+i)^n – 1}/i –{n}]
A = G(A/G,i%,n)
P = G(P/G,i%,n)
F = G(F/G,i%,n)

EXAMPLES ON ARITHMETIC GRADIENT
 Find the value of each of the following:
 (A/G,14.53%,23)
 (A/G,31%,50)
 (P/G,12%,10)
 (P/G,15.6%,35)
 (F/G,7.8%,21)
 F/G,12.5%,18)

 Compute the value of the amount of C.
C
0
50
100
150
i =10%

SOLUTION
C = G(P/G, i%, n)
G = 50, i = 10%, n =4 , (P/G, i%, n) =4.378
C = 50(4.378) = 218.91
0
50
C
100
150
i =10%

 Compute the value of the amount of F
50
F
200
100
150
i =
10%

 Mr. Marcelo Santos, author of Para Sa
Hopeless Romantic was offered he following
alternatives by Star Cinema for the rights to
make his novel into a movie.
 A single lump sum payment of P 500,000 or
 An initial payment of P 250,000 plus 2% of the
movie’s gross receipts for the next 5 years estimated
to be as follows:
After the 5th year, the author will not receive further
royalties. If money is worth 14%, which alternative
should he select. Disregard income tax
considerations.
End of year Gross Receipts
In Millions
2% of Gross
Receipts in
Thousands
1 P 10 P200
2 8 160
3 6 120
4 4 80
5 2 40

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