The use of applications of many formulae became more of a...
Transcript of The use of applications of many formulae became more of a...
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Thought to fonder:
The use of applications
of many formulae
became more of a
convenience rather
than a necessity.
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Recall:Two principal types of problem
occur:
The problem involving a single
payment; and
the problem dealing with series
of equal payment.
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Game:
Reveal the title
of our lesson for
today.
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Chapter 4Mathematics of Annuities
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Lesson 4.1Basic Concepts in
Annuity
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Annuity
It is a series of equal payments made at equal
intervals of time.
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Annuity
The word “annuity” implies
etymologically annual payments.
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Challenge
Can you give
general examples of
annuity in business
transactions?
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Examples of Annuity
Interest
Payments on
Bonds
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Examples of Annuity
Premiums on
Life Insurance
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Examples of Annuity
Depreciation
of Funds
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Examples of Annuity
Sinking
Funds
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Three
MainClasses of Annuities
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Annuity Certain
It is an annuity
whose term begins
and ends at fixed
dates.
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Examples of Annuity
Certain
Mortgage
payment of
house
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Examples of Annuity
Certain
Car
Loans
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Examples of Annuity
Certain
Installment
Plans
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Examples of Annuity
Certain
Loans
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Perpetuity
It is an annuity whose
term begins at a fixed
date but continues
forever/indefinitely.
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Examples of Perpetuity
Fixed coupon
payments on
permanently invested
(irredeemable) sums of
money
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Examples of Perpetuity
Scholarships paid
perpetually from
an endowment
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Contingent Annuity
It is an annuity when the date of
either the first payment or the last
payment of an annuity is
uncertain, depending upon some
events whose dates of
occurrence cannot be foretold.
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Example of
Contingent Annuity
Life insurance that
cease when the
person insured dies
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Classifications
ofAnnuities Certain
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Ordinary Annuity
It is one in which payments
are made at the end of each
rent period, the first payment
being due one period
hence.
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Examples of Ordinary
Annuity
Salaries
Stock Dividends
Pension Plan
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Annuity Due
It is one in which the
payments are made at the
beginning of each rent
period, the first payment
being due now.
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Examples of Annuity
DueLease Agreements
(House Rent, Car Rent
etc.)
Educational Insurance
Plan
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Deferred AnnuityIt is an ordinary annuity
whose term begins a given
number of periods from now,
the payments being made at
the end of each period after
the term begins.
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Examples of Deferred Annuity
SSS Salary Loan
PAG-IBIG Salary Loan
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Basic Concepts
in
Annuity
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Periodic PaymentThe size of each
payment.
The interval between
two successive
payments.
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Term of the AnnuityThe interval between the
beginning of the first rent
period and the end of the
last rent period is called
the term of the annuity.
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Final Value of an Annuity
The sum of the
compound amount of
all payments,
compounded to the
end of the term.
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Present Value of an Annuity
The lump sum
required at the
beginning.
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Lesson 4.2The Amount or Final
Value of an Ordinary
Annuity
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Note:The periodic payments
of an ordinary annuity
are made at the end of
each rent period.
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Final Value of Annuity
The amount, or final value of
an annuity is the sum of the
compound amount of the
payments that accumulated
to the end of the term.
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DerivationThe Formula for
Finding the Final
Value of an Annuity
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𝑺𝒏:It is the final value
of an n-payment
ordinary annuity.
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𝑹:It is the size of
each annuity
payment.
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𝒊:It is the interest rate per
compounding period (which
for ordinary annuity, equals
the interest rate per payment
interval).
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𝒏:It is the number
of payments in
the annuity.
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𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟏:𝑺𝒏 = 𝑹(𝟏 + 𝒊)𝒏−𝟏 +𝑹(𝟏 + 𝒊)𝒏−𝟐 + 𝑹(𝟏 + 𝒊)𝒏−𝟑 +⋯+𝑹 𝟏 + 𝒊 𝟑 + 𝑹 𝟏 + 𝒊 𝟐 +𝑹 𝟏 + 𝒊 𝟏 + 𝑹
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𝑻𝒉𝒐𝒖𝒈𝒉𝒕 𝒕𝒐 𝑭𝒐𝒏𝒅𝒆𝒓:
What can you say
to the formula? Is it
convenient to use?
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𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟐:
𝑺𝒏 = 𝑹(𝟏+𝒊)𝒏−𝟏
𝒊
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𝑬𝒙𝒂𝒎𝒑𝒍𝒆 𝟒. 𝟏:Find the amount of an
annuity of P7,000.00 payableat the end of each 6 months
for 15 years, if money isworth 4% compounded
semi−annually.
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𝑭𝒊𝒏𝒂𝒍 𝑨𝒏𝒔𝒘𝒆𝒓:
The amount of
an annuity is
P283,976.55.
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𝑬𝒙𝒂𝒎𝒑𝒍𝒆 𝟒. 𝟐:Mrs. Dolly Mirabueno has been
contributing P460.00 at the end of each
quarter for the past 18 quarters to a
savings plan that earns 9%
compounded quarterly. What amount
will she accumulate if she continues
with the plan for another year?
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𝑭𝒊𝒏𝒂𝒍 𝑨𝒏𝒔𝒘𝒆𝒓:The amount she will
accumulate if she
continues the plan for
another year is
P12,911.12.
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Math
RunnersThe Finding the Final
Value of an Annuity
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Math Runners
Problem (For 5 minutes, for 5
points and for first 10
pairs only)
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Mr. Poppeye spends P850.00
per month on cigarettes. If
he stops smoking and invests
the same amount in a plan
paying 15% compounded
monthly, how much will he
have after 4 years?
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He will have
P55,444.13
after 4 years.
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𝑳𝒆𝒕′𝒔 𝑷𝒓𝒂𝒄𝒕𝒊𝒄𝒆: Solve the following
problem.1. Mr. Ledesma saves P15,000.00 each half-
year and invests it at 9% converted semi-
annually. Find the amount of his savings at
the end of 6 years.
2. Ms. Tipay deposits P8,000.00 semi-annually
in a bank that pays interest at 7 ½%
converted semi-annually. How much is her
credit at the end of 7 years?
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𝑨𝒔𝒔𝒊𝒈𝒏𝒎𝒆𝒏𝒕: Solve the following
problem.
1. To buy a new car in 5 years, a man
deposits P9,000.00 monthly in a bank paying
8 ½% interest converted monthly. How much
will he have after this period?
2. Ms. O deposits P600.00 every 6 months for
3 years. If the interest is at 7% compounded
semiannually, how much does Ms. O have in
the bank for 3 years?
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Lesson 4.3The Present Value of
an Ordinary Annuity
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Present Value of an Ordinary
Annuity
The present value of an annuity is
that single sum of money which if
invested now at a given rate, will
amount to the sum of the compound
amount of the payments at the end
of the term of the annuity.
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Present Value of an Ordinary
Annuity
It may also be defined
as the sum of the present
values of all payments of
the annuity.
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Present Value of an Ordinary
Annuity
The present value of an
annuity is sometimes
referred to as the “cash
equivalent of the annuity.”
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Present Value of an Ordinary
Annuity𝑨𝒏 = 𝑷𝒓𝒆𝒔𝒆𝒏𝒕 𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝒂𝒏 𝑶𝒓𝒅𝒊𝒏𝒂𝒓𝒚 𝑨𝒏𝒏𝒖𝒊𝒕𝒚
𝑺𝒏 = 𝑭𝒊𝒏𝒂𝒍 𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝒂𝒏 𝒏 −𝒑𝒂𝒚𝒎𝒆𝒏𝒕 𝒐𝒓𝒅𝒊𝒏𝒂𝒓𝒚 𝒔𝒊𝒎𝒑𝒍𝒆 𝒂𝒏𝒏𝒖𝒊𝒕𝒚
𝑹 = 𝒕𝒉𝒆 𝒔𝒊𝒛𝒆 𝒐𝒇 𝒆𝒂𝒄𝒉 𝒂𝒏𝒏𝒖𝒊𝒕𝒚 𝒑𝒂𝒚𝒎𝒆𝒏𝒕
𝒊 = 𝒕𝒉𝒆 𝒊𝒏𝒕𝒆𝒓𝒆𝒔𝒕 𝒓𝒂𝒕𝒆 𝒑𝒆𝒓 𝒄𝒐𝒎𝒑𝒐𝒖𝒏𝒅𝒊𝒏𝒈 𝒑𝒆𝒓𝒊𝒐𝒅
(𝒘𝒉𝒊𝒄𝒉, 𝒇𝒐𝒓 𝒐𝒓𝒅𝒊𝒏𝒂𝒓𝒚 𝒂𝒏𝒏𝒖𝒊𝒕𝒊𝒆𝒔,𝒆𝒒𝒖𝒂𝒍𝒔 𝒕𝒉𝒆 𝒊𝒏𝒕𝒆𝒓𝒆𝒔𝒕 𝒓𝒂𝒕𝒆 𝒑𝒆𝒓 𝒑𝒂𝒚𝒎𝒆𝒏𝒕 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍)
𝒏 = 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒑𝒂𝒚𝒎𝒆𝒏𝒕𝒔 𝒊𝒏 𝒕𝒉𝒆 𝒂𝒏𝒏𝒖𝒊𝒕𝒚
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Using:
𝑷 =𝑺
(𝟏+𝒊)𝒏𝒐𝒓
𝑷 = 𝑺(𝟏 + 𝒊)−𝒏
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Formula:
𝑨𝒏 = 𝑹𝟏−(𝟏+𝒊)−𝒏
𝒊
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟑
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Example 4.3:
Find the present value of
an annuity of P8,000.00
payable per annum for 35
years, if money is worth 6%
(m=1).
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Final Answer:This means that paying
P115,985.97 now is just
as good as making 35
annual payments of
P8,000.00 each.
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Example 4.4:A certain investment pays back
P3,200.00 at the end of every six months
for 15 years. At the end of 15 years, the
investment pays back P45,000.00 in
addition to the regular P3,200.00. What is
the present value of all of the payments
if money is can earn 4% compounded
semiannually?
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Final Answer:
The combined
present value is
P96,511.85.
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Let′s Practice: Solve the following
problems:(1) What is the present value of an annuity of
P20,500.00 per annum if the term of the
annuity is 15 years and money is worth 9.5%?
(2) A house is being sold allowing the buyer
to make payments of P5,000.00 a month for
12 years “like paying rent or rent to own”.
Find the equivalent price now if money is
worth 9% converted monthly.
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Assignment 4.2: Solve the following
problem:De La Salle University hires a new dean for
their De La Salle University-Lipa branch. The
contract states that if the new dean works for
10 years, then he will receive a retirement
benefit of P80,000.00 at the end of each
quarter period for 5 years. Find the lump sum
the college could deposit today to satisfy the
retirement contact if fund can be invested at
12% compounded quarterly.
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Lesson 4.4The Periodic
Payment of an
Ordinary Annuity
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Periodic Payment (R)The periodic payment of an ordinary
annuity is the every payment that is
equal which is being made at equal
interval within a series of payment. In
other words, it is the size of each
annuity payment.
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Periodic Payment (R)
We will denote
period payment
using majuscule
letter R.
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Two Formulae Used in Computing for
Periodic Payment (R) of an Ordinary
AnnuityThe periodic payment (R) when
Final Value of the Ordinary Annuity
is given
The period payment (R) when
Present Value of the Ordinary
Annuity is given
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1. The Periodic Payment (R) when Final
Value of the Ordinary Annuity is given
This is used whenever we want
to compute for the period
payment (R) when final value ofthe ordinary annuity (𝑺𝒏) is given
and not the present value of theannuity (𝑨𝒏).
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Derivation of Formula:𝑺𝒏 = 𝑭𝒊𝒏𝒂𝒍 𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝒂𝒏 𝒏 −
𝒑𝒂𝒚𝒎𝒆𝒏𝒕 𝒐𝒓𝒅𝒊𝒏𝒂𝒓𝒚 𝒔𝒊𝒎𝒑𝒍𝒆 𝒂𝒏𝒏𝒖𝒊𝒕𝒚
𝑹 = 𝒕𝒉𝒆 𝒔𝒊𝒛𝒆 𝒐𝒇 𝒆𝒂𝒄𝒉 𝒂𝒏𝒏𝒖𝒊𝒕𝒚 𝒑𝒂𝒚𝒎𝒆𝒏𝒕
𝒊 = 𝒕𝒉𝒆 𝒊𝒏𝒕𝒆𝒓𝒆𝒔𝒕 𝒓𝒂𝒕𝒆𝒑𝒆𝒓 𝒄𝒐𝒎𝒑𝒐𝒖𝒏𝒅𝒊𝒏𝒈 𝒑𝒆𝒓𝒊𝒐𝒅
(𝒘𝒉𝒊𝒄𝒉, 𝒇𝒐𝒓 𝒐𝒓𝒅𝒊𝒏𝒂𝒓𝒚 𝒂𝒏𝒏𝒖𝒊𝒕𝒊𝒆𝒔,𝒆𝒒𝒖𝒂𝒍𝒔 𝒕𝒉𝒆 𝒊𝒏𝒕𝒆𝒓𝒆𝒔𝒕 𝒓𝒂𝒕𝒆 𝒑𝒆𝒓 𝒑𝒂𝒚𝒎𝒆𝒏𝒕 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍)
𝒏 = 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒑𝒂𝒚𝒎𝒆𝒏𝒕𝒔 𝒊𝒏 𝒕𝒉𝒆 𝒂𝒏𝒏𝒖𝒊𝒕𝒚
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Using Multiplicative Inverse:
𝑹 =𝑺𝒏𝒊
(𝟏+𝒊)𝒏−𝟏
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟒
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Example 4.5:How much must be set aside
semi-annually so as to have a
fund of P200,000.00 at the end
of 12 years with interest at the
rate of 12% converted semi-
annually?
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1. The Periodic Payment (R) when Present
Value of the Ordinary Annuity is given
This is used whenever we want
to compute for the period
payment (R) when final value ofthe ordinary annuity (𝑺𝒏) is not
given and the present value ofthe annuity (𝑨𝒏) is given.
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Derivation of Formula:𝑨𝒏 = 𝑷𝒓𝒆𝒔𝒆𝒏𝒕 𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝒂𝒏 𝑶𝒓𝒅𝒊𝒏𝒂𝒓𝒚 𝑨𝒏𝒏𝒖𝒊𝒕𝒚
𝑹 = 𝒕𝒉𝒆 𝒔𝒊𝒛𝒆 𝒐𝒇 𝒆𝒂𝒄𝒉 𝒂𝒏𝒏𝒖𝒊𝒕𝒚 𝒑𝒂𝒚𝒎𝒆𝒏𝒕
𝒊 = 𝒕𝒉𝒆 𝒊𝒏𝒕𝒆𝒓𝒆𝒔𝒕 𝒓𝒂𝒕𝒆 𝒑𝒆𝒓𝒄𝒐𝒎𝒑𝒐𝒖𝒏𝒅𝒊𝒏𝒈 𝒑𝒆𝒓𝒊𝒐𝒅
(𝒘𝒉𝒊𝒄𝒉, 𝒇𝒐𝒓 𝒐𝒓𝒅𝒊𝒏𝒂𝒓𝒚 𝒂𝒏𝒏𝒖𝒊𝒕𝒊𝒆𝒔,𝒆𝒒𝒖𝒂𝒍𝒔 𝒕𝒉𝒆 𝒊𝒏𝒕𝒆𝒓𝒆𝒔𝒕 𝒓𝒂𝒕𝒆 𝒑𝒆𝒓 𝒑𝒂𝒚𝒎𝒆𝒏𝒕 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍)
𝒏 = 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒑𝒂𝒚𝒎𝒆𝒏𝒕𝒔 𝒊𝒏 𝒕𝒉𝒆 𝒂𝒏𝒏𝒖𝒊𝒕𝒚
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Using Multiplicative Inverse:
𝑹 =𝑨𝒏𝒊
𝟏−(𝟏+𝒊)−𝒏
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟓
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Example 4.5:If money is invested at 6%
compounded quarterly, find
the quarterly rent of an
annuity for 15 years if its
present value is P120,000.00.
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Final Answer:The quarterly rent
of an annuity for 15
years is
P3,047.21.
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Let′s Practice: Solve the following
problems:(1) Mr. Clean wishes to accumulate
P500,000.00 in 10 years. How much should he
invest each quarter at 9 ½% compounded
quarterly?
(2) Mr. Bean borrows P26,000.00. He agrees
to pay the principal and interest by paying a
sum each year for 5 years. Find his annual
payment if he pays interest at 7%
compounded annually.
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Assignment 4.4: Solve the following
problems:A spinster sold a piece of property for P1,300,000.00. A
down payment of P500,000.00 was made and the
remainder was to be repaid in equal quarter
installments, the first due 3 months after the date of sale.
The interest was 15% compounded quarterly, and the
debt was to be amortized in 7 years.
What quarterly payment is required?
What will be the total amount of payment?
How much interest will be paid?
What is the total cost of the property?
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Lesson 4.5The Term (t) of an
Ordinary Annuity
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Term of an Ordinary Annuity (t)
The term of an ordinary
annuity is defined as the
interval between the
beginning of the first rent
period and the end of the last
rent period.
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Two Formulae to be Used When Computing for
Term (t) of an Ordinary Annuity
The term of an Ordinary Annuity
(t) when Final Value of the
Ordinary Annuity is given
The term of an Ordinary Annuity
(t) when Present Value of the
Ordinary Annuity is given
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The Term of an Ordinary Annuity (t) when Final
Value of the Ordinary Annuity is given
This is used whenever we want
to compute for the term of an
ordinary annuity (t) when final value of the ordinary annuity (𝐒𝐧)
is given and not the present value of the annuity (𝐀𝐧).
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Formula:
𝒕 =𝒍𝒐𝒈
𝑺𝒏𝒊
𝑹+𝟏
𝒎𝒍𝒐𝒈(𝟏+𝒊)
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟔
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Example 4.7:How long will it take for annual
year-end contributions of
P57,000.00 to an Educational
Savings Plan to grow to
P1,000,000.00 if the Educational
Savings Plan earns 13%
compounded annually?
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Final Answer:
It will take
9.72 years or 10
years.
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The Term of an Ordinary Annuity (t) when
Present Value of the Ordinary Annuity is given
This is used whenever we want
to compute for the term of an
ordinary annuity (t) when present value of the ordinary annuity (𝑨𝒏)
is given and not the final value of the annuity (𝑺𝒏).
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Formula:
𝒕 =𝒍𝒐𝒈 𝟏−
𝑨𝒏𝒊
𝑹
−𝒎𝒍𝒐𝒈(𝟏+𝒊)
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟕
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Example 4.8:Einstein and Newton are discussing
the terms of a P270,000.00 home
improvement loan with their bank’s
lending officer. The interest rate of the
loan will be 18% compounded
quarterly. How long will it take to
repay the loan if the quarterly
payments are P25,000.00?
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Final Answer:
It will take 3.78
years or 4 years to
repay the loan.
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Lesson 4.6The interest rate (j) of
an Ordinary Annuity
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Two Formulae Used in Computing the Interest
Rate (j) of an Ordinary Annuity
The interest rate (j) when Present
Value of the Ordinary Annuity is
given
The interest rate (j) when Final
Value of the Ordinary Annuity is
given
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The Interest Rate (j) when Present Value of the
Ordinary Annuity is Given
This process of computing for
interest rate (j) can be used whenever the present value (𝑨
𝒏),
periodic rent (R), number of
payments (n) and periodic
interval (m) are given.
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Formula:
𝒋 = 𝒎𝒊
Formula 4.8
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Quadratic Formula:
𝒙 =−𝒃± 𝒃𝟐−𝟒𝒂𝒄
𝟐𝒂
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Formula:
𝒊 =−𝒃± 𝒃𝟐−𝟒𝒂𝒄
𝟐𝒂
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟗
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where:
𝒂 = 𝒏𝟐 − 𝟏𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟏𝟎
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where:
𝒃 = 𝟔 𝒏 + 𝟏𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟏𝟏
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where:
𝒄 = 𝟏𝟐 𝟏 −𝒏𝑹
𝑨𝒏
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟏𝟐
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Example 4.9:A loan of P8,838.51 is to be
discharged by making 15
semiannual payments of P940.00
each. At what rate compounded
semiannually is the interest
charged on the loan?
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Example 4.9:
i = 6.47%, hence
the interest charged
on the loan is
12.94%.
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The Term of an Ordinary Annuity (t) when Final
Value of the Ordinary Annuity is given
This process of computing for
interest rate (j) can be used whenever the final value (𝑺
𝒏),
periodic rent (R), number of
payments (n) and periodic
interval (m) are given.
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Quadratic Formula:
𝒙 =−𝒃± 𝒃𝟐−𝟒𝒂𝒄
𝟐𝒂
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Formula:
𝒊 =−𝒃± 𝒃𝟐−𝟒𝒂𝒄
𝟐𝒂
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟗
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where:
𝒂 = 𝒏𝟐 − 𝟏𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟏𝟎
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where:
𝒃 = −𝟔 𝒏 − 𝟏𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟏𝟑
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where:
𝒄 = 𝟏𝟐 𝟏 −𝒏𝑹
𝑺𝒏
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟏𝟒
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Example 4.10:Payments of P300.00 each
made every year. At what
rate compounded monthly
will these payments amount
to P8,336.42 in 2 years?
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Formula:
i = 1.25%, hence
the rate
compounded
monthly will be 15%.
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Let′s Practice: Solve the following problems:
(1) Madam Red wants to accumulate
P67,000.00 by depositing P4,000.00 in a fund
at the end of each month. The fund pays 21%
converted monthly. How long will it take to
do this?
(2) What is the rate compounded quarterly
of an annuity of P1,400.00 payable at the
end of each quarter for 3 years if its present
value if P13,528.67?
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Assignment 4.5: Solve the following problems.
(1) An air-conditioning unit can be purchased for
P14,627.46 or for P5,000.00 down payment and an
installment of P700.00 every two months for 3
years. Find the interest rate compounded
bimonthly.
(2) As soon as Madam Chantal has saved
P22,500.00, she intends to invest the money. If he
can save P650.00 every 3 months and invest it at
15% compounded quarterly, find the number of
P650.00 deposits he must make and the total
amount of the final deposit.