Thought to fonder:

The use of applications

of many formulae

became more of a

convenience rather

than a necessity.

Recall:Two principal types of problem

occur:

The problem involving a single

payment; and

the problem dealing with series

of equal payment.

Game:

Reveal the title

of our lesson for

today.

Chapter 4Mathematics of Annuities

Lesson 4.1Basic Concepts in

Annuity

Annuity

It is a series of equal payments made at equal

intervals of time.

Annuity

The word “annuity” implies

etymologically annual payments.

Challenge

Can you give

general examples of

annuity in business

transactions?

Examples of Annuity

Interest

Payments on

Bonds

Examples of Annuity

Premiums on

Life Insurance

Examples of Annuity

Depreciation

of Funds

Examples of Annuity

Sinking

Funds

Three

MainClasses of Annuities

Annuity Certain

It is an annuity

whose term begins

and ends at fixed

dates.

Examples of Annuity

Certain

Mortgage

payment of

house

Examples of Annuity

Certain

Car

Loans

Examples of Annuity

Certain

Installment

Plans

Examples of Annuity

Certain

Loans

Perpetuity

It is an annuity whose

term begins at a fixed

date but continues

forever/indefinitely.

Examples of Perpetuity

Fixed coupon

payments on

permanently invested

(irredeemable) sums of

money

Examples of Perpetuity

Scholarships paid

perpetually from

an endowment

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.

Example of

Contingent Annuity

Life insurance that

cease when the

person insured dies

Classifications

ofAnnuities Certain

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.

Examples of Ordinary

Annuity

Salaries

Stock Dividends

Pension Plan

Annuity Due

It is one in which the

payments are made at the

beginning of each rent

period, the first payment

being due now.

Examples of Annuity

DueLease Agreements

(House Rent, Car Rent

etc.)

Educational Insurance

Plan

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.

Examples of Deferred Annuity

SSS Salary Loan

PAG-IBIG Salary Loan

Basic Concepts

in

Annuity

Periodic PaymentThe size of each

payment.

The interval between

two successive

payments.

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.

Final Value of an Annuity

The sum of the

compound amount of

all payments,

compounded to the

end of the term.

Present Value of an Annuity

The lump sum

required at the

beginning.

Lesson 4.2The Amount or Final

Value of an Ordinary

Annuity

Note:The periodic payments

of an ordinary annuity

are made at the end of

each rent period.

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.

DerivationThe Formula for

Finding the Final

Value of an Annuity

𝑺𝒏:It is the final value

of an n-payment

ordinary annuity.

𝑹:It is the size of

each annuity

payment.

𝒊:It is the interest rate per

compounding period (which

for ordinary annuity, equals

the interest rate per payment

interval).

𝒏:It is the number

of payments in

the annuity.

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

𝑻𝒉𝒐𝒖𝒈𝒉𝒕 𝒕𝒐 𝑭𝒐𝒏𝒅𝒆𝒓:

What can you say

to the formula? Is it

convenient to use?

𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟐:

𝑺𝒏 = 𝑹(𝟏+𝒊)𝒏−𝟏

𝒊

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.

𝑭𝒊𝒏𝒂𝒍 𝑨𝒏𝒔𝒘𝒆𝒓:

The amount of

an annuity is

P283,976.55.

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?

𝑭𝒊𝒏𝒂𝒍 𝑨𝒏𝒔𝒘𝒆𝒓:The amount she will

accumulate if she

continues the plan for

another year is

P12,911.12.

Math

RunnersThe Finding the Final

Value of an Annuity

Math Runners

Problem (For 5 minutes, for 5

points and for first 10

pairs only)

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?

He will have

P55,444.13

after 4 years.

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?

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?

Lesson 4.3The Present Value of

an Ordinary Annuity

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.

Present Value of an Ordinary

Annuity

It may also be defined

as the sum of the present

values of all payments of

the annuity.

Present Value of an Ordinary

Annuity

The present value of an

annuity is sometimes

referred to as the “cash

equivalent of the annuity.”

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

Using:

𝑷 =𝑺

(𝟏+𝒊)𝒏𝒐𝒓

𝑷 = 𝑺(𝟏 + 𝒊)−𝒏

Formula:

𝑨𝒏 = 𝑹𝟏−(𝟏+𝒊)−𝒏

𝒊

𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟑

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).

Final Answer:This means that paying

P115,985.97 now is just

as good as making 35

annual payments of

P8,000.00 each.

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?

Final Answer:

The combined

present value is

P96,511.85.

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.

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.

Lesson 4.4The Periodic

Payment of an

Ordinary Annuity

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.

Periodic Payment (R)

We will denote

period payment

using majuscule

letter R.

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

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).

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

Using Multiplicative Inverse:

𝑹 =𝑺𝒏𝒊

(𝟏+𝒊)𝒏−𝟏

𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟒

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?

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.

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

Using Multiplicative Inverse:

𝑹 =𝑨𝒏𝒊

𝟏−(𝟏+𝒊)−𝒏

𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟓

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.

Final Answer:The quarterly rent

of an annuity for 15

years is

P3,047.21.

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.

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?

Lesson 4.5The Term (t) of an

Ordinary Annuity

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.

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

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).

Formula:

𝒕 =𝒍𝒐𝒈

𝑺𝒏𝒊

𝑹+𝟏

𝒎𝒍𝒐𝒈(𝟏+𝒊)

𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟔

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?

Final Answer:

It will take

9.72 years or 10

years.

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).

Formula:

𝒕 =𝒍𝒐𝒈 𝟏−

𝑨𝒏𝒊

𝑹

−𝒎𝒍𝒐𝒈(𝟏+𝒊)

𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟕

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?

Final Answer:

It will take 3.78

years or 4 years to

repay the loan.

Lesson 4.6The interest rate (j) of

an Ordinary Annuity

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

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.

Formula:

𝒋 = 𝒎𝒊

Formula 4.8

Quadratic Formula:

𝒙 =−𝒃± 𝒃𝟐−𝟒𝒂𝒄

𝟐𝒂

Formula:

𝒊 =−𝒃± 𝒃𝟐−𝟒𝒂𝒄

𝟐𝒂

𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟗

where:

𝒂 = 𝒏𝟐 − 𝟏𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟏𝟎

where:

𝒃 = 𝟔 𝒏 + 𝟏𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟏𝟏

where:

𝒄 = 𝟏𝟐 𝟏 −𝒏𝑹

𝑨𝒏

𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟏𝟐

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?

Example 4.9:

i = 6.47%, hence

the interest charged

on the loan is

12.94%.

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.

Quadratic Formula:

𝒙 =−𝒃± 𝒃𝟐−𝟒𝒂𝒄

𝟐𝒂

Formula:

𝒊 =−𝒃± 𝒃𝟐−𝟒𝒂𝒄

𝟐𝒂

𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟗

where:

𝒂 = 𝒏𝟐 − 𝟏𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟏𝟎

where:

𝒃 = −𝟔 𝒏 − 𝟏𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟏𝟑

where:

𝒄 = 𝟏𝟐 𝟏 −𝒏𝑹

𝑺𝒏

𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟒. 𝟏𝟒

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?

Formula:

i = 1.25%, hence

the rate

compounded

monthly will be 15%.

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?

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.