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Al-Imam Muhammad Ibn Saud Islamic

University

College of Economics and Administration

Sciences

Department of Finance and Investment

Financial MathematicsCourse

FIN 118Unit course

11Number Unit

Long Term Annuities

Amortization & sinking

Funds

Unit Subject

Dr. Lotfi Ben Jedidia

Dr. Imed Medhioub

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we will see in this unit

Long term ordinary annuity

Long term annuity due

Amortization & sinking Funds

Some real life examples

2

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Learning Outcomes

3

At the end of this chapter, you should be able to:

1. Understand what is meant by long term

annuities.

2. Calculate Present and future value long term

annuities for the case of ordinary and annuity due.

3. Calculate the single payment for real life

examples in the case of long term annuities

( Amortization & sinking Funds).

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Introduction As we say in unit 10, Annuity is a series of equal cash

payments or deposits. These regular equal paymentscan be planned for short term, intermediate or longterm periods.

In this unit we are interested to the calculation of long

term payments or deposits.

Two basic questions can be exposed with annuities: Calculate how much money will be accumulated if we

consider an annuity plan for long time (30 years forexample) How to calculate the periodic payments or deposits

in order to obtain a specific amount on a given timeperiod (calculate monthly payments for a mortgage

loan for example). 4

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Long Term Ordinary Annuity As we have seen in unit 10, the same formula is

applied to the case of long term annuities.

The future and present value of an ordinary annuityare given respectively by

if payments are made annually.

If payments are made non annually (more than oncein year) we must introduce in the previous formulathe number of time per year. So, we have thefollowing formula

t

i

tn

t

i

tn PMTFVA

11 .

,

i

iPMTFVA

n

n

11

i

iPMTPVA

n

n

11

t

i

tn

t

i

tn PMTPVA

.

,

11

5

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Example1You plan to deposit in an account $5000 atthe end of each year for the next 25 years.If the account pays 6% annually, what is the

value of your deposits at the end of 25 years?Solution

56.274322$06.0

106.015000

25

25

FVA

Long Term Ordinary Annuity

6

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Example2

You plan to deposit, in an account, $100 at theend of each month for the next 25 years. Ifthe account pays 6% annually, what is the

value of your deposits at the end of 25 years?Solution

Long term Ordinary Annuity

396.69299$111001206.0

)12)(25(1206.0

12,25

FVA

7

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Example3

You plan to withdraw at the end of each yearfrom an account $5000 for the next 25years. If the account pays 6% annually, what

is the amount that you must deposit today inorder to guarantee these withdrawals?

Solution

78.63916$

06.0

06.0115000

25

25

PVA

Long Term Ordinary Annuity

8

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Example4

You plan to withdraw at the end of eachmonth $100 from an account of the next 25years. If the account pays 6% annually, what

is the amount that you must deposit today inorder to guarantee these withdrawals?

Solution

Long term Ordinary Annuity

686.15520$

11100

1206.0

)12)(25(

1206.0

12,25

PVA

9

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Long Term Annuity Due As we have seen in unit 10, the same formula is

applied to the case of long term annuities.

The future and present value of an annuity dueare given respectively by

if payments are made annually. If payments are made non annually (more than

once in year) we must introduce in the previous

formula the number of time per year. So, we havethe following formula

t

iPMTFVA

t

i

tn

t

i

tn 111

.

,

i

i

iPMTFVA

n

n

1

11 ii

iPMTPVA

n

n

111

t

iPMTPVA

t

i

tn

t

i

tn 111

.

,

10

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Example1

You plan to deposit in an account $5000 atthe beginning of each year for the next 25years. If the account pays 6% annually, what is

the value of your deposits at the end of 25years?Solution

914.290781$

06.0106.0

106.015000

25

25

FVA

Long Term Annuity Due

11

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Example2

You plan to deposit in an account $100 at thebeginning of each month for the next 25 years.If the account pays 6% annually, what is the

value of your deposits at the end of 25 years?Solution

Long term Annuity Due

893.69645$

111

100 1206.0

1206.0

)12)(25(

12

06.0

12,25

FVA

12

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Example3

You plan to withdraw from an account $5000at the beginning of each year for the next 25years. If the account pays 6% annually, what

is the amount that you must deposit today inorder to guarantee these withdrawals?

Solution

787.67751$

06.0106.0

06.0115000

25

25

PVA

Long Term Annuity Due

13

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Example4

You plan to withdraw from an account $100 atthe beginning of each month for the next 25years. If the account pays 6% annually, what

is the amount that you must deposit today inorder to guarantee these withdrawals?

Solution

Long term Annuity Due

289.15598$

111

1001206.0

1206.0

)12)(25(

1206.0

12,25

PVA

14

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Amortization &Sinking Funds

When a lender pays a debt (includinginterest) by making periodic payments atregular intervals, the debt is said to beamortized.

When a payment is made to an investmentfund each period at a fixed interest rate to

yield a predetermined future value, thepayment is called a sinking fund.

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Ordinary Amortization Formula

Ordinary Sinking Fund Payment

11

tn

t

i

t

i

FVAPMT

tn

ti

t

i

PVAPMT

11

Amortization &Sinking Funds

Non-annualAnnual

n

i

iPVAPMT

11

Non-annualAnnual

11n

i

iFVAPMT

16

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Real life example: Loan AmortizationExample

Suppose you want to borrow money to buy a house. Youare considering a 7-year or a 25-year loan.A bank offers different interest rates, reflecting thedifferences in risks of intermediate-term and long-

term lending.For the 7-year loan, the annual interest rate is 5.25%compounded monthly.For the 25-year loan, the annual interest rate is

6.75% compounded monthly. If you borrow $150000,what would be your monthly payments for each type ofloan?

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Real life example: Loan Amortization

SolutionFirst Scenario: 7-year loan

Second Scenario: 25-year loan

751.2137$11150000 )12)(7(120525.0

120525.0

PMT

367.1036$

11

150000)12)(25(

120675.0

120675.0

PMT

18

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Example 1

Suppose you decide to use a sinking fund to save$150 000 for a house. If you plan to make 300monthly payments (25 years) and you receive

6.75% interest per annum, what is the requiredpayment for an ordinary annuity?

Solution

617.192$

11

150000)12)(25(

120675.0

120675.0

PMT

Real life example: Sinking Fund

19

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Example 2

Suppose you use a sinking fund to save $50 000for a car. If you plan on 60 monthly payments(five years) and you receive 5% interest per

annum, what is the required payment for anordinary annuity?

Solution

228.735$

11

50000)12)(5(

1205.0

12

05.0

PMT

Real life example: Sinking Fund

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Time to Review !

Long term annuities is an extension tothe ordinary and annuity due.

Making periodic payments to repay a

debt, including the principal and interest,is called amortization.

A fund into which periodic payments

necessary to realise a given sum of moneyin the future are made is called sinkingfund.

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we will see in the next unit

Whats a bond

Different types of bonds

Bond valuation