Ecf1120 Module 02

7/27/2019 Ecf1120 Module 02

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Module 2

Multiple cash flows

Learning objectives

This topic expands on the multiple cash flows calculation to include the differenttypes of annuities. After completing this module, you should be able to:

• handle situations with multiple cash flows

• identify the different types of annuities

• calculate present and future values of ordinary and annuity due

• calculate repayments on loans

In this topic you will also be applying time lines effectively to calculate solutionsto present and future value problems.

Introduction

In the earlier module, we looked at what we should do when faced with more thanone cash flow. You do the same as you do with a single cash flow. The formuladoes not change and all you need to do is to treat each cash flow separately. Inthis topic we also look at how we can handle cash flows that repeat. That is, thesame dollar value occurring over regular intervals.

7/27/2019 Ecf1120 Module 02


Recap on multiple cash flows

Activity 2.1

An investment promises a return of $100 in year 1, $200 in year 2 and $300 inyear 3. If interest rate is 10% per annum compounding annually, what is the

present value of the investment?

The present value of the three cash flows can be calculated by bringing back totime 0, the individual cash flow using the present value formula:

58.481

39.22529.16591.90

)1.01(

300

)1.01(

200

)1.01(

100321

=++=

++

++

+= PV

Notationally:

3

3

2

2

1

1

)1()1()1( i

CF

i

CF

i

CF PV

+

+

+

+

+

=

In general:

n

n

i

CF

i

CF

i

CF PV

)1(...

)1()1( 2

2

1

1

+

++

+

+

+

=

100 200 300

0 321

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Annuities

In the above example, the individual cash flows were of different values. Therecan be situations when the cash flow repeats itself at a regular interval of time.

We call this an annuity. An annuity is a stream of equal amounts that occur at aregular interval in time.

Key words: StreamEqual amountsRegular interval

Types of annuities

Before we can find the future value or the present value of annuities, we must firstascertain the type of annuity we are dealing with. The type of annuity depends onwhen the first payment of the annuity falls in time.

There are primarily three types of annuities – ordinary annuity (also called annuitycertain), annuity due and deferred annuity.

Ordinary: First payment falls at the end of the first period

Due: First payment falls at the beginning of the first period

Deferred: First payment falls after the first period

The annuities will appear on the time line as below:

Let us now look at each type of annuity in detail.

PMT PMT PMT

0 321 4 5

PMT PMT Ordinary annuity

PMTPMTPMTPMTPMT Annuity due

PMTPMT PMT PMT Deferred annuity

7/27/2019 Ecf1120 Module 02


Ordinary annuity

The ordinary annuity is also called annuity certain. For an ordinary annuity, thefirst payment starts at the beginning of the first period. Examples of ordinaryannuities include mortgage repayments, car loans and insurance premiums.

Here is an example of how we deal with an ordinary annuity.

Say, we have an annuity of $100 that starts at the end of the first year for 4 yearswith i = 12% per annum. If we need to calculate the future value of the cashflows, we can take the individual cash flow up to time 4 as follows:

93.477

)12.1(100)12.1(100)12.1(100100321

=+++= FV

The above problem can be interpreted as a geometric progression. Using theformula for the sum of a geometric progression, we can summarise a formula thatis more convenient for the above problem. You do not need to know thederivation of the formula, but you need to be able to interpret and apply theformula. The formula for the future value of an ordinary annuity is as follows:

The accumulated value (future value):

( )

−+=

i

i PMT FV

n

11

100 100 100

0 321 4

100

7/27/2019 Ecf1120 Module 02


Activity 2.2

To provide for the ultimate purchase of a home, a man deposits $3,000 at the end

of every 6 months at 14% p.a. compounded semi-annually. At the end of theeighth year how much does he have to his credit?

Remember, it is always a good practice to draw a time line!

PMT =3000 n = 8 x 2 = 16 I = 0.14/2 = 0.07

(Note that because the interest is compounding semi-annually, we divide interest by 2 and multiply n by 2).

Alternatively,

* Please read the section on setting the calculator at the end of this module.

3000 3000 3000

0 321 … 16

… 3000

( )

−+=

i

i PMT FV

n

11

( )

16.664,83

07.0

107.013000

16

=

−+= FV

3,000 PMT

7 I/YR

16 N

FV = -83,664.16

7/27/2019 Ecf1120 Module 02


Annuity due

An annuity due is a stream of equal payments that starts at the beginning of thefirst period. Rent is an example of an annuity due.

Finding the future value of an annuity due is similar to that of an ordinary annuity.We use the formula for the sum of a geometric progression to arrive at theformula.

Since the payment starts at the beginning of the first period instead of the end, theformula for the accumulated value is slightly different to the one derived

previously. Once again, there is no need for you to learn how to derive theformula.

The formula for the accumulated value of an annuity due is:

Activity 2.3

Starting today, you wish to make quarterly deposits of $200 into an investmentthat is earning 9% p.a. compounding quarterly. You intend to keep up identicalregular deposits for 5 years. What sum do you accumulate at the end of the 5years?

PMT = 200 I = 0.09/4 = 0.025 n = 5x4 = 20

Alternatively,

( )

−++=

i

ii PMT FV

n

11)1(

200 200 200

0 1921 … 20

…200

( )

−++=

i

ii PMT FV

n

11)1(

( )

41.5094

0225.0

10225.01)0225.01(200

20

=

−++= FV

7/27/2019 Ecf1120 Module 02


Remember, this is an annuity due, therefore your calculator has to be set in the begin mode:

Note that your calculator will display the word Begin. It is important to makesure that this word does not appear when you are doing the ordinary annuity

problem. To get rid of the begin mode, press the yellow button followed by theBEG/END button. This will get rid of the word begin in your display.

Present value of an ordinary annuity

Finding the present value of an annuity is similar to finding the future value. For present value of an annuity we have to discount each cash flow back to year 0.For example, if we are looking for the present value of a series of four payments

of $100 starting at the end of the first period if interest rate is 12% per annum:

75.303

4)12.1(100

3)12.1(100

2)12.1(100

1)12.1(100

=

−+

−+

−+

−= PV

If we were to use geometric progression, we would arrive at the formula:

+−=

−

i

i PMT PV

n)1(1

Activity 2.4

YELLOW

BEG/END

200 PMT

2.25 I/YR

20 N

FV = -5094.41

100 100 100

0 321 4

100

7/27/2019 Ecf1120 Module 02


What is the present value of a sand quarry that will yield an annual end of year income of $30,000 for the next 16 years and then be worthless if money is worth15% per annum compounded annually?

PMT =30000 n = 16 I = 15

Alternatively,

Activity 2.5

In a contest the winner is allowed an option of $10,000 at the end of each year for

the next 20 years or $80,000 in cash now. If money is worth 13% per annum tothe contest winner, which is the greater amount?

The natural tendency is to multiply the $10,000 by 20 to get $200,000 andcompare that with the $80,000. This is wrong! Remember, we cannot add,multiply, subtract or divide money across time. The correct way of comparing thetwo sums is to bring the annuity of $10,000 to the same point in time as the$80,000.

30000 30000 30000

0 321 … 16

… 30000

( )

+−=

−

i

i PMT PV

n

11

( )

05.627,178

15.0

15.01130000

16

=

+−= PV

30,000 PMT

15 I/YR

16 N

PV = -178,627.05

0 …21 20

80,000

7/27/2019 Ecf1120 Module 02


Present value of 10,000 annuity

A = 10,000 i = 0.13 n = 20

Using the formula or calculator will give you:

PV = $70,247.52

Therefore, the larger amount is $80,000 today!!

Amortisation

To amortise a loan is to pay off the loan with regular equal payments so that theloan balance is zero. Here, we know the loan amount (PV), the number of

payments needed and the interest rate. We need to calculate the payment (PMT)that would be required to discharge the loan.

To calculate the repayment, we simply rearrange the present value formula above:

The calculator functions need for an amortisation is the similar to those we usedfor the ordinary annuity calculations. We know PV, i and n, we look for PMT:

( )

( )

+−=

+−=

−

−

i

i

PV PMT

i

i PMT PV

n

n

11

11

10000 10000 … 10000

7/27/2019 Ecf1120 Module 02


Activity 2.6

A businessman has incurred a debt of $30,000 that is to be repaid monthly

payments. Find the monthly payment necessary to cancel the debt in three yearsif the loan is charged at 12% p.a. compounding monthly.

PV = 30,000 I = 0.12/12=0.01 n = 3 x 12 = 36

( )

43.99601.0

)01.1(1

000,30

11

36

=

−=

+−=

−

−

i

i

PV PMT

n

Activity 2.7

When visiting a finance company for a loan of $5000 you may be quoted therepayments per month over three years as $200. What is the nominal andeffective rates of interest?

PV = 5000 PMT= -200 n = 36

Unfortunately, we cannot use the formula to work out the solution to this problemas the interest rate appears both in the numerator as well as the denominator. Wecan employ interpolation but this will take too long. So, the best way to work outthe answer is to use the calculator.

Alternatively,

Thus, the interest is 2.12% per month

Nominal annual rate = 2.12 x12 = 25.44%

Effective annual rate

-200 PMT

5000 PMT

36 N

I/YR = 2.12

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2863.0

112

2544.01

11

12

=

−

+=

−

+=

m

m

i E

Activity 2.8

You have taken out a loan for $180,000 at 6% per annum compounding daily.The term of the loan is 25 years. What is your monthly repayment? What is your fortnightly repayment?

First we need to work out the effective interest rate for a month:

005012.0

1365

06.01

12

365

=

−

+= E

( )

33.1161

005012.0)005012.1(1

000,180

11

300

=

−

=

+−=

−

−

i

i

PV PMT

n

Hence, the monthly payment is 1161.33.

See if you can calculate the fortnightly payments:

7/27/2019 Ecf1120 Module 02


Present value of an annuity due

Finding the present value of an annuity due is similar to what we have done for anordinary annuity. The formula for the present value of an annuity due is given

below. Remember, if you are using the calculator, make sure that you activate theBEGIN mode for annuity due problems.

The formula is:

Activity 2.9

Instead of paying $7000 a year rent at the beginning of each year for the next 10years, a renter decides to buy a house. If money is worth 12% p.a. what would be

the cash value equivalent to 10 years rent?

PMT = 7000 I=0.12 n=10

PV = 44,297.72

( )

+−+=

−

i

ii PMT PV

n

11)1(

( )

+−+=

−

i

ii PMT PV

n

11)1(

7/27/2019 Ecf1120 Module 02


Formulas covered

Future value of an ordinary annuity

Future value of an annuity due

Present value of an ordinary annuity

+−=

−

i

i PMT PV

n)1(1

Present value of an annuity due

Reflecting on what you have learned

Write a brief description on the following:

Ordinary annuity

Annuity Due

Amortisation

( )

−+=

i

i PMT FV

n

11

( )

−++=

i

ii PMT FV

n

11)1(

( )

+−+=

−

i

ii PMT PV

n

11)1(

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Ecf1120 Module 02