ZULFIQAR HASAN 1

ZULFIQAR HASAN 2

Topic Contents

What is EAR, What are the importance of EAR, How EAR can be calculated, Annuities, Ordinary and Annuity Due, Amortization, Sinking Fund, Types of Loan, Amortization Schedule

ZULFIQAR HASAN 3

Formula For FV, PV, EAR and Annuities

11

m

m

iEAR 1 xeEAR

i)(1

or

i)(1i

1- i 1 PMT

FVAnFVAn

FVAn

OrdinaryDueAnnuity

n

DueAnnuity

m

i 1

mi

1- mi

1 PMT

mn

DueAnnuity FVAn

i

1- i 1 PMT

n

FVAnAnnuityOridinary

mi

1- mi

1 PMT

mn

AnnuityOrdinary FVAn

)1(PVAn PVAn,

)1( PMT PVAn

OrdinaryDueAnnuity

ii 1

1 - 1

DueAnnuity

n

ior

i

ii 1

1 - 1

Ordinary

n

PMT PVAn

m

im

i 1

1 - 1

Ordinary

mn

PMT PVAn

PMT = Payment or Installment amount, PVAn = PV of AnnuityFVAn = FV of Annuity, n = Year, m = Period, i = Interest Rate

ZULFIQAR HASAN 4

The annual rate of interest actually paid or earned is called Effective Annual Rate (EAR).The effective interest rate, effective annual interest rate, annual equivalent rate (AER) or simply effective rate is the interest rate on a loan or financial product restated from the nominal interest rate as an interest rate with annual compound interest payable in arrears. It is used to compare the annual interest between loans with different compounding terms (daily, monthly, annually, or other).

11

m

m

iEAR 1 xeEAR

For daily, monthly, annually….. If interest is paid continuously

11

m

m

iEAR 1 xeEAR

ZULFIQAR HASAN 5

Example 01: Find out the EAR for a nominal rate of 10%, compounded semiannually?

%25.10

1025.0

11025.1

1)05.1(

1%)51(

12

%101

11

2

2

2

m

m

iEAR

ZULFIQAR HASAN 6

Example 02: Comparing EARsConsider the following interest rates quoted by three banks:Bank A: 15%, compounded daily (365 or 360 days in a year)

Bank B: 15.5%, compounded quarterlyBank C: 16%, compounded annually

16% 1 - 1

0.16 1 EAR

16.42% 1 - 4

0.155 1 EAR

16.18% 1 - 365

0.15 1 EAR

1

CBank

4

Bank B

365

ABank

Which is the best bank? For a saver, Bank B offers the best (highest) interest rate. For a borrower, Bank C offers the best (lowest) interest rate.The highest NIR (Nominal Interest Rate) is not necessarily the best.Compounding during the year can lead to a significant difference between the NIR and the EAR.

ZULFIQAR HASAN 7

Practice 01: EAR

Practice 02 EAR: What is the Effective Annual INTEREST rate (EAR) for a sum of money compounded at the rate of 15% annually.

1. On a quarterly basis? 2.On a monthly basis?3. On a weekly basis? 4.On a daily basis? 5.On a continuous basis?

Assume a 360 day year. 1EARbasis Continous

xe

Fred Moreno wishes to find the effective annual rate associated with an 8% nominal annual rate (i0.08) when interest is compounded (1) annually (m=1); (2) semiannually (m=2); and (3) quarterly (m=4)

a. 8% b. 8.16% c. 8.24%

ZULFIQAR HASAN 8

Practice 03: TVM and EAR Delia Martin has $10000 that she can deposit in any of three accounts for a 3-year period. Bank A compounds interest on an annual basis, bank B compounds interest twice each year, and Bank C compounds interest each quarter. All three banks have a stated annual interest rate of 4%.

1.What amount would Ms. Martin have at the end of the third year, leaving all interest paid on deposit, in each Bank?

2.What effective annual rate (EAR) would she earn in each of the banks?

3.On the basis of your findings in parts 01 and 02, which bank should Ms. Martin deal with? Why?

4.If a fourth bank (Bank D), also with a 4% stated interest rate, compounds interest continuously, how much would Ms. Martin have at the end of the third Year? Does this alternative change your recommendation in part c? Explain why or why not.

ZULFIQAR HASAN 9

An annuity is a series of equal, periodic payments. Followings are the types of annuities:1. Ordinary Annuity:

An ordinary annuity is a series of constant cash flows that occur at the end of each period for some fixed number of periods. Examples include consumer loans and home mortgages.

2. Annuity Due: An annuity for which the cash flow occurs at the beginning of each period is called Annuity Due.

3. Perpetuity: A perpetuity is an annuity in which the cash flows continue forever.

Preparation at Home: What are the differences between Ordinary annuity and Annuity Due? Which annuity gives the higher value?

ZULFIQAR HASAN 10

i

1- i 1 PMT

n

FVAnOridinary

Future Value of An Annuity

The compounding term is called the future value interest factor for annuities (FVIFA).

i)(1

or

i)(1i

1- i 1 PMT

FVAnFVAn

FVAn

OrdinaryDueAnnuity

n

DueAnnuity

mi

1- mi

1 PMT

mn

FVAnOridinary

i

1- i 1 PMT

n

FVAnOridinary

i)(1

or

i)(1i

1- i 1 PMT

FVAnFVAn

FVAn

OrdinaryDueAnnuity

n

DueAnnuity

mi

1- mi

1 PMT

mn

FVAnOridinary

i

1- i 1 PMT

n

FVAnOridinary

ZULFIQAR HASAN 11

Example 01: FV of Annuity For Annual Deposit

Mr. Anisur Rahman wishes to determine how much money he will have at the end of 5 years if he deposits Tk 1000 at the end of each of the next 5 years? Given that the annual interest rate is 7%?

5750.74Tk

74)1000(5.750Tk 0.07

.4025517310 1000Tk

0.07

1 - .4025517311 1000Tk

0.07

1 - 07.1 1000Tk

7%

1 - 7%1 1000Tk FV

5

5

ZULFIQAR HASAN 12

Example 02: FV of Annuity for multiple depositMr. Anisur Rahman has opened a DPS at Luminous Cooperatives on the following conditions:Duration: 05 YearsPayment modes: MonthlyInstallment : Tk 1000Interest Rate: 14%Determine the amount of his DPS he will get after 05 years from Luminous Cooperatives. What will be the value if he deposits at the beginning of the month?

86194.88Tk

194879.86 1000Tk

0.0116667

00561.1 1000Tk

0.0116667

1- .00561372 1000Tk

0.0116667

1- 116667 1.0 1000Tk

1.16667%

1- 1.16667% 1 1000Tk

1214%

1- 12

14% 1

1000Tk

mi

1- mi

1 PMT

60

60

215

mn

FVAnOridinary

Due: Tk 87200.49

ZULFIQAR HASAN 13

Practice 01: FV of Annuity for multiple deposit

Sadia Rahman has opened a DPS at DBBL on the following conditions:

Duration: 10 Years

Payment modes: Monthly

Installment : Tk 2000

Interest Rate: 12%

Determine the amount of his DPS she will get after 10 years from Dutch Bangla Bank Limited

mi

1- mi

1 PMT

mn

FVAnOridinary

ZULFIQAR HASAN 14

Examples of Future Value of Annuity

Example 03: What is the future value at the end of 3 years of an ordinary annuity of $100 at 12% per annum?

$337.44

4)$100(3.374 12%

1 - 12%1 $100 FV

3

Example 04: What is the future value $200 deposited at the end of every year for 10 years if the interest rate is 6% per annum?

636.20 $2

13.181 $200 0.06

1 - 1.06 $200 FV

10

ZULFIQAR HASAN 15

Practice 05: Future Value of Annuity) from the Text)

For each case in the following table, answer the following questions that Follow:

Case Amount of Annuity Interest rate YearA 2500 8% 10B 500 12% 6C 30000 20% 5D 11500 9% 8E 6000 14% 30

1. Calculate the future value of the annuity assuming that it isa) An Ordinary annuityb) An Annuity Due

2. Compare the findings in part (a) and (b). All else being identical, which type of annuity is preferable? Explain Why?

ZULFIQAR HASAN 16

Practice 06: Future Value of Annuity (from the Text)

Ramesh Abdul wishes to choose the better of two equally costly cash flow streams: annuity X and annuity Y. X is an annuity due with a cash inflow of $9,000 for each of 6 years. Y is an ordinary annuity with a cash inflow of $10,000 for each of 6 years. Assume that Ramesh can earn 15% on his investments.

a.On a purely subjective basis, which annuity do you think is more attractive? Why?

b.Find the future value at the end of year 6, FVA6, for both annuity X and annuity Y.

c. Use your finding in part b to indicate which annuity is more attractive. Why?

d.Compare your finding to your subjective response in part a.

ZULFIQAR HASAN 17

Practice 07: Future Value Of A Retirement Annuity

An insurance agent is trying to sell you an immediate-retirement annuity, which for a single amount paid today will provide you with $12,000 at the end of each year for the next 25 years. You currently earn 9% on low-risk investments comparable to the retirement annuity. Ignoring taxes, what is the most you would pay for this annuity?

$117,870.96

ZULFIQAR HASAN 18

Practice 08: Future Value Of A Retirement Annuity

To supplement your planned retirement in exactly 42 years, you estimate that you need to accumulate $220,000 by the end of 42 years from today. You plan to make equal annual end-of-year deposits into an account paying 8% annual interest.

a. How large must the annual deposits be to create the $220,000 fund by the end of 42 years?

b. If you can afford to deposit only $600 per year into the account, how much will you have accumulated by the end of the 42nd year?

a. PMT = $723.10 b. $182,546.40

ZULFIQAR HASAN 19

Present Value of An Ordinary Annuity

The discounting term is called the present value interest factor for annuities (PVIFA).

)1(PVAn PVAn,

)1( PMT PVAn

OrdinaryDueAnnuity

ii 1

1 - 1

DueAnnuity

n

ior

i

ii 1

1 - 1

Ordinary

n

PMT PVAn

Present Value of An Annuity Due

m

im

i 1

1 - 1

Ordinary

mn

PMT PVAn

ZULFIQAR HASAN 20

Examples

Examples 03. What is the present value of an ordinary annuity of $100 per year at 12% per annum for three years?

$240.18 8)$100(2.401

12%1

- 1

$100 PVAn3

%12

1

Examples 04. You borrow $7 500 to buy a car and agree to repay the loan by way of equal monthly repayments over 5 years. The current interest rate is 12% per annum, compounded monthly. What is the amount of each monthly repayment?

$166.83

44.96 500 $7 PMT

0.011.01

1 - 1

PMT 500 $760

ZULFIQAR HASAN 21

Practice 04 : Finding the Periodic Payment (PMT)

m

im

i 1

1 - 1

mn

PVAn PMT

$955.66

Suppose you borrow $100,000 to buy a new house. If the mortgage interest rate is 8% on a 15-year mortgage, how much would your MONTHLY payments (Installment) be?

$955.62. PMT

100000$ PMT

PVAn PMT

12

0.0812

0.08 1

1 - 1

m

im

i 1

1 - 1

2151

mn

ZULFIQAR HASAN 22

Practice 03: Present Value of Annuity (from the Text)For each case in the following table, answer the following questions that Follow:

Case Amount of Annuity Interest rate YearA 2500 8% 10B 500 12% 6C 30000 20% 5D 11500 9% 8E 6000 14% 30

1. Calculate the Present value of the annuity assuming that it isa) An Ordinary annuityb) An Annuity Due

2. Compare the findings in part (a) and (b). All else being identical, which type of annuity is preferable? Explain Why?

ZULFIQAR HASAN 23

Multiple Period In A YearExamples 05: What is the present value of an ordinary annuity of $50 paid every 6 months at 12% per annum for 3 years?

A. $240.18 B. $245.87 C. $253.79 D. $279.12

$245.87 )$50(4.9173

212%

212%

1

1 - 1

$50 PVAn

23

Examples 06: You will receive $500 at the end of each of the next 5 years. The current interest rate is 9% per annum. What is the present value of this series of cash flows?

944.85 $1 3.8897 $500

0.091.09

1 - 1

$500 PVAn 5

ZULFIQAR HASAN 24

Growing Annuity & Growing Perpetuity• A growing stream is one in which each successive cash flow

is larger than the previous one. A common problem is one in which the cash flows grow by some fixed percentage

• A growing annuity is an annuity in which the cash flows grow at a constant rate g:

2

2 3 1

1

(1 ) (1 ) (1 )...

(1 ) (1 ) (1 ) (1 )

11

1

n

n

N

C C g C g C gPV

R R R R

C g

R g R

A growing perpetuity is an annuity where the cash flows continue indefinitely:

2

2 3

11

1

(1 ) (1 ) (1 )...

(1 ) (1 ) (1 ) (1 )

(1 )

(1 )

tt

tt

C C g C g C gPV

R R R R

C g C

R R g

ZULFIQAR HASAN 25

Types of Loans

1. A pure discount loan is a loan where the borrower receives money today and repays a single slump sum in the future.

2. An interest only loan requires the borrower to pay interest each period and to repay the entire principal at some point in the future.

3. An amortized loan requires the borrower to repay both the principal and interest over time.

ZULFIQAR HASAN 26

Example: Simple Amortized Loan Schedule

Suppose a business takes out Tk 5000, five-year loan at 9%. The agreement calls for the borrower to pay the interest on the loan balance each year and to reduce the loan balance each year by Tk 1000. Please make an amortization Schedule.

Tk 500Tk 1350Tk 6350Totals

0100090109010005

10001000180118020004

20001000270127030003

30001000360136040002

Tk 4000Tk 1000Tk 450Tk 1450Tk 50001

Ending Balance

Principal Paid

Interest Paid

Total Payment

Beginning Balance

Year

ZULFIQAR HASAN 27

Practice: Simple Amortized Loan Schedule

Suppose a business takes out a $10000, five-year loan at 10%. The agreement calls for the borrower to pay the interest on the loan balance each year and to reduce the loan balance each year by $2000. Please make an amortization Schedule.

Year Beginning Balance

Total Payment

Interest Paid

Principal Paid

Ending Balance

1 $10000 $3000 $1000 $2000 8000

2 8000

3

4

5

Totals