1. BBA(Hons.), MBA(Finance), London
Topic # 03
EAR,
Annuities &
Loan Management
Zulfiqar Hasan
BBA(Hons.), MBA(Finance), London
Associate Professor
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
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3. Formula For FV, PV, EAR and Annuities
PMT = Payment or Installment amount, PVAn = PV of Annuity
FVAn = FV of Annuity, n = Year, m = Period, i = Interest Rate
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4. Effective Annual Interest Rate (EAR)
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).
For daily, monthly, annually…..
If interest is paid continuously
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5. Example 01: EAR
Example 01: Find out the EAR for a nominal rate of 10%, compounded semiannually?
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6. Example 02: Comparing EARs
Consider 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 quarterly
Bank C: 16%, compounded annually
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
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%
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?
On a weekly basis? 4.On a daily basis?
5.On a continuous basis?
Assume a 360 day year.
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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%.
What amount would Ms. Martin have at the end of the third year, leaving all interest paid on deposit, in each Bank?
What effective annual rate (EAR) would she earn in each of the banks?
On the basis of your findings in parts 01 and 02, which bank should Ms. Martin deal with? Why?
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.
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9. Annuities
An annuity is a series of equal, periodic payments.
Followings are the types of annuities:
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.
Annuity Due:
An annuity for which the cash flow occurs at the beginning of each period is called Annuity Due.
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?
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10. Future Value of An Annuity
The compounding term is called the future value interest factor for annuities (FVIFA).
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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%?
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12. Example 02: FV of Annuity for multiple deposit
Mr. Anisur Rahman has opened a DPS at Luminous Cooperatives on the following conditions:
Duration: 05 Years
Payment modes: Monthly
Installment : Tk 1000
Interest 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?
Due: Tk
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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
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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?
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?
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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
Year A
2500 8% 10 B
500
12% 6 C
30000
20% 5 D
11500 9% 8 E
6000
14% 30
Calculate the future value of the annuity assuming that it is
An Ordinary annuity
An Annuity Due
Compare the findings in part (a) and (b). All else being identical, which type of annuity is preferable? Explain Why?
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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.
On a purely subjective basis, which annuity do you think is more attractive? Why?
Find the future value at the end of year 6, FVA6, for both annuity X and annuity Y.
Use your finding in part b to indicate which annuity is more attractive. Why?
Compare your finding to your subjective response in part a.
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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
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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
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19. Present Value of An Ordinary Annuity
Present Value of An Annuity Due
The discounting term is called the present value interest factor for annuities (PVIFA).
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20. Examples
Examples 03. What is the present value of an ordinary annuity of $100 per year at 12% per annum for three years?
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?
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21. Practice 04 : Finding the Periodic Payment (PMT)
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?
$955.66
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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
Year A
2500 8% 10 B
500
12% 6 C
30000
20% 5 D
11500 9% 8 E
6000
14% 30
Calculate the Present value of the annuity assuming that it is
An Ordinary annuity
An Annuity Due
Compare the findings in part (a) and (b). All else being identical, which type of annuity is preferable? Explain Why?
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23. Multiple Period In A Year
Examples 05: What is the present value of an ordinary annuity of $50 paid every 6 months at 12% per annum for 3 years?
A. $ B. $ C. $ D. $279.12
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?
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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:
A growing perpetuity is an annuity where the cash flows continue indefinitely:
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25. Types of Loans
A pure discount loan is a loan where the borrower receives money today and repays a single slump sum in the future.
An interest only loan requires the borrower to pay interest each period and to repay the entire principal at some point in the future.
An amortized loan requires the borrower to repay both the principal and interest over time.
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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 Please make an amortization Schedule.
Year
Beginning Balance
Total Payment
Interest Paid
Principal Paid
Ending Balance 1
Tk 5000
Tk 1450
Tk 450
Tk 1000
Tk 4000 2
4000
1360
360
1000
3000 3
3000
1270
270
1000
2000 4
2000
1180
180
1000
1000 5
1000
1090 90
1000
Totals
Tk 6350
Tk 1350
Tk 500
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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 3 4 5
Totals
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