1. UNDERSTANDING MONEY MANAGEMENT
CHAPTER 3

2. Nominal and Effective Interest Rates
Equivalence Calculations using Effective Interest Rates
Debt Management

3. Focus
1. If payments occur more frequently than annual, how do you calculate economic equivalence?
If interest period is other than annual, how do you calculate economic equivalence?
How are commercial loans structured?

4. Nominal Versus Effective Interest Rates
Nominal Interest Rate
Interest rate quoted based on an annual period
Effective Interest Rate
Actual interest earned or paid in a year or some other time period

5. 18 % Compounded Monthly
Nominal
interest rate
Interest
period
Annual
Percentage
Rate (APR)

6. In words, What It Really Means?
18% Compounded Monthly
What It Really Means?
Interest rate per month ( i ) = 18% / 12 = 1.5%
Number of interest periods per year (N ) = 12
In words,
If you borrowed money, Bank will charge 1.5% interest each month on your unpaid balance.
If you deposited money, You will earn 1.5% interest each month on your remaining balance.

7. 18% compounded monthly
Question: Suppose that you invest $1000 for one year at 18% compounded monthly. How much interest would you earn?
Solution:
Your gain=195.62
?= 18 %
Annual Percentage Yield
18%
= 1.5%

8. Effective Annual Interest Rate
r = nominal interest rate per year
ia = effective annual interest rate
M = number of interest periods per year

9. Compounding Semi-annually Compounding Quarterly
Nominal and Effective Interest Rates with Different Compounding Periods
Effective Rates
Nominal Rate
Compounding Annually
Compounding Semi-annually
Compounding Quarterly
Compounding Monthly
Compounding Daily 4%
4.00%
4.04%
4.06%
4.07%
4.08% 5
5.00
5.06
5.09
5.12
5.13 6
6.00
6.09
6.14
6.17
6.18 7
7.00
7.12
7.19
7.23
7.25 8
8.00
8.16
8.24
8.30
8.33 9
9.00
9.20
9.31
9.38
9.42 10
10.00
10.25
10.38
10.47
10.52 11
11.00
11.30
11.46
11.57
11.62 12
12.00
12.36
12.55
12.68
12.74

10. PRACTICE PROBLEM
If your credit card calculates the interest based on 12.5% APR, what is your monthly interest rate and annual effective interest rate (APY), respectively? Your current outstanding balance is $2,000 and skips payments for 2 months. What would be the total balance 2 months from now?
SOLUTION Monthly Interest Rate Effective Interest Rate ia = ( )12 – 1 = 13.24% Total Outstanding Balance F = B2 = $2,000(F/P, %, 2)= $2,041.88

11. PRACTICE PROBLEM
Suppose your savings account pays 9% interest compounded quarterly. If you deposit $10,000 for one year, how much would you have?

12. EFFECTIVE ANNUAL INTEREST RATES ( 9% COMPOUNDED QUARTERLY )
First quarter
Base amount + Interest (2.25%)
$10,000 + $225
Second quarter
= New base amount + Interest (2.25%)
= $10,225 + $230.06
Third quarter
= $10, $235.24
Fourth quarter
= New base amount + Interest (2.25 %)
Value after one year
= $10, $240.53
= $10,930.83

13. EFFECTIVE INTEREST RATE BASED ON PAYMENT PERIOD ( i )
C = number of interest periods per payment period
K = number of payment periods per year
C K = M, total number of interest periods per year
r / K = nominal interest rate per payment period
r / C K = nominal interest rate per interest period

14. 4th Qtr One-year 1st Qtr 2nd Qtr 3rd Qtr
12% compounded monthly Payment Period = Quarter (transaction occur) Compounding Period = Month
One-year
1st Qtr
2nd Qtr
3rd Qtr
4th Qtr 1% 1% 1%
C=3
K=4
Effective interest rate per quarter
Effective annual interest rate

16. where C K is number of compounding periods per year.
Effective Interest Rate per Payment Period with Continuous Compounding Although continuous compounding sounds impressive, in practice it results in almost the same effective yield as daily compounding.
where C K is number of compounding periods per year.
Continuous compounding =>

17. Case 1: 8% compounded quarterly
Payment Period = Quarter
Interest Period = Quarterly
** i=r/4
1st Q
2nd Q
3rd Q
4th Q
1 interest period
Given r = 8%,
K = 4 payments per year
C = 1 interest period per quarter
M = 4 interest periods per year

18. Example: 3.2 Effective Rate per Payment Period
Case 2: 8% compounded monthly
Payment Period = Quarter
Interest Period = Monthly
** Not equal to r/4
1st Q
2nd Q
3rd Q
4th Q
3 interest periods
Given r = 8%,
K = 4 payments per year
C = 3 interest periods per quarter
M = 12 interest periods per year Find: i

19. Example: 3.3 Calculating Effective Interest Rate
Case 3: 8% compounded weekly
Payment Period = Quarter
Interest Period = Weekly
1st Q
2nd Q
3rd Q
4th Q
13 interest periods
1 year =52 Weeks
Given r = 8%,
K = 4 payments per year
C = 13 interest periods per quarter
M = 52 interest periods per year

20. Case 4: 8% Compounded Continuously
Payment Period = Quarter
Interest Period = Continuously
1st Q
2nd Q
3rd Q
4th Q
 interest periods
Given r = 8%,
K = 4 payments per year

21. Effective interest rate per quarter Case 1 Case 2 Case 3 Case 4
8% compounded quarterly
8% compounded monthly
8% compounded weekly
8% compounded continuously
Payments occur quarterly
2.000% per quarter
2.013% per quarter
2.0186% per quarter
2.0201% per quarter

22. Equivalence Analysis Using Effective Interest Rates
Step 1: Identify the payment period (annual, quarter, month, week, etc.)
Step 2: Identify the interest period (annually, quarterly, monthly, etc.)
Step 3: Find the effective interest rate ( i ) that covers the payment period.

23. Case 1: When Payment Periods and Compounding periods coincide
Step 1: Identify the number of compounding (or payment) periods (M = K)
per year.
Step 2: Compute the effective interest rate per payment period ( i )
i = r / M
Step 3: Determine the total number of payment periods (N)
N = M x (number of years)
Step 4: Use i and N in the appropriate formulas.

24. Example 3.4
Calculating
Auto Loan Payments

25. Example 3.4 Calculating Auto Loan Payments
Given:
Invoice Price = $21,599
Sales tax at 4% = $21,599 (0.04) = $863.96
Dealer’s freight = $21,599 (0.01) = $215.99
Total purchase price = $22,678.95
Down payment = $2, What is left is: $20,000
Dealer’s interest rate = 8.5% APR
Length of financing = 48 months
Find:
What would be the monthly payment?
After the 25th payment, you want to pay off the remaining loan in a lump-sum amount. What is the required amount of this lump-sum?

26. Solution: Payment Period = Interest Period
$20,000 48 A
Given: P = $20,000, r = 8.5% per year
K = 12 payments per year
N = 48 payment periods
Find: A
Step 1: M = 12
Step 2: i = r / M = 8.5% / 12 = % per month
Step 3: N = (12)(4) = 48 months
Step 4: A = $20,000(A/P, %,48) = $492.97

27. Step 2: i = r / M = 8.5% / 12 = 0.7083% per month
$20,000 48 1 2 25 24
$492.97
$492.97
25 payments that
were already made
23 payments that are
still outstanding
Step 1: M = 12
Step 2: i = r / M = 8.5% / 12 = % per month
Step 3: N = (12)(4) = 48 months
Step 4: A = $20,000(A/P, %,48) = $492.97

28. $20,000 48 1 2 25 24
$492.97
$492.97
25 payments that
were already made
23 payments that are
still outstanding
P = B25 = $ (P/A, %, 23) = $10, lump sum
Total payment for 48 months = 48 x = $23,662.56

29. PRACTICE PROBLEM
You have a habit of drinking a cup of coffee ($2.00 a cup) on the way to work every morning for 30 years.
If you put the money in the bank for the same period, how much would you have, assuming your accounts earns 5% interest compounded daily.
NOTE: Assume you drink a cup of coffee every day including weekends.

30. SOLUTION
Payment period: Daily
Compounding period: Daily

31. Case 2: When Payment Periods Differ from Compounding Periods
Step 1: Identify the following parameters
M = No. of compounding periods per year
K = No. of payment periods per year
C = No. of interest periods per payment period
Step 2: Compute the effective interest rate per payment period
For discrete compounding
For continuous compounding
Step 3: Find the total no. of payment periods N = K x (no. of years)
Step 4: Use i and N in the appropriate equivalence formula

34. Figure 3.9 Equivalence calculation for continuous compounding

35. Figure 3.10 Cash flow diagram for a deposit series with changing interest rates

37. PRACTICE PROBLEM
A series of receiving equal quarterly payments of $5,000 for 10 years is equivalent to what present amount at an interest rate of 9% compounded ….
Quarterly
Monthly
Continuously

38. SOLUTION
A = $5,000
40 Quarters

39. (a) QUARTERLY A = $5,000 Payment period : Quarterly
Interest Period: Quarterly
40 Quarters

40. (b) MONTHLY Payment period : Quarterly A = $5,000
Interest Period: Monthly
A = $5,000
40 Quarters

41. (c) CONTINUOUSLY A = $5,000 Payment period : Quarterly
Interest Period: Continuously
40 Quarters

42. Example 3.7 Amortized Loan
Suppose you secure a home improvement loan in the amount of $5,000 from a local bank. The loan officer gives you the following loan terms:
Contract amount = $5,000
Contract period = 24 months
Annual percentage rate = 12%
Monthly installment = $235.37

43. Example 3.7 Amortized Loan
Given: P = $5,000, i = 12% APR, N = 24 months Find: A
A = $5,000(A/P, 1%, 24) = $235.37

44. Loan Repayment Schedule

45. Loan Repayment Schedule

46. Interest Paid = 235.3647*24 – 5000 = 648.817 P No Payment
Princ Payment
Interest Payment
Loan Balance 1
50.000 2
48.146 3
46.274 4
44.383 5
42.473 6
40.544 7
38.596 8
36.628 9
34.641 10
32.634 11
30.606 12
28.559 13
26.491 14
24.402 15
22.292 16
20.162 17
18.010 18
15.836 19
13.641 20
11.423 21
9.184 22
6.922 23
4.638 24
2.330
0.000
Sum
Interest Paid = *24 – 5000 =

47. Calculating the Remaining Loan Balance after Making the nth Payment

48. Example 3.7 Computing the Outstanding Loan Balance after making the 6th payment

49. Loan Repayment Schedule

50. PRACTICE PROBLEM Consider the 7th payment ($235.37)
(a) How much is the interest payment in the 7th period?
(b) What is the amount of principal payment in the 7th period?

51. Solution i = 1% per month $5,000 A = $235.37 Interest payment = ?
i = 1% per month
Interest payment = ?
Principal payment = ?

52. SOLUTION

53. SUMMARY
Financial institutions often quote (quotation) interest rate based on an APR.
In all financial analysis, we need to convert the APR into an appropriate effective interest rate based on a payment period.
When payment period and interest period differ, calculate an effective interest rate that covers the payment period. Then use the appropriate interest formulas to determine the equivalent values

54. CHAPTER 3 PRACTICE PROBLEMS

55. 1) A loan offers money at 1. 5% per month compounded monthly
1) A loan offers money at 1.5% per month compounded monthly. (a) What is the nominal interest rate? (b) What is the effective annual interest rate?

56. SOLUTION # 1

57. 2) Find the effective interest rate per payment period for an interest rate of 8% compounded monthly for each of the given payment schedule:
(a) monthly (b) quarterly (c) semiannual (d) annual

59. 3) What will be the amount accumulated at the end of the time by each of the given present investments?
(a) $6,000 in 12 years at 8% compounded semi-annually.
(b) $14,500 in 18 years at 6% compounded quarterly.
(c) $12,500 in seven years at 8% compounded monthly.

61. 4) What is the future worth of each of the given series of payments?
(a) $12,000 at the end of each six-month period for 12 years at 8% compounded semiannually.
(b) $8,000 at the end of each quarter for six years at 12% compounded quarterly.
(c) $6,000 at the end of each month for 5 years at 6% compounded monthly.

63. 5) What equal series of payments must be paid into a sinking fund in order to accumulate each given amount?
(a) $1,700 in 10 years at 8% compound semiannually when payments are semiannual.
(b) $9,000 in 6 years at 3% compounded quarterly when payments are quarterly.
(c) $4,000 in 2 years at 12% compounded monthly when payments are monthly.

65. 6) What is the present worth of each of the given series of payments?
(a) $2,700 at the end of each six-month period for 10 years at 8% compounded semiannually.
(b) $10,000 at the end of each quarter for five years at 12% compounded quarterly.
(c) $14,000 at the end of each month for eight years at 6% compounded monthly.

67. 7) Ahmed deposits $4,000 in a savings account that pays 6% interest compounded monthly. Three years later, he deposits $5,000. Two years after the $5,000 deposit he makes another deposit in the amount of $7,000. Four years after the $7,000 deposit, half of the accumulated money is transferred to a fund that pays 8% interest compounded quarterly. How much money will be in each account six years after the transfer?

69. 8) A couple is planning to finance its three-year-old son’s college education. The couple can deposit money at 12% compounded quarterly. What quarterly deposit must be made from the son’s 3rd birthday to his 18th birthday in order to provide $30,000 on each birthday from the 18th to 21st? (Note that the last deposit is made on the day of first withdrawal.)

71. 9) Suppose you borrowed $12,000 at an interest rate of 8%, compounded monthly and to be paid monthly over 36 months. At the end of first year (after 12 payments), you want to negotiate with the bank to pay off the remainder of the loan in eight equal quarterly payments. What is the amount of this quarterly payment, if the interest rate and compounding frequency remain the same?

73. 10) Talha is considering the purchase of a used car. The price, including the title and taxes, is $10,500. Talha is able to make a $2,500 down payment. The balance, $8,000, will be borrowed from his credit union at an interest rate of 10% compounded daily. The loan should be paid in 48 equal monthly payments. Compute the monthly payment. What is the total amount of interest Talha has to pay over the life of the loan?

74. Total interest = 203*48 – 8000 = 1744

75. 11) A lender requires that monthly mortgage payments be no more than one third of gross monthly income, with a maximum term of 20 years. If you can make only a 20% down payment, what is the minimum monthly income you would need in order to purchase a $420,000 house when the interest rate is 12% compounded monthly?