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UNDERSTANDING MONEY MANAGEMENT CHAPTER 3

UNDERSTANDING MONEY MANAGEMENT CHAPTER 3. Nominal and Effective Interest Rates Equivalence Calculations using Effective Interest Rates Debt Management. Focus. 1. If payments occur more frequently than annual, how do you calculate economic equivalence?

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UNDERSTANDING MONEY MANAGEMENT CHAPTER 3

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  1. UNDERSTANDING MONEY MANAGEMENT CHAPTER 3

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

  3. Focus • 1.If paymentsoccur 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 Interest Rate Interest rate quoted based on an annual period Effective Interest Rate Actualinterest earned or paid in a year or some other time period NominalVersusEffective Interest Rates

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

  6. 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. Nominal and Effective Interest Rates with Different Compounding Periods

  10. 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? PRACTICE PROBLEM SOLUTION Monthly Interest Rate  Effective Interest Rate ia=(1+0.010417)12 – 1 = 13.24% Total Outstanding Balance F=B2= $2,000(F/P, 1.0417%, 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 )

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

  15. Effective Interest Rate per Payment Period with Continuous CompoundingAlthough 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 =>

  16. 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

  17. 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

  18. 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

  19. 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

  20. Effective interest rate per quarter

  21. 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.

  22. 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 iand N in the appropriate formulas.

  23. Example 3.4 Calculating Auto Loan Payments

  24. 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 =$2,678.95 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?

  25. Solution:Payment Period =Interest Period $20,000 48 1 2 3 4 0 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 = 0.7083% per month Step 3: N = (12)(4) = 48 months Step 4: A = $20,000(A/P, 0.7083%,48) =$492.97

  26. $20,000 48 1 2 24 25 0 $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 = 0.7083% per month Step 3: N = (12)(4) = 48 months Step 4: A = $20,000(A/P, 0.7083%,48) =$492.97

  27. $20,000 48 1 2 24 25 0 $492.97 $492.97 25 payments that were already made 23 payments that are still outstanding P = B25= $492.97 (P/A, 0.7083%, 23) =$10,428.96 lump sum Total payment for 48 months = 48 x 492.97 =$23,662.56

  28. 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.

  29. SOLUTION • Payment period: Daily • Compounding period: Daily

  30. 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

  31. Figure 3.9 Equivalence calculation for continuous compounding

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

  33. 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

  34. SOLUTION A = $5,000 0 1 2 40 Quarters

  35. (a) QUARTERLY • Payment period : Quarterly • Interest Period: Quarterly A = $5,000 0 1 2 40 Quarters

  36. (b) MONTHLY • Payment period : Quarterly • Interest Period: Monthly A = $5,000 0 1 2 40 Quarters

  37. (c) CONTINUOUSLY A = $5,000 • Payment period : Quarterly • Interest Period: Continuously 0 1 2 40 Quarters

  38. 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

  39. 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

  40. Loan Repayment Schedule

  41. Loan Repayment Schedule

  42. Interest Paid = 235.3647*24 – 5000 = 648.817

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

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

  45. Loan Repayment Schedule

  46. 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?

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