Effective Money Management: Understanding Interest Rates & Loans

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Delve into Chapter 3 of money management to grasp concepts like economic equivalence calculation for non-annual payments, structuring commercial loans, and debt management. Understand the nuances between nominal & effective interest rates, APR, and the impact of compounding on interest earned. Gain insights into managing your finances effectively.

  • Finance
  • Money Management
  • Interest Rates
  • Loans
  • Debt
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  1. UNDERSTANDING MONEY MANAGEMENT CHAPTER 3

  2. Main Issues to Consider 1. If payments occur more frequently than annual, how do you calculate economic equivalence? 2. If interest period is other than annual, how do you calculate economic equivalence? 3. How are commercial loans structured? 4. How should you manage your debt?

  3. 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

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

  5. 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, Bank will charge 1.5% interest each month on your unpaid balance, if you borrowed money. You will earn 1.5% interest each month on your remaining balance, if you deposited money.

  6. Effective Annual Interest Earned Question: Suppose that you invest $1 for 1 year at 18% compounded monthly. How much interest would you earn? Solution: + = i F 1 ( 1 $ = + 12 12 ) 1 ( 1 $ . 0 015 ) = $1.1956 0.1956 or 19.56% 18% = ai 1.5%

  7. Relationship Between Nominal and Effective Interest Rates 18% 1.5% 18% compounded monthly or 1.5% per month for 12 months or 19.56% compounded annually

  8. Effective Annual Interest Rate (Yield) M = + 1 ( / ) 1 i r M a r = nominal interest rate per year ia = effective annual interest rate M = number of interest periods per year

  9. Effective Annual Interest Rates (9% compounded quarterly) Base amount + Interest (2.25%) $10,000 + $225 First quarter = New base amount + Interest (2.25%) = $10,225 +$230.06 Second quarter = New base amount + Interest (2.25%) = $10,455.06 +$235.24 Third quarter = New base amount + Interest (2.25 %) = Value after one year = $10,690.30 + $240.53 = $10,930.83 Fourth quarter

  10. Effective Interest Rates with Different Compounding Periods Effective Rates Compounding Annually Compounding Semi-annually Compounding Quarterly Compounding Monthly Compounding Daily Nominal Rate 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

  11. Example Find the compounding period for each CD (certificate of deposit) Type of certificate Interest rate (APR) Annual Percentage yield (APY) 2.25 3.1 3.4 3.5 Minimum required to open 1-year 2-year 3-year 4-year 2.23 3.06 3.35 3.45 $500 $500 $500 $500

  12. Example Type of certificate Interest rate (APR) Annual Percentage yield (APY) 2.25 3.1 3.4 3.5 Minimum required to open 1-year 2-year 3-year 4-year 2.23 3.06 3.35 3.45 $500 $500 $500 $500 Find the compounding period for each CD (certificate of deposit) = + M 1 ( / ) 1 i r M a = + M . 0 035 1 ( 0345 . / ) 1 M = + M . 1 035 = M 1 ( 0345 . / ) M 6

  13. Calculating Effective Interest Rates Based on Payment Periods Payment period Interest period Effective interest rate per payment period Month 0 1 2 3 0 1 2 3

  14. Effective Interest Rate per Payment Period (i) + = r i C 1 [ / ] 1 CK C = number of interest periods per payment period K = number of payment periods per year CK = total number of interest periods per year, or M r/K = nominal interest rate per payment period

  15. Relationship between i and ia 12% compounded monthly Payment Period = Quarter; Compounding Period = Month K=4, C=3 1st Qtr 2nd Qtr 3rd Qtr 4th Qtr 1% 1% 1% 3.030 % One-year Effective interest rate per quarter 12 . 0 1 ( + = i = 3 / 3 * 4 ) 1 . 3 030 % Effective annual interest rate i i a = ( 1 = + + = ) 12 ( 1 001 003030 . . ) 1 1268% 1268% . a = 1 4 .

  16. Example Suppose you make quarterly deposits into a saving account that earns 8% compounded monthly, compute the effective interest rate per quarter and per year Given r = 8%, K = 4 payments periods per year C = 3 interest periods per quarter = = = + + C [ [ 2 013% 1 1 r CK / . 0 08 ] 1 i 3 / ( )( )] 3 4 per quarter 1 . 2.013% per quarter ia= {1+.2013)4 -1 = 8.3%

  17. Effective Interest Rate per Payment Period with Continuous Compounding = + C 1 [ / ] 1 i r CK where CK = number of compounding periods per year C continuous compounding =>

  18. Effective Interest Rate per Payment Period with Continuous Compounding

  19. Effective Interest Rate with Continuous Compounding

  20. Case 0: 8% compounded quarterly Payment Period = Quarter Interest Period = Quarterly 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 = = = + + C [ [ 2 000% 1 1 r CK / . 0 08 ] 1 4 1 i 1 / ( )( )] per quarter 1 .

  21. Case 1: 8% compounded monthly Payment Period = Quarter Interest Period = Monthly 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 = = = + + C [ [ 2 013% 1 1 r CK / . 0 08 ] 1 i 3 / ( )( )] 3 4 per quarter 1 .

  22. Case 2: 8% compounded weekly Payment Period = Quarter Interest Period = Weekly 1st Q 2nd Q 3rd Q 4th Q 13 interest periods Given r = 8%, K = 4 payments per year C = 13 interest periods per quarter = = = + + )( )] 13 4 per quarter C [ [ 2 0186% 1 1 r CK / . 0 08 ] 1 i 13 / ( 1 .

  23. Case 3: 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 = / r K 1 i e = . 0 02 1 e = 0201 . 2 per % quarter

  24. Summary: Effective Interest Rate per Quarter Case 0 Case 1 Case 2 Case 3 8% compounded quarterly Payments occur quarterly 8% compounded monthly Payments occur quarterly 8% compounded weekly Payments occur quarterly 8% compounded continuously Payments occur quarterly 2.000% per quarter 2.013% per quarter 2.0186% per quarter 2.0201% per quarter

  25. 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, respectively? Your current outstanding balance is $2,000 and skips payments for 2 months. What would be the total balance 2 months from now?

  26. Solution 12.5% 12 Monthly Interest Rate: i = = 1.0417% Annual Effective Interest Rate: ai = + 12 (1 0.010417) 13.24% 1 = Total Outstanding Balance: = = = $2,000( / ,1.0417%,2) $2,041.88 F B F P 2

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

  28. Solution (a) Interest rate per quarter: 9% 4 (b) Annual effective interest rate: (1 0.0225) (c) Balance at the end of one year (after 4 quarters) $10,000( $10,000( $10,9 = = = 2.25% i = + = 4 1 9.31% i a = / ,2.25%,4) / ,9.31%,1) F P 31 F F P =

  29. Practice Problem Suppose your savings account pays 9% interest compounded quarterly. If you deposit $10,000 for one year, how much would you have? / 1 [ + = r i K=1 C ] 1 CK r= 9% C=4 4 = + = 1 [ 09 . / 4 ] 1 * 1 09308 . ai F=10000(1+.09308)=10930.8

  30. Equivalence Analysis using Effective Interest Rates

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

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

  33. Example 3.4 Calculating Auto Loan Payments

  34. Financial Data: 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 Dealer s interest rate = 8.5% APR Length of financing = 48 months Find: the monthly payment (A)

  35. Solution: Payment Period = Interest Period Monthly $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

  36. Process of Calculating the remaining Balance of the Auto Loan

  37. Dollars Down in Drain What three levels of coffee drinkers who bought drinks every day for 50 years at $2.00 a cup would have if they had instead banked that money each week: Level of drinker Would have had 1 cup a day $193,517 2 cups a day $387,034 3 cups a day $580,552 Note: Assumes constant price per cup, the money banked weekly and an annual interest rate of 5.5% compounded weekly

  38. Sample Calculation: One Cup per Day Step 1: Determine the effective interest rate per payment period: Payment period = weekly 5.5% interest compounded weekly i = 5.5%/52 = 0.10577% per week Step 2: Compute the equivalence value Weekly deposit amount Total number of deposit periods N = (52 weeks/yr.)(50 years) = 2600 weeks F = $14.00 (F/A, 0.10577%, 2600) = $193,517 A = $2.00 x 7 = $14.00 per week

  39. Case II: When Payment Periods Differ from Compounding Periods Step 1: Identify the following parameters K = No. of payment periods C = No. of interest periods per payment period Step 2: Compute the effective interest rate per payment period For discrete compounding 1 [ = i + C / ] 1 r CK For continuous compounding = / r K 1 i e Step 3: Find the total no. of payment periods N = K (no. of years) Step 4: Use i and N in the appropriate equivalence formula

  40. Example 3.5 Discrete Case: Quarterly deposits with Monthly compounding Step 1: K = 4 payment periods/year C = 3 interest periods per quarter Step 2: [1 0.12/(3)(4)] 3.030% = i = + 3 1 Step 3: N = 4(3) = 12 Step 4: F = $1,000 (F/A, 3.030%, 12) = $14,216.24

  41. Continuous Case: Quarterly deposits with Continuous compounding Step 1: K = 4 payment periods/year C = interest periods per quarter Step 2: Step 3: N = 4(3) = 12 Step 4: F = $1,000 (F/A, 3.045%, 12) = $14,228.37

  42. Practice Problem A series of equal quarterly payments of $5,000 for 10 years is equivalent to what present amount at an interest rate of 9% compounded (a) quarterly (b) monthly (c) continuously A = $5,000 0 1 2 40 Quarters

  43. (a) Quarterly Payment period : Quarterly Interest Period: Quarterly A = $5,000 0 40 1 2 9% 4 40 quarters $5,000( / ,2.25%,40) $130,968 = = 2.25% per quarter i = = = N P P P A

  44. (b) Monthly Payment period : Quarterly Interest Period: Monthly 9% 0.75% per month 12 (1 0.0075) = + = = = A = $5,000 0 1 2 40 = = i = 3 2.267% per quarter i p 40 quarters $5,000( / ,2.267%,40) $130,586 N P P P A

  45. (c) Continuously Payment period : Quarterly Interest Period: Continuously A = $5,000 0 40 1 2 = = = = = 0.09/4 1 2.276% per quarter 40 quarters $5,000( / ,2.276%,40) $130,384 i e N P P A P

  46. Debt Management

  47. Credit Card Debt

  48. Revisit Example 3.4 Financing an Automobile $20,000 48 1 2 24 25 0 Given: APR = 8.5%, N = 48 months, and P = $20,000 Find: A A = $20,000(A/P,8.5%/12,48) = $492.97

  49. Suppose you want to pay off the remaining loan in lump sum right after making the 25th payment. How much would this lump be? $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 = $492.97 (P/A, 0.7083%, 23) = $10,428.96

  50. Example 3.7 Determine a Loans Balance, Principal, and Interest = = $5,000( / ,1%,24) $235.37 A A P

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