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Chapter 12

Chapter 12. Compound Interest and Present Value. #12. Compound Interest and Present Value. Learning Unit Objectives. Compound Interest (Future Value) – The Big Picture. LU12.1. Compare simple interest with compound interest

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Chapter 12

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  1. Chapter 12 Compound Interest and Present Value

  2. #12 Compound Interest and Present Value Learning Unit Objectives Compound Interest (Future Value) – The Big Picture LU12.1 • Compare simple interest with compound interest • Calculate the compound amount and interest manually and by table lookup • Explain and compute the effective rate

  3. #12 Compound Interest and Present Value Learning Unit Objectives Present Value -- The Big Picture LU12.2 • Compare present value (PV) with compound interest (FV) • Compute present value by table lookup • Check the present value answer by compounding

  4. Compounding • Compounding: The process of calculating the interest periodically over the life of the loan (or an investment). • After each calculation, the interest is added to the loan, and starts to accrue additional interest for the next period based on the adjusted principal (equal to the previous principal plus the interest).

  5. Compound Interest • Compound interest: Interest on the principal of the loan, plus the interest on all the accrued interests (the interests of all previous periods).

  6. Future Value (Compound Amount) • Future value (or Compound amount): The final amount of the loan or the investment at the end of the last period. • Refer to the next slide to explore $1 will grow in the value of at 8% in 4 consecutive years. $1

  7. Figure 12.1 Future Value of $1 at 8% for Four Periods Compounding goes from present value to future value Future Value After 4 periods $1 is worth $1.36 After 3 periods $1 is worth $1.26 After 2 periods $1 is worth $1.17 After 1 period $1 is worth $1.08 Present value $1.2597 $1.3605 $1.1664 $1.08 $1.00 Number of periods

  8. Growth of $1 in 4 Years at 8% • Present: $1 • After 1 year: $1.08 (end of year 1) • After 2 years: $1.17 (end of year 2) • After 3 years: $1.26 (end of year 3) • After 4 years: $1.36 (end of year 4) • Future value of $1 at 8% in 4 years: $1.36

  9. Figure 12.1 Future Value of $1 at 8% for Four Periods Manual Calculation

  10. Present Value vs Future Value • Present value: The value of money as of today. • Future value (or Compound amount): The final amount of the money, loan or investment at the end of the last period.

  11. Compounding Interest (Future Value) Compound interest - the interest on the principal plus the interest of prior periods Compounding - involves the calculation of interest periodically over the life of the loan or investment Future value (compound amount) - is the final amount of the loan or investment at the end of the last period Present value - the value of a loan or investment today

  12. Compounding Terms Compounding Periods Interested Calculated Compounding Annually Once a year Compounding Semiannually Every 6 months Compounding Quarterly Every 3 months Compounding Monthly Every month Compounding Daily Every day

  13. Simple Interest Formula Recall Chapter 10 Simple Interest (I) = Principal (P) x Rate (R) x Time (T) Stated as a Percent Stated as a Percent

  14. Simple Versus Compound Interest Simple Compounded Al Jones deposited $1,000 in a savings account for 5 years at an annual interest rate of 10%. What is Al’s simple interest and maturity value? Al Jones deposited $1,000 in a savings account for 5 years at an annual compounded rate of 10%. What is Al’s interest and compounded amount? I = P x R x T I = $1,000 x .10 x 5 I = $500 MV = $1,000 + $500 MV = $1,500 Interest: $1,610.51 - $1,000 = $610.51

  15. Calculating Compound Amount & Interests • Manual Method • (As in the previous slide) • 2. Look-up Method from a Table • Use the formula: • Principal x Table factor = Compound Amount • (Future Value) • How to find Table factor: • * Define the number of periods of interest • * Define the appropriate rate for each period.

  16. Tools for Calculating Compound Interest Number of periods(N) Number of years multiplied the number of times the interest is compounded per year Rate for each period(R) Annual interest rate divided by the number of times the interest is compounded per year If you compounded $100 for 3 years at 6% annually, semiannually, or quarterly What is N and R? Periods Rate Annually: 6% / 1 = 6% Semiannually: 6% / 2 = 3% Quarterly: 6% / 4 = 1.5% Annually: 3 yrsx 1 = 3 Semiannually: 3 yrs x 2 = 6 Quarterly: 3 yrs x 4 = 12

  17. Calculating Compound Amount by Table Lookup Step 4. Multiply the table factor by the amount of the loan. Step 3. Go down the period column of the table to the number desired; look across the row to find the rate. At the intersection is the table factor Step 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year Step 1. Find the periods: Years multiplied by number of times interest is compounded in 1 year

  18. Table 12.1 - Future Value of $1 at Compound Interest

  19. Calculating Compound Amount by Table Lookup Steve Smith deposited $80 in a savings account for 4 years at an annual compounded rate of 8%. What is Steve’s interest and compounded amount? N = 4 x 1 = 4 R = 8% = 8% 1 Table Factor = 1.3605 Compounded Amount: $80 x 1.3605 = $108.84 I = $108.84 - $80 = $28.84

  20. Practice John deposits $1,000 in his savings account that pays 6% interest compounded quarterly. What will be the balance of his account at the end of 6 years? Step1: Calculate the numbers of periods: Periods = 4 x 6 years = 24 periods. Step 2: Calculate the appropriate period rate: Rate = 6% / 4 = 1.50% Step 3: Locate the table factor: 24 periods, at 1.5% Look up Table factor =1.4295 Step 4: Use the formula: Principal x Table factor = $1,000 x 1.4295 = $1,4295

  21. Problem 12-13: 4% 2 = 2% (Period rate) Solution: Loan: $25,000 7 years at 4% interest compounded semiannually. 7 years x 2 = 14 periods $25,000.00 x 1.3195 = $32,987.50 Compound amount (Future value) at the end of 7 years Lookup table factor Loan amount

  22. Problem 12-15: 8% 4 10% 2 = 2% = 5% Solution: Which bank provides higher compound amount? Four Rivers 4 years x 4 = 16 periods Mystic 4 years x 2 = 8 periods $10,000 x 1.4775 = $14,775 - 10,000 $ 4,775 $10,000 X 1.3728 = $13,728 -10,000 $ 3,728 Lookup table factor Lookup table factor

  23. Problem 12-16: 12% 2 = 6% Solution: Compound amount at end of year 4 Lookup table factor 3 years x 2 = 6 periods $20,000 x 1.4185 = $28,370 Add extra amount for year 5) +30,000 Original deposit: $20,000 $58,370 $58,370 x 1.4185 = $82,797.85 Lookup table factor Total amount of deposit at the beginning of year 5.

  24. Nominal and Effective Rates (APY) of Interest Nominal Rate (Stated Rate) - The rate on which the bank calculates interest. Truth in Savings LawAnnual Percentage Yield Annual Percentage Yield Formula Effective Rate = Interest for 1 year (APY) Principal

  25. Calculating Effective Rate APY Blue, 8% compounded quarterly Periods = 4 (4 x 1) Percent = 8% = 2% 4 Principal = $8,000 Table 12.1 lookup: 4 periods, 2% 1.0824 x $8,000 Less $8,659.20 $8,000.00 659.20 APY 659.20 = .0824 $8,000 = 8.24% It has a greater APY when the frequency of compounding increases. Sun, 8% compounded semiannually Periods = 2 (2 x 1) Percent = 8% = 4% 2 Principal = $8,000 Table 12.1 lookup: 2 periods, 4% 1.0816 x $8,000 Less $8,652.80 $8,000.00 652.80 APY 652.80 = .0816 $8,000 = 8.16%

  26. Figure 12.3 - Nominal and Effective Rates (APY) of Interest Compared Beginning Nominal rate Compounding End Effective rate balance of interest period balance (APY) of interest Annual Semiannual Quarterly Daily $1,060.00 $1,060.90 $1,061.40 $1,061.80 6.00 6.09% 6.14% 6.18% $1,000 + 6% Daily compounding provides the highest effective rate (APY) of interest.

  27. Table 12.2 - Compounding Interest Daily

  28. Compounding Interest Daily Calculate what $2,000 compounded daily for 7 years will grow to at 6% Use “Compounding Interest Daily Table N = 7 R = 6% Factor 1.5219 $2,000 x 1.5219 = $3,043.80

  29. Figure 12.4 Present Value of $1 at 8% for Four Periods Present value goes from the future value to the present value Future Value $1.0000 $.9259 Present value $.8573 $.7938 $.7350 Number of periods

  30. Calculating Present Value by Table Lookup Step 4. Multiply the table factor by the future value. This is the present value. Step 3. Go down the period column of the table to the number desired; look across the row to find the rate. At the intersection is the table factor. Step 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year Step 1. Find the periods: Years multiplied by number of times interest is compounded in 1 year

  31. Table 12.3 - Present Value of $1 at End Period

  32. Comparing Compound Interest (FV) Table 12.1 with Present Value (PV) Table 12.3 Compound value Table 12.1 Present value Table 12.3 Table Present FutureTable Future Present 12.1Value Value12.3 Value Value 1.3605 x $80 = $108.840.7350 x $108.84 = $80.00 (N = 4, R = 8) (N = 4, R = 8) We know the present dollar amount and find what the dollar amount is worth in the future We know the future dollar amount and find what the dollar amount is worth in the present

  33. Problem 12-25: 12% 2 = 6% Solution: Compounding 5 years x 2 = 10 periods Present value 10 periods 6% Future value: $15,000 Calculate present value $15,000 x 0.55 = $8,376 Present value OR Calculate future value: $10,000 x 1.7908 = $17,908 Yes.

  34. Calculating Present Value Amount by Table Lookup Steve Smith needs $108.84 in 4 years. His bank offers 8% interest compounded annually. How much money must Steve put in the bank today (present) to reach his goal in 4 years? N = 4 x 1 = 4 R = 8% = 8% 1 Table Factor = 0.7350 Compounded Amount: $108.84 x 0.7350 = $80.00 Invest Today

  35. Problem 12-27: 6% 2 = 3% Find present value of a future amount (cost of college tuition): Solution: 8 years x 2 = 16 periods Present value of $6,000 is: $6,000 x 0.6232 = $3,739.20 Lookup table factor

  36. Homework

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