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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 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

  5. 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

  6. 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

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

  8. 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 x 1 = 3 Semiannually: 3 x 2 = 6 Quarterly: 3 x 4 = 12

  9. 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

  10. 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

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

  12. 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

  13. 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 Effective Rate = Interest for 1 year (APY) Principal

  14. 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% 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%

  15. 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%

  16. Table 12.2 - Compounding Interest Daily

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

  18. 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

  19. 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

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

  21. 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 = .7350 Compounded Amount: $108.84 x .7350 = $80.00 Invest Today

  22. 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 Future Table Future Present 12.1 Value Value 12.3 Value Value 1.3605 x $80 = $108.84 .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

  23. Problem 12-13: 4% 2 = 2% Solution: 7 years x 2 = 14 periods $25,000.00 x 1.3195 = $32,987.50

  24. Problem 12-15: 8% 4 10% 2 = 2% = 5% Solution: 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

  25. Problem 12-16: 12% 2 = 6% Solution: 3 years x 2 = 6 periods $20,000 x 1.4185 = $28,370 +30,000 $58,370 $58,370 x 1.4185 = $82,797.85

  26. Problem 12-25: 12% 2 = 6% Solution: Compounding 5 years x 2 = 10 periods $10,000 x 1.7908 = $17,908 Present value 10 periods $15,000 x .55 6% OR Yes.

  27. Problem 12-27: 6% 2 = 3% Solution: 8 years x 2 = 16 periods $6,000 x .6232 = $3,739.20

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