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4.2 exponential functions and compound interests

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4.2 exponential functions and compound interests

1. 1. The Exponential Functions http://www.lahc.edu/math/precalculus/math_260a.html
2. 2. The Exponential Functions http://www.lahc.edu/math/precalculus/math_260a.html
3. 3. The Exponential Functions The meaning positive integral exponents such as x2 is clear.
4. 4. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N  b 1 The Exponential Functions K N The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
5. 5. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N  b 1 Example A. 80 = 8 = The Exponential Functions K N 3 2 3 2 8 –2 = 8 = The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
6. 6. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N  b 1 Example A. 80 = 1 8 = The Exponential Functions K N 3 2 3 2 8 –2 = 8 = The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
7. 7. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N  b 1 Example A. 80 = 1 8 = 82 1 64 1 The Exponential Functions K N 3 2 3 2 8 –2 = = 8 = The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
8. 8. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N  b 1 Example A. 80 = 1 8 = (  8 ) = 4 3 2 82 1 64 1 The Exponential Functions K N 3 2 3 2 8 –2 = = 8 = The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
9. 9. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N  b 1 Example A. 80 = 1 8 = (  8 ) = 4 3 2 82 1 3 2  8 1 64 1 The Exponential Functions K N 3 2 3 2 8 –2 = = 8 = ( ) = 1/4 The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
10. 10. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N  b 1 Example A. 80 = 1 8 = (  8 ) = 4 3 2 82 1 3 2  8 1 64 1 The Exponential Functions K N 3 2 3 2 Decimal exponents are well defined since decimals may be represented as reduced fractions. 8 –2 = = 8 = ( ) = 1/4 The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
11. 11. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N  b 1 Example A. 80 = 1 8 = (  8 ) = 4 3 2 82 1 3 2  8 1 64 1 The Exponential Functions K N 3 2 3 2 Decimal exponents are well defined since decimals may be represented as reduced fractions. b. 101.22 = 8 –2 = = 8 = ( ) = 1/4 Example B. a. 91.50 = The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
12. 12. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N  b 1 Example A. 80 = 1 8 = (  8 ) = 4 3 2 82 1 3 2  8 1 64 1 The Exponential Functions K N 3 2 3 2 Decimal exponents are well defined since decimals may be represented as reduced fractions. b. 101.22 = 8 –2 = = 8 = ( ) = 1/4 Example B. a. 91.50 = 9 = 3 2 The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
13. 13. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N  b 1 Example A. 80 = 1 8 = (  8 ) = 4 3 2 82 1 3 2  8 1 64 1 The Exponential Functions K N 3 2 3 2 Decimal exponents are well defined since decimals may be represented as reduced fractions. b. 101.22 = 8 –2 = = 8 = ( ) = 1/4 Example B. a. 91.50 = 9 = (9 ) = 27 3 2 3 The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
14. 14. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N  b 1 Example A. 80 = 1 8 = (  8 ) = 4 3 2 82 1 3 2  8 1 64 1 The Exponential Functions K N 3 2 3 2 Decimal exponents are well defined since decimals may be represented as reduced fractions. b. 101.22 = 10 61 50 8 –2 = = 8 = ( ) = 1/4 Example B. a. 91.50 = 9 = (9 ) = 27 3 2 3 The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
15. 15. b0 = 1 b–K = b = ( b ) b = ( ) K N K N bK 1 K N  b 1 Example A. 80 = 1 8 = (  8 ) = 4 3 2 82 1 3 2  8 1 64 1 The Exponential Functions K N 3 2 3 2 Decimal exponents are well defined since decimals may be represented as reduced fractions. b. 101.22 = 10 = ( 10 )  16.59586…. 61 50 50 61 8 –2 = = 8 = ( ) = 1/4 Example B. a. 91.50 = 9 = (9 ) = 27 3 2 3 The meaning positive integral exponents such as x2 is clear. Below are the rules for other special exponents:
16. 16. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result.   3.14159.. 10  Example C. The Exponential Functions
17. 17. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result.   3.14159.. 3.1 3.14 3.141 3.1415 10  Example C.  The Exponential Functions
18. 18. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result.   3.14159.. 3.1 3.14 3.141 3.1415 10  10 Example C. 31 10  The Exponential Functions ≈1258.9..
19. 19. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result.   3.14159.. 3.1 3.14 3.141 3.1415 10  10 10 Example C. 31 10 314 100  The Exponential Functions ≈1258.9.. ≈1380.3..
20. 20. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result.   3.14159.. 3.1 3.14 3.141 3.1415 10  10 10 10 10 Example C. 31 10 314 100 3141 1000 31415 10000  The Exponential Functions ≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
21. 21. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result.   3.14159.. 3.1 3.14 3.141 3.1415 10  10 10 10 10 10≈1385.45.. Example C. 31 10 314 100 3141 1000 31415 10000  The Exponential Functions ≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
22. 22. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result.   3.14159.. 3.1 3.14 3.141 3.1415 10  10 10 10 10 10≈1385.45.. Example C. 31 10 314 100 3141 1000 31415 10000  The Exponential Functions Hence exponential functions or functions of the form f(x) = bx (b > 0 and b  1) are defined for all real numbers x. ≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
23. 23. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result.   3.14159.. 3.1 3.14 3.141 3.1415 10  10 10 10 10 10≈1385.45.. Example C. 31 10 314 100 3141 1000 31415 10000  The Exponential Functions Hence exponential functions or functions of the form f(x) = bx (b > 0 and b  1) are defined for all real numbers x. ≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1.. Exponential functions show up in finance, bio science, computer science and physical sciences.
24. 24. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result.   3.14159.. 3.1 3.14 3.141 3.1415 10  10 10 10 10 10≈1385.45.. Example C. 31 10 314 100 3141 1000 31415 10000  The Exponential Functions Hence exponential functions or functions of the form f(x) = bx (b > 0 and b  1) are defined for all real numbers x. ≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1.. Exponential functions show up in finance, bio science, computer science and physical sciences. The most used exponential functions are y = 10x, y = ex and y = 2x.
25. 25. For a real-number-exponent such as , we approximate the real number with fractions then use the fractional powers to approximate the result.   3.14159.. 3.1 3.14 3.141 3.1415 10  10 10 10 10 10≈1385.45.. Example C. 31 10 314 100 3141 1000 31415 10000  The Exponential Functions Hence exponential functions or functions of the form f(x) = bx (b > 0 and b  1) are defined for all real numbers x. ≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1.. Exponential functions show up in finance, bio science, computer science and physical sciences. The most used exponential functions are y = 10x, y = ex and y = 2x. Let’s use \$ growth as applications below.
26. 26. Example D. We deposit \$1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? Compound Interest
27. 27. Example D. We deposit \$1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i)
28. 28. Example D. We deposit \$1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = \$1010. Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i)
29. 29. Example D. We deposit \$1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = \$1010. After 2 months: 1010(1 + 0.01) Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i)
30. 30. Example D. We deposit \$1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = \$1010. After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01) Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i)
31. 31. Example D. We deposit \$1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = \$1010. After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01) = 1000(1 + 0.01)2 = \$1020.10 Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i)
32. 32. Example D. We deposit \$1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = \$1010. After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01) = 1000(1 + 0.01)2 = \$1020.10 Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i) after 2 periods A = P(1 + i)(1 + i) = P(1 + i)2
33. 33. Example D. We deposit \$1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = \$1010. After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01) = 1000(1 + 0.01)2 = \$1020.10 After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01) = 1000(1 + 0.01)3 = \$1030.30 Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i) after 2 periods A = P(1 + i)(1 + i) = P(1 + i)2
34. 34. Example D. We deposit \$1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = \$1010. After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01) = 1000(1 + 0.01)2 = \$1020.10 After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01) = 1000(1 + 0.01)3 = \$1030.30 Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i) after 2 periods A = P(1 + i)(1 + i) = P(1 + i)2 after 3 periods A = P(1 + i)2(1 + i) = P(1 + i)3
35. 35. Example D. We deposit \$1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = \$1010. After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01) = 1000(1 + 0.01)2 = \$1020.10 After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01) = 1000(1 + 0.01)3 = \$1030.30 After 4 months: 1030(1 + 0.01) = 1000(1 + 0.01)3(1 + 0.01) = 1000(1 + 0.01)4 = \$1040.60 Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i) after 2 periods A = P(1 + i)(1 + i) = P(1 + i)2 after 3 periods A = P(1 + i)2(1 + i) = P(1 + i)3
36. 36. Example D. We deposit \$1,000 in an account that gives 1% interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months? After 1 month: 1000(1 + 0.01) = \$1010. After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01) = 1000(1 + 0.01)2 = \$1020.10 After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01) = 1000(1 + 0.01)3 = \$1030.30 After 4 months: 1030(1 + 0.01) = 1000(1 + 0.01)3(1 + 0.01) = 1000(1 + 0.01)4 = \$1040.60 Compound Interest Let P = principal, i = (periodic) interest rate, A = accumulation after 1 period A = P(1 + i) after 2 periods A = P(1 + i)(1 + i) = P(1 + i)2 after 3 periods A = P(1 + i)2(1 + i) = P(1 + i)3 Continue the pattern, after N periods, we obtain the exponential periodic-compound formula (PINA): P(1 + i)N = A.
37. 37. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation The PINA Formula (Periodic Interest)
38. 38. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest)
39. 39. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. 0 1 2 3 Nth periodN–1
40. 40. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward
41. 41. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward P(1 + i)
42. 42. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward P(1 + i) P(1 + i) 2
43. 43. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward P(1 + i) P(1 + i) 2 P(1 + i) 3
44. 44. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1
45. 45. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N
46. 46. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N Example E. \$1,000 is in an account that has a monthly interest rate of 1%. How much will be there after 60 years?
47. 47. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N Example E. \$1,000 is in an account that has a monthly interest rate of 1%. How much will be there after 60 years? We have P = \$1,000, i = 1% = 0.01, N =
48. 48. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N Example E. \$1,000 is in an account that has a monthly interest rate of 1%. How much will be there after 60 years? We have P = \$1,000, i = 1% = 0.01, N = 60 *12 = 720 months
49. 49. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N Example E. \$1,000 is in an account that has a monthly interest rate of 1%. How much will be there after 60 years? We have P = \$1,000, i = 1% = 0.01, N = 60 *12 = 720 months so by PINA, there will be 1000(1 + 0.01) 720
50. 50. Compound Interest Let P = principal i = (periodic) interest rate, N = number of periods A = accumulation then P(1 + i) N = A The PINA Formula (Periodic Interest) We use the following time line to see what is happening. P 0 1 2 3 Nth periodN–1 Rule: Multiply (1 + i) each period forward P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N Example E. \$1,000 is in an account that has a monthly interest rate of 1%. How much will be there after 60 years? We have P = \$1,000, i = 1% = 0.01, N = 60 *12 = 720 months so by PINA, there will be 1000(1 + 0.01) 720 = \$1,292,376.71 after 60 years.
51. 51. Compound Interest In practice, compound interests are usually quoted in annual interest rate r and the frequency f, the number of times of compounding in one year, so the periodic rate i = . r f
52. 52. Example F. We open an account with annual rate r = 9%, compounded monthly, i.e. 12 times a year. Compound Interest In practice, compound interests are usually quoted in annual interest rate r and the frequency f, the number of times of compounding in one year, so the periodic rate i = . r f
53. 53. Example F. We open an account with annual rate r = 9%, compounded monthly, i.e. 12 times a year. We have r = 9% = 0.09 for one year, and f = 12 is the numbers of times of compounding in one year, Compound Interest In practice, compound interests are usually quoted in annual interest rate r and the frequency f, the number of times of compounding in one year, so the periodic rate i = . r f
54. 54. Example F. We open an account with annual rate r = 9%, compounded monthly, i.e. 12 times a year. We have r = 9% = 0.09 for one year, and f = 12 is the numbers of times of compounding in one year, so the periodic or monthly rate i = 0.09/12. Compound Interest In practice, compound interests are usually quoted in annual interest rate r and the frequency f, the number of times of compounding in one year, so the periodic rate i = . r f
55. 55. Example F. We open an account with annual rate r = 9%, compounded monthly, i.e. 12 times a year. After 40 years the total return is \$250,000, what was the initial principal? We have r = 9% = 0.09 for one year, and f = 12 is the numbers of times of compounding in one year, so the periodic or monthly rate i = 0.09/12. Compound Interest In practice, compound interests are usually quoted in annual interest rate r and the frequency f, the number of times of compounding in one year, so the periodic rate i = . r f
56. 56. Example F. We open an account with annual rate r = 9%, compounded monthly, i.e. 12 times a year. After 40 years the total return is \$250,000, what was the initial principal? We have r = 9% = 0.09 for one year, and f = 12 is the numbers of times of compounding in one year, so the periodic or monthly rate i = 0.09/12. After 40 years, i.e. N = 40(12) = 480 months the return A = 250,000 Compound Interest In practice, compound interests are usually quoted in annual interest rate r and the frequency f, the number of times of compounding in one year, so the periodic rate i = . r f
57. 57. Example F. We open an account with annual rate r = 9%, compounded monthly, i.e. 12 times a year. After 40 years the total return is \$250,000, what was the initial principal? We have r = 9% = 0.09 for one year, and f = 12 is the numbers of times of compounding in one year, so the periodic or monthly rate i = 0.09/12. After 40 years, i.e. N = 40(12) = 480 months the return A = 250,000, so by PINA: Compound Interest P (1 + ) 480 = 250,0000.09 12 In practice, compound interests are usually quoted in annual interest rate r and the frequency f, the number of times of compounding in one year, so the periodic rate i = . r f
58. 58. Example F. We open an account with annual rate r = 9%, compounded monthly, i.e. 12 times a year. After 40 years the total return is \$250,000, what was the initial principal? We have r = 9% = 0.09 for one year, and f = 12 is the numbers of times of compounding in one year, so the periodic or monthly rate i = 0.09/12. After 40 years, i.e. N = 40(12) = 480 months the return A = 250,000, so by PINA: Compound Interest P (1 + ) 480 = 250,0000.09 12 or (1 + ) 480 P = 250,000 0.09 12 In practice, compound interests are usually quoted in annual interest rate r and the frequency f, the number of times of compounding in one year, so the periodic rate i = . r f
59. 59. Example F. We open an account with annual rate r = 9%, compounded monthly, i.e. 12 times a year. After 40 years the total return is \$250,000, what was the initial principal? We have r = 9% = 0.09 for one year, and f = 12 is the numbers of times of compounding in one year, so the periodic or monthly rate i = 0.09/12. After 40 years, i.e. N = 40(12) = 480 months the return A = 250,000, so by PINA: Compound Interest P (1 + ) 480 = 250,0000.09 12 or (1 + ) 480 P = 250,000 0.09 12 P = \$6,923.31 by calculator Hence the initial deposit in \$6,923.31. In practice, compound interests are usually quoted in annual interest rate r and the frequency f, the number of times of compounding in one year, so the periodic rate i = . r f
60. 60. The graphs shown here are the different returns with r = 20% with different compounding frequency. Compounded return on \$1,000 with annual interest rate r = 20% (Wikipedia) Compound Interest
61. 61. The graphs shown here are the different returns with r = 20% with different compounding frequency. We observe that I. the more frequently we compound, the bigger the return Compounded return on \$1,000 with annual interest rate r = 20% (Wikipedia) Compound Interest
62. 62. The graphs shown here are the different returns with r = 20% with different compounding frequency. We observe that I. the more frequently we compound, the bigger the return II. but the returns do not go above the blue-line the continuous compound return, which is the next topic. Compounded return on \$1,000 with annual interest rate r = 20% (Wikipedia) Compound Interest