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# 63 continuous compound interest

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### 63 continuous compound interest

1. 1. Continuous Compound Interest
2. 2. Continuous Compound InterestIn the last section, we gave the formula for the return ofperiodic compound interest.
3. 3. Continuous Compound InterestIn the last section, we gave the formula for the return ofperiodic compound interest. LetP = principal,i = periodic rate,N = total number of periodsA = accumulated valuethen A = P(1 + i )N
4. 4. Continuous Compound InterestIn the last section, we gave the formula for the return ofperiodic compound interest. LetP = principal,i = periodic rate,N = total number of periodsA = accumulated valuethen A = P(1 + i )NExample A. We deposited \$1000 in an account with annualcompound interest rate r = 8%, compounded 4 times a year.How much will be there after 20 years?
5. 5. Continuous Compound InterestIn the last section, we gave the formula for the return ofperiodic compound interest. LetP = principal,i = periodic rate,N = total number of periodsA = accumulated valuethen A = P(1 + i )NExample A. We deposited \$1000 in an account with annualcompound interest rate r = 8%, compounded 4 times a year.How much will be there after 20 years?P = 1000, yearly rate is 0.08,
6. 6. Continuous Compound InterestIn the last section, we gave the formula for the return ofperiodic compound interest. LetP = principal,i = periodic rate,N = total number of periodsA = accumulated valuethen A = P(1 + i )NExample A. We deposited \$1000 in an account with annualcompound interest rate r = 8%, compounded 4 times a year.How much will be there after 20 years? 0.08P = 1000, yearly rate is 0.08, so i = 4 = 0.02,
7. 7. Continuous Compound InterestIn the last section, we gave the formula for the return ofperiodic compound interest. LetP = principal,i = periodic rate,N = total number of periodsA = accumulated valuethen A = P(1 + i )NExample A. We deposited \$1000 in an account with annualcompound interest rate r = 8%, compounded 4 times a year.How much will be there after 20 years? 0.08P = 1000, yearly rate is 0.08, so i = 4 = 0.02, in 20 years,N = (20 years)(4 times per years) = 80 periods
8. 8. Continuous Compound InterestIn the last section, we gave the formula for the return ofperiodic compound interest. LetP = principal,i = periodic rate,N = total number of periodsA = accumulated valuethen A = P(1 + i )NExample A. We deposited \$1000 in an account with annualcompound interest rate r = 8%, compounded 4 times a year.How much will be there after 20 years? 0.08P = 1000, yearly rate is 0.08, so i = 4 = 0.02, in 20 years,N = (20 years)(4 times per years) = 80 periodsHence A = 1000(1 + 0.02 )80 4875.44 \$
9. 9. Continuous Compound InterestIn the last section, we gave the formula for the return ofperiodic compound interest. LetP = principal,i = periodic rate,N = total number of periodsA = accumulated valuethen A = P(1 + i )NExample A. We deposited \$1000 in an account with annualcompound interest rate r = 8%, compounded 4 times a year.How much will be there after 20 years? 0.08P = 1000, yearly rate is 0.08, so i = 4 = 0.02, in 20 years,N = (20 years)(4 times per years) = 80 periodsHence A = 1000(1 + 0.02 )80 4875.44 \$What happens if we keep everything the same but compoundmore often, that is, increase K, the number of periods?
10. 10. Continuous Compound InterestExample B. We deposited \$1000 in an account with annualcompound interest rate r = 8%. How much will be there after20 years if its compounded 100 times a year? 1000 times ayear? 10000 times a year?
11. 11. Continuous Compound InterestExample B. We deposited \$1000 in an account with annualcompound interest rate r = 8%. How much will be there after20 years if its compounded 100 times a year? 1000 times ayear? 10000 times a year?P = 1000, r = 0.08, T = 20,
12. 12. Continuous Compound InterestExample B. We deposited \$1000 in an account with annualcompound interest rate r = 8%. How much will be there after20 years if its compounded 100 times a year? 1000 times ayear? 10000 times a year?P = 1000, r = 0.08, T = 20,For 100 times a year, i = 0.08 = 0.0008, 100
13. 13. Continuous Compound InterestExample B. We deposited \$1000 in an account with annualcompound interest rate r = 8%. How much will be there after20 years if its compounded 100 times a year? 1000 times ayear? 10000 times a year?P = 1000, r = 0.08, T = 20,For 100 times a year, i = 0.08 = 0.0008, 100N = (20 years)(100 times per years) = 2000
14. 14. Continuous Compound InterestExample B. We deposited \$1000 in an account with annualcompound interest rate r = 8%. How much will be there after20 years if its compounded 100 times a year? 1000 times ayear? 10000 times a year?P = 1000, r = 0.08, T = 20,For 100 times a year, i = 0.08 = 0.0008, 100N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000
15. 15. Continuous Compound InterestExample B. We deposited \$1000 in an account with annualcompound interest rate r = 8%. How much will be there after20 years if its compounded 100 times a year? 1000 times ayear? 10000 times a year?P = 1000, r = 0.08, T = 20,For 100 times a year, i = 0.08 = 0.0008, 100N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 \$
16. 16. Continuous Compound InterestExample B. We deposited \$1000 in an account with annualcompound interest rate r = 8%. How much will be there after20 years if its compounded 100 times a year? 1000 times ayear? 10000 times a year?P = 1000, r = 0.08, T = 20,For 100 times a year, i = 0.08 = 0.0008, 100N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 \$ 0.08For 1000 times a year, i = 1000 = 0.00008,
17. 17. Continuous Compound InterestExample B. We deposited \$1000 in an account with annualcompound interest rate r = 8%. How much will be there after20 years if its compounded 100 times a year? 1000 times ayear? 10000 times a year?P = 1000, r = 0.08, T = 20,For 100 times a year, i = 0.08 = 0.0008, 100N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 \$ 0.08For 1000 times a year, i = 1000 = 0.00008,N = (20 years)(1000 times per years) = 20000
18. 18. Continuous Compound InterestExample B. We deposited \$1000 in an account with annualcompound interest rate r = 8%. How much will be there after20 years if its compounded 100 times a year? 1000 times ayear? 10000 times a year?P = 1000, r = 0.08, T = 20,For 100 times a year, i = 0.08 = 0.0008, 100N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 \$ 0.08For 1000 times a year, i = 1000 = 0.00008,N = (20 years)(1000 times per years) = 20000Hence A = 1000(1 + 0.00008 )20000
19. 19. Continuous Compound InterestExample B. We deposited \$1000 in an account with annualcompound interest rate r = 8%. How much will be there after20 years if its compounded 100 times a year? 1000 times ayear? 10000 times a year?P = 1000, r = 0.08, T = 20,For 100 times a year, i = 0.08 = 0.0008, 100N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 \$ 0.08For 1000 times a year, i = 1000 = 0.00008,N = (20 years)(1000 times per years) = 20000Hence A = 1000(1 + 0.00008 )20000 4952.72 \$
20. 20. Continuous Compound InterestExample B. We deposited \$1000 in an account with annualcompound interest rate r = 8%. How much will be there after20 years if its compounded 100 times a year? 1000 times ayear? 10000 times a year?P = 1000, r = 0.08, T = 20,For 100 times a year, i = 0.08 = 0.0008, 100N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 \$ 0.08For 1000 times a year, i = 1000 = 0.00008,N = (20 years)(1000 times per years) = 20000Hence A = 1000(1 + 0.00008 )20000 4952.72 \$ 0.08For 10000 times a year, i = 10000= 0.000008,
21. 21. Continuous Compound InterestExample B. We deposited \$1000 in an account with annualcompound interest rate r = 8%. How much will be there after20 years if its compounded 100 times a year? 1000 times ayear? 10000 times a year?P = 1000, r = 0.08, T = 20,For 100 times a year, i = 0.08 = 0.0008, 100N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 \$ 0.08For 1000 times a year, i = 1000 = 0.00008,N = (20 years)(1000 times per years) = 20000Hence A = 1000(1 + 0.00008 )20000 4952.72 \$ 0.08For 10000 times a year, i = 10000= 0.000008,N = (20 years)(10000 times per years) = 200000
22. 22. Continuous Compound InterestExample B. We deposited \$1000 in an account with annualcompound interest rate r = 8%. How much will be there after20 years if its compounded 100 times a year? 1000 times ayear? 10000 times a year?P = 1000, r = 0.08, T = 20,For 100 times a year, i = 0.08 = 0.0008, 100N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 \$ 0.08For 1000 times a year, i = 1000 = 0.00008,N = (20 years)(1000 times per years) = 20000Hence A = 1000(1 + 0.00008 )20000 4952.72 \$ 0.08For 10000 times a year, i = 10000= 0.000008,N = (20 years)(10000 times per years) = 200000Hence A = 1000(1 + 0.000008 )200000
23. 23. Continuous Compound InterestExample B. We deposited \$1000 in an account with annualcompound interest fvHow much will be there after 20 years ifits compounded 100 times a year? 1000 times a year?10000 times a year?P = 1000, r = 0.08, T = 20,For 100 times a year, i = 0.08 = 0.0008, 100N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 \$ 0.08For 1000 times a year, i = 1000 = 0.00008,N = (20 years)(1000 times per years) = 20000Hence A = 1000(1 + 0.00008 )20000 4952.72 \$ 0.08For 10000 times a year, i = 10000= 0.000008,N = (20 years)(10000 times per years) = 200000Hence A = 1000(1 + 0.000008 )200000 4953.00 \$
24. 24. Continuous Compound InterestWe list the results below as the number compounded per yearK gets larger and larger.
25. 25. Continuous Compound InterestWe list the results below as the number compounded per yearK gets larger and larger. 4 times a year 4875.44 \$
26. 26. Continuous Compound InterestWe list the results below as the number compounded per yearK gets larger and larger. 4 times a year 4875.44 \$100 times a year 4949.87 \$
27. 27. Continuous Compound InterestWe list the results below as the number compounded per yearK gets larger and larger. 4 times a year 4875.44 \$100 times a year 4949.87 \$1000 times a year 4952.72 \$10000 times a year 4953.00 \$
28. 28. Continuous Compound InterestWe list the results below as the number compounded per yearK gets larger and larger. 4 times a year 4875.44 \$100 times a year 4949.87 \$1000 times a year 4952.72 \$10000 times a year 4953.00 \$
29. 29. Continuous Compound InterestWe list the results below as the number compounded per yearK gets larger and larger. 4 times a year 4875.44 \$100 times a year 4949.87 \$1000 times a year 4952.72 \$10000 times a year 4953.00 \$
30. 30. Continuous Compound InterestWe list the results below as the number compounded per yearK gets larger and larger. 4 times a year 4875.44 \$100 times a year 4949.87 \$1000 times a year 4952.72 \$10000 times a year 4953.00 \$ 4953.03 \$
31. 31. Continuous Compound InterestWe list the results below as the number compounded per yearK gets larger and larger. 4 times a year 4875.44 \$100 times a year 4949.87 \$1000 times a year 4952.72 \$10000 times a year 4953.00 \$ 4953.03 \$We call this amount the continuously compounded return.
32. 32. Continuous Compound InterestWe list the results below as the number compounded per yearK gets larger and larger. 4 times a year 4875.44 \$100 times a year 4949.87 \$1000 times a year 4952.72 \$10000 times a year 4953.00 \$ 4953.03 \$We call this amount the continuously compounded return.This way of compounding is called compounded continuously.
33. 33. Continuous Compound InterestWe list the results below as the number compounded per yearK gets larger and larger. 4 times a year 4875.44 \$100 times a year 4949.87 \$1000 times a year 4952.72 \$10000 times a year 4953.00 \$ 4953.03 \$We call this amount the continuously compounded return.This way of compounding is called compounded continuously.The reason we want to compute interest this way is becausethe formula for computing continously compound return iseasy to manipulate mathematically.
34. 34. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)
35. 35. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828…
36. 36. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828… There is no “f” because it’s compounded continuously
37. 37. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828…Example C. We deposited \$1000 in an account compoundedcontinuously.a. if r = 8%, how much will be there after 20 years?
38. 38. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828…Example C. We deposited \$1000 in an account compoundedcontinuously.a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20.
39. 39. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828…Example C. We deposited \$1000 in an account compoundedcontinuously.a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20
40. 40. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828…Example C. We deposited \$1000 in an account compoundedcontinuously.a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6
41. 41. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828…Example C. We deposited \$1000 in an account compoundedcontinuously.a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6 4953.03\$
42. 42. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828…Example C. We deposited \$1000 in an account compoundedcontinuously.a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6 4953.03\$b. If r = 12%, how much will be there after 20 years?
43. 43. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828…Example C. We deposited \$1000 in an account compoundedcontinuously.a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6 4953.03\$b. If r = 12%, how much will be there after 20 years? r = 12%, A = 1000*e0.12*20
44. 44. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828…Example C. We deposited \$1000 in an account compoundedcontinuously.a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6 4953.03\$b. If r = 12%, how much will be there after 20 years? r = 12%, A = 1000*e0.12*20 = 1000e 2.4
45. 45. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828…Example C. We deposited \$1000 in an account compoundedcontinuously.a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6 4953.03\$b. If r = 12%, how much will be there after 20 years? r = 12%, A = 1000*e0.12*20 = 1000e 2.4 11023.18\$
46. 46. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828…Example C. We deposited \$1000 in an account compoundedcontinuously.a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6 4953.03\$b. If r = 12%, how much will be there after 20 years? r = 12%, A = 1000*e0.12*20 = 1000e 2.4 11023.18\$c. If r = 16%, how much will be there after 20 years?
47. 47. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828…Example C. We deposited \$1000 in an account compoundedcontinuously.a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6 4953.03\$b. If r = 12%, how much will be there after 20 years? r = 12%, A = 1000*e0.12*20 = 1000e 2.4 11023.18\$c. If r = 16%, how much will be there after 20 years? r = 16%, A = 1000*e0.16*20
48. 48. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828…Example C. We deposited \$1000 in an account compoundedcontinuously.a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6 4953.03\$b. If r = 12%, how much will be there after 20 years? r = 12%, A = 1000*e0.12*20 = 1000e 2.4 11023.18\$c. If r = 16%, how much will be there after 20 years? r = 16%, A = 1000*e0.16*20 = 1000*e 3.2
49. 49. Continuous Compound InterestFormula for Continuously Compounded Return (Perta)Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828…Example C. We deposited \$1000 in an account compoundedcontinuously.a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6 4953.03\$b. If r = 12%, how much will be there after 20 years? r = 12%, A = 1000*e0.12*20 = 1000e 2.4 11023.18\$c. If r = 16%, how much will be there after 20 years? r = 16%, A = 1000*e0.16*20 = 1000*e 3.2 24532.53\$
50. 50. Continuous Compound InterestAbout the Number e
51. 51. Continuous Compound InterestAbout the Number eJust as the number π, the number e 2.71828… occupies aspecial place in mathematics.
52. 52. Continuous Compound InterestAbout the Number eJust as the number π, the number e 2.71828… occupies aspecial place in mathematics. Where as π 3.14156… is ageometric constant–the ratio of the circumference to thediameter of a circle, e is derived from calculations.
53. 53. Continuous Compound InterestAbout the Number eJust as the number π, the number e 2.71828… occupies aspecial place in mathematics. Where as π 3.14156… is ageometric constant–the ratio of the circumference to thediameter of a circle, e is derived from calculations.For example, the following sequence of numbers zoom–in onthe number, ( 2 )1, ( 3 )2, ( 4 )3, ( 5 )4, … 1 4 2.71828… 2 3
54. 54. Continuous Compound InterestAbout the Number eJust as the number π, the number e 2.71828… occupies aspecial place in mathematics. Where as π 3.14156… is ageometric constant–the ratio of the circumference to thediameter of a circle, e is derived from calculations.For example, the following sequence of numbers zoom–in onthe number, ( 2 )1, ( 3 )2, ( 4 )3, ( 5 )4, … 1 4 2.71828…which is 2 3the same as ( 2.71828…)
55. 55. Continuous Compound InterestAbout the Number eJust as the number π, the number e 2.71828… occupies aspecial place in mathematics. Where as π 3.14156… is ageometric constant–the ratio of the circumference to thediameter of a circle, e is derived from calculations.For example, the following sequence of numbers zoom–in onthe number, ( 2 )1, ( 3 )2, ( 4 )3, ( 5 )4, … 1 4 2.71828…which is 2 3the same as ( 2.71828…)
56. 56. Continuous Compound InterestAbout the Number eJust as the number π, the number e 2.71828… occupies aspecial place in mathematics. Where as π 3.14156… is ageometric constant–the ratio of the circumference to thediameter of a circle, e is derived from calculations.For example, the following sequence of numbers zoom–in onthe number, ( 2 )1, ( 3 )2, ( 4 )3, ( 5 )4, … 1 4 2.71828…which is 2 3the same as ( 2.71828…)This number emerges often in the calculation of problems inphysical science, natural science, finance and in mathematics.
57. 57. Continuous Compound InterestAbout the Number eJust as the number π, the number e 2.71828… occupies aspecial place in mathematics. Where as π 3.14156… is ageometric constant–the ratio of the circumference to thediameter of a circle, e is derived from calculations.For example, the following sequence of numbers zoom–in onthe number, ( 2 )1, ( 3 )2, ( 4 )3, ( 5 )4, … 1 4 2.71828…which is 2 3the same as ( 2.71828…)This number emerges often in the calculation of problems inphysical science, natural science, finance and in mathematics.Because of its importance, the irrational number 2.71828…is named as “e” and it’s called the “natural” base number.http://www.ndt-ed.org/EducationResources/Math/Math-e.htmhttp://en.wikipedia.org/wiki/E_%28mathematical_constant%29
58. 58. Continuous Compound InterestWith a fixed interest rate r, utilizing the Prffta–formula,we conclude that the more often we compound, the higher thereturn would be.
59. 59. Continuous Compound InterestWith a fixed interest rate r, utilizing the Prffta–formula,we conclude that the more often we compound, the higher thereturn would be. However the continuously compounded returnsets the “ceiling”or the “limit” as howmuch the returnscould be regardlesshow often wecompound, as shownhere. Compounded return on \$1,000 with annual interest rate r = 20% (Wikipedia)
60. 60. Continuous Compound InterestWith a fixed interest rate r, utilizing the Prffta–formula,we conclude that the more often we compound, the higher thereturn would be. However the continuously compounded returnsets the “ceiling”or the “limit” as howmuch the returnscould be regardlesshow often wecompound, as shownhere. We may think ofthe continuous –compound ascompounding withinfinite frequencyhence yielding morereturn than all other Compounded return on \$1,000 with annual interest rate r = 20% (Wikipedia)methods.
61. 61. Continuous Compound InterestGrowth and Decay
62. 62. Continuous Compound InterestGrowth and DecayIn all the interest examples we have the interest rate r is positive,and the return A = Perx grows larger as time x gets larger.
63. 63. Continuous Compound InterestGrowth and DecayIn all the interest examples we have the interest rate r is positive,and the return A = Perx grows larger as time x gets larger.We call an expansion that may be modeled by A = Perxwith r > 0 as “ an exponential growths with growth rate r”.
64. 64. Continuous Compound InterestGrowth and DecayIn all the interest examples we have the interest rate r is positive,and the return A = Perx grows larger as time x gets larger.We call an expansion that may be modeled by A = Perxwith r > 0 as “ an exponential growths with growth rate r”.For example,y = e1x has the growth rate ofr = 1 or 100%.
65. 65. Continuous Compound InterestGrowth and DecayIn all the interest examples we have the interest rate r is positive,and the return A = Perx grows larger as time x gets larger.We call an expansion that may be modeled by A = Perxwith r > 0 as “ an exponential growths with growth rate r”.For example, y=ex An Exponential Growthy = e1x has the growth rate ofr = 1 or 100%. Exponentialgrowths are rapid expansionscompared to other expansion–processes as shown here. y = 100x y = x3
66. 66. Continuous Compound InterestGrowth and DecayIn all the interest examples we have the interest rate r is positive,and the return A = Perx grows larger as time x gets larger.We call an expansion that may be modeled by A = Perxwith r > 0 as “ an exponential growths with growth rate r”.For example, y=ex An Exponential Growthy = e1x has the growth rate ofr = 1 or 100%. Exponentialgrowths are rapid expansionscompared to other expansion–processes as shown here.The world population may be y = 100x y=x 3modeled with an exponentialgrowth with r ≈ 1.1 % or 0.011as of 2011.
67. 67. Continuous Compound InterestGrowth and DecayIn all the interest examples we have the interest rate r is positive,and the return A = Perx grows larger as time x gets larger.We call an expansion that may be modeled by A = Perxwith r > 0 as “ an exponential growths with growth rate r”.For example, y=ex An Exponential Growthy = e1x has the growth rate ofr = 1 or 100%. Exponentialgrowths are rapid expansionscompared to other expansion–processes as shown here.The world population may be y = 100x y=x 3modeled with an exponentialgrowth with r ≈ 1.1 % or 0.011as of 2011. However, this rate is dropping but it’s unclear howfast this growth rate is shrinking.
68. 68. Continuous Compound InterestGrowth and DecayIn all the interest examples we have the interest rate r is positive,and the return A = Perx grows larger as time x gets larger.We call an expansion that may be modeled by A = Perxwith r > 0 as “ an exponential growths with growth rate r”.For example, x y=e An Exponential Growthy = e1x has the growth rate ofr = 1 or 100%. Exponentialgrowths are rapid expansionscompared to other expansion–processes as shown here.The world population may be y = 100x 3 y=xmodeled with an exponentialgrowth with r ≈ 1.1 % or 0.011as of 2011. However, this rate is dropping but it’s unclear howfast this growth rate is shrinking. For more information:(http://en.wikipedia.org/wiki/World_population)
69. 69. Continuous Compound InterestIf the rate r is negative, or that r < 0 then the return A = Perxgrows smaller as time x gets larger.
70. 70. Continuous Compound InterestIf the rate r is negative, or that r < 0 then the return A = Perxgrows smaller as time x gets larger.We call a contraction that may be modeled A = Perxwith r < 0 as “an exponential decay at the rate | r |”.
71. 71. Continuous Compound InterestIf the rate r is negative, or that r < 0 then the return A = Perxgrows smaller as time x gets larger.We call a contraction that may be modeled A = Perxwith r < 0 as “an exponential decay at the rate | r |”.For example, An Exponential Decay –xy = e–1x has the decay or y=econtraction rate of r = 1 or 100%.
72. 72. Continuous Compound InterestIf the rate r is negative, or that r < 0 then the return A = Perxgrows smaller as time x gets larger.We call a contraction that may be modeled A = Perxwith r < 0 as “an exponential decay at the rate | r |”.For example, An Exponential Decay –xy = e–1x has the decay or y=econtraction rate of r = 1 or 100%.In finance, shrinking values iscalled “depreciation” or”devaluation”.
73. 73. Continuous Compound InterestIf the rate r is negative, or that r < 0 then the return A = Perxgrows smaller as time x gets larger.We call a contraction that may be modeled A = Perxwith r < 0 as “an exponential decay at the rate | r |”.For example, An Exponential Decay –xy = e–1x has the decay or y=econtraction rate of r = 1 or 100%.In finance, shrinking values iscalled “depreciation” or”devaluation”. For example,a currency that is depreciatingat a rate of 4% annually may bemodeled by A = Pe –0.04xwhere x is the number of years elapsed.
74. 74. Continuous Compound InterestIf the rate r is negative, or that r < 0 then the return A = Perxgrows smaller as time x gets larger.We call a contraction that may be modeled A = Perxwith r < 0 as “an exponential decay at the rate | r |”.For example, An Exponential Decay –xy = e–1x has the decay or y=econtraction rate of r = 1 or 100%.In finance, shrinking values iscalled “depreciation” or”devaluation”. For example,a currency that is depreciatingat a rate of 4% annually may bemodeled by A = Pe –0.04xwhere x is the number of years elapsed.Hence if P = \$1, after 5 years, its purchasing power is1*e–0.04(5) = \$0.82 or 82 cents.
75. 75. Continuous Compound InterestIf the rate r is negative, or that r < 0 then the return A = Perxgrows smaller as time x gets larger.We call a contraction that may be modeled A = Perxwith r < 0 as “an exponential decay at the rate | r |”.For example, An Exponential Decay –xy = e–1x has the decay or y=econtraction rate of r = 1 or 100%.In finance, shrinking values iscalled “depreciation” or”devaluation”. For example,a currency that is depreciatingat a rate of 4% annually may bemodeled by A = Pe –0.04xwhere x is the number of years elapsed.Hence if P = \$1, after 5 years, its purchasing power is1*e–0.04(5) = \$0.82 or 82 cents. For more information:http://math.ucsd.edu/~wgarner/math4c/textbook/chapter4/expgrowthdecay.htm