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# 2.3 continuous compound interests

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### 2.3 continuous compound interests

1. 1. Continuous Compound Interest
2. 2. In the last section, we have the PINA-formula for the return of periodic compound interest. Continuous Compound Interest
3. 3. P = principal, i = periodic rate, N = total number of periods A = accumulated value then P(1 + i )N = A In the last section, we have the PINA-formula for the return of periodic compound interest. Let Continuous Compound Interest
4. 4. Example A. We deposited \$1000 in an account with annual compound interest rater = 8%, compounded 4 times a year. How much will be there after 20 years? P = principal, i = periodic rate, N = total number of periods A = accumulated value then P(1 + i )N = A In the last section, we have the PINA-formula for the return of periodic compound interest. Let Continuous Compound Interest
5. 5. Example A. We deposited \$1000 in an account with annual compound interest rater = 8%, compounded 4 times a year. How much will be there after 20 years? P = 1000, yearly rate is 0.08, P = principal, i = periodic rate, N = total number of periods A = accumulated value then P(1 + i )N = A In the last section, we have the PINA-formula for the return of periodic compound interest. Let Continuous Compound Interest
6. 6. Example A. We deposited \$1000 in an account with annual compound interest rater = 8%, compounded 4 times a year. How much will be there after 20 years? 4 0.08 P = 1000, yearly rate is 0.08, so i = = 0.02, P = principal, i = periodic rate, N = total number of periods A = accumulated value then P(1 + i )N = A In the last section, we have the PINA-formula for the return of periodic compound interest. Let Continuous Compound Interest
7. 7. Example A. We deposited \$1000 in an account with annual compound interest rater = 8%, compounded 4 times a year. How much will be there after 20 years? 4 0.08 P = 1000, yearly rate is 0.08, so i = = 0.02, in 20 years, N = (20 years)(4 times per years) = 80 periods P = principal, i = periodic rate, N = total number of periods A = accumulated value then P(1 + i )N = A In the last section, we have the PINA-formula for the return of periodic compound interest. Let Continuous Compound Interest
8. 8. Example A. We deposited \$1000 in an account with annual compound interest rater = 8%, compounded 4 times a year. How much will be there after 20 years? 4 0.08 P = 1000, yearly rate is 0.08, so i = = 0.02, in 20 years, N = (20 years)(4 times per years) = 80 periods Hence A = 1000(1 + 0.02 )80  4875.44 \$ P = principal, i = periodic rate, N = total number of periods A = accumulated value then P(1 + i )N = A In the last section, we have the PINA-formula for the return of periodic compound interest. Let Continuous Compound Interest
9. 9. Example A. We deposited \$1000 in an account with annual compound interest rater = 8%, compounded 4 times a year. How much will be there after 20 years? 4 0.08 P = 1000, yearly rate is 0.08, so i = = 0.02, in 20 years, N = (20 years)(4 times per years) = 80 periods Hence A = 1000(1 + 0.02 )80  4875.44 \$ P = principal, i = periodic rate, N = total number of periods A = accumulated value then P(1 + i )N = A In the last section, we have the PINA-formula for the return of periodic compound interest. Let What happen if we keep everything the same but compound more often, that is, increase N, the number of periods? Continuous Compound Interest
10. 10. Example B. We deposited \$1000 in an account with annual compound interest rater = 8%. How much will be there after 20 years if it's compounded 100 times a year? 1000 times a year? 10000 times a year? Continuous Compound Interest
11. 11. Example B. We deposited \$1000 in an account with annual compound interest rater = 8%. How much will be there after 20 years if it's compounded 100 times a year? 1000 times a year? 10000 times a year? P = 1000, r = 0.08, T = 20, Continuous Compound Interest
12. 12. Example B. We deposited \$1000 in an account with annual compound interest rater = 8%. How much will be there after 20 years if it's 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, 100 0.08 i = = 0.0008, Continuous Compound Interest
13. 13. Example B. We deposited \$1000 in an account with annual compound interest rater = 8%. How much will be there after 20 years if it's 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, 100 0.08 i = = 0.0008, N = (20 years)(100 times per years) = 2000 Continuous Compound Interest
14. 14. Example B. We deposited \$1000 in an account with annual compound interest rater = 8%. How much will be there after 20 years if it's 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, 100 0.08 i = = 0.0008, N = (20 years)(100 times per years) = 2000 Hence A = 1000(1 + 0.0008 )2000 Continuous Compound Interest
15. 15. Example B. We deposited \$1000 in an account with annual compound interest rater = 8%. How much will be there after 20 years if it's 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, 100 0.08 i = = 0.0008, N = (20 years)(100 times per years) = 2000 Hence A = 1000(1 + 0.0008 )2000  4949.87 \$ Continuous Compound Interest
16. 16. Example B. We deposited \$1000 in an account with annual compound interest rater = 8%. How much will be there after 20 years if it's 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, 100 0.08 i = = 0.0008, N = (20 years)(100 times per years) = 2000 Hence A = 1000(1 + 0.0008 )2000  4949.87 \$ For 1000 times a year, 1000 0.08 i = = 0.00008, Continuous Compound Interest
17. 17. Example B. We deposited \$1000 in an account with annual compound interest rater = 8%. How much will be there after 20 years if it's 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, 100 0.08 i = = 0.0008, N = (20 years)(100 times per years) = 2000 Hence A = 1000(1 + 0.0008 )2000  4949.87 \$ For 1000 times a year, 1000 0.08 i = = 0.00008, N = (20 years)(1000 times per years) = 20000 Continuous Compound Interest
18. 18. Example B. We deposited \$1000 in an account with annual compound interest rater = 8%. How much will be there after 20 years if it's 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, 100 0.08 i = = 0.0008, N = (20 years)(100 times per years) = 2000 Hence A = 1000(1 + 0.0008 )2000  4949.87 \$ For 1000 times a year, 1000 0.08 i = = 0.00008, N = (20 years)(1000 times per years) = 20000 Hence A = 1000(1 + 0.00008 )20000 Continuous Compound Interest
19. 19. Example B. We deposited \$1000 in an account with annual compound interest rater = 8%. How much will be there after 20 years if it's 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, 100 0.08 i = = 0.0008, N = (20 years)(100 times per years) = 2000 Hence A = 1000(1 + 0.0008 )2000  4949.87 \$ For 1000 times a year, 1000 0.08 i = = 0.00008, N = (20 years)(1000 times per years) = 20000 Hence A = 1000(1 + 0.00008 )20000  4952.72 \$ Continuous Compound Interest
20. 20. Example B. We deposited \$1000 in an account with annual compound interest rater = 8%. How much will be there after 20 years if it's 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, 100 0.08 i = = 0.0008, N = (20 years)(100 times per years) = 2000 Hence A = 1000(1 + 0.0008 )2000  4949.87 \$ For 1000 times a year, 1000 0.08 i = = 0.00008, N = (20 years)(1000 times per years) = 20000 Hence A = 1000(1 + 0.00008 )20000  4952.72 \$ For 10000 times a year, 10000 0.08 i = = 0.000008, Continuous Compound Interest
21. 21. Example B. We deposited \$1000 in an account with annual compound interest rater = 8%. How much will be there after 20 years if it's 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, 100 0.08 i = = 0.0008, N = (20 years)(100 times per years) = 2000 Hence A = 1000(1 + 0.0008 )2000  4949.87 \$ For 1000 times a year, 1000 0.08 i = = 0.00008, N = (20 years)(1000 times per years) = 20000 Hence A = 1000(1 + 0.00008 )20000  4952.72 \$ For 10000 times a year, 10000 0.08 i = = 0.000008, N = (20 years)(10000 times per years) = 200000 Continuous Compound Interest
22. 22. Example B. We deposited \$1000 in an account with annual compound interest rater = 8%. How much will be there after 20 years if it's 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, 100 0.08 i = = 0.0008, N = (20 years)(100 times per years) = 2000 Hence A = 1000(1 + 0.0008 )2000  4949.87 \$ For 1000 times a year, 1000 0.08 i = = 0.00008, N = (20 years)(1000 times per years) = 20000 Hence A = 1000(1 + 0.00008 )20000  4952.72 \$ For 10000 times a year, 10000 0.08 i = = 0.000008, N = (20 years)(10000 times per years) = 200000 Hence A = 1000(1 + 0.000008 )200000 Continuous Compound Interest
23. 23. Example B. We deposited \$1000 in an account with annual compound interest rater = 8%. How much will be there after 20 years if it's 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, 100 0.08 i = = 0.0008, N = (20 years)(100 times per years) = 2000 Hence A = 1000(1 + 0.0008 )2000  4949.87 \$ For 1000 times a year, 1000 0.08 i = = 0.00008, N = (20 years)(1000 times per years) = 20000 Hence A = 1000(1 + 0.00008 )20000  4952.72 \$ For 10000 times a year, 10000 0.08 i = = 0.000008, N = (20 years)(10000 times per years) = 200000 Hence A = 1000(1 + 0.000008 )200000  4953.00 \$ Continuous Compound Interest
24. 24. We list the results below as the number compounded per year K gets larger and larger. Continuous Compound Interest
25. 25. We list the results below as the number compounded per year K gets larger and larger. 4 times a year 4875.44 \$ Continuous Compound Interest
26. 26. We list the results below as the number compounded per year K gets larger and larger. 100 times a year 4949.87 \$ 4 times a year 4875.44 \$ Continuous Compound Interest
27. 27. We list the results below as the number compounded per year K gets larger and larger. 10000 times a year 4953.00 \$ 1000 times a year 4952.72 \$ 100 times a year 4949.87 \$ 4 times a year 4875.44 \$ Continuous Compound Interest
28. 28. We list the results below as the number compounded per year K gets larger and larger. 10000 times a year 4953.00 \$ 1000 times a year 4952.72 \$ 100 times a year 4949.87 \$ 4 times a year 4875.44 \$ Continuous Compound Interest
29. 29. We list the results below as the number compounded per year K gets larger and larger. 10000 times a year 4953.00 \$ 1000 times a year 4952.72 \$ 100 times a year 4949.87 \$ 4 times a year 4875.44 \$  Continuous Compound Interest
30. 30. We list the results below as the number compounded per year K gets larger and larger. 10000 times a year 4953.00 \$ 1000 times a year 4952.72 \$ 100 times a year 4949.87 \$ 4 times a year 4875.44 \$  4953.03 \$ Continuous Compound Interest
31. 31. We list the results below as the number compounded per year K gets larger and larger. 10000 times a year 4953.00 \$ 1000 times a year 4952.72 \$ 100 times a year 4949.87 \$ 4 times a year 4875.44 \$  4953.03 \$ We call this amount the continuously compounded return. Continuous Compound Interest
32. 32. We list the results below as the number compounded per year K gets larger and larger. 10000 times a year 4953.00 \$ 1000 times a year 4952.72 \$ 100 times a year 4949.87 \$ 4 times a year 4875.44 \$  4953.03 \$ We call this amount the continuously compounded return. This way of compounding is called compounded continuously. Continuous Compound Interest
33. 33. We list the results below as the number compounded per year K gets larger and larger. 10000 times a year 4953.00 \$ 1000 times a year 4952.72 \$ 100 times a year 4949.87 \$ 4 times a year 4875.44 \$  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 because the formula for computing continously compound return is easy to manipulate mathematically. Continuous Compound Interest
34. 34. Continuous Compound Interest
35. 35. Continuous Compound Interest Formula for Continuously Compounded Return (PERTA)
36. 36. Continuous Compound Interest Formula for Continuously Compounded Return (PERTA) Let P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value,
37. 37. Continuous Compound Interest Formula 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 (PERTA) where e  2.71828..
38. 38. Example C. a. We deposited \$1000 in an account compounded continuously. a. if r = 8%, how much will be there after 20 years? Continuous Compound Interest Formula 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 (PERTA) where e  2.71828..
39. 39. Example C. a. We deposited \$1000 in an account compounded continuously. a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. Continuous Compound Interest Formula 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 (PERTA) where e  2.71828..
40. 40. Example C. a. We deposited \$1000 in an account compounded continuously. 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 Continuous Compound Interest Formula 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 (PERTA) where e  2.71828..
41. 41. Example C. a. We deposited \$1000 in an account compounded continuously. 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 Continuous Compound Interest Formula 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 (PERTA) where e  2.71828..
42. 42. Example C. a. We deposited \$1000 in an account compounded continuously. 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\$ Continuous Compound Interest Formula 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 (PERTA) where e  2.71828..
43. 43. Example C. a. We deposited \$1000 in an account compounded continuously. 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? Continuous Compound Interest Formula 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 (PERTA) where e  2.71828..
44. 44. Example C. a. We deposited \$1000 in an account compounded continuously. 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 Continuous Compound Interest Formula 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 (PERTA) where e  2.71828..
45. 45. Example C. a. We deposited \$1000 in an account compounded continuously. 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 Continuous Compound Interest Formula 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 (PERTA) where e  2.71828..
46. 46. Example C. a. We deposited \$1000 in an account compounded continuously. 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\$ Continuous Compound Interest Formula 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 (PERTA) where e  2.71828..
47. 47. Example C. a. We deposited \$1000 in an account compounded continuously. 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? Continuous Compound Interest Formula 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 (PERTA) where e  2.71828..
48. 48. Example C. a. We deposited \$1000 in an account compounded continuously. 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 Continuous Compound Interest Formula 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 (PERTA) where e  2.71828..
49. 49. Example C. a. We deposited \$1000 in an account compounded continuously. 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 Continuous Compound Interest Formula 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 (PERTA) where e  2.71828..
50. 50. Example C. a. We deposited \$1000 in an account compounded continuously. Formula 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 (PERTA) where e  2.71828.. 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\$ Continuous Compound Interest
51. 51. The Exponential Functions Find the variable given the following terms of continuous compound interest accounts Pert = A (Perta) HW A 1. P = \$1,000, r = 1%, t = 30 P = principal, r = continuous interest annual rate, t = the number of years, A = accumulated (future) value. Find A in the following problems. 2. P = \$1,000, r = ½ %, t = 10 3. P = \$1,000, r = 2%, t = 10 4. P = \$1,000,r = 3/4%, t = 30 5. P = \$1,000, r = 1%, t = 30 6. P = \$1,000, r = 2%, t = 40 7. P = \$1,000, r = 3%, t = 40 8. P = \$1,000, r = 4%, t = 100
52. 52. The Exponential Functions HW B 1. A = \$1,000, r = 1%, t = 30 P = principal, r = continuous interest annual rate , t = the number of years, A = accumulated (future) value. Find P in the following problems. 2. A = \$1,000, r = ½ %, t = 10 3. A = \$1,000, r = 2%, t = 10 4. A = \$1,000, r = 3/4%, t = 30 5. A = \$1,000, r = 1%, t = 30 6. A = \$1,000, r = 2%, t = 40 7. A = \$1,000, r = 3%, t = 40 8. A = \$1,000, r = 4%, t = 100 Find the variable given the following terms of continuous compound interest accounts Pert = A (Perta)