2.3 continuous compound interests
TRANSCRIPT
Continuous Compound Interest
We have the PINA formula for the return of periodic compound interest from the last section.
Continuous Compound Interest
P = principal,i = periodic rate, N = total number of periods A = accumulated valuethen P(1 + i )N = A
We have the PINA formula for the return of periodic compound interest from the last section. Let
Continuous Compound Interest
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?
We have the PINA formula for the return of periodic compound interest from the last section. Let
Continuous Compound Interest
P = principal,i = periodic rate, N = total number of periods A = accumulated valuethen P(1 + i )N = A
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,
We have the PINA formula for the return of periodic compound interest from the last section. Let
Continuous Compound Interest
P = principal,i = periodic rate, N = total number of periods A = accumulated valuethen P(1 + i )N = A
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?
40.08
P = 1000, yearly rate is 0.08, so i = = 0.02,
We have the PINA formula for the return of periodic compound interest from the last section. Let
Continuous Compound Interest
P = principal,i = periodic rate, N = total number of periods A = accumulated valuethen P(1 + i )N = A
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?
40.08
P = 1000, yearly rate is 0.08, so i = = 0.02, in 20 years,N = (20 years)(4 times per years) = 80 periods
We have the PINA formula for the return of periodic compound interest from the last section. Let
Continuous Compound Interest
P = principal,i = periodic rate, N = total number of periods A = accumulated valuethen P(1 + i )N = A
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?
40.08
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 $
We have the PINA formula for the return of periodic compound interest from the last section. Let
Continuous Compound Interest
P = principal,i = periodic rate, N = total number of periods A = accumulated valuethen P(1 + i )N = A
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?
40.08
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 $
We have the PINA formula for the return of periodic compound interest from the last section. Let
What happen if we keep everything the same but compound more often, that is, increase K, the number of periods?
Continuous Compound Interest
P = principal,i = periodic rate, N = total number of periods A = accumulated valuethen P(1 + i )N = A
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
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
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, 1000.08 i = = 0.0008,
Continuous Compound Interest
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, 1000.08 i = = 0.0008,
N = (20 years)(100 times per years) = 2000
Continuous Compound Interest
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, 1000.08 i = = 0.0008,
N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000
Continuous Compound Interest
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, 1000.08 i = = 0.0008,
N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 $
Continuous Compound Interest
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, 1000.08 i = = 0.0008,
N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 $
For 1000 times a year, 10000.08 i = = 0.00008,
Continuous Compound Interest
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, 1000.08 i = = 0.0008,
N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 $
For 1000 times a year, 10000.08 i = = 0.00008,
N = (20 years)(1000 times per years) = 20000
Continuous Compound Interest
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, 1000.08 i = = 0.0008,
N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 $
For 1000 times a year, 10000.08 i = = 0.00008,
N = (20 years)(1000 times per years) = 20000Hence A = 1000(1 + 0.00008 )20000
Continuous Compound Interest
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, 1000.08 i = = 0.0008,
N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 $
For 1000 times a year, 10000.08 i = = 0.00008,
N = (20 years)(1000 times per years) = 20000Hence A = 1000(1 + 0.00008 )20000 4952.72 $
Continuous Compound Interest
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, 1000.08 i = = 0.0008,
N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 $
For 1000 times a year, 10000.08 i = = 0.00008,
N = (20 years)(1000 times per years) = 20000Hence A = 1000(1 + 0.00008 )20000 4952.72 $
For 10000 times a year, 100000.08 i = = 0.000008,
Continuous Compound Interest
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, 1000.08 i = = 0.0008,
N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 $
For 1000 times a year, 10000.08 i = = 0.00008,
N = (20 years)(1000 times per years) = 20000Hence A = 1000(1 + 0.00008 )20000 4952.72 $
For 10000 times a year, 100000.08 i = = 0.000008,
N = (20 years)(10000 times per years) = 200000
Continuous Compound Interest
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, 1000.08 i = = 0.0008,
N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 $
For 1000 times a year, 10000.08 i = = 0.00008,
N = (20 years)(1000 times per years) = 20000Hence A = 1000(1 + 0.00008 )20000 4952.72 $
For 10000 times a year, 100000.08 i = = 0.000008,
N = (20 years)(10000 times per years) = 200000Hence A = 1000(1 + 0.000008 )200000
Continuous Compound Interest
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, 1000.08 i = = 0.0008,
N = (20 years)(100 times per years) = 2000Hence A = 1000(1 + 0.0008 )2000 4949.87 $
For 1000 times a year, 10000.08 i = = 0.00008,
N = (20 years)(1000 times per years) = 20000Hence A = 1000(1 + 0.00008 )20000 4952.72 $
For 10000 times a year, 100000.08 i = = 0.000008,
N = (20 years)(10000 times per years) = 200000Hence A = 1000(1 + 0.000008 )200000 4953.00 $
Continuous Compound Interest
We list the results below as the number compounded per yearK gets larger and larger.
Continuous Compound Interest
We list the results below as the number compounded per yearK gets larger and larger.
4 times a year 4875.44 $
Continuous Compound Interest
We list the results below as the number compounded per yearK gets larger and larger.
100 times a year 4949.87 $ 4 times a year 4875.44 $
Continuous Compound Interest
We list the results below as the number compounded per yearK 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
We list the results below as the number compounded per yearK 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
We list the results below as the number compounded per yearK 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
We list the results below as the number compounded per yearK 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
We list the results below as the number compounded per yearK 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
We list the results below as the number compounded per yearK 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
We list the results below as the number compounded per yearK 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 becausethe formula for computing continously compound return is easy to manipulate mathematically.
Continuous Compound Interest
The Perta-Formula for Continuously Compounded ReturnContinuous Compound Interest
The Perta-Formula for Continuously Compounded ReturnLet P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828..
Continuous Compound Interest
Example C. a. We deposited $1000 in an account compounded continuously.
The Perta-Formula for Continuously Compounded ReturnLet P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828..
a. if r = 8%, how much will be there after 20 years?
b. If r = 12%, how much will be there after 20 years?
c. If r = 16%, how much will be there after 20 years?
Continuous Compound Interest
Example C. a. We deposited $1000 in an account compounded continuously.
The Perta-Formula for Continuously Compounded ReturnLet P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828..
a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20.
b. If r = 12%, how much will be there after 20 years?
c. If r = 16%, how much will be there after 20 years?
Continuous Compound Interest
Example C. a. We deposited $1000 in an account compounded continuously.
The Perta-Formula for Continuously Compounded ReturnLet P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A 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
b. If r = 12%, how much will be there after 20 years?
c. If r = 16%, how much will be there after 20 years?
Continuous Compound Interest
Example C. a. We deposited $1000 in an account compounded continuously.
The Perta-Formula for Continuously Compounded ReturnLet P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828..
a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6 4953.03$
b. If r = 12%, how much will be there after 20 years?
c. If r = 16%, how much will be there after 20 years?
Continuous Compound Interest
Example C. a. We deposited $1000 in an account compounded continuously.
The Perta-Formula for Continuously Compounded ReturnLet P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828..
a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6 4953.03$
b. If r = 12%, how much will be there after 20 years?r = 12%, A = 1000*e0.12*20
c. If r = 16%, how much will be there after 20 years?
Continuous Compound Interest
Example C. a. We deposited $1000 in an account compounded continuously.
The Perta-Formula for Continuously Compounded ReturnLet P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828..
a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6 4953.03$
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?
Continuous Compound Interest
Example C. a. We deposited $1000 in an account compounded continuously.
The Perta-Formula for Continuously Compounded ReturnLet P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828..
a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6 4953.03$
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
Continuous Compound Interest
Example C. a. We deposited $1000 in an account compounded continuously.
The Perta-Formula for Continuously Compounded ReturnLet P = principal r = annual interest rate (compound continuously) t = number of year A = accumulated value, then Per*t = A where e 2.71828..
a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20. So the continuously compounded return is A = 1000*e0.08*20 = 1000*e1.6 4953.03$
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
Continuous Compound InterestAbout the Number e
Just as the number π, the number e 2.71828… occupies a special place in mathematics.
Continuous Compound InterestAbout the Number e
Just as the number π, the number e 2.71828… occupies a special place in mathematics. Where as π 3.14156… is a geometric constant–the ratio of the circumference to the diameter of a circle, e is derived from calculations.
Continuous Compound InterestAbout the Number e
Just as the number π, the number e 2.71828… occupies a special place in mathematics. Where as π 3.14156… is a geometric constant–the ratio of the circumference to the diameter of a circle, e is derived from calculations. For example, the following sequence of numbers zoom–in on the number,
( )1,2 1 … ( )4,
5 4
( )3,4 3
( )2,3 2 2.71828…
Continuous Compound InterestAbout the Number e
Just as the number π, the number e 2.71828… occupies a special place in mathematics. Where as π 3.14156… is a geometric constant–the ratio of the circumference to the diameter of a circle, e is derived from calculations. For example, the following sequence of numbers zoom–in on the number,
( 2.71828…)the same as
( )1,2 1 … ( )4,
5 4
( )3,4 3
( )2,3 2 2.71828…which is
Continuous Compound InterestAbout the Number e
Just as the number π, the number e 2.71828… occupies a special place in mathematics. Where as π 3.14156… is a geometric constant–the ratio of the circumference to the diameter of a circle, e is derived from calculations. For example, the following sequence of numbers zoom–in on the number,
( 2.71828…)the same as
( )1,2 1 … ( )4,
5 4
( )3,4 3
( )2,3 2 2.71828…which is
Continuous Compound InterestAbout the Number e
Just as the number π, the number e 2.71828… occupies a special place in mathematics. Where as π 3.14156… is a geometric constant–the ratio of the circumference to the diameter of a circle, e is derived from calculations. For example, the following sequence of numbers zoom–in on the number,
This number emerges often in the calculation of problems in physical science, natural science, finance and in mathematics.
( 2.71828…)the same as
( )1,2 1 … ( )4,
5 4
( )3,4 3
( )2,3 2 2.71828…which is
Continuous Compound InterestAbout the Number e
Just as the number π, the number e 2.71828… occupies a special place in mathematics. Where as π 3.14156… is a geometric constant–the ratio of the circumference to the diameter of a circle, e is derived from calculations. For example, the following sequence of numbers zoom–in on the number,
http://en.wikipedia.org/wiki/E_%28mathematical_constant%29
This number emerges often in the calculation of problems in physical 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.
( 2.71828…)the same as
http://www.ndt-ed.org/EducationResources/Math/Math-e.htm
( )1,2 1 … ( )4,
5 4
( )3,4 3
( )2,3 2 2.71828…which is
Continuous Compound InterestAbout the Number e