63 continuous compound interest

Continuous Compound Interest

In the last section, we gave the formula for the return of periodic compound interest.


P = principal,i = periodic rate, N = total number of periods A = accumulated valuethen A = P(1 + i )N

In the last section, we gave the formula for the return of periodic compound interest. Let


Example A. We deposited $1000 in an account with annual compound interest rate r = 8%, compounded 4 times a year. How much will be there after 20 years?





P = 1000, yearly rate is 0.08,





40.08

P = 1000, yearly rate is 0.08, so i = = 0.02,





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





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 $





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 $



What happens if we keep everything the same but compound more often, that is, increase K, the number of periods?


Example B. We deposited $1000 in an account with annual compound interest rate r = 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,



P = 1000, r = 0.08, T = 20,

For 100 times a year, 1000.08 i = = 0.0008,



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



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



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 $



P = 1000, r = 0.08, T = 20,

For 100 times a year, 1000.08 i = = 0.0008,


For 1000 times a year, 10000.08 i = = 0.00008,



P = 1000, r = 0.08, T = 20,

For 100 times a year, 1000.08 i = = 0.0008,


For 1000 times a year, 10000.08 i = = 0.00008,




P = 1000, r = 0.08, T = 20,

For 100 times a year, 1000.08 i = = 0.0008,


For 1000 times a year, 10000.08 i = = 0.00008,




P = 1000, r = 0.08, T = 20,

For 100 times a year, 1000.08 i = = 0.0008,


For 1000 times a year, 10000.08 i = = 0.00008,




P = 1000, r = 0.08, T = 20,

For 100 times a year, 1000.08 i = = 0.0008,


For 1000 times a year, 10000.08 i = = 0.00008,


For 10000 times a year, 100000.08 i = = 0.000008,



P = 1000, r = 0.08, T = 20,

For 100 times a year, 1000.08 i = = 0.0008,


For 1000 times a year, 10000.08 i = = 0.00008,


For 10000 times a year, 100000.08 i = = 0.000008,




P = 1000, r = 0.08, T = 20,

For 100 times a year, 1000.08 i = = 0.0008,


For 1000 times a year, 10000.08 i = = 0.00008,


For 10000 times a year, 100000.08 i = = 0.000008,



Example B. We deposited $1000 in an account with annual compound interest fvHow 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,


For 1000 times a year, 10000.08 i = = 0.00008,


For 10000 times a year, 100000.08 i = = 0.000008,



We 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 $ 4 times a year 4875.44 $



10000 times a year 4953.00 $

1000 times a year 4952.72 $




10000 times a year 4953.00 $

1000 times a year 4952.72 $




10000 times a year 4953.00 $

1000 times a year 4952.72 $




10000 times a year 4953.00 $

1000 times a year 4952.72 $


4953.03 $



10000 times a year 4953.00 $

1000 times a year 4952.72 $


4953.03 $

We call this amount the continuously compounded return.



10000 times a year 4953.00 $

1000 times a year 4952.72 $


4953.03 $

We call this amount the continuously compounded return.This way of compounding is called compounded continuously.



10000 times a year 4953.00 $

1000 times a year 4952.72 $


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.


Formula for Continuously Compounded Return (Perta)Continuous Compound Interest

Continuous Compound InterestFormula for Continuously Compounded Return (Perta)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 where e 2.71828…


There is no “f” because it’s 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 where e 2.71828…

Example C. We deposited $1000 in an account compounded continuously.

a. if r = 8%, how much will be there after 20 years?

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…


a. if r = 8%, how much will be there after 20 years? P = 1000, r = 0.08, t = 20.



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



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



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?




b. If r = 12%, how much will be there after 20 years?r = 12%, A = 1000*e0.12*20




b. If r = 12%, how much will be there after 20 years?r = 12%, A = 1000*e0.12*20 = 1000e 2.4




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?





c. If r = 16%, how much will be there after 20 years?r = 16%, A = 1000*e0.16*20





c. If r = 16%, how much will be there after 20 years?r = 16%, A = 1000*e0.16*20 = 1000*e 3.2





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 InterestAbout the Number e

Just as the number π, the number e 2.71828… occupies a special place in mathematics.


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.


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…



( 2.71828…)the same as

( )1,2 1 … ( )4,

5 4

( )3,4 3

( )2,3 2 2.71828…which is




( )1,2 1 … ( )4,

5 4

( )3,4 3

( )2,3 2 2.71828…which is



This number emerges often in the calculation of problems in physical science, natural science, finance and in mathematics.


( )1,2 1 … ( )4,

5 4

( )3,4 3

( )2,3 2 2.71828…which is



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.


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


With a fixed interest rate r, utilizing the Prffta–formula, we conclude that the more often we compound, the higher the return would be.


Compounded return on $1,000 with annual interest rate r = 20% (Wikipedia)

With a fixed interest rate r, utilizing the Prffta–formula, we conclude that the more often we compound, the higher the return would be. However the continuously compounded returnsets the “ceiling” or the “limit” as how much the returns could be regardless how often we compound, as shown here.


Compounded return on $1,000 with annual interest rate r = 20% (Wikipedia)

With a fixed interest rate r, utilizing the Prffta–formula, we conclude that the more often we compound, the higher the return would be. However the continuously compounded returnsets the “ceiling” or the “limit” as how much the returns could be regardless how often we compound, as shown here. We may think of the continuous –compound as compounding with infinite frequency hence yielding more return than all other methods.


Continuous Compound InterestGrowth and Decay

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.

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 = Perx

with r > 0 as “ an exponential growths with growth rate r”.


y = e1x has the growth rate of r = 1 or 100%.

Growth 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 = Perx

with r > 0 as “ an exponential growths with growth rate r”. For example,


y = e1x has the growth rate of r = 1 or 100%. Exponential growths are rapid expansions compared to other expansion– processes as shown here.

y = x3y = 100x

y = ex



An Exponential Growth



y = x3y = 100x

y = ex

The world population may be modeled with an exponential growth with r ≈ 1.1 % or 0.011 as of 2011.






y = x3y = 100x

y = ex

The world population may be modeled with an exponential growth with r ≈ 1.1 % or 0.011 as of 2011. However, this rate is dropping but it’s unclear how fast this growth rate is shrinking.






y = x3y = 100x

y = ex

The world population may be modeled with an exponential growth with r ≈ 1.1 % or 0.011 as of 2011. However, this rate is dropping but it’s unclear how fast this growth rate is shrinking. For more information:(http://en.wikipedia.org/wiki/World_population)




Continuous Compound InterestIf the rate r is negative, or that r < 0 then the return A = Perx grows smaller as time x gets larger.

Continuous Compound InterestIf the rate r is negative, or that r < 0 then the return A = Perx grows smaller as time x gets larger. We call a contraction that may be modeled A = Perx

with r < 0 as “an exponential decay at the rate | r |”.


with r < 0 as “an exponential decay at the rate | r |”. For example,y = e–1x has the decay or contraction rate of r = 1 or 100%.

y = e–x An Exponential Decay


with r < 0 as “an exponential decay at the rate | r |”. For example,y = e–1x has the decay or contraction rate of r = 1 or 100%. In finance, shrinking values is called “depreciation” or”devaluation”.



with r < 0 as “an exponential decay at the rate | r |”. For example,y = e–1x has the decay or contraction rate of r = 1 or 100%. In finance, shrinking values is called “depreciation” or”devaluation”. For example, a currency that is depreciating at a rate of 4% annually may be modeled by A = Pe –0.04x

where x is the number of years elapsed.




where x is the number of years elapsed. Hence if P = $1, after 5 years, its purchasing power is 1*e–0.04(5) = $0.82 or 82 cents.




where x is the number of years elapsed. Hence if P = $1, after 5 years, its purchasing power is 1*e–0.04(5) = $0.82 or 82 cents. For more information:


http://math.ucsd.edu/~wgarner/math4c/textbook/chapter4/expgrowthdecay.htm

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63 continuous compound interest