13.4. Continuous Interest http://www.ck12.org
FV=PV( 1 +i)t= 100 ( 1 + 0. 12 )^1 = 112
For twice per year,k= 2 :
FV=PV( 1 +i)t= 100 ( 1 +^0. 212 )^2 = 112. 36
For twelve times per year,k= 12 :
FV=PV( 1 +i)t= 100 ( 1 +^012.^12 )^12 ≈ 112. 68
At this point Carol might notice that while she more than doubled the number of compounding periods, she did not
more than double the extra pennies. The growth slows down and approaches the continuously compounded growth
result.
For continuously compounding interest :
FV=PV·ert= 100 ·e^0.^12 ·^1 ≈ 112. 75
No matter how small Clever Carol might convince her bank to compound the 12%, the most she can earn is around
12.75 in interest.
Vocabulary
A continuously compounding interest rateis the rate of growth proportional to the amount of money in the
account at every instantaneous moment in time. It is equivalent to infinitely many but infinitely small compounding
periods.
Guided Practice
- What is the future value of $500 invested for 8 years at a continuously compounding rate of 9%?
- What is the continuously compounding rate which grows $27 into $99 in just 4 years?
- What amount invested at 3% continuously compounding yields $9,000,000 after 200 years?
Answers:
1.FV= 500 e^8 ·^0.^09 ≈ 1027. 22 - 99= 27 e^4 r
Solving forryields:r= 0. 3248 = 32 .48% - 9, 000 , 000 =PVe^200 ·^0.^03
Solving forPVyields:PV=$22, 308. 77
Practice
For problems 1-10, find the missing value in each row using the continuously compounding interest formula.
TABLE13.6:
Problem Number PV FV t r- $1,000 7 1.5%
- $1,575 $2,250 5
- $4,500 $5,500 3%
- $10,000 12 2%
- $1,670 $3,490 10