SOLUTION: This is a question given on my assignment and I have no idea where to go with it. Given information: A loan company offers money at 1.8% per month, compounded monthly and the nomin

Question 227831: This is a question given on my assignment and I have no idea where to go with it. Given information: A loan company offers money at 1.8% per month, compounded monthly and the nominal rate is 21.6%. How many years will it take an investment to triple itself if the nominal interest is compounded continuously?
Found 2 solutions by nerdybill, Theo:
Answer by nerdybill(7384) (Show Source): You can put this solution on YOUR website!
A loan company offers money at 1.8% per month, compounded monthly and the nominal rate is 21.6%. How many years will it take an investment to triple itself if the nominal interest is compounded continuously?
.
Use the "continuous compound interest" formula:
A = Pe^(rt)
.
Let P = our initial investment
then
3P = Pe^(.018*12*t)
3 = e^(.018*12*t)
ln 3 = .018*12*t
ln 3/(.018*12) = t
5.086 years = t

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
Loan company offers 1.8% per month.

i = interest rate per time period = .018

Nominal rate is 21.6% per year / 12 = 1.8% which is the monthly rate.

What this is telling you is that the annual interest rate is 21.6% per year and
if you divide that by 12 you get the monthly interest rate.

You want to know how long it will take if the nominal interest rate is compounded continuously.

Continuous compounding is another formula that assumes compounding is at very small intervals, like perhaps every second, or every milli-second, or even every microsecond.

Continuous compounding is based on calculus and takes the limit of the compounding periods as they approach zero. It's more often than compounding per month, or per day, or per minute, or per second, or ........

The following hyperlink should help you to understand.

continuous compounding formula with example

The formula is:

A = Pe^rt

where:

A = Future Value of the Investment.
P = Present Amount to be invested.
e = 2.718281828... which is a scientific constant identified by the letter e.
r = annual, or nominal, interest rate.
t = number of years.

In your problem:

A = 3 if your initial investment is 1.
P = 1
e = 2.718281828...
r = 21.6% = .216
t = x

You are trying to find x which is the amount of time it will take for the investment to triple.

If you calculate x in years, then the formula stands as stated.

Your formula becomes:

3 = 1*e^(.216*x)

Take the log of both sides of this equation to get:

log(3) = log(e^(.216*x) = .216*x*log(e)

Divide both sides of this equation by .216*log(e), and you get:

log(3)/(.216*log)e) = x

Solve using your calculator to get:

x = 5.086168003

Plug this value of x into your original equation to get:

3 = 1*e^(.216*5.086168003) = 3

The calculation is good.

Do the same equation using interest rate of 1.8% per month and you should get the same answer because the interest rate is being compounded continuously which is as fine as you can get.

Your original equation becomes:

3 = 1*e^(.018*y)

I used y to represent number of months to distinguish it from x which represented number of years.

Take the log of both sides of this equation to get:

log(3) = log(e^(.018*y) = .018*y*log(e)

Divide both sides of this equation by .018*log(e), and you get:

log(3)/(.018*log)e) = y

Solve using your calculator to get:

y = 61.03401604

That equals the number of months it will take for your money to triple.

Compare that to the value of x derived above and you will see that they are equal.

x = 5.086168003 = number of years to triple your money.

y = 61.03401604 = number of months to triple your money.

Divide y by 12 and you get 5.086168003 which is the same as x.

Compare this to:

21.6% compounded yearly equals 5.6175812122 years.

21.6% / 12 = 1.8% compounded monthly equals 61.58168893 months / 12 equals 5.121807411 years.

21.6% / 365 = .059178082% compounded daily equals 1,857.000573 days / 365 = 5.087672803 years.

21.6% / 8,760 = .0002465753% compounded hourly equals 44,555.38101 hours / 8,760 = 5.086230709 years.

We're getting close to the continuous compounding rate each time the number of compounding periods gets greater.

I could compound every minute and every second and would get even closer but I'll stop here because I think you probably have the idea by now.

If you look at the formula:

A = Pe^rt

If you divide r by the same amount that you multiply t by, the exponent the exponent of e stays the same which means you will always get the same answer.

.216 * 5.086168003 = 1.098612289

.018 * 61.03401604 = 1.098612289

That's why your answer was the same whether we used interest rate per month or interest rate per year.

Answer to your question is:

It will take 5.086168003 years to triple your money.

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