# reverse interest calculator

I need an on-lline calculator to give me the effective yearly interest earned. I will enter initial dollar amount, dollar amount earned, and term in months. As in
\$1200, earned \$350, and term was 22 months.

Thanks,
P

I couldn’t find one online but if you have a scientific calculator, there’s a fairly easy way to do it yourself.

A is the Amount you have now (in your example, \$1550)
P is your starting Principle (\$1200)
e is simply e, which is about 2.7 but you won’t have to worry about that.
r is the interest rate (what you want to know)
t is the time (22 months)

1. Take the natural log of each side.
lnA = lnPe[sup]rt[/sup]

2. Multiplying two numbers is equivalent to adding their logs, so:
lnA = lnP + lne[sup]rt[/sup]

3. The natural log of e[sup]x[/sup] is just x (trust me), so:
lnA = lnP + rt (remember I said you wouldn’t need to worry about e?)

4. Now just calculate the natural logs (on a scientific calculator it’s the ‘ln’ button)
lnA = 7.34601
lnP = 7.09007

and find rt

## 7.34601 - 7.09007

0.25594 = rt

Since t = 22 months, divide this number by 22 to get the monthly interest (1.16%) or by (22/12) to get the annual interest (13.96%)

Basically, just take the natural log of the current amount and the starting principle, subtract them, and the answer will be equal to the interest rate multiplied by the time.

Answer: it depends on how you want to consider the interest compounded. Credit cards compound daily (well, daily based on average daily balance), for example, which does correspond to an APR in terms of DPR*365.

If you want to consider it only compounded yearly, or continuously, the answer is different. Here is a general formula for anything but continuously compounded interest:

P = principle
F = final total
I = 1+R (that is, 1 + PercentageRate/100)
t = time compounded

I[sup]t[/sup]P = F

I’m assuming you want to solve for R = I - 1. Derivation follows:
I[sup]t[/sup]P = F
I[sup]t[/sup] = F/P
t ln(I) = ln(F/P)
ln(I) = [ln(F/P)]/t
I = e[sup][ln(F/P)]/t[/sup]
R = e[sup][ln(F/P)]/t[/sup] - 1

Which I whipped up in an excel spreadsheet.

Compounded monthly, your monthly interest rate over 22 months was
0.011701266 or about 1.17%, corresponding to an APR of 14.042%.

Compunded yearly: 0.149814099, or about 14.981% APR.

Compounded daily: 0.000382539, or about 13.963% APR.

Here’s a pretty good compound interest calculator:
http://www.1728.com/compint.htm

You would be solving for RATE

The principal is \$1200
The years = 1.83333 (22 months)
The Total is 1550 (1200 + 350)

The answer is 14.98 % (if compounded yearly.

If you wanted to convert this to other compounding methods go to:
http://www.1728.com/yieldint.htm

Enter 14.98, click “calculate” scroll to section 2 and you will see this equals

14.457454988 Semi-Annually
14.205219656 Quarterly
14.040303859 Monthly
13.961470914 Daily
13.958801421 Continuously

The monthly and daily amounts agree with erislover’s calculations.

Thanks all of you. This one is the best. It works without taking my scentific cacu out of the drawer

Paul
Gee thanks.
By the way, that’s my website and I’m glad you like that calculator.