Interest of 12% per year compounded monthly is roughly equivalent to an interest rate of 12.68% per year compunded yearly when using the formula: A=p(1+r/n)^nt
If you solve the problem the two are equal; how can you derive 12.68% compounded yearly from 12% per year compounded monthly?
Let the rate compounded yearly be i
then (1+i)^1 = (1+.12/12)^12
1+i = 1.01^12
1+i = 1.126825
i = .1268..
so the annual rate is 12.68%
Thank you very much, that was most helpful.
To derive the equivalent interest rate of 12.68% compounded yearly from 12% compounded monthly, we can use the concept of Effective Annual Rate (EAR). The EAR is the actual interest rate earned or paid over a year when compounding occurs more frequently than once a year.
To begin, let's break down the given information:
Annual interest rate (APR) = 12% = 0.12
Compounding frequency per year (n) = 12 (monthly compounding)
Time period in years (t) = 1
Now, let's substitute these values into the formula for compound interest:
A = P(1 + r/n)^(nt)
For the monthly compounding case, we have:
A = P(1 + 0.12/12)^(12*1)
A = P(1.01)^12
By solving this equation, we find the multiplication factor of (1.01)^12 is approximately 1.12682503.
So, if you invest $1 at 12% per year compounded monthly, after one year, it will grow to approximately $1.12682503.
To find the equivalent interest rate compounded yearly, we need to make the growth factor equal to (1 + r), where r represents the annual interest rate.
(1 + r) = 1.12682503
r = 1.12682503 - 1
By calculating the value of (1.12682503 - 1), we find that r is approximately 0.1268 or 12.68%.
Therefore, an interest rate of 12% per year compounded monthly is roughly equivalent to an interest rate of 12.68% per year compounded yearly.