For the following question, I would like to know why the method to solve is the following instead of taking into account of the monthly compounding:
Q) If an inestment of $4,000 will grow to $6,520 in four years with monthly compounding, the effective annual interest rate will be:
A) N = 4, PMT = 0, PV = -4,000, FV = 6,520
I = 12.99%
I thought N = 4* 12 months = 48, calculate for i, then multiply by 12.
Thanks in advance!
Your approach gives you a nominal annual rate, not an _ effective _ annual rate.
However, if you use n = 48, calculate i, then compound i for four twelve periods, you’ll get the correct answer (which, not coincidentally, is 12.99%).
If you compound an effective rate, your result will be an effective rate. Always.
You’re welcome.
I understand the nominal (stated rate) vs. effective rate (takes into the compounding), but I am still unsure of the calculation portion.
If I use n = 48, interest = 1.023 or 1.023%. So if I multiply that by 12, then it is 12.27% which you are saying is the nominal rate? I think I understand this part.
From interest = 1.023%, how do I get to 12.99%? Thanks!
Good.
My mistake: I’d written four above; I meant twelve. (I’ve corrected it.)
(1 + 1.023%)¹² – 1 = 12.99%.
(6520/4000)^(1/4) - 1 = 0.129918 (effective annual rate, i.e if compounding only happens once a year)
ok what are you given?
PV, FV after four years, and PMT = 0.
if you plug thses values with N=48, you will get monthly rate with monthly compounding.
which is 1.023 or 0.01023 in decimal. To get effective rate you use below
EAR = (1+ 0.0123)^12 -1
which give 0.1299 or 12.99%
if you don’t want to use above method, you can think of this as below.
If your 4000 PV grew to 6520 in 4 years, at what yearly compounding rate would they have grown? That is after all Effective annual rate.
If you plug all values, but with N=4, that calculates the rate at which your money would have grown with yearly compounding…
Think of the Effective annual rate as the percentage change in $1 over a year’s time. In this case, you are earning 1.23% per month. So, calculate the FV of $1 for 12 periods at 1.23%, and you get $1.1299 ==>therefore, you have an EFFECTIVE annual rate of 12.99%.
the 12.3% is the NOMINAL rate - that’s merely a convention where you take the periodic rate and multiply times the number of periods in a year. It’s what you would have earned if the interest were merely deposited in an account that received no “interest on interest” except once a year (i.e. with annual compounding)