If an investment of $4000 will grow to $6520 in four years with monthly compounding, the effective annual interest rate will be closest to:
I used (1+r/12)^48 to get the interest rate of 12.3%. But the answer is 13.0%. The solutions says “The question asks for the effective annual rate and gives the beginning and ending values. Monthly compounding is not relevant”. Why is this?
Effective annual means compounding is applied only once a year: 12.277% is the nominal rate compounded monthly.
(6520/4000)^(1/4) - 1 = 12.992%
And…
(1 + 12.277%/12) ^ 12 = 1.12992, so 12.277% compounded monthly produces the same ending dollar amount as 12.992% compounded annually, i.e. effective annual rate.
The problem is that you wrote (1 + _ r/12 _)^48; by using _ r/12 _ you’ve made r a nominal annual rate, not an effective annual rate.
You either have to use (1 + r)^48, where r then is an effective monthly rate, then compound it for 12 months to get an effective annual rate, or (1 + r)^4, to get the effective annual rate directly. Either way, the EAR is 13%.
That’s 12.277% _ nominal _, annual.
Yes.