Effective annual interest rate

[hei.so]

If an investment of $4000 will grow to $6520 in four years with monthly compounding, the effective annual interest rate will be closest to:

1. 11.2%
2. 12.3%
3. 13.0%

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?

 

[breadmaker]

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%

 

[breadmaker]

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.

 

[S2000magician]

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%.

 

[S2000magician]

breadmaker wrote: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.

That’s 12.277% nominal, annual.

 

[S2000magician]

edupristine wrote:As far as I understand, the simplest way to get effective annual interest rate is to use the one suggested by breadmaker i.e.
EAY = (6520/4000)^0.25 - 1 = 12.992%.

Yes.