Stated Vs Effective Rate

[Mizi]

Which of the following is the most accurate statement about stated and effective annual interest rates?
A)The stated rate adjusts for the frequency of compounding.
B)The stated annual interest rate is used to find the effective annual rate.
C)So long as interest is compounded more than once a year, the stated annual rate will
always be more than the effective rate.

Can someone explain why C is wrong.

 

[S2000magician]

6% (stated) compounded twice per year gives:
(1 + 6%/2)² − 1 = 6.09% (effective)
By the way, none of those statements by themselves is necessarily true. The author of that question isn’t half as clever as he thinks himself.
Where did you get this question?

 

[Harrogath]

The effective rate results from compounding periods in a year, so A is wrong.
Due above conclusion, effective annual interest rate can be equal or higher than stated annual interest rate (look at S2000magician formula), so C is wrong, because there can be 1 or more compounding periods in a year.
Option B is correct because we use the stated rate to find the effective rate given the compound periods.

 

[Toc Xoan]

Option B is correct when you follow the formula above

 

[busprof]

I’ll put on my nit-picking hat.
B is not necessarily correct. The stated rate *can* be used to find the effective rate (if you also know the compounding frequency). However, you don’t necessarily need a stated rate to calculate the effective rate. As long as you have a periodic rate, you can calculate an effective rate. Having the stated rate and the frequency of compounding allows you to get the effective rate. But I regularly give students the information to calcualte a periodic rate (e.g. a monthly mortgage problem with term, payments, and initial balance) and ask them to calcualte the effective rate.
And C wouldn’t hold with negative rates. A perverse example, but “nitpicky”.
But under reasonable assumptions, B is the obvious choice.

 

[S2000magician]

busprof wrote:I’ll put on my nit-picking hat.
And C wouldn’t hold with negative rates. A perverse example, but “nitpicky”.

Your nit-picking hat must be heavy, causing your head to tip back: you’re leading with your chin here.
Redo my calculation, above, with a −6% nominal rate instead of +6%: the effective rate is −5.91%, and, as you know, −5.91% > −6.00%.
The effective rate is equal to the stated rate when the interest rate is zero, but is greater than the state rate for all non-zero interest rates. But the effective rate is never less than the stated rate for compounding more than once per period.
A is ambiguous: what does it mean to “adjust” for the frequency of compounding? If the effective rate is held constant and the number of compounding periods changes, the nominal rate changes.
B is also ambiguous: you can use the nominal rate to determine the effective rate only if you also have the number of compounding periods; having the nominal rate by itself is insufficient.
As I wrote earlier, the author of this question isn’t nearly as clever as he thinks he is.
This, unfortunately, is a prevalent problem with authors of practice questions. They don’t spend remotely as much time as they should checking and rechecking their questions and answers, and the candidates end up with junk like this.

 

[Harrogath]

S2000magician wrote:

busprof wrote:I’ll put on my nit-picking hat.
And C wouldn’t hold with negative rates. A perverse example, but “nitpicky”.

Your nit-picking hat must be heavy, causing your head to tip back: you’re leading with your chin here.
Redo my calculation, above, with a −6% nominal rate instead of +6%: the effective rate is −5.91%, and, as you know, −5.91% > −6.00%.
The effective rate is equal to the stated rate when the interest rate is zero, but is greater than the state rate for all non-zero interest rates. But the effective rate is never less than the stated rate for compounding more than once per period.
A is ambiguous: what does it mean to “adjust” for the frequency of compounding? If the effective rate is held constant and the number of compounding periods changes, the nominal rate changes.
B is also ambiguous: you can use the nominal rate to determine the effective rate only if you also have the number of compounding periods; having the nominal rate by itself is insufficient.
As I wrote earlier, the author of this question isn’t nearly as clever as he thinks he is.
This, unfortunately, is a prevalent problem with authors of practice questions. They don’t spend remotely as much time as they should checking and rechecking their questions and answers, and the candidates end up with junk like this.

Well, option B is clear for me because what I read is “Tomatoes are used to make salads”, obviously salads does not include tomatoes only, but other vegetables too. However, at the end, is correct to state that tomatoes are used to make salads.
Option A is some kind of clear too due that effective rates adjust for compounding periods the stated rates. Relatively tricky but we can catch it.
I’m not sure what you mean with negative interest rates, but C is clear too, stated rates can be equal or lower than effective rates, not higher, so C is wrong.