# Effective annual interest Question

The stated (quoted) annual interest rate on an automobile loan is 10%. The effective annual rate (EAR) of the loan is 10.47%. The frequency of compound- ing per year for the loan is closest to:

1. A quarterly.
2. B monthly.
3. C weekly.

B is correct. EAR = (1 + Periodic interest rate) m − 1.

The solution is found iteratively by substituting the possible frequency of compound-ing until the EAR is 10.47%.
For weekly compounding, (1 + 0.10/52)52 − 1 = 0.10506 = 10.51%.
For monthly compounding, (1 + 0.10/12)12 − 1 = 0.10471 = 10.47%.
For quarterly compounding, (1 + 0.10/4)4 − 1 = 0.10381 = 10.38%.

The answer here shown by trial and error method. Is there any alternative method to find the (frequency) answer to this question ? Such as using logarithm ? Thank you so much

Possible alternatives:

1. ICONV on the BAII will give you the frequency quite quickly. I am not sure if the HP12 has a similar function.

2. Use the s2000magician method of trying b first. If your answer is too low, you haven’t compounded enough and need a higher frequency; if your answer is too high, you have compounded too much and need a lower frequency. In this case, (b) is the answer and you’re done!!!

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You could equate the formula for the effective rate of interest for an m-compounded interest rate with the given effective interest rate, and take the logarithms.
you’d get, assuming i didn’t make a dumb mistake,
m * ln(1 + r annual / m) = ln (1 + r effective)
in this case,
m * ln (1 + .10/m) = ln 1.1047
you have m inside and outside the logarithm, so this approach takes longer and is no better than the approach used by the solution manual.

frankly, you’re better off just using the solution manual method or @breadmaker 's method. it’s far more efficient that way. it’s also another reason to get to know your calculator.

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Thank you it seems that you can’t directly compute the frequency (C/Y) on the BA2 II plus calculator, but I could just use trial and error in the INCOV function to find the answer.

You are correct: I thought ICONV would allow calculation of C/Y. It does save time though because it does the grunt calculations for you!

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