# Effective Annual Rate Calculation: Tricky periods and payout frequencies

For calculating the Effective Annual Rate (EAR) from various stated interest rates, I'm using the formula:

$$EAR= \left(1+\frac{r}{p}\right)^{pt}-1$$

where,
$p$ = no. of payouts in a period,
$t$ = no. of compounding periods (typically years),
$r$ = stated interest rate

So, if I have \$$10,000 to invest for one year and the scheme says it will pay annual interest at 10%, compounded quarterly, then for this situation: p = 4 (since it is compounded quarterly), t = 1 (I'm investing it for one year only), r = 0.1 (@ 10%) and my EAR is calcutalted as$$EAR=\left(1+\frac { 0.1 }{ 4 }\right)^{ 4\cdot1 }-1= 10.38\%$$Questions: What will be the values in the EAR equation for the following scenarios (I need to see the values to plug into the equation, not results, which I can obtain from a financial calculator): 1. Period is 9 months instead of one year, nominal rate is 10% per 9 months, and compounding is done thrice in the period. 2. Period is 7 months instead of one year, nominal rate is 10% per 7 months, and compounding is done every 14 months (i.e. double period compounding) 3. Period if 1 year, stated rate in 10% per year, but compounding is done every 2 years, not every year. Thank you. - Your questions can be answered if how you use the terms is made more explicit. Take Question 1. Do you want the effective annual rate? What does period 9 months mean? Is the nominal rate of 10% per 9 months, or is it for 1 year? If it is for 1 year then for effective annual rate the period is irrelevant. It will be enough if you trace 1 dollar. And then you will find the answer yourself! – André Nicolas Mar 1 '12 at 17:45 Question edited. – IntelligentMoron Mar 1 '12 at 18:53 ## 1 Answer Here is the calculation for your first question. Start with a dollar. The nominal rate is 0.10 per 9 months, which I will take as meaning \frac{3}{4} of a year. So the interest rate is \frac{0.10}{3} per third of 9 months, compounded every 3 months. So if we start with 1 dollar, after 3 months we have \left(1+\frac{0.10}{3}\right)^1, after 6 months we have \left(1+\frac{0.10}{3}\right)^2, after 9 months we have \left(1+\frac{0.10}{3}\right)^4. Finally, after one year we have \left(1+\frac{0.10}{3}\right)^4. Thus the effective annual interest rate is$$\left(1+\frac{0.10}{3}\right)^4-1.$$My calculator gives about 0.1401494, a little bit over 14%. The calculation for your second question is mathematically very similar, but feels a little strange because of the unusual compounding. The nominal interest rate is 0.10 per 7 months, compounded every 14 months. So in 14 months, 1 dollar grows to \left(1+\frac{0.10}{1/2}\right). (I am using this somewhat strange way of putting things, instead of writing 1+0.20, so that you can fit it into the pattern of the formula.) Now 1 year is the fraction \frac{12}{14} of the compounding period. So in one year, 1 dollar grows to \left(1+\frac{0.10}{1/2}\right)^{12/14}, so the effective annual rate is$$\left(1+\frac{0.10}{1/2}\right)^{12/14}-1.$$The calculator gives an answer of about 0.1691484. The third question is the same, except easier. The effective annual rate is$$\left(1+\frac{0.10}{1/2}\right)^{1/2}-1.

I hope these calculations are enough to tell you what's going on. Typically, that is in fact not how things are done. The usual way is to determine the "force of interest" and then use the exponential function $e^x$.

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perfectly explained, and understood. Thank you! – IntelligentMoron Mar 2 '12 at 14:04
Such a beautiful explanation! I hope you are teaching somewhere. Thank you very much indeed! – user83908 Jun 26 '13 at 4:47