HOW FAST MONEY DOUBLES  RULE OF 72
A quick and easy way to see how many years it takes money to double is divide the interest rate received into 72. A rate of 6% divided into 72 produces an answer of 12 years, 8% takes 9 years to double and so.
ANNUAL PERCENTAGE YIELD (APY) VS. ANNUAL INTEREST RATE
APY reflects the true annual compounded yield earned on an interest earning account. As examples, a 4.0% interest rate would also have an APY of 4.0% if interest was compounded annually, but the APY would be 4.06% if interest were compounded quarterly. Why
Compounding divides up the interest rate into smaller bits representing the way you are compounding. If you were compounding quarterly the annual interest rate is divided into quarters or fourths, so the annual rate of 4.0% is divided by 4 meaning you earn 1% for that quarter. At the end of the quarter you now have 101% of what you started with representing the 100% principal and the 1% interest earned. When you begin the second quarter you will again earn 1%, but you earn on both the original principal and the interest already earned. 101% times a 1% rate means you earn 1.01% for the second quarter. This is added to the 101% giving you 102.01%. This process of earning interest on both principal and interest repeats in the third and fourth quarter, so at the end of the year you have 104.06% and that .06% reflects extra interest earned The formula to convert a nominal or annual rate into an annual percentage rate is: ER = [1 + (i/n)]^{n}  1
Say that you are comparing a fixed rate annuity to a certificate of deposit. The fixed rate annuity has an annual rate of 6.1% compounded annually, and the certificate of deposit has a nominal rate of 6%, compounded monthly. Because the fixed annuity has only one compounding period the fixed annuity’s nominal and effective rate are the same — 6.1%. However, because the CD compounds monthly there are twelve compounding periods.
The CD’s effective rate is 6.17% computed as follows: ER = [1 + (.06/12)]^{12} – 1 = 1.0617 – 1 = .0617 or 6.17%
COMPOUND INTEREST VS. SIMPLE INTEREST
Compound interest works a little differently. Compound interest multiples the previous balance  in this case $1.09, by the interest rate (9%), and adds the numbers together to determine a new value.
So, $1.09 multiplied by 9% produces not 9 cents but 9.81 cents. This is added to the previous balance (1.09 + .0981) to produce a second year total of $1.1881. A short hand way to figure the results of compounding is to put a one in front of the interest rate ($1.0 x 1.09 = $1.09). If you are only looking at a few periods there is not a lot of difference between the effects of simple and compound interest. However, the benefits of compound interest become proportionately greater with each passing period.
Rate of ReturnIn an attempt to compare the annual return of different investments some people will divide the total return by the number of years involved. However, unless the investments are of equal duration, the results may lead to the wrong conclusion.
Say that Investment A returned 30% in threeyears, Investment B returned 60% in sixyears, and Investment C returned 90% in nineyears. If you divide each investment’s total return by the number of years in the period, the answer for each one is 10%. But, the actual annual rate of return for Investment A is 9.13%, the rate of return for Investment B is 8.15%, and the rate of return for Investment C is 7.18%. The formula to calculate the honest rate of return for an investment without irregular cash flows is this: i = [(FV/PV)^{(1/n)} – 1]
If the Present Value is $10,000, the Future Value is $15,000, and the number of periods is 5. The equation would be:
i = [(15,000/10,000)^{(1/5)} – 1] = 1.0845 – 1 = .0845 or 8.45%
How Long $100,000 Lasts – Earning % / Withdrawing $ Each Year

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