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Math Help - Annualized interest rates help

  1. #1
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    Annualized interest rates help

    Hi, so I was looking up annualized interest rates because they're used a lot in the news like when reporters talk about "annualized GDP". Investopedia (Annualized Rate Definition | Investopedia) says that it is basically multiplying the monthly return on assets by 12. But that's kind of weird because if you keep that asset in for a year, your pile of money will grow exponentially. Why does Investopedia do that?
    The formula that Ready Ratios (Annualized Rate) uses makes a lot more sense to me because you put your money in for n periods and watch it grow. Then you take away the 1 to take away your initial investment to see just the growth caused by the interest rate.
    Which is the "real" formula for calculating the annualized interest rate?
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  2. #2
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    Re: Annualized interest rates help

    First, let's get our vocabulary straight. What we are really talking about mathematically is rate of change; in mathematical terms, an interest rate is merely one example of a rate of change. When a rate of change is positive, it is also frequently called a rate of growth.

    The mathematically correct formula for figuring out what the overall rate of change over n periods will be on the assumption that the rate of change for the first period remains constant over the next n - 1 periods is:

    $(1 + z)^n -1,$ where z is the rate of change for the first period.

    However, $if\ |z| \approx 0\ and\ n \le 12,\ (1 + z)^n -1 \approx n * z.$

    Let's see how the approximation works for z = 0.25% and n = 12.

    The answer using the correct formula is $(1 + 0.0025)^{12} - 1 = 1.0025^{12} - 1 \approx 1.0304 - 1 = 0.304 = 3.04\%.$

    The answer using the approximation formula is $12 * 0.0025 = 0.03 = 3.00\%.$

    For many practical purposes, the approximation formula is just as reliable as the correct formula.
    Last edited by JeffM; June 28th 2014 at 08:52 AM.
    Thanks from mathbeginner97
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  3. #3
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    Re: Annualized interest rates help

    IF the interest is "compounded monthly" then "r percent monthly" is (very slightly) greater than "12r percent annually". But if it is simple interest then "r percent monthly" is precisely "12r percent annually".
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