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  1. #1
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    Question formula for future value of immediate annuity

    How to find rate of interest in the formula for future value of immediate annuity, if future value, number of periods and amount of annuity (i-e fixed amount payable regularly at equal intervals) are known?
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    Re: formula for future value of immediate annuity

    you know (hopefully) the formula for the future value in terms of interest, annuity amount, number of periods.



    in principle this gives you an equation you can solve for the interest rate. However this is a polynomial of degree n, and i know of no general formula for the result. Unless you have been taught how to solve polynimials of the relevent size you will need to use numerical methods.

    it is easy to show that i is the IRR of the annuity so you can use whatever numerical method you were taught to calculate an IRR.
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    Re: formula for future value of immediate annuity

    Quote Originally Posted by Vinod View Post
    How to find rate of interest in the formula for future value of immediate annuity, if future value, number of periods and amount of annuity (i-e fixed amount payable regularly at equal intervals) are known?
    As stated by SpringFan, the future value of annuity equation can only be solved for R the payment, N the number of periods and FVA the future value of annuity. To solve for i the rate of interest one can use a number of methods:

    1. The simplest of these methods would be use linear interpolation but this will only provide an approximate of the actual rate of interest

    2. Make use of mathematical techniques such as Newton Raphson Method or the Secant Method.

    Let us start of with linear interpolation, assume that we know that payments are in amount of $1000, number of periods are 10 and future value of annuity is $12578. To use linear interpolation we would need to find two rates one at which FVA is lower than $12578 and one at which FVA is higher than $12578. From there we can use linear interpolation to find the approximate rate of interest

    FVA = R FVIFA(i%,n)
    FVA = R [(1+i)^n - 1]/i

    Let us say we use 4% as the rate of interest

    FVA = 1000 FVIFA(4%,10)
    FVA = 1000 [(1+4%)^10 - 1]/4%
    FVA = 1000 x 12.0061071
    FVA = $12006

    At 4% the FVA we found $12006 is less than the actual FVA of $12578, thus we will use a rate higher than 4% to make the FVA above $12578

    FVA = 1000 FVIFA(6%,10)
    FVA = 1000 [(1+6%)^10 - 1]/6%
    FVA = 1000 x 13.1807949
    FVA = $13181

    At 6% the FVA we calculated $13181 is higher than the actual FVA of $12578, now we can use linear interpolation to approximate the rate of interest

    i = iL + [(iH-iL)(FVALower-FVA)] / [FVALower-FVAUpper]
    i = 4% + [(6%-4%)(12006-12578)] / [12006-13181]
    i = 0.04 + [(0.02)(-569)] / [-1175]
    i = 0.04 + [-11.38] / [-1175]
    i = 0.04 + 0.00968511
    i = 0.049685
    i = 4.97%

    Thus the rate of interest is in close approximation of 4.97%

    But as I said this is an approximation, to find the actual rate one can use the Newton Raphson method.

    With NR method you have to first define the equation as a function say f(x) and then to find its differential. From there on there is an iterative procedure that looks for convergence. If the values converge we assume to have to find the rate of interest. If the values do not converge we restart the procedure by selecting a different seed value. This Future Value Annuity calculator finds one of the four results from annuity payment, number of periods, future value of annuity and rate of interest when you provide the other three values. It too uses the Newton Raphson method to find rate of interest

    Using the online calculator we find that the actual rate of interest is 5%
    Last edited by dexteronline; September 23rd 2011 at 07:23 PM.
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    Re: formula for future value of immediate annuity

    Hello,i read your reply. Would you show me example where Newton Raphson method is used?
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    Re: formula for future value of immediate annuity

    Quote Originally Posted by Vinod View Post
    Hello,i read your reply. Would you show me example where Newton Raphson method is used?
    This online calculator is used to solve for interest rate in annuity. Here is how the Newton Raphson Method produces the RATE for an annuity with future worth of 12577.89 where Payment R is $1000 for 10 Years.

    Here is the solution you can try out the rate in annuity calculator for yourself

    f(x) = -12577.89 + 1000 { (1 + X)^10 - 1 } / X
    f'(x) = -12577.89 + 1000{ 10 X (1 + X)^9 - (1 + X)^10 + 1} / X^2
    x0 = 0.1
    f(x0) = 3359.5346
    f'(x0) = 63842.6331
    x1 = 0.1 - 3359.5346/63842.6331 = 0.0473778815441
    Error Bound = 0.0473778815441 - 0.1 = 0.052622 > 0.000001
    x1 = 0.0473778815441
    f(x1) = -152.8715
    f'(x1) = 45319.6038
    x2 = 0.0473778815441 - -152.8715/45319.6038 = 0.0507510689824
    Error Bound = 0.0507510689824 - 0.0473778815441 = 0.003373 > 0.000001
    x2 = 0.0507510689824
    f(x2) = 44.184
    f'(x2) = 46364.0167
    x3 = 0.0507510689824 - 44.184/46364.0167 = 0.0497980885861
    Error Bound = 0.0497980885861 - 0.0507510689824 = 0.000953 > 0.000001
    x3 = 0.0497980885861
    f(x3) = -11.8449
    f'(x3) = 46067.1187
    x4 = 0.0497980885861 - -11.8449/46067.1187 = 0.0500552112083
    Error Bound = 0.0500552112083 - 0.0497980885861 = 0.000257 > 0.000001
    x4 = 0.0500552112083
    f(x4) = 3.2443
    f'(x4) = 46147.0816
    x5 = 0.0500552112083 - 3.2443/46147.0816 = 0.0499849069157
    Error Bound = 0.0499849069157 - 0.0500552112083 = 7.0E-5 > 0.000001
    x5 = 0.0499849069157
    f(x5) = -0.8835
    f'(x5) = 46125.2071
    x6 = 0.0499849069157 - -0.8835/46125.2071 = 0.0500040615327
    Error Bound = 0.0500040615327 - 0.0499849069157 = 1.9E-5 > 0.000001
    x6 = 0.0500040615327
    f(x6) = 0.241
    f'(x6) = 46131.1661
    x7 = 0.0500040615327 - 0.241/46131.1661 = 0.0499988376948
    Error Bound = 0.0499988376948 - 0.0500040615327 = 5.0E-6 > 0.000001
    x7 = 0.0499988376948
    f(x7) = -0.0657
    f'(x7) = 46129.5409
    x8 = 0.0499988376948 - -0.0657/46129.5409 = 0.0500002619581
    Error Bound = 0.0500002619581 - 0.0499988376948 = 1.0E-6 > 0.000001
    x8 = 0.0500002619581
    f(x8) = 0.0179
    f'(x8) = 46129.984
    x9 = 0.0500002619581 - 0.0179/46129.984 = 0.0499998736089
    Error Bound = 0.0499998736089 - 0.0500002619581 = 0 < 0.000001
    RATE = x9 = 0.0499998736089 or 5%
    Last edited by dexteronline; September 24th 2011 at 02:39 AM. Reason: fixed a typo
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  6. #6
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    Re: formula for future value of immediate annuity

    Quote Originally Posted by Vinod View Post
    How to find rate of interest in the formula for future value of immediate annuity, if future value, number of periods and amount of annuity (i-e fixed amount payable regularly at equal intervals) are known?
    Just noticed your use of the term immediate annuity
    I would assume that is used to mean an annuity where payments are made at the start of each period, this is called an annuity due and its formula is slightly different from the one I have listed earlier for ordinary annuity where payments or deposits are made at the end of each period

    I am bit tired at this time and there are power outages here every two hours so I get you the answer for finding rate when you have an annuity due

    The formula for finding future value of the two types of annuities I discussed are as follow, and to find the rate i, or payment R, or number of periods n differ between the two

    Future value of ordinary annuity
    FVA = R x FVIFA(i%,n)
    FVA = R x [(1+i)^n - 1]/i

    Future value of annuity ude
    FVAD = R x FVIFAD(i%,n)
    FVAD = R x (1+i)[(1+i)^n - 1]/i
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    Re: formula for future value of immediate annuity

    "immediate annuity" does not necessarily mean "annuity due". for example it is possible to have an "immediate annuity payable in arrears", and these are regularly sold as retirment products in the UK.
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