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?
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.
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%
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%
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