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?

Printable View

- Sep 22nd 2011, 06:12 AMVinodformula 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?

- Sep 22nd 2011, 07:16 AMSpringFan25Re: 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. - Sep 23rd 2011, 07:48 PMdexteronlineRe: formula for future value of immediate annuity
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% - Sep 23rd 2011, 11:06 PMVinodRe: formula for future value of immediate annuity
Hello,i read your reply. Would you show me example where Newton Raphson method is used?

- Sep 24th 2011, 02:49 AMdexteronlineRe: formula for future value of immediate annuity
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%** - Sep 24th 2011, 03:57 AMdexteronlineRe: formula for future value of immediate annuity
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 - Sep 24th 2011, 07:10 AMSpringFan25Re: 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.