Re: solving CAGR challenge

On Mon, 28 Jan 2008 14:32:28 -0800 (PST), pete <ptgallant@xxxxxxxxx>
wrote:

On Jan 25, 5:07 pm, quasi <qu...@xxxxxxxx> wrote:
On Fri, 25 Jan 2008 13:38:55 -0800 (PST), pete <ptgall...@xxxxxxxxx>
wrote:

I want to rearrange the following formula to solve for R where

V=(S*((1+R)^(N+1)-1)/(R))-S.

where:
S = 10
V = 39.93375
N = 3

I know the answer is .15.

Essentially, if I compound a starting amount of10 at a rate of 15% for
3 years, the cummulative sum is 39.93375.   How do I solve for the
rate if I am given the cummulative sum, the period in years, and the
starting amount?

For the numbers you specified, you can solve exactly using elementary
algebra.

Make sure to use rational numbers instead of decimals.

Simply substitute the given numbers in the equation, get all the terms
on one side, zero on the other side, then factor.

quasi

Can you walk me through the elementary algebra steps to get to a
formula that starts R=____
My algebra is rusty and I'm having trouble rearranging the exponential
portions of the equation.

Thanks

Pete

There is no simple algebraic formula of the form "R = ".

Rewriting your formula slightly, we have

[ (1+R)^(N+1) - 1 ] / R = (V + S) / S

For integer values of N > 0, the left hand side is a polynomial in R
of degree N. Here are the polynomials for the first few values of N:

N = 1: R + 2

N = 2: R^2 + 3 R + 3

N = 3: R^3 + 4 R^2 + 6 R + 4

N = 4: R^4 + 5 R^3 + 10 R^2 + 10 R + 5

Thus, given values for V, S, and N, the basic problem to find the
roots of a polynomial of degree N. There is no simple "formula" for
doing this for values of N > 4. Indeed, finding the solution "by
hand" for N > 2 is a task to be avoided. In general, numerical
methods are needed to solve for R.

For the values you gave above, namely

S = 10
V = 39.93375
N = 3

the polynomial is

R^3 + 4 R^2 + 6 R + 4 = (39.93375 + 10) / 10

R^3 + 4 R^2 + 6 R - 0.993375 = 0

This polynomial has one real root, R = 3/20 = 0.15

.

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