# Calculating duration and interest Rate for an investment

I'm trying to build a calculator to determine different Factors for specific Investments. I have a Formula to calculate the Grand Total, but I can't see any way to get either the Duration or the interest rate from this, as Long as there is both an Initial Investment and a monthly Payment. Unfortunatly I am no native Speaker, so I'm probably not searching for the correct phrases.

K: Grand Total
p: partial Interest(monthly)
d: Duration(in Months)
iv: Initial Investment
mv: monthly Investment

K(d) = (mv * (1 + p) * ((1 + p)^d - 1) / p) + iv * (1 + p)^d

Do you know a formula or other method to calculate either the Duration or the interest rate, assuming all other Factors are known?

## 1 Answer

Let us use standard actuarial notation in this problem, it is a little easier to read:

It seems to me that you are attempting to determine the future value of both a lump sump and an annuity. This represents the two parts of your equation. Let $A(t)$ represent the accumulated value (or future value) of your initial investment and $Ps_{n|i}$ the future value of a series of equal monthly payments.

The grand total, or in your case K, should be equal to

$K = A(t) + s_{n|i}$

Using the formulas for both of the values above will result in:

$K = R(1+i)^n + P \dfrac{(1+i)^n - 1}{i}$, where K is the future value of your cash flows, R is the initial deposit, i is the monthly interest rate, n is the number of payment periods and P is the size of each monthly payment.

Your formula provided above would be correct if the first monthly deposit was made at time n = 0.

Is there anything else you need?

• The Formula is correct yes :). My Problem is that I don't see a way to use it to calculate either the Duration or Interest Rate. – Andy Feb 19 '14 at 15:08
• That would be only possible with advanced mathematical software, since you are often solving for polynomials of over a hundred degrees. For example a financial calculator? – David L Feb 19 '14 at 15:12
• I'm trying to build a simple financial calculator. Yes. Do you know where I could find Information about useable algorithms for such Problems? – Andy Feb 19 '14 at 15:18