The general opinion is that investments with high internal rate of return are more profitable than other investments. In a few situations, a project with a lower internal rate of return can be more efficient. The internal rate of return requires initial income streams and capital, a series of amounts of money than come and go at specific times through which to finance further projects. This is the investment; at year 0, this value is less than zero and represents the cost of the investment.

We assume that P is the amount of money invested and p are the additional deposits made each period of time. There are n periods of time (months, years). The number of periods is n and "i" is the interest rate for each period of time. The IRR formula is:

T =-P(1+i)^n-[p(1+i)((1+i)^n-1)/i]

T is the future value of the investment. Remember that P and p must be less than zero, respecting the standard of representing cash paid out as less than zero and cash received as greater than zero. Now we must find the p of the amount of P in n periods of time. The formula is:

      -P i(1+i)^n
p = ------------
       (1+i) ^ n - 1

We must also find the numbers using:

     log (-p)-log(-Pi -p)
n = ----------------------
               log (1+i)

Where "I" is the interest rate for each period of time. To find the annual rate, we can use the formula:

i =(r + 1) 1/12-1

In this formula, r is the rate. We have also the formula for NPV:

                         N
NVP(C,t,d)=Sum C[i]/(1+d)^t[i]
                         i=0

In this formula, C[i] is the cash flow (C[0] the first, C[N] is the last cash flow. "d" is the discount rate, t[i] is the time between the first cash flow and the "i" cash flow. The unit time can be a month, a year or any other time period. This must be constant during the calculus. We also know that NVP(C,t,IRR)=0. Now we realize that there is no analytical solution for IRR. The solution must be calculated iteratively. You must have a probable value for IRR, plug this value into the NVP formula. Repeat the calculus until NVP is very close to zero.

The situation is simple when we have only one initial investment. The formulas are now:

(initial-value)-(final-value)/(1+irr)^T=0
(initial-value)*(1_irr)^T-(final-value)=0
(initial-value)*(1+irr)^T=(final-value)
(1+irr)^T=(final-value)/(initial-value)

The final formula is intuitive and easy to use:

IRR=(final-value)/(initial-value) ^(1/T)-1

There are also some programs to calculate IRR on the Internet. Sometimes IRR is referred as ERR (economic rate of return). IRR or ERR are the rate of growth an investment is able to produce. A higher IRR value would provide a greater chance of substantial growth, high IRR is a good indication towards an investments future development. Executives use IRR when they must take major decisions about their company’s future investments. Some specialist prefer the modified MIRR, offered by excel, as well as the IRR function form Excel too.