IRR calculations are quite difficult when the number of cash flows exceed 5, however for number of cash flows limited to a count of 3 one may use the Quadratic formula to find the two IRR solutions. Here the cash flows take the form of a Quadration equation that is solved to find the two roots of the 2nd degree polynomial. Here me Abraham A. will take you on a guided lesson that will show the various steps required in solving the quadratic equation to find the two internal rate of return values.
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IRR quadratic formula is explained and illustrated with example calculation using the IRR quadratic formula. IRR equation rewritten as a quadratic equation helps find IRR.
I will show you that IRR Equation for three cash flows may be written as the Quadratic equation thus allowing us to use Quadratic Formula to find the roots of the IRR or Quadratic equation in turn getting us the solution set for internal rate of return
Quadratic equation is of the form listed below, it is a polynomial of degree 2, where a,b and c are constants and x is the variable. Although we have used the letter x to denote the variable however any other letter may be interchanged to represent the variable.
So how does a quadratic equation help us represent the IRR equation you ask. Let us have a look
As you can see the equation above, the IRR equation sets equal the sum of discounted cash flows less the initial cash outlay. Here the denominator is the discount factor, and if we set 1+i to equal q, we can rewrite the IRR equation with 3 cash flows in form of the quadratic equation. We do this be multiplying both sides with q2 thus eliminating the denominators and reducing the IRR equation to the quadratic equation.
Quadratic equations may be solved mathematically with an algebraic formula titled Quadratic Formula. This formula provides a solution set for the roots of the quadratic equation. The solution set for a quadratic equation may consists of two solutions, one solution or none at all. What follows below is the quadratic formula that solves the IRR quadratic equation. This looks a bit different from the original quadratic formula, this is due to the fact that the sign of second order term in the IRR polynominal is negative.
Let us use the Quadratic formula to find internal rate of return where we have three net cash flows. Let us work with a hypothetical investment where our initial cash outlay is $10,000 and we are expect two net cash flows at the end of each of the next two years in the amounts of $5,000 and $7,000.
Here are free cash flows look like -10000 5000 and 7000. Using the quadratic equation as explained earlier, we can rewrite the IRR equation as a quadratic equation such as
-10000q2+5000q+7000 = 0
Have a look at the step by solution to find the roots of this IRR quadratic equation as listed below.
The Quadratic formula finds two roots of the IRR quadratic equation one of which is -0.6232125 and the second one which is 1.1232125. Since on the outset we let q = 1 + i, thus using substituting q in this equation we are left with two rates one equals -1.6232125 (-162.32%) and the second one which equals 0.1232125 (12.32%). We can ignore the negative rate and conclude that for our investment the internal rate of return equals 12.32%
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