Financial Simulation Modeling in Excel

APPENDIX B

Newton-Raphson Method

The Newton-Raphson Method is an iterative procedure used to determine the root of an equation. In some simple situations the root is easy to find. Consider the following equation (equation B.1):

Clearly the root of f(x), the value of x such that f(x) = 0, is when x = 3. A slightly more complicated example is a generic quadratic equation (equation B.2):

where the solution is the quadratic equation in the form of (equation B.3):

However, what happens when we need to compute the root of an arbitrary function that does not have an analytical solution? For example, how would we find the root of the following function (equation B.4)?

The answer in this case is not so easily determined. The simplest solution is to guess a possible range that might contain the answer and then step through each value until the answer is found. This brute-force method, however, is highly unreliable and extremely slow. A more robust and faster solution is to use the Newton-Raphson iterative method.

The method requires the analyst to start with a guess, x_o, and then the process ...