Taylor approximation formula
- Taylor approximation formula
We know that when the yield on a bond changes, the bond's price will also change. What investors need to know is by how much. In finance, we often approximate a complex function by other simpler functions. The Taylor approximation formula does this by employing derivatives, that is, rates of change.
Geometrically, the derivative is defined as the slope of a line, that can be measured by the change in the y-coordinate (Price) caused by a one unit change in the x-coordinate (Yield). In mathematical notation, this would be:
[
_at_point_a_is_given_by_the_Newton_gradient_and_is_defined_as_follows:)
[
_is_the_difference_between_the_value_of_function_f(x)_at_point_a_+_h_(where_h_is_a_small_increment)_and_point_a,_divided_by_the_small_increment_h._)
For the purpose of bond analysis, all we need is the general rule for obtaining the derivative of a non-linear function of the form:
[
_=_ax-n_._In_this_case,_the_derivative_of_the_function_is_given_by:_f)
Hence, we bring down the negative power to multiply ax and then subtract 1 from the power.
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