Overview
This section shows how to derive the formula for the duration.
Starting with the formula for yield to maturity:
{% P = \sum e^{-y \times t_i} C_i %}
where {% y %} is the yield to maturity. Next, take the differential:
{% dP = - \sum (t_i e^{-y \times t_i} C_i) \; dy %}
or dividing by {% P %}
{% dP/P = - \sum (t_i e^{-y \times t_i} C_i / P) \; dy %}
This formula is the Macauley duration. It differs from the duration quoted above in that we used the yield to maturity here instead of the individual yields
as functions of time. However, the calculation goes the same in the individual case if we assume that {% dy_t %} is the same for each value of {% t %}.