Fixed Income Risk - Duration

Overview


Duration was initially conceived as a measure of the "effective maturity" of a bond. A bonds maturity only specifies the date of final, principal payment of the bond. This ignores all the payments made prior to the maturity date. The MaCauley duration was initially conceived as a weighted average of the amount of time to each payment in the bonds contract. This means that while the bonds formal maturity is the date of the bonds last payment, the duration would be less than that date.

It was discovered that with a minor adjustment, the duration could be used to calculate an approximate bond price when the bonds yield changes by a small amount.
{% \frac{dP}{dy} = -D \times P %}
where {% D %} is the bonds duration.

Duration


The most prominent measure of fixed income risk is the Duration of a series of cash flows. It is used to approximate the change in the price of a bond (or series of cash flows) as follows
{% \Delta P / P \approx -D \times \Delta y %}
where
{% D = \sum t w_t %}
and
{% w_t = (C_t/ e^{y \times t}) / P = e^{-y \times t} C_t / P %}
(see nawalhka)

This makes the duration a weighted average of the cash flows. Notice that we use the yield to maturity here, not the individual discount rates for each cash flow. (see yield to maturity)

Price - Yield Relationship



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Topics


Duration API


The duration library provides methods for doing duration calculations.

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