Hedging Fixed Income Risk
Overview
Hedging refers to the process of trading instruments in
a portfolio in order to make the portfolio immune to
fluctuations in the interest rate curve.
As a general rule, a portfolio of long only fixed income
instruments is only hedged if all the instruments are
floating rate instruments. This is because any simple bond will
lose value as rates go up, and gain if rates go down, so a portfolio
with these types of bonds will lost value as rates go up. The only
instrument which does not lose value is a floating rate instrument.
Therefore in general, you will only hedge a portfolio that
contains both assets and liabilities, or a portfolio with
optionality.
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 %} is the duration and is computed as
{% D = \sum t w_t %}
and
{% w_t = (C_t/ e^{y(t) t}) / P %}
and {% \Delta y %} is the change in the yield to maturity.
This makes the duration a weighted average of the cash flows.
The weights are the present values of the cash flows divided by the total
price.
Portfolio Duration
The duration of a portfolio is the weighted average of the durations of the underlying bonds, where the weights
are the present values of each bond.
{% Duration_{portfolio} = \frac{1}{Portfolio \: PV} \sum w_i \times Duration_i %}
The additive nature of the porfolio duration makes it easy to hedge.
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