The interest rate curve is one of the primary determinants of the value of a fixed income instrument. One of the primary goals of a fixed income attribution is to determine how changes in the curve has changed the market value of a security, or portfolio of securities.
The value of a security can be derived from the curve using standard present value considerations. However, the challenge is that the curve represents an infinite set of points, but we only do an attribution on a finite set of factors. This requires some sort of model that fits the curve using only a finite set of factors. These fitted factors then become the inputs to the sensitivity analysis.
Parameterization
There are multiple ways to build a whole curve from a set of parameters. See curve models for more information.
- Bootstrap Rates with Interpolation
- the curve is bootstrapped at a set of common points on the curve, such as 3 month, 6 month, 1 year, 5 year, 10 year and 30 year.
Then the rest of the curve is interpolated by using linear or cubic splines. In this example, each bootstrapped rate becomes
an input into the curve which is used to price the security, and then becomes an attribution factor.
{% value = f(3 month, 6 month, 1 year, 5 year, 10 year, 30 year, ....) %}
- Nelson Segel
- (or other parameterized curve models). The Nelson Segel curve has four parameters that are fit the a set
of bootstrapped rates. Once the parameters have been fit for any day, the result is a curve that closely matches
fixed income prices. From day to day, the fitted parameters will change. These parameters become the inputs to the
attribution. This has the advantage that some parameters more closely represent aspects of the curve such as curvature
which is not represented in the bootstrap rates above.
{% value = f(\alpha_1, \alpha_2, \alpha_3, \beta, ....) %}