Market Risk Analyzer Interpolation - Why does interpolation matters?

 Market Risk Analyzer Interpolation - Why does interpolation matters?

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.

In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate (i.e., estimate) the value of that function for an intermediate value of the independent variable.

Interpolation serves to determine a value for an interest term of a yield curve (for which no interest rate exists) from the existing interest rates. The interpolation function is used in the following cases:

• Interpolation of annual grid values from the grid values of a yield curve type of the yield category par rate for the structuring of the yield curve.  Note Annual grid values, meaning grid points such as 1 year, 2 years that are not defined as reference interest rate (grid point) of the yield curve are needed for the calculation of the zero bond discounting factors.
• Interpolation of interest rates with an interest date different from that of the grid points of a yield curve type when accessing the yield curve.

For the yield category par rate, the par rate is interpolated on the required interest date according to the chosen interpolation procedure of the yield curve type and calculated from the interpolated par rate of the zero coupon and the zero bond discounting factor. For the yield category zero bond yield, first the zero bond rate is interpolated according to the interpolation procedure and from this, the zero bond discounting factor is calculated. In this case, there is no calculation of a par rate.

Exception: Continuous compounding zero interpolation is used.

When continuous compounding zero interpolation is activated, zero bond rates are calculated independently of the yield category in the zero bond discounting factors, using continuous compounding and interest calculation method ACT/365. The equivalent yield curve calculated without continuous compounding is used as the initial yield curve. In other words, at the grid points, a yield curve with continuous compounding zero interpolation is identical to the equivalent yield curve without continuous compounding. It is always linear interpolation that is used to calculate the continuous compounding zero bond rates, irrespective of which interpolation procedure is applied. The sections Zero Bond Discounting Factors and Forward Rates describe how par rates and zero coupon rates are calculated (in the interest calculation method of the yield curve) from interpolated continuous compounding zero bond rates.

Features

If the term of a required interest rate is before the first or after the last grid point, the first or the last grid point is transferred (extrapolation). If the required interest date is between two grid points of the yield curve, depending on the interpolation parameter of the yield curve type, either linear interpolation or cubic spline interpolation is performed.

1. Linear interpolation calculates a required value on the basis of two given values on the line between the two given values. The interpolation is performed as follows

Pt: interpolated interest rate at time t

Pt-1: lower grid value at time t-1

Pt+1: upper grid value at time t+1

dt: term of the required interest rate in days

dt-1 : term of the lower grid value in days

dt+1: term of the upper grid value in days

2. Calculation of the number of days is always based on the interest calculation method of the yield curve. Exception: Continuous compounding zero interpolation is used.
3. The cubic spline interpolation is used along with linear interpolation to achieve “smoother” yield curves “Smooth” means that with cubic spline interpolation for the procedure implemented here, there is constant differentiability, whereas with linear interpolation, the resulting curve is constant. The yield curve resulting from the cubic spline interpolation retains the feature that in the case of monotonous initial data (for example, a normal yield curve), the monotony remains.
Cubic spline interpolation uses parts of third degree polynomials that are linked to the grid points by suitable conditions in such a way that continuous differentiability of the yield curve is ensured. Contrary to linear interpolation, in the case of cubic spline interpolation, all grid points are included in the calculation of an interpolated value.
Cubic spline interpolation provides better interpolation values than linear interpolation. This means, however, that cubic spline interpolation is more complex and more extensive and that it therefore requires longer terms than linear interpolation.
Note

For the graphical display of a yield curve, the grid points are linked in linear fashion, independently of the interpolation category of the yield curve. This means it is not possible to view cubic spline interpolation graphically.

Example

Interest rates of the grid values:

Interest calculation method: act/360

Interpolated annual grid value for the term of 2 years (731 days):

• Linear interpolation:
• Cubic spline interpolation: P731 = 0.062503419