## What is Interpolation?

Interpolation is a method of constructing new data points within the range of a discrete set of known data points.

Typically, yield curve data is made available showing interest rates for several corresponding durations as shown in **Table 1** below. We must then interpolate if we want to find rates for durations not present in the original data.

With the sample data shown in Table 1 below, we would need to interpolate if we wanted to determine the rate for a 20 month duration.

## What Interpolation methods are available within PrecisionLender?

Interpolation method settings are located in Funding Packages. PrecisionLender offers three interpolation methods when dealing with yield curves:

**Piecewise Linear**

- A simple line is drawn between each data point on the curve
- https://en.wikipedia.org/wiki/Linear_interpolation

**Cubic Spline**

**Monotone Convex**

- https://en.wikipedia.org/wiki/Monotone_cubic_interpolation
- Preferred when using Caps and Floors functionality

If your bank is using the Caps and Floors functionality, we recommend using the Monotone Convex interpolation method. This is because the Caps and Floors functionality relies on forward rate estimates which are sensitive to artifacts within the interpolated curve (kinks).

**Table 1. Example yield curve spot rates along with interpolated values:**

Duration | Rate | Piecewise Linear | Cubic Spline | Monotone Convex |
---|---|---|---|---|

0 | 0.2980% | 0.2980% | 0.2980% | 0.2620% |

1 | 0.2640% | 0.2640% | 0.2640% | 0.2640% |

2 | 0.2680% | 0.2680% | 0.2680% | 0.2680% |

3 | 0.2820% | 0.2820% | 0.2820% | 0.2820% |

4 | 0.2900% | 0.2900% | 0.2900% | 0.2900% |

5 | 0.3010% | 0.3010% | 0.3010% | 0.3010% |

6 | 0.3070% | 0.3070% | 0.3070% | 0.3070% |

7 | 0.3260% | 0.3260% | 0.3260% | 0.3260% |

8 | 0.3390% | 0.3390% | 0.3390% | 0.3390% |

9 | 0.3540% | 0.3540% | 0.3540% | 0.3540% |

10 | 0.3760% | 0.3760% | 0.3760% | 0.3760% |

11 | 0.3900% | 0.3900% | 0.3900% | 0.3900% |

12 | 0.4070% | 0.4070% | 0.4070% | 0.4070% |

0.4475% | 0.4349% | 0.4385% | ||

0.4880% | 0.4721% | 0.4820% | ||

0.5285% | 0.5155% | 0.5293% | ||

0.5690% | 0.5618% | 0.5744% | ||

0.6095% | 0.6077% | 0.6144% | ||

18 | 0.6500% | 0.6500% | 0.6500% | 0.6500% |

0.6802% | 0.6863% | 0.6818% | ||

0.7103% | 0.7177% | 0.7107% | ||

0.7405% | 0.7459% | 0.7383% | ||

0.7707% | 0.7730% | 0.7666% | ||

0.8008% | 0.8007% | 0.7970% | ||

24 | 0.8310% | 0.8310% | 0.8310% | 0.8310% |

0.8707% | 0.8652% | 0.8683% | ||

0.9103% | 0.9029% | 0.9074% | ||

0.9500% | 0.9431% | 0.9476% | ||

0.9897% | 0.9848% | 0.9883% | ||

1.0293% | 1.0271% | 1.0289% |

## Graphical differences between methods

The example image below shows the yield curve spot rates (blue markers) along with the three interpolation methods. All of the interpolated lines pass through each and every data point. However, between the data points, there can be slight differences between the methods (see inset). Although slight, these differences can impact the monthly forward rates used within PrecisionLender.

All other interpolations used throughout PL (including Liquidity Curve, Obligor Loss and Obligor Capital Rates) are calculated using the piecewise linear method.

## More Information about the math behind these interpolation methods

- Methods for constructing a Yield Curve (Hagan & West)
- A Brief Comparison of Interpolation Methods for Yield Curve Construction