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Interest Rate Curves

The EDHECinfra approach to computing the value of unlisted infrastructure companies equity (or debt) requires using a term structure of interest rates at the time of valuation and to the relevant investment horizon. Interest rate curves (or term structures) are needed both when extracting observed risk premia from secondary market transactions and later when computing asset values

All asset values are computed using a discounted cash flow (DCF) approach, requiring an estimate of the 'risk-free' rate of interest along with the infrastructure equity risk premia at the time of valuation. This last point plays an important role in the achievement of fair market value estimates. IFRS 13 guidance requires using the most relevant market inputs at the time of the asset valuation.

Example

e.g. if observing a deal IRR or pricing the equity of a wind power generation company with a 20-year life in Spain in the 3rd quarter of 2012, a 20-year term structure of the Spanish government bond yields in that quarter is needed and estimated.

To this end, government bond yield curves are computed for each valuation date (Quarter end date) since 2000Q1 for all horizons up to 100 years. This is done for each country included in the Index Universe by interpolating observable market rates as described below.

Raw Data

Raw interest rate data is sourced from DataStream® using the following the rules:

1. Government bond yield benchmark for short ( $\begin{array}{l}\leq\end{array}$24 months), medium (> 24 months $\begin{array}{l}\leq\end{array}$ 7-year) and long-term maturities (> 7 years).
2. ON (overnight) or TN (tomorrow next) rates, for yields below 1 year maturity
3. Input data that does change more than 3 consecutive quarters is dropped to avoid stale inputs

Model

The yield curve is given by the following function (Nelson-Siegel-Svensson or NSS model)

Where $\begin{array}{l}m\end{array}$ is maturity, $\begin{array}{l}\lambda_1\end{array}$ and $\begin{array}{l}\lambda_2\end{array}$ control the curve shape and $\begin{array}{l}b_1\end{array}$ is the long-term rate.

Calibration

The short-term rates are first linear-interpolated to have 1, 3, 6 and 12 months rates. These interpolated rates are used in the calibration. For the long-term extrapolation, the following constraints are added:

1. Add a punishment in the target function to avoid non-monotone extrapolation.
2. The yield at the maximum horizon (100 years) is required to should fall within 1% of market yields for the longest available horizon (typically, 15 to 30 years)
3. Linear interpolation is used between the longest horizon (100 years) from the longest market data available (15-30 years) if extrapolated rates are not monotone.

In period of data instability (e.g. 2008) the two parameters  $\begin{array}{l}\lambda_1\end{array}$ and $\begin{array}{l}\lambda_2\end{array}$  can also be calibrated by hand to prevent unstable model calibrations.

Examples: 1999 to 2020

Germany Yield Curves

click image to see animation

France Yield Curves

click image to see animation