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Methodology for Risk-free Discount Rates and CPI Assumptions for Accounting Valuation Purposes

Appendix 2  Forward Rate Yield Curve Fitting Methodology

The purpose of this appendix is to describe the process of converting quoted government stock yields to a smoothed forward rate yield curve to be used as a basis for the short-term risk-free rates.

Short-Term Interest Rates

The available sources of information are the Overnight Cash Rate (OCR) and Treasury Bills.

Construction of zero Coupon Portfolio

Government stocks are decomposed into maturity and individual coupon payments to produce a set of equivalent zero coupon stocks maturing on the 15th of the month.

Bootstrapping

Starting at the short end of the yield curve a forward rate is determined for the shortest stock, equal to the spot rate for the whole period up until the first stock matures. For the period between the first stock and the second stock a forward rate is determined so that the second stock market value is equaled using the previous forward rate as well. The forward rates are applied to the deconstructed cash flows already determined. This process is repeated solving for each successive forward rate until all stocks have been valued.

Curve Fitting and Interpolation

The fitting process will ideally be able to allow for anomalous prices for a particular stock. This is most likely to occur for a stock with less available on the market.

The proposed process is to fit a curve of forward rates to the zero coupon portfolio of available stock. The parameters of the fitted curve are determined by solving to minimize the least squares differences of the resulting fitted market values with the actual market values.

This process is equivalent to weighting the yields by the amount available in the market, which excludes the amounts held by the Reserve Bank of New Zealand (RBNZ) and the Earthquake Commission (which is not usually traded). This means that implied forward rates automatically give less weight to those stocks that is less market information.

The curve fitted is a cubic spline on the forward rates with 4 knots. This is fairly standard methodology with enough flexibility to fit most yield curves. There is some judgment involved in selecting the position of the knots, but this also gives a little flexibility to cope with any anomalies that may be present in the yield curve without changing the fundamental principles.

The smoothing process in itself is not particularly important as unsmoothed forward rates could be used with a minimal effect on the overall liability; however the critical factor in the smoothing is how it copes with any market anomalies, particularly at the longer end of the curve, or anywhere there are gaps between maturities.

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