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

8  Bridging the short and long-term rates

8.1  Introduction

8.1.1 This section sets out the Treasury's view of how to bridge the short-term market rates to the long-term assumed rate. This includes a decision of the appropriate period over which to smooth the “bridge”.

8.1.2 We recognise that this “bridge” is one of the most subjective areas of the methodology. The accounting standards do not contemplate this requirement; however, subsequent research and discussion papers outline the broad principles of extrapolation.

8.1.3 We have mainly relied on the principles in a number of actuarial technical papers identified in section 9 of this paper. These papers propose appropriate technical methods to join the short and long-term rates for valuation purposes.

8.2  Summary

8.2.1 The government stock yield curve currently finishes at 15 May 2021. This is currently 11 years. In future the longest duration for government stock is likely to range between 10 and 12 years.

8.2.2 The long-term nominal risk-free rate has been set at 6% under this methodology for accounting valuations.

8.2.3 In our view the most robust place to start the smoothing is the calendar date of the last stock. The selection of the end date (ie, where the 6% long-term rate starts on the hypothetical yield curve) would be guided by:

  • the difference between the long-term rate and the rate at the end of the yield curve (ie, if this difference is greater the smoothing period may need to be longer), and
  • forward rates on bank SWAPS at that duration to the extent that they are reliable.

8.2.4 The difference between short and long-term risk-free discount rates should be smoothed. This is consistent with principle in the March 2010 CEIOPS paper that proposes that extrapolated rates should follow a smooth path from the entry point to the unconditional ultimate long-term forward rate.

8.2.5 At present, the difference between the long-term rate and the rate at the end of the yield curve is quite small and the current swap forward rates reduce at long durations. Therefore, we conclude that at present 5 years is an appropriate period to smooth over.

8.2.6 It is appropriate, in our view, to lock this 5 year period in for a reasonable time and not change it in response to minor market fluctuations. For future proofing we prefer to express this as a period which results in a fixed date (15 May 2026), dependent on the maturity date of the longest bond. This corresponds to option 4 described below.

8.3  Analysis

8.3.1 In reviewing the literature we have identified a range of views over when the long-term rate should start:

  • at the end of the yield curve
  • from 20 years onwards (example in the paper from CEIOPS)
  • somewhere in between.

8.3.2 However, to narrow the choices we have focused on how the interpolation should be done between the short and long-term rates under five viable options.

8.4  Interpolation between Market and Long-Term Rates

8.4.1 There are a number of ways to interpolate between the short and long-term rates, including:

  • no interpolation, a step in the rates at the end of the yield curve
  • starting the interpolation before the end of the yield curve
  • interpolation from the end of the yield to a fixed duration (eg, 16 years)
  • interpolation from the end of the yield for a fixed period (eg, end of yield curve plus 5 years), and
  • interpolation from the end of the yield at a fixed slope (eg, -0.1% p.a).

8.4.2 All of these approaches will have advantages and disadvantages and the final selection is a matter of judgement. A combination of the last two is another option, with the fixed period being modified if the slope becomes excessive.

8.4.3 Linear interpolation should be adequate for all the options above, as it is unlikely that there will be advantage in using a more complex interpolation method.

8.4.4 The interpolation should attempt to be consistent with bank SWAP rates where they are available. However, long duration bank SWAPS are subject to scarcity premiums due to excess demand over supply so the yields may consequently be artificially low at the end of the curve.

8.4.5 If these bank SWAP rates, or any other observable rates, are significantly out of line with the resulting curve, then consideration should be given to any adjustment required. The rates should be adjusted for any scarcity or risk premium or any other adjustments that may required. The weight given to other observable rates needs to account for the uncertainty in the rates. Investigation would also be required to determine the pricing basis and level of trading underlying these rates. For example, in some instances the longer duration bank SWAP curve is not a genuine market observation, but is generated by extrapolating the forward rate from shorter durations. Currently the 15 year SWAP rate is consistent with the smoothed yield curve, but we would give a relatively low weighting to this information.

8.4.6 Under option 1, a step in the rates has the advantage that it results in the long-term real return immediately after the end of the yield curve. This will produce smooth spot rates and is simple. However, it potentially ignores additional market information from SWAP rates. The Treasury has decided not to use this as it ignores information from bank SWAP rates and will produce “odd looking” forward rates, although the spot rates will be smooth.

8.4.7 Under option 2, starting the interpolation before the end of the yield curve arguably ignores some market data to the end of the curve. For that reason it is not our preferred option.

8.4.8 Under option 3, interpolation for a fixed duration looks reasonable at face value, however the interpolation period will change as the duration of the longest stock changes with time. As we are looking to provide some future proofing we have ruled out this option.

8.4.9 Under option 4, interpolation to a fixed date keeps the interpolation period the same. However it still requires the selection of an arbitrary period to smooth over.

8.4.10 Under option 5, interpolation at a fixed slope is also attractive however it also requires the selection of an arbitrary slope.

8.4.11 Of all the options, the last two are the most reasonable for our methodology. We recognize that both options require an arbitrary assumption about either a time period or slope. Even so, we have elected to use option 4 as it provides the most reasonable and pragmatic framework in our “bridging” methodology.

8.4.12 The choice of the period to be smoothed over is slightly arbitrary. There are arguments that the long-term rate should be smoothed to at 20 year duration, which would currently give a smoothing period of 9 years. This may be the case if only nominal yields were relevant; however because of the importance we have placed on real risk-free rates, we believe that a shorter smoothing period is more appropriate.

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