In the European Union financial regulation requires that life and pension (L&P) companies use the Smith and Wilson (2000) model for the term structure of risk-free interest rates when valuing their liabilities and long term guarantees. Some key features of this model are that it allows for a perfect fit to market observed bond prices, and that its extrapolated long rates converge towards a constant level, the Ultimate Forward Rate (UFR). Both this level and the rate at which convergence towards it takes place are directly specified via parameters of the model. Since the Smith–Wilson model is not one of finance theory’s standard term structure models, we introduce the model and summarize its most important mathematical properties. We also describe how the European Solvency II regulation came to embrace this particular model. The paper moves on to document how the regulation also imposes quite detailed and tight restrictions on how the Smith–Wilson model should be parameterized and applied. We argue that many of these implementation instructions – one of which is the regulator’s specification of a very high UFR – seem biased in the same direction and that this could indicate a systematic attempt to “lift” the term structure curve up and away from its true location whereby artificially high discount rates are induced. The result of the bias is not only significant undervaluation of L&P liabilities but also a peculiar contradiction of the Solvency II overall objective of enhancing financial stability and of protecting policyholders via the promotion of economic valuation in accordance with market consistent principles. The paper’s analysis is accompanied by valuation illustrations based on data on the liability composition of an actual medium-sized Danish pension fund. The results suggest that the undervaluation of liabilities resulting from use of the regulation compliant Smith–Wilson model can be massive compared to results obtained from some alternative and more freely calibrated models.

An analysis of the Solvency II regulatory framework’s Smith-Wilson model for the term structure of risk-free interest rates

What Does a Term Structure Model Imply About Very Long-Term Discount Rates?

We extrapolate interest rate yield curves for the purpose of discounting very long-dated pension liabilities and insurance contracts. The extrapolation uses a no-arbitrage term structure model estimated on liquid euro swap instruments with maturities between 5 and 20 years. The extrapolation towards maturities up to 100 years appears mainly driven by the near unit root of the level factor under the risk neutral measure. In a no-arbitrage term structure model this leads to a strong convexity effect at the very long end of the yields curve and a very low ultimate forward rate (UFR). Our estimates use Bayesian methods with an informative prior on mean reversion parameters and the unconditional means. As a puzzle for term structure models we find that our extrapolated curves are generally above observed swap rates at maturities in the 20-50 years range.

What Does a Term Structure Model Imply About Very Long-Term Interest Rates?

/pdf/long-term-extrapolation-and-hedging-of-the-south-african-15ez4vpjdh.pdf

Long term extrapolation and hedging of the South African yield curve

Mathematical models form the basis of decisions made in the fields of economics, finance, medicine and climatology. Given that things tend not to follow a mathematical model in real life, there's almost always a mismatch between model-based decisions and their desired effect. This dissertation examines alternative models that are statistically plausible. It also seeks to find an investment strategy that is less sensitive to deviations from the current model. With life expectancy continuing to rise, model-based interest rates will have a considerable impact on the policies developed by pension funds and insurance companies over the next century or so. The amount of interest determines how much money should be set aside to cover future pay-outs. Given the current pension debate, this dissertation identifies the uncertainties associated with the interest term structure.

Model uncertainty : the effect on robustness, estimation and stochastic optimisation

Many life insurance companies and pension funds have been involved in selling products which offer policyholders an element of protection against long term interest rate risk. Typical products, and product features, that offer protection against long term interest rate risk include conventional annuities, defined benefit pensions and guaranteed annuity options. These products and features typically provide protection at terms for which no interest rate instruments are tradable and the associated long term interest rate risks can be very large. Due to the lack of liquidity in the swap and bond market beyond 30 years, companies (and pension funds) often use fairly simplistic strategies to hedge these risks (such as investing in the longest available government bond). This paper examines some alternative approaches to dealing with long term interest rate risk. Specifically, the paper examines some common approaches that are used to forecast the yield curve beyond 30 years. The hedging implications of the various forecasting approaches are considered, and the performance of each approach is assessed in light of historical evidence. KEYWORDS Hedging; Forecasting; Long Term Interest Rate Risk Management; South Africa; Derivatives; Yield Curve CONTACT DETAILS Michael Thomas mithomas@deloitte.co.uk Eben Mare eben.mare@up.ac.za

Long Term Forecasting and Hedging of the South African Yield Curve

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Long Term Forecasting and Hedging of the South African Yield Curve