Smith–Wilson method

The Smith–Wilson method is a method for extrapolating forward rates. It is recommended by EIOPA to extrapolate interest rates. It was introduced in 2000 by A. Smith and T. Wilson for Bacon & Woodrow.

Mathematical formulation

Let UFR be some ultimate forward rate and $u_{i}$ be the time to the i'th maturity. Then $P(t)$ defines the price of a zero-coupon bond at time t.

$P(t)=e^{-UFR\cdot t}+\sum _{j=1}^{N}\xi _{j}\cdot W(t,u_{j})$

Where $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$

and the symmetric W matrix is $W=(W(u_{i},u_{j}))_{i=1,...,N:j=1,...,N}$

and $p=(P(u_{1}),...,P(u_{N}))^{T}$, $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$, $\xi =W^{-1}(p-\mu )$.

References

• A Technical Note on the Smith-Wilson Method, The Financial Supervisory Authority of Norway, (1 July 2010)
• Lagerås, Andreas & Lindholm, Mathias. (2016). Issues with the Smith-Wilson method. Insurance: Mathematics and Economics. 71. 10.1016/j.insmatheco.2016.08.009.
• Smith, A. and Wilson, T. (2000). Fitting Yield Curves with Long Term Constraints. Research report, Bacon & Woodrow.
• Technical documentation of the methodology to derive EIOPA's risk-free interest rate term structures