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