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 {\displaystyle u_{i}} be the time to the i'th maturity. Then P ( t ) {\displaystyle P(t)} defines the price of a zero-coupon bond at time t.
P ( t ) = e − U F R ⋅ t + ∑ j = 1 N ξ j ⋅ W ( t , u j ) {\displaystyle P(t)=e^{-UFR\cdot t}+\sum _{j=1}^{N}\xi _{j}\cdot W(t,u_{j})}
Where
W ( t , u j ) = e − U F R ⋅ ( t + u j ) ⋅ ( α ⋅ min ( t , u j ) − 0.5 e − α ⋅ max ( t , u j ) ⋅ ( e α ⋅ min ( t , u j ) − e − α ⋅ min ( t , u j ) ) ) {\displaystyle 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 {\displaystyle W=(W(u_{i},u_{j}))_{i=1,...,N:j=1,...,N}}
and
p = ( P ( u 1 ) , . . . , P ( u N ) ) T {\displaystyle p=(P(u_{1}),...,P(u_{N}))^{T}} ,
μ = ( e − U F R ⋅ u 1 , . . . , e − U F R ⋅ u N ) T {\displaystyle \mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}} ,
ξ = W − 1 ( p − μ ) {\displaystyle \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