Finance: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 [math]\displaystyle{ u_i }[/math] be the time to the i'th maturity. Then [math]\displaystyle{ P(t) }[/math] defines the price of a zero-coupon bond at time t.
[math]\displaystyle{ P(t) = e^{-UFR\cdot t} + \sum_{j=1}^N \xi_j \cdot W(t, u_j) }[/math]
Where [math]\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)})) }[/math]
and the symmetric W matrix is [math]\displaystyle{ W = (W(u_i, u_j))_{i=1,...,N:j=1,...,N} }[/math]
and [math]\displaystyle{ p = (P(u_1), ..., P(u_N))^T }[/math], [math]\displaystyle{ \mu = (e^{-UFR\cdot u_1}, ..., e^{-UFR\cdot u_N})^T }[/math], [math]\displaystyle{ \xi = W^{-1}(p-\mu) }[/math].
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
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