SmithWilson¶
The main space in the SmithWilson model
SmithWilson
is the only space in the SmithWilson model.
The SmithWilson method is used for extrapolating riskfree interest rates under the Solvency II framework. The method is described in “QIS 5 Riskfree interest rates – Extrapolation method”, a technical paper issued by CEIOPS (the predecessor of EIOPA). The technical paper is available on EIOPA’s web site. Cells in this space are named consistently with the mathematical symbols in the technical paper.
References

log
¶ log function from the standard math library

exp
¶ exp function from the standard math library

N
¶ Number of durations for which the observed spot rates are available

spot_rates
¶ List of the observed spot rates (annual compound)

UFR
¶ The ultimate forward rate (continuous compound)

alpha
¶ The convergence parameter \(\alpha\)
Cells

Zerocoupon bond prices extrapolated by the SmithWilson method. 

The extrapolated annual compound sport rates. 

The Wilson functions. 

The 

Observed zerocoupon bond prices at time \(u_i\). 

The 

Ultimate Forward Rate (UFR) discount factors 
The 


Time (\(u_i\)) 

The \(\zeta_i\) parameters fitted to the observed spot rates. 
The 

u
(i)[source]¶ Time (\(u_i\))
u()
is a series of discrete time points.i
is a 1based index. By default,u()
just returnsi
.For i = 1, …,
N
,u()
corresponds to \(u_i\) in the technical paper, The time to maturities of the observed zerocoupon bond prices.spot_rates
, the observed spot rates must exist at eachu(i)
from i = 1 toN
. Note thatspot_rates
is 0based, so the sport rate atu(i)
isspot_rates[i1]
. Parameters
i – The time index (1, 2, …)

m
(i)[source]¶ Observed zerocoupon bond prices at time \(u_i\).
m()
is calculated fromspot_rates
as \((1 + spot\_rates[i1])^{u_i}\) Parameters
i (int) – Time index (1, 2, …,
N
)

mu
(i)[source]¶ Ultimate Forward Rate (UFR) discount factors
mu()
is defined as \(e^{UFR\cdot u_i}\). Parameters
i (int) – Time index (1, 2, …)

W
(i, j)[source]¶ The Wilson functions.
W()
corresponds to formula (2) on page 16 in the technical paper defined as:\[W(t, u_j)= \ e^{UFR\cdot (t+u_j)}\cdot \ \left\{ \ \alpha\cdot\min(t, u_j) \ 0.5\cdot e^{\alpha\cdot\max(t, u_j)}\cdot( \ e^{\alpha\cdot\min(t, u_j)} \ e^{\alpha\cdot\min(t, u_j)} \ ) \ \right\}\]where \(t = u_i\).

zeta_vector
()[source]¶ The
zeta()
vector.zeta_vector()
returns \(\zeta\) parameters calculated by formula (5) on page 17 in the technical paper, which is\[\bf \zeta= W^{1}(p\mu)\]

zeta
(i)[source]¶ The \(\zeta_i\) parameters fitted to the observed spot rates.
 Parameters
i (int) – Time index (1, 2, …,
N
)

P
(i)[source]¶ Zerocoupon bond prices extrapolated by the SmithWilson method.
P()
corresponds to formula (1) on page 16 or formula (6) on page 18 in the technical paper, defined as:\[P(t) = e^{UFR\cdot t}+\sum_{j=1}^{N}\zeta_{j}\cdot W(t, u_j)\]substituting \(t\) with \(u_i\).
The values of
P()
fori=1, ..., N
should be the same as the values ofm()
, the observed bond prices. Parameters
i (int) – Time index (1, 2, …)

R
(i)[source]¶ The extrapolated annual compound sport rates.
R()
corresponds to \(R_t\) defined as:\[R_t = \left(\frac{1}{P(t)}\right)^\left(\frac{1}{t}\right)1\]on page 18 in the technical paper, substituting \(t\) with \(u_i\).
The values of
R()
fori=1,...,N
should be same as the values of the observed spot ratesspot_rates
for 0, … ,N1. Parameters
i (int) – Time index (1, 2, …)