# Bisection method that finds the optimal parameter α for the Smith & Wilson algorithm#

This repository has an implementation for a simple bisection method that finds the optimal parameter α for the Smith & Wilson algorithm often used in insurance to interpolate/extrapolate rates or yields.

The implementation is based on Technical documentation of the Methodology to derive EIOPA’s risk-free interest rate term structure and Wiki on Bisection method

## Problem#

Before using the Smith & Wilson algorithm, the user needs to provide the convergence speed parameter α. This parameter needs to be calibrated primarily so that that the extrapolated result matches the desired long-term behaviour.

## Solution#

By transforming the minimization problem at the point of convergence into a problem of finding a root of the shifted function g(α) - τ, this repository implements a simple bisection algorithm to find the optimal α.

### Input#

• The minimum allowed value of the convergence speed parameter α.

• The maximum allowed value of the convergence speed parameter α.

• Maturities of bonds, observed on the market and provided as output.

• Zero-coupon rates, for which the user wishes to calibrate the algorithm. Each rate belongs to an observable zero-coupon bond with a known maturity.

• The ultimate forward rate towards which the user wishes the resulting curve to converge.

• Allowed difference between the given ultimate forward rate and the resulting curve.

• The numeric precision of the calculation. Higher the precision, more accurate the estimation of the root.

• The maximum number of iterations allowed. This is to prevent an infinite loop in case the method does not converge to a solution.

### Output#

• Optimal value of the parameter α (if the bisection method converged).

Note that to be consistent with EIOPA’s recommandations, the lower bound of the interval should be set to 0.05.

## Getting started#

```import numpy as np
from smith_wilson_funcs import SWCalibrate, SWExtrapolate
from bisection_alpha import Galfa as Galfa
from bisection_alpha import BisectionAlpha as BisectionAlpha

# maturities of bonds observed on the market
M_Obs = np.transpose(np.array([1, 2, 4, 5, 6, 7]))

# yields observed on the market
r_Obs = np.transpose(np.array([0.01, 0.02, 0.03, 0.032, 0.035, 0.04]))

# ultimate forward rate
ufr = 0.04
# Numeric precision of the optimisation
Precision = 0.0000000001

# Targeted distance between the extrapolated curve and the ultimate forward rate at the convergence point
Tau = 0.0001 # 1 basis point

# Examples of a call to Galfa and BisectionAlpha
print("Example in the documentation for Galfa: "+ str(Galfa(M_Obs, r_Obs, ufr, 0.15, Tau)))
print("Example in the documentation for BisectionAlpha: "+ str(BisectionAlpha(0.05, 0.5, M_Obs, r_Obs, ufr, Tau, Precision, 1000)))
```

Note that this implementation use functions `SWCalibrate()` from the Smith & Wilson algorithm algorithm. They are duplicated to this repository for completeness.

If there are any inconsistencies or suggestions, raise an issue, or contact the original authors, or start a discussion at lifelib Discussions.