Approximation of heavy models
using Radial Basis Functions
Graeme Alexander (Deloitte)
Jeremy Levesley (Leicester)
www.le.ac.uk
The problem
• Calculate Value at Risk
• Need to determine 0.5th percentile of insurer’s
net assets in one year
• Net assets = f(R1,R2,R3,...Rn)
• Many firms have previously calculated the
percentiles of univariate distns, and aggregated
using correlation matrix / copula approach
Moving to Solvency II
• For internal model approach, strongly encouraged to
calculate the whole distribution of Net Assets, not just
the percentile
• It is a simple matter to generate 100,000 simulations of
(R1,R2,..Rn)
• However, evaluating f(r1,r2,..rn) for a single
realisation of the risk vector using the “heavy model”
can take hours!!
• Common approach: Run the heavy models on a small
number of points, and interpolate to obtain estimator
function fE(r1, r2, ..,rn), known as a “lite model”
Splines
Linear spline
approximation to sin(x)
Combination of hat
functions
Cubic Splines
Cubic spline
approximation to sin(x)
Combination of B-splines
Radial basis function approximation
t1 < t2 <
• A basis function f
• Set of points
< tn
• Approximation
n
sn (t) = åaif (t - ti )
i=1
More generally
sn ( x) 
   x  y 
y
y Y
Data Y
x
x y
Gaussian
 ( r )  exp(  c r )
2
2
y
How to compute coefficients
Interpolation
s n ( x )  f ( x ),
xY.
Linear Equations
   y1  y1

  y 2  y 1



  y n  y1


  y1  y 2
  y2  y2




  yn  y2





  y1  y n     1 
 f ( y1 ) 
  

  y 2  y n   2
f ( y2 )
 

     

  

  y n  y n   n   f ( y n ) 
An Example - annuity
•
Difficult to test our interpolation on real-life data due to the length of time it
takes to run heavy models
•
So let’s take a simple product, a single life annuity, £1 payable p.a.
•
Assume just two risk factors, discount rate and mortality
•
Assume a constant rate of mortality 1/T in each future year. Thus, the cash flows
are:
(T-1)/T at the end of year 1, (T-2)/T at end of year 2,1 / T at end of year T-1
T 1
PV 

 1 
t 1

t 
t
 (1  disc )
T 
 1

1  disc 
1
 

1

2 
T 
disc
T
.
disc
(
1

disc
)



•Allow T and disc to vary stochastically
disc~ N (8%, 2.5%2)
T ~ N (20,9)
An Example - annuity
• We used 10 fitting points.
• It turns out that the polynomial function (order 3) performs slightly better
than the RBF
99.5th percentile of liability:
Actual = 9.27
RBF (Gaussian) estimate = 8.86, error = 4%
Polynomial estimate = 9.25, error = 0.19%
Annuity – how good was the fit
What if there is a discontinuity?
Chart shows liabilities against T, for fixed
disc=8%: Was fitted using “norm” function.
Unlikely to arise in practice, though.
However....
Choice of polynomial or RBF
• Choice of appropriate polynomial terms is
problematic. High degree polynomials are famously
unstable (Gibb’s phenomena)
• Choice of RBF is related to the “smoothness of the
data” – see difference between Gaussian and norm
function. This requires some user input, but does
not require other experimentation.
• RBF is adaptable to the placement of new points
near to where error is being observed in
approximation. This is not robust with polynomial
approximation.
With profits
•
The realistic balance sheet includes a “cost of guarantees”
•
For example, suppose there is a guaranteed sum assured on the assets, equal
to £500.
•
Crudely, we can model the cost of guarantees as a put option on the asset
share.
Assume that:
Asset Share is £1,000
Strike price (guarantee) is £500
Assets ~ N (1000, 3002), disc~ N (8%, 2.5%2)
This time the radial basis function (“norm”) does better:
Actual = £83.53
RBF estimate = £74.6, error = 11%
Polynomial estimate = £1,735, error =
1978%
With profits
 Polynomial has difficulty coping with
the particular behaviour shown
 Also, the fitting problem is prone to
becoming singular
 RBF (using “norm”) does much better
Smoothing splines
• If the data is noisy
• Minimise
l ( ) 

( f ( y )  s Y ( y ))  l  ( s Y ).
2
y Y
 ( g )  smoothness
measure
of g
• Choice of l is crucial
l  0 , least squares
l   , interpolat ion if enough degrees of freedom
Summary
• It is worthwhile to explore the use of radial basis
functions for approximation.
• They are good in high dimensions, and adapt easily to
the local shape of the surface.
• Polynomials are good where the surface is close to a
polynomial in reality
• They are also difficult to implement in high dimensions.
• There are different RBFs and different approximation
processes depending on the nature and reliability of the
data.