approximation of heavy models using radial basis functions
DESCRIPTION
Approximation of heavy models using Radial Basis Functions. Graeme Alexander (Deloitte) Jeremy Levesley (Leicester). The problem. Calculate Value at Risk Need to determine 0 .5 th percentile of insurer’s net assets in one year Net assets = f(R1,R2,R3,... Rn ) - PowerPoint PPT PresentationTRANSCRIPT
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Approximation of heavy models using Radial Basis FunctionsGraeme Alexander (Deloitte)Jeremy Levesley (Leicester)
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”
How to compute coefficients Interpolation
Linear Equations
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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
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• Allow T and disc to vary stochasticallydisc~ 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 = 9.27RBF (Gaussian) estimate = 8.86, error = 4%Polynomial estimate = 9.25, error = 0.19%
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 £1,000Strike price (guarantee) is £500Assets ~ N (1000, 3002), disc~ N (8%, 2.5%2)
This time the radial basis function (“norm”) does better:
Actual = £83.53RBF 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
• Choice of l is crucial
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of measure smoothness)(
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freedom of degreesenough ifion interpolat ,squaresleast ,0
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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.