Parameter Uncertainty and Residual Estimation Risk

• Author: Andreas Tsanakas,Valeria Bignozzi
• DOI: http://doi.org/10.1111/jori.12075
• Date: 01 December 2016
• Published date: 01 December 2016

©2015 The Journal of Risk and Insurance. Vol.83, No. 4, 949–978 (2016).

DOI: 10.1111/jori.12075

Parameter Uncertainty and Residual

Estimation Risk

Valeria Bignozzi

Andreas Tsanakas

Abstract

The notion of residual estimation risk is introduced to quantify the impact

of parameter uncertainty on capital adequacy, for a given risk measure and

capital estimation procedure. Residual risk equals the risk measure applied

to the difference between a random loss and the corresponding capital esti-

mator. Modified estimation procedures are proposed, based on parametric

bootstrapping and predictive distributions, which compensate the impact of

parameter uncertainty and lead to higher capital requirements. In the par-

ticular case of location-scale families, the analysis simplifies and a capital

estimator can always be found that leads to a residual risk of exactly zero.

Introduction

Insurance decisions, such as pricing, reserving, and capital setting, are informed by

the outputs of statistical risk models. Such models are typically parametric. However,

the true values of parameters are in principle unknown and must be estimated from

samples of relevant observations. Such samples are often very small and statistical er-

ror means that the estimated parameter values can diverge substantially from the true

values. The potential for error in estimated parameters, termed parameter uncertainty,

introduces possible error into insurance decisions based on model outputs. For exam-

ple, if the estimated frequency used to model claims from a natural hazard is lower

than the true one, insurance policies may be underpriced and a portfolio of such poli-

cies undercapitalized. It is conventional to view parameter uncertainty in the context

of an otherwise correctly specified risk model. If the model is not known with certainty,

we talk of model uncertainty.

Decisions sensitive to tails of distributions, for which limited information resides in

available data, are more sensitive to parameter error. Thus, there is particular focus on

applications where the extremes of loss distributions areof interest, for example, when

Valeria Bignozzi is at the School of Economics and Management, University of Firenze,

Italy. Bignozzi can be contacted via e-mail: valeria.bignozzi@unifi.it. Andreas Tsanakas is at

the Cass Business School, City University London. Tsanakas can be contacted via e-mail:

a.tsanakas.1@city.ac.uk.This is a preprint of an article accepted for publication in the Journal of

Risk and Insurance, ©2004 the American Risk and Insurance Association.

949

950 The Journal of Risk and Insurance

setting capital by a tail risk measure like Value-at-Risk (VaR) and Tail-Value-at-Risk

(TVaR), or when pricing high reinsurance layers.

Investigations by insurance practitioners have shown that the impact of parameter

uncertainty in realistic modeling applications can indeed be very substantial (see

Mata, 2000; Borowicz and Norman, 2008). High sensitivity to parameter error has

also been demonstrated in the context of credit risk modeling by McNeil, Frey, and

Embrechts(2005) and in a banking context by Jorion (1996), who argues that confidence

bands should be reported alongside estimated VaRs. Cont, Deguest, and Scandolo

(2010) study risk measurement procedures and their sensitivity to changes in data

used for estimation, with reference to the notion of qualitative robustness.

In an early response to parameter uncertainty, Venezian (1983) recommends an ex-

plicit increase to insurer capital, with the adjustment reflecting estimation volatility.

Allowance for parameter uncertainty is now often made in actuarial risk models;

see Cairns, Blake, and Dowd (2006) on stochastic mortality and longevity bond pricing

and Verrall and England (2006) on stochastic claims reserving. A broadly applicable

response to parameter uncertainty is to work with predictive distributions, arising

as weighted averages of distributions with different parameter values. Cairns (2000)

argues in favor of a fully Bayesian approach to capture parameter (as well as model)

uncertainty. Predictive distributions tend to be more volatile than distributions that

are derived via point estimates of parameters and thus can lead to more conservative

decisions (see also Landsman and Tsanakas,2012). Gerrard and Tsanakas (2011) show

that the increase in VaR,which using a predictive distribution implies, is appropriate

for restoring a frequentist failure probability to its required nominal level.

In the present contribution, we also study risk measurement procedures, but from a

perspective that is complementary to that of Cont, Deguest, and Scandolo (2010). We

introduce a criterion for assessing, in monetary units, the impact of parameter un-

certainty on capital adequacy. We assume that the required capital for an insurance

company is calculated by applying to a random loss variable a risk measure that is

positive homogeneous, translation invariant, and law invariant. Commonly used risk

measures, such as VaRor TVaR, satisfy these properties. A random sample for the loss

is available and the estimated capital is a function of that sample; we call this function

the capital estimator. To assess the effectiveness of a capital-setting procedure, the risk

measure is applied to the difference between the loss (a random variable representing

process variability) minus the capital estimator (a random variable reflecting variabil-

ity due to estimation). The result of this calculation we call residual estimation risk.Ifthe

distribution of the loss is known, the residual risk is zero. In general, residual estima-

tion risk is not equal to zero and can be viewed as the (deterministic) amount of capital

that needs to be added to the capital estimator such that the total position becomes

acceptable. When the sample size grows, the capital estimator typically converges to

the unknown required capital and the residual estimation risk thus goes to zero.

This approach can in principle be used to assess the effectiveness of any estimation

procedure, even though our focus is on parametric models. In particular,model uncer-

tainty can also be assessed via residual estimation risk, if model selection is data driven