My 3-month hiatus has ended thanks to the Canadian long weekend. In these past few months I haven't had much time time to commit to posting new content, primarily because I started a new job. This has put my research interests on hold for the time being. So while I will likely not post as frequently as I have in the past, I will be providing more content when time permits. When my schedule reverts back to manageable hours, I will continue with weekly updates (when possible).

The purpose of this post is to introduce the use of a new and alternative measure, *residual estimation risk*. To motivate the definition, consider a firm that attempts to estimate the loss of capital in one of their portfolios. This quantity may be known as the capital estimator. The firm acknowledges that the capital estimator is a mere estimation, and in lieu of this, would like to determine the inherent risk in relying on such an estimate. This inherent risk can be viewed as the quantity that remains after the portfolio incurs realized losses. This residual amount is specifically the quantity that the firm aims to measure; the residual estimation risk (RER). Initially introduced by Bignozzi and Tsanakas in [1, 2, 3], the authors define RER as any risk measure applied to the difference between actual losses and a capital estimator. They provide thorough results on how to estimate RER under parametric and empirical loss distributions, eliminate RER by adjusting the capital estimator via bootstrapping, as well as how to explicitly calculate parameter uncertainty, model uncertainty, and model risk under various distributional assumptions.

One can formally define RER as:

Above, is a random variable representing an actual loss, is the capital estimator, and is any risk measure satisfying the following properties:

i. Monotonicity

ii. Translation Invariance

iii. Positive Homogeneity:

iv. Law Invariance:

can be thought of as a model or function that takes historical loss data and outputs an estimate of the capital loss. In such a case, . If perfectly estimates the true loss of from the historical loss data , then . Thus, one can see that if , then by the properties above, .

Some common risk measures that satisfy these properties are Value-At-Risk (VAR), Expected Shortfall (ES), and Restricted Value-At-Risk (RVAR). Following the expression above, RER can be interpreted as any risk measure applied to the absolute error distribution of model outputs and actual outcomes. Equivalently, RER is the additional amount of that should be added to the current estimates to safely cover historical levels of. This is mathematically equivalent to

As such, RER is sensitive to the shape of the absolute error distribution and selection of risk measure. From the above expression, one can infer the following. If RER is negative, the model estimates are sufficient in covering actuals that may result from the true distribution. In this case, one may conclude that the model used to generateproduces conservative enough estimates. Conversely, if RER is positive, the model estimates are not sufficient in covering exposures that may result from the true distribution. In this case, one may conclude that the model used to generate produces conservative enough estimates. Conversely, if RER is positive, the model estimates are not sufficient in covering exposures that may result from the true distribution. In this case, we may conclude that the model does not produce conservative enough estimates of . Since the distribution of underestimates the risk of the true distribution, it should be interpreted as a warning. For if the historical distribution is any indication of the future, use of these estimates may continuously underestimate the portfolio (at some pre-specified level of confidence). When RER is zero, we have completely eliminated any excess risk; i.e. the model estimates exactly cover the actuals that may result from the true distribution. It can be said that when RER=0, the resulting distribution of model outputs is *optimal. *

Simple risk measure functions can be produced in R to calculate RER. In the functions below, the argument x represents the difference between the actual loss and capital estimator.

VAR <- function(x,p){ x=sort(x) return(x[floor(length(x)*p)]) } TVAR <- function(x,p){ x=sort(x) return(mean(x[floor(length(x)*p):length(x)])) } RVAR <- function(x,p,q){ x=sort(x) return(mean(x[floor(length(x)*p):floor(length(x)*q)])) }

Or in SAS:

*Value-at-Risk Function; %MACRO VAR(dataset= , target= , prob= ); proc sort data=&dataset out=sorted_target; by ⌖ run; proc sql noprint; select count(&target) into :number from sorted_target; quit; data sorted_target; set sorted_target; index=_N_; run; proc sql print; select &target from sorted_target where index=floor(&number*&prob); drop table sorted_target; quit; %MEND; *Expected Shortfall Function; %MACRO ES(dataset= , target= , prob= ); proc sort data=&dataset out=sorted_target; by ⌖ run; proc sql noprint; select count(&target) into :number from sorted_target; quit; data sorted_target; set sorted_target; index=_N_; run; proc sql print; select mean(&target) from sorted_target where index > floor(&number*&prob); drop table sorted_target; quit; %MEND; *Restricted Value-at-Risk Function; %MACRO RVAR(dataset= , target= , prob1= , prob2= ); proc sort data=&dataset out=sorted_target; by ⌖ run; proc sql noprint; select count(&target) into :number from sorted_target; quit; data sorted_target; set sorted_target; index=_N_; run; proc sql print; select mean(&target) from sorted_target where (index > floor(&number*&prob1)) and (index < floor(&number*&prob2)); drop table sorted_target; quit; %MEND;

In a subsequent post we will implement the above description of RER and demonstrate it's practical use.

[1]. Bignozzi, Valeria and Tsanakas, Andreas, *Model Uncertainty in Risk Capital Measurement* (2013). Available at SSRN: http://dx.doi.org/10.2139/ssrn.2334797

[2]. Bignozzi, Valeria and Tsanakas, Andreas, *Parameter Uncertainty and Residual Estimation Risk* (2014). This is a preprint of an article accepted for publication in the Journal of Risk and Insurance. Available at SSRN: http://dx.doi.org/10.2139/ssrn.2158779

[3]. Bignozzi, Valeria and Tsanakas, Andreas, *Residual Estimation Risk* (2012). Submitted for publication. Available at ETH-Zurich: http://www.math.ethz.ch/u/bvaleria/residualrisk