### Abstract

The European Parliamentís Solvency II Directive, scheduled to come into effect on 1 January 2016, introduces new regulation for insurance. This aims to establish a consistently improved level of policyholder protection via a three-pillared process. The first pillar contains quantitative requirements for the insurance industry relating to Technical Provisions and the Solvency Capital Requirement. The reserve risk is a substantial contributor to the insurance risk and is addressed by the quantitative requirements.

We demonstrate ways that ICRFS-Plus™ can be used to fulfil the quantitative requirements for non-life reserve risk in particular, in the context of the European Commissionís Quantitative Impact Study (2010). This includes a standard formula with undertaking specific parameters and a partial Solvency II Internal Model.

**Keywords** Case Development Result, One-Year Risk Horizon, Solvency II
Standard Formula, Solvency II Internal Model, Solvency Capital Requirement

#### 1. Introduction

Insolvency risk is an inherent part of insurance business. Insurance risk, and reserve risk in particular is not hedgeable. The price of this risk cannot be determined by the market as there is no open market for insurance liabilities. For Solvency II Quantitative Requirements (European Commission et al. (2007a, 2009)), the risk is characterised by

- the risk profile: the distribution of basic own funds,
- the risk measure: value-at-risk, applied to Solvency Capital Requirements (SCR),
- risk tolerance: set at 99.5% with a one year time horizon (1-in-200-year distress event),
- risk margins: determined by using the Cost-of-Capital method.

The paper is organised as follows.

- We discuss important details from the Quantitative Impact Study.
- We describe the CDR, and how the standard deviation given ICRFS-Plus™ can be used with standard method 2.
- We discuss how the standard deviation from the aggregate of LoBs can be plugged into a partial internal model that extends standard method 2.
- We discuss a simulation-based internal model for non-life risk implemented in ICRFS-Plus™, which is free from some of the limitations imposed by the CDR approach.
- A case study is presented that illustrates method 2 and the Internal Models.

#### 2. The Quantitative Impact Study

The *Quantitative Impact Study*, QIS5 (European Commission et al. (2010))
addresses Quantitative Requirements in reserve risk for Solvency II by presenting three standard methods which can be used to estimate the standard
deviation of the reserve risk for a line of business (LoB):

*the proportionality principle*- the variance of the best estimate for claims outstanding in one year plus the incremental claims paid over the same year is taken to be proportional to the current best estimate for claims outstanding (European Commission et al. (2010), SCR.10.40.)- an undertaking-specific standard deviation based on
*a third party estimate*(European Commission et al. (2010), SCR.10.48.) - an undertaking-specific standard deviation based on
*the Chain Ladder method*(European Commission et al. (2010), SCR.10.54.)

The standard methods 2 and 3 are known as Merz-Wüthrich approach (Merz and Wüthrich (2008)). We are sceptical about the proportionality principle which underlies method 1, see Munroe et al. (2015). We are also sceptical of the Chain Ladder method, see Barnett et al. (2005), Barnett and Zehnwirth (2008). Fortunately, method 2 allows for an undertaking-speciÖc estimate of the standard deviation.

The standard deviation for reserve risk for each LoB is one of the source components for the QIS5 standard formula for SCR. The standard formula for SCR is invariant to the method that produces the standard deviation.

The Claims Development Result (*CDR*) concept is essential for standard methods 2 and 3 with undertaking-specific parameters, as the standard
deviation of *CDR* quantifies the standard deviation of reserve risk.

Apart from the standard formulas, the regulation also allows for an Internal Model that estimates capital requirements for non-life risk, subject
to supervisory approval. This model is only a *partial* internal model, as it
covers non-life risk (reserve and premium risk) only.

#### 3. Claims development result

The Claims development result (*CDR*) was introduced by Wüthrich et al.
(2008) and is commonly used in the Solvency II literature. *CDR* is a random
variable that represents the difference between the expected ultimate loss at
inception (or at a given time, *t*) and one year later. It is commonly said that
the *CDR*(1) represents movement of an economic balance sheet between now
and one year ahead.

Its distribution is linked with the Solvency Capital Requirement (*SCR*)
that is nominally set to the 99.5th quantile of the basic own funds distribution
limited to the year in question.

In fact, with two years in run-off, the CDR at inception could be deÖned
as a function of L_{1}, the next year's loss, by the formula:

CDR(1) = E(L_{1}) L_{1} + E(L_{2}) E(L_{2}|L_{1}),

where L_{t} is the loss in the future year t, and conditioning L_{2}|L_{1} indicates that CDR(1) is based on the joint distribution f(L_{1}, L_{2}) = f(L_{1})f(L_{2}|L_{1}).

This leads to (in-line with CDR-centric papers)

SCR^{cdr} = VaR_{0.5%}(CDR(1)) (1)

which corresponds to

SCR^{cdr} = VaR_{99.5%}(L_{1}) + E(L_{2}|L_{1} = λ_{1}) E(L_{2}),

where VaR_{99.5%}(L_{1}) is the 99.5% percentile of L_{1} above its best estimate,
E(L_{1}), and

λ_{1} = E(L_{1}) + VaR_{99.5%}(L_{1}),

see Appendix B in Munroe et al. (2015) for technicalities.

Generalizing for *n* years in run-off is easy

CDR(1) = E(L_{1}) L_{1} + Σn t=2 (E(L_{t}) E(L_{t}|L_{1})) (2)

= E(L) E(L|L_{1}), (3)

SCR^{cdr} = VaR_{99.5%}(L_{1}1) + Σn t=2 (E(L_{t}|L_{1} = λ1) E(L_{t})) (4)

where L is the ultimate loss

L = Σn t=1 L_{t}(5)

In the QIS5 standard formula for *SCR*, **S.D.(CDR(1))** is the proxy for **SCR**
and (4) is not used, see Appendix A. SCR.9.16. in particular.

The *CDR* has zero mean. Taking expectation with respect to L_{1} from
both sides of

E(CDR(1)) = 0 (6)

The *variance of* **CDR(1)** *is essential to the Merz-W¸thrich approach*.
Taking the variance of both sides of (3),

Var(CDR(1)) = V ar(E(L|L_{1})) (7)

(7)
as E(L) is a constant and E(L|L_{1}) is a random variable.

By the law of total variance

Var(L) = E(Var(L|L_{1})) + V ar(E(L|L_{1})). (8)

Substituting (7) into above,

Var(L) = E(Var(L|L_{1})) + V ar(CDR(1)). (9)

As L|L_{1} is a random variable, E(V ar(L|L_{1})) is greater than zero, so the
variance of CDR(1) is less than the variance of the ultimate loss L

Var(CDR(1)) < Var(L). (10)

The variance of the expected ultimate loss conditional on the first future year loss is less than the unconditional variance of the ultimate loss.

Generalizing of *CDR*(1) and its properties for *CDR(t)* is straightforward.
In the following we focus on *CDR*(1).