# BEL best estimate liability, deduced from the mean loss as a present value of mean loss,

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1 Calculations of SCR, Technical Provisions, Market Value Margin (MVM) using the Cost of Capital method, for a one-year risk horizon, based on an ICRFS-Plus single composite model and associated forecast scenarios for the aggregate of multiple LOBs Introduction There are many interpretations of the Solvency II Capital Requirements (SCR) and Market Value Margins (MVM) in the context of long tail liabilities. In particular, there are several CEIOPS' Quantitative Impact Studies (QIS) and there are so-called Internal models. Furthermore, as with a Balance Sheet, there are two sides in the Solvency II regulation, liabilities and assets. The approach presented here can be categorised as an Internal model, focusing on liabilities. Here the Internal model is based on a single composite model for all long tail liability LOBs. From a composite model for multiple LOBs all necessary details can be deduced - mean loss, risk capital for a particular quantile (based on VaR/T-VaR), and MVM using the Cost of Capital method. In the context of this memo, the following definitions hold, BEL best estimate liability, deduced from the mean loss as a present value of mean loss, RC risk capital to cover non-hedgeable (that is, insurance) risk; defined from VaR or T-VaR for a particular quantile (e.g. 99.5%) of the predictive aggregate loss distribution; defined for each future calendar period until run-off, SCR - Solvency Capital Requirement - the risk capital required for the first year; RC(1). MVM(k) present value cost of holding risk capital RC; defined for each future calendar year k until run-off, MVM (or Total MVM) - the sum of MVM for all run-off periods, Insurance liabilities fair value - the sum of BEL and total MVM. This quantity is also referred as Technical Provision (TP). TP = BEL + MVM In each year k we have TP(k) = BEL(k) + MVM(k) The aggregate TP is the sum of the individual year TPs, including Present Value discounting, since these sums will only be required at the start of their respective years. For the purpose of Insurance Liabilities fair value calculation and Solvency II we assume, Insurer does not borrow the capital, it raises it instead. That is, SCR is raised at inception (evaluation date), RC(k) for k > 1 is raised in year k.

2 In one case out of 200 (99.5%) insurer is ruined. The liabilities fair value is relevant to the evaluation date. This quantity is assumed to be funded from premiums; the total MVM is estimated at inception (evaluation date). RC and the associated MVM can be computed for one year risk horizon and ultimate risk horizon. The Solvency II regime favours the former. In order to compute MVM using the Cost of Capital approach and SCR for both one year risk horizon and ultimate risk horizon for the aggregate of all long tail LOBs and each LOB the following critical information is required: Probability distributions of paid losses (liability stream) by calendar year (k=1,..,n) and their correlations, for each LOB and the aggregate of all LOBs. Probability distributions of total reserves for each LOB and the aggregate of all LOBs. Probability distributions of the aggregate paid losses from calendar year k to calendar year n for each LOB and the aggregate of all LOBs. This is required for each k raging from 1 to n, where complete run-off is achieved at the ultimate calendar year n. The above mentioned distributions enable the calculation of, amongst other statistics, the VaR q (k) for the paid losses (total loss) in calendar year k, at percentile q, for each LOB and for the aggregate of all LOBs. For a discussion of the ultimate risk horizon calculations see One year risk horizon With the one year risk horizon, the risk capital fund is raised at the beginning of each year and is released at the end of each year. At the conclusion of each year the unused portion of the fund is returned to the capital provider together with the interest earned at the risk free rate, and the MVM, the risk premium paid for access to the fund. If we temporarily ignore the correlations between the loss distributions for the future calendar years, the amount of the risk capital RC(k) raised at the beginning of year k is VaR 99.5% (k). The VaR 99.5% (k) is a quantity calculable from the distribution of losses for calendar year k, these distributions are available to us "now" at the beginning of year 1. We refer to these as the unconditional distributions. Assuming the spread above the risk free rate is s, and the risk free rate is d, the MVM for year k is given by the Cost of Capital method as (1) MVM(k) = VaR 99.5% (k), where s = spread. In this formula the factor

3 represents the spread*present value factor. The MVM is paid to the risk capital provider at the conclusion of the year and so MVM for year k can be considered at accumulate interest at the riskfree rate for k years. Present value adjustment for the BEL differs from this by half a year. Payments of losses occur roughly evenly throughout the year and so we treat them for discounting purposes as if they all occur exactly half way through the year. The present value factor for BEL is accordingly. In practice a more detailed analysis may treat the d as a random variable conditional on k, in this case the present value factor could be a mean value computed by simulation from the relevant rate models. Recall, the Technical Provision (TP), the sum the insurer must hold, is the sum of BEL and MVM. (2) Total MVM =

5 we refer to this quantity as ΔTP(k). Now, TP(k) = BEL(k) + MVM(k), and both of these quantities are computed from the distribution of the losses in year k, L k. Example: two year run-off after first year is in distress In a simple example where the run-off period is two years, and all risk capital is invested and assumed to get (at a minimum) the risk free rate, the formulas are very straightforward and can be summarised as follows: MVM(2) = (VaR 99.5 (2) * s) * PV(2) ΔVaR 99.5 (2) = VaR 99.5 (2 ξ) - VaR 99.5 (2) ΔMVM(2) = ΔVaR 99.5 (2) * s * PV(1) BEL(2) = E[L 2 ]*PV(1.5) ΔBEL(2) = (E[L 2 ξ] - E[L 2 ]) * PV(0.5) ΔTP(2) = ΔBEL(2) + ΔMVM(2) MVM(1) = s * SCR * PV(1); where SCR = VaR 99.5 (1) + ΔTP(2) MVM = MVM(1) + MVM(2)

6 Now, if the first year is in distress then: VaR 99.5 (1) is consumed. MVM(1) grows to SCR * s which is returned to the capital providers along with the risk free rate, d, on the total risk capital. Total amount returned is: SCR(d+s) ΔBEL(2) is used to adjust BEL(2) at the beginning of year 2; E[L 2 ξ] * PV(0.5) is available at the beginning of year 2. ΔMVM(2) is used to adjust MVM(2) at the beginning of year 2; MVM(2) / PV(1) + ΔVaR 99.5 (2) * s * PV(1) is available at the beginning of year 2. Any return on ΔBEL(2) and ΔMVM(2) in the first year is returned to the capital providers as part of the risk free rate calculated on the SCR which is why the PV adjustment is one year less than for the MVMs and BELs respectively. If the risk capital is assumed not to be invested, then the equations change so that MVM is higher as risk free return is added to it,

7 MVM(2) = (VaR 99.5 (2) * (s + d)) * PV(2) ΔMVM(2) = ΔVaR 99.5 (2) * (s + d) * PV(1) MVM(1) = (s + d) * SCR * PV(1); where SCR = VaR 99.5 (1) + ΔTP(2) MVM = MVM(1) + MVM(2) Unless specified otherwise, we assume that risk capital including VaR component earns interest at the risk-free rate. General run-off formulation Generally speaking, BEL(k) = and MVM(k) = VaR 99.5% ( L k ), so ΔTP(k) = +, where s = spread. So the total upgrade of the provisions for years 2+, given that year 1 is in distress at the 99.5 th percentile is, ΔTP =.

8 This amount that must be available from the risk fund at the end of year 1 so that the health of the reserves can be re-established after a highly distressed year. As far as the component of the SCR that is needed at the end of year 1, we do not apply discounting because all the interest on the risk capital fund must be returned to the capital provider. Thus our Solvency Risk Capital (SCR) at the beginning of year 1 is given by, SCR = VaR 99.5% (1). MVM(1) =, but MVM(k) = VaR 99.5% (L k ) as before, for k 2. based on the unconditional distribution, Formula (2) above, still holds, but when the correlated nature of the calendar year loss distributions is taken into account the value of MVM(1) is adjusted so that we are prepared for distributional changes that may be required at the beginning of year 2. If parameter uncertainty is excluded, then there is no correlation between calendar years. In this situation the SCR reverts to the naïve assumption given initially - that is: SCR=VaR(1), and MVM(k) = MVM(k ξ). Let us consider two scenarios. Scenario 1: First year is in a distress situation The amount VaR 99.5% (1) is exhausted, MVM(1)*(1+d) (plus the risk free rate on the SCR) is returned to the capital providers, and ΔTP(k) for k>2 is allocated to year k, so that the total provisions are augmented to their conditional values. At the beginning of year two VaR 99.5% (2 ξ) is raised from the capital providers. Scenario 2: First year is not in a distress situation The amount MVM(1)*(1+d) (plus the risk free rate on the SCR) is released to the capital providers

9 plus the minimum of (SCR+Mean Loss - Loss), and SCR. As illustrated by the above scenarios, the spread above the risk free rate used by the Cost of Capital method should not be confused with the expected excess return over the risk free return the capital providers receives. The expected return on the risk capital must be adjusted by the expected consumption of the risk capital. Consistent estimates of SCR, Technical Provisions and MVM for one-year risk horizon on updating In the section on Variation in Estimates of Ultimates conditions for consistency of prior (accident) year ultimates on updating are discussed. It is mentioned that only models and forecast assumptions based on the PTF and MPTF modelling frameworks achieve the necessary conditions of consistency on updating. Under the same conditions SCR, TP and MVM calculations are also consistent on updating from year to year. For example, if a forecast scenario assumes a calendar trend of 10% +- 2% for next calendar year followed by 5% +- 1% thereafter, then, SCR, TP and MVM are consistent on updating one year hence provided the next year's observed paid losses falls on the assumed 10% +- 2% trend line and the subsequent calendar trend is set to (assumed to be) 5% +- 1%. Note that for a long tail liability LOB the parameter uncertainty reduces on updating as the model is re-estimated with more data. Fungibility of Lines of Business and Calendar Years By default, we assume that the there is no fungibility between calendar years, but LOBs are fungible. That is, a surplus in one LOB can supplement a deficit in another LOB. In practice, this is not necessarily the case for legal or other reasons. Thus, we provide additional options to describe fungibility as follows: * All lines are fungible, calendar years going forward are fungible (default in ICRFS-Plus ) * Lines of business are not fungible, calendar years going forward are fungible * All lines are fungible, calendar years are not fungible * No fungibility either by Line of Business or by calendar year The above options are important as they describe the level of diversification afforded for writing multiple Lines of Business - for MVM and Risk Capital. Fungibility applies to the risk capital fund and whether surpluses supplement the risk capital fund (which lowers the required risk capital) or whether the risk fund has to stand on its own. For more details, see Proportionality principle and Standard Formula Quantitative Impact Studies QIS4 and QIS5 run by CEIOPS postulate the principle of proportionality

10 in the Standard Formula (see for example paragraph SCR.1.3 of Technical specifications for QIS5). This principle assumes VaR is proportional to the BEL for the future years. According to the QIS5, the insurer should assess whether the proportionality principle is applicable. Based on the research conducted in Insureware, proportionality does not hold for long-tail non-life insurance risk, manifesting the need for the Solvency II Internal model. Using one year risk horizon metrics module in ICRFS-Plus, the assertion of inapplicability of the proportionality principle can easily be verified. One year risk horizon metrics module ICRFS-Plus estimates SCR and MVM for one year risk horizon, based on the identified composite model for multiple LOBs with the associated forecast scenarios. One year risk horizon metrics module performs the following operations: runs the identified composite model with the associated forecasting scenario that predicts lognormal distributions for each cell and their correlations for each LOB; based on forecast results, computes BEL as the sum of present value mean losses; uses the PALD module to simulate from the predicted lognormals to obtain samples of the aggregates by calendar year for each LOB and the aggregate of all LOBs; Approximately 12 million simulation paths generated; computes VaR(k) for each calendar year k = 1 to n, for the aggregate of all LOBs; computes MVM(k) as the present value of spread multiplied by VaR(k); selects 10,000 of distress simulation paths ξ that fall into the 99.5% quantile interval for the first future calendar period; Based on the set of simulation paths ξ, calculates ΔBEL(k) and VaR(k ξ) for each calendar year k = 2 to n; computes ΔMVM(k) as spread multiplied by present value of difference between VaR(k) and conditional VaR(k ξ) for each calendar year k = 2 to n; computes ΔTP(k) as the sum of ΔBEL(k) and ΔMVM(k) for each calendar year k = 2 to n; computes ΔTP as the sum of ΔTP(k); calculates SCR as VaR(1) plus ΔTP; computes MVM as the sum of MVM(k) plus spread multiplied by present value of ΔTP (MVM(1) is an unadjusted value in this context); computes TP as the sum of BEL and MVM

11 Calculating MVMs using the Cost of Capital method is a doable task provided risk capital allocation by calendar year is available from the predictive distributions for each calendar year (and their correlations). It is possible to customise the approach presented (e.g. integrate Insurer margin; implement tax adjustments; use Percentiles method instead of Cost of Capital method to calculate MVM etc.). But it is hard to develop a meaningful approach without predictive loss distributions that are produced by ICRFS-Plus composite model for multiple LOBs. References: Ernst & Young s report Market Value Margins for Insurance Liabilities in Financial Reporting and Solvency Applications, The Ernst & Young s report describes the Cost of Capital method and contains some valuable examples, - Appendix D, Tables 30 and 32, and Appendix E referring to ICRFS-Plus.

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