气候压力测试下一致的股权风险中性估值

1January 2024 Consistent equity risk-neutralvaluation under climate stress tests2January 20243.In the third setting we consider we propose to differentiate the volatility structure of each sector index by assigning to eachequity risk-factor a proper volatility structure.In experiments 2 and 3, we can assess the materiality of the sector-based calibration.1.Finally, we propose to perform some sensitivity regarding3 above.We will modify the volatility structure obtained in experiment3 by applying some shocks on the volatility parameters, based on the discussion in the study “ReturnVolatility,Correlation,andHedging ofGreen andBrownStocks:IsThere aRole forClimateRiskFactors?” (SeeLBGF in References.)Becausemostclimatestress tests do not specify how volatilities are impacted, this fourth step would aim to assess the materialityof stressing the volatility assumption.To lead this work, the stochasticscenarioshave been generated by Milliman Economic Scenario Generator.1Theasset-liability management (ALM)run hasbeen performed by Agile ModelMilliman Agile ALM.2Equity paths and sector-based indices:SettingsRisk-neutral modelling consistsofsimulating the future flows of an asset in a market-consistent way. Assuming a complete marketwith no arbitrage opportunities, therisk-neutral probability is unique and makes the discounted values of assets martingales. Risk-neutral economic scenario generators are used to evaluate theBest Estimate(BE) of life insurance liabilities. In this work, we haveusedtherisk-neutral model volatility to generate equity paths is the Black-Scholes model with timevarying.DATAAs mentioned above, we perform our study based on the Euro Stoxx 50(STX). In run 1, only one equity riskfactor is simulated,and its volatility structure is calibrated to STX implied volatilities ofat-the-money (ATM)European call options quoted onthemarketas of31March 2023, over maturities 1 to 20.Becauseruns 2 to 4 are associated with sector-based calibration, we also need dataregarding the composition of the Euro Stoxx 50 per sector of activity, usingthe index’sStatistical Classification ofEconomicActivities in the European Community (NACE)codes. To conduct this study, we used the stocks making up the Euro Stoxx 50index, which comprises the50largest market capitalisations in the Eurozone.The shock applied for this study comes from theEIOPA table drawn up in April 2022 for the IORP stress tests(seeEIOPA in References). The Euro Stoxx 50 comprises 13different sectors, defined by the NACE code groupings drawn up by EIOPA. The equity portfolio of the virtual company we haveconsidered is thus composed of 13 assets in the ALM runs.We have tested four approaches to integrating climate risk intorisk-neutral equity scenarios. The four methods differ in theirgranularity, and we will introduce them from the most general to the most detailed.SETTING NO. 1The first approach is a standard one, in which the equities of the undertaking portfolio are all mapped to a single equity risk factorthat is the Euro Stoxx 50. In this approach, the volatility structure is identical for all sectors, corresponding to thatof the EuroStoxx 50.The equity factor, whose value at time𝑡is denoted by𝑆𝑡,is modelled using Black-Scholes model with time-varying volatility. Intherisk-neutral universe, this equity evolves as:d𝑆𝑡𝑆𝑡=𝑟𝑡d𝑡+𝜎(𝑡)d𝑊𝑡𝑆,where𝑟𝑡is the (time-𝑡) value of the short risk-free rate and𝑊𝑆is a Brownian motion leading the evolution of the equity factor.The function𝑡↦𝜎(𝑡)is piecewise constant and is calibrated to market data. More precisely, it is determined using Euro Stoxx 50implied volatilities of ATM European call options quoted onthemarket as of31March 2023, over maturities 1 to 20.1Seehttps://www.milliman.com/products/economic-scenario-generator.2Seehttps://www.milliman.com/en/products/milliman-agile-alm. Consistent equity risk-neutralvaluation under climate stress testsSETTING NO. 2In the second approach proposed, we classify the50undertakings composing the Euro Stoxx 50 according toitssector of activity(usingitsNACE codes). We obtained then 13 groups of stocks, as depicted inthe tablein Figure1, that will be modelled andsimulated by Milliman ESG. NACE indexing is a standardised classification system for economic activities attheEuropean level.NACE codes can be found either on the European Commission websites3or in information on economic activities reported bycompanies in their annual reports(seeReportsin References). Where a company has several attributed NACE codes, we haveproceeded as follows:We preferably identify the main activity of the company and attribute the NACE code of it to the undertaking;If not possible,because the various activities are equally weighted or because the names of the activities cited in the annual report correspondto several NACE codes, weassign to the undertaking the NACE code associated with the most unfavourable EIOPA shock.FIGURE1:ANALYSIS OF THE EURO STOXX 50 BYNACE CODENACE CODESA01A0