Solvency Capital Requirement for Non-Life Premium Risk Calculation Technique

A PRICING TECHNIQUE TO CALCULATE A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT THE SOLVENCY CAPITAL REQUIREMENT FOR NON FOR NON- -LIFE PREMIUM RISK LIFE PREMIUM RISK TRONCONI ANDREA TRONCONI ANDREA Torino, Torino, 4 4 Dicembre 2014 Dicembre 2014 a andrea.tronconi@uniroma1.it ndrea.tronconi@uniroma1.it

AGENDA INTRODUCTION MODEL FEATURES PROJECTION OF THE PORTFOLIO ESTIMATION OF THE FREQUENCY PARAMETER CONCLUSIONS A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 2

INTRODUCTION 1/2 PREMIUM RISK The Premium Risk derives from fluctuations in timing of frequency and severity, of insured events, which ensure that the premiums income will be not enough to pay future claims. In this context the perception is a lack of connection with actuarial best practices of pricing. PROJECTION OF INCOME & COST RISK AREAS ULR t 97% 415 101% 102% 100% Average Premium t Delta Frequency Delta Average Severity Delta Frequency; Delta Average Severity; ULR t+1 Average Premium. Average Premium t+1 Average Cost t+1 400 415 -3,5% Delta A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 3

INTRODUCTION 2/2 FACTORS CONNECTED WITH ??+1 PREMIUM RISK SCR CALCULATION Renewal process Discount trend New Business Tariff structure ?????= ???99.5% ? ? ? ? distribution of the total claim amount Simulating approach probability distribution of ? Instead of ? ? we will use ??+1, which is an estimation of the next year premiums income to derive the BUT . Renewal process and New Business not only effect the future premiums level but even frequency and severity. A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 4

AGENDA INTRODUCTION MODEL FEATURES PROJECTION OF THE PORTFOLIO ESTIMATION OF THE FREQUENCY PARAMETER CONCLUSIONS A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 5

MODEL FEATURES 1/3 COLLECTIVE APPROACH ? ? = ?? ?=1 ?? amount of a single claim ? number of claims: FOCUS OF THE PRESENTATION DUE TO THE CARD SYSTEM ??? ??? ??? ??? ??+ ??+ ?? ?? ? = ?? ?? ?? ?? ?=1 ?=1 ?=1 ?=1 A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 6

MODEL FEATURES 2/3 ATTRITIONAL VS LARGE CLAIMS For each claim types we will consider a new segmentation (attritional / large). Focusing on the NC claims: ??? ?? ????? ?? ??? ???+ (1 ???) ?? ?? ???= ?=1 Being ? the treshold between large and attritional : ??> ?) ??? 1 ? ???= 1 ??? ? = ?( ?? 0 A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 7

MODEL FEATURES 3/3 FREQUENCY ???~???????(???,????+ ???,????) ???,???? is an external parameter which considers a market change in the overall frequency of NC claims. How to derive the future expected level of ???,????? STARTING POINTS Policies that have at least one day of coverage during the solvency period considered (the level of expected frequency is connected to the features of the policies in portfolio) Multivariated models that explain the frequency (built and used by the actuarial department to construct the tariff) A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 8

AGENDA INTRODUCTION MODEL FEATURES PROJECTION OF THE PORTFOLIO ESTIMATION OF THE FREQUENCY PARAMETER CONCLUSIONS A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 9

PROJECTION OF THE PORTFOLIO 1/3 WHAT SHOULD BE CONSIDERED? Renewal process Company probability of renew (maybe depending on the agency) New business Connection with the business plan: is the company going to open some agencies somewhere? How many new business policies are expected? Assumption: the features of the new business policy are equal to the ones of the last year new business policies Future tariff Discount trend A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 10

PROJECTION OF THE PORTFOLIO 2/3 EXAMPLE OF PROJECTION Jan Feb Mar Apr May Jun Jul Ago Sep Oct Nov Dec TIMELINE Age 27 Car Type Audi A3 NC 5% Age 28 Car Type Audi A3 NC 4.5% POLICY 1 Age 50 Car Type VW Golf NC 1.5% Age 51 Car Type VW Golf NC 1.7% POLICY 2 Age 18 Car Type VW UP NC 7% POLICY 3 NB Age 33 Car Type BMW S1 NC 5% POLICY 4 A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 11

PROJECTION OF THE PORTFOLIO 3/3 EXPECTED FREQUENCY Having the frequency ante and post renew we can calculate, for each profile, the expected frequency of the year and then the ???,????of the whole portfolio: NC,A 5% 1.50% NC,P 4.50% 1.70% 7% Policy Exposure A 1 2 3 "Missing" "Missing" 4 5/12 Exposure P 9/12 4/12 9/12 "Missing" "Missing" nNC,YEAR 4.63% 1.57% 5.25% 2.08% 3/12 8/12 5% 4???,???? ?????? ?? ???????? ? ?= ???,????= WHERE ? =???,??????????+ ???,?????????? ???,???? A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 12

AGENDA INTRODUCTION MODEL FEATURES PROJECTION OF THE PORTFOLIO ESTIMATION OF THE FREQUENCY PARAMETER CONCLUSIONS A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 13

ESTIMATION OF THE FREQUENCY PARAMETER 1/4 FOR POLICY N.1 WE HAVE THAT Jan Feb Mar Apr May Jun Jul Ago Sep Oct Nov Dec TIMELINE Age 27 Car Type Audi A3 NC 5% Age 28 Car Type Audi A3 NC 4.5% POLICY 1 HOW TO CALCULATE ???? We use the pricing multivariate model that explain the NC frequency . GLM GLM models are the best practice in the tariff process cause take in account correlations between variables. In the following slides there are some interesting results A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 14

ESTIMATION OF THE FREQUENCY PARAMETER 2/4 GRAPH 1 - CAR AGE 1.1 1 0.9 Relativities 0.8 0.7 0.6 0.5 0.4 0 5 10 15 20 25 30 Car Age A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 15

ESTIMATION OF THE FREQUENCY PARAMETER 3/4 GRAPH 2 - AGE 2 1.8 1.6 1.4 Relativities 1.2 1 0.8 0.6 0.4 0.2 0 0 20 40 60 80 100 120 Age of the policy holder A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 16

ESTIMATION OF THE FREQUENCY PARAMETER 4/4 GRAPH 3 - NUMBER OF CLAIMS 3 2.5 2 Relativities 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 Number of Claims A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 17

AGENDA INTRODUCTION MODEL FEATURES PROJECTION OF THE PORTFOLIO ESTIMATION OF THE FREQUENCY PARAMETER CONCLUSIONS A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 18

CONCLUSIONS SCR CALCULATION Replicating this approach for each claim types we frequency parameters that we need to start the simulations, and finally we reach the distribution of ?. This process allows to calculate the percentile of ? and the SCR. MAIN FEATURES calculate the Includes the CARD System Considers the real risk profile of the company Takes into account strategies Is strictly connected to actuarial pricing technique Considers future tariff and the discount trend in the calculation of ??+1 Can easily include structures the Company WHAT IF All Company Board to take decisions, answering to fundamental question: What will happen in terms profitability, loss ratio, SCR, size of portfolio if .? of these aspects helps the the following reinsurance of expected A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 19

REFERENCES C.D. Daykin, T. Pentikainen, M. Pesonen, Practical Risk Theory for Actuaries , 1993; ANIA, CARD, convenzione tra assicuratori per il risarcimento Diretto , 2013; Towers Watson, Practitioner s Guide to Generalized Linear Models ,2004; EIOPA, QIS5 Technical Specifications ,2010; FINMA, Technical Document on the Swiss Solvency Test , 2007; Swiss Federal Office of Private Insurance, White Paper of the Swiss Solvency Test , 2004. A PRICING TECHNIQUE TO CALCULATE THE SOLVENCY CAPITAL REQUIREMENT FOR NON-LIFE PREMIUM RISK 20