Inside the Solvency 2 Black Box Net Asset Values and Solvency Capital Requirements with a Least Squares Monte Carlo Approach

Transcription

1 Inside the Solvency 2 Black Box Net Asset Values and Solvency Capital Requirements with a Least Squares Monte Carlo Approach Anthony Floryszczak SMABTP Group in collaboration with Olivier Le Courtois (EM Lyon Business School) and Mohamed Majri (SMABTP Group) 1

2 Life Insurance Company Liabilities We consider guaranteed with-profit contracts ( Contrats en Euros avec Participation aux Bénéfices ) with Guaranteed amount Guaranteed rate Participating bonus The study can be readily extended to unit-linked contracts ( Contrats en Unités de Compte ) 2

3 Surrender and Mortality Risks The mortality table selected is the S1PFL table produced by the Institute and Faculty of Actuaries. The surrender model is as described in the figure : 50% 45% 40% 35% Lapse rate 30% 25% 20% 15% 10% 5% 0% 10% 8% 6% 4% 2% 0% 2% 4% 6% 8% 10% Market Rate Crediting Rate 3

4 Life Insurance Company Assets The assets of the life insurance business are invested in stocks Remark : geometric Brownian motion assumption Government bonds Remark : autoregressive (CIR) assumption for the dynamics of the risk-free rate cash Remark : credit risk is neglected in all of our study 4

5 Definitions Net Net Asset Value : NAV = A BE DT where : A are the Assets of the Life Insurance company BE is the Best Estimate value of Liabilities DT are the Deferred Taxes (assumed null hereafter) 5

6 Definitions Note that the Best Estimate can be decomposed into BE = GBE + FDB where : GBE is the Guaranteed Best Estimate FDB is the value of Future Discretionary Benefits 6

7 Definitions We can now define the Gross Net Asset Value as follows : NAV = NAV + FDB so that NAV = A GBE DT The gross NAV is easier to compute than the net NAV because it does not include the FDB. 7

8 Definitions Solvency Capital Requirement : SCR = NAV 0 NAV 99.5% e r 1 where NAV 0 is the current NAV NAV 99.5% is the NAV in the 99.5% worst case scenario Ex. : the 50 th worst NAV out of simulated values Remark : these values are simulated in the real world one year from now and must be discounted at the rate r 1 8

9 Nested Simulations Thus, the goal is to obtain a set of values of NAV : { NAV(ω p, t = 1) ; p = 1,..., N p } Each NAV corresponds to a different state ω p of primary, real-world, parameters among N p ones. Specifically, the state of the world ω 99.5% comprises the values of equity and rate that produce NAV 99.5%. 9

10 Nested Simulations For each ω p, NAV is a conditional expectation estimated by means of N s secondary, market-consistent simulations. NAV(ω p, t = 1) = 1 N s N s T s=1 i=2 CF p i,s e rp i,s i Goal : valuation of explicit or implicit contingent claims linking the insurer to its policyholders. 10

11 Nested Simulations Nested simulations are nearly impossible to achieve in practice. (N p = ) (N s = 2 500) (30 years) (2 state variables) equals 1.5 billion random numbers 11

12 The LSMC Approach The least squares Monte Carlo approach approximates NAV by a polynomial function of the state variables, namely equity and rate here. For example : NAV(ω p, t = 1) = αr(ω p ) + βr(ω p ) ǫs(ω p ) + ζs(ω p ) γr(ω p )S(ω p ) + δr(ω p ) 2 S(ω p ) + where S is the equity value and r is the risk-free rate value. 12

13 LSMC with no interactions 30 NAV Equity Yield Risk Free Rate 13

14 Checking LSMC with no interactions 30 NAV Equity Yield Risk Free Rate 14

15 LSMC with interactions 30 NAV Equity Yield Risk Free Rate 15

16 Checking LSMC with interactions 30 NAV Equity Yield Risk Free Rate 16

17 LSMC with no interaction terms NAV SoS net NAV LSMC net NAV Rank 17

18 LSMC with interaction terms NAV SoS net NAV LSMC net NAV Rank 18

19 An Alternative Approach We look for a generic approach that can confirm the validity of the LSMC method. 19

20 Gross and Net NAVs Sim. gross NAV SoS net NAV 30 NAV Risk Free Rate 20

21 Gross and Net NAVs Sim. gross NAV SoS net NAV NAV Equity Yield

22 Scenario-Based Method Consider the ordered net NAVs : { and the ordered gross NAVs : { } NAV ˆ φ(p) } NAV ˆ ψ(p) The scenario-based consists in setting : φ = ψ Therefore, the net NAV at the 99.5% level can be directly computed from the gross NAV at the same level. The algorithm requires N p values of gross NAVs instead of net NAVs. 22

23 Scenario-Based Method Algorithm : Simulate N p gross NAVs Order them Extract the state of the world (r, S) that generates the 99.5% worst gross NAV value Compute the net NAV in this state of the world using secondary simulations. This is NAV 99.5% 23

24 Scenario-Based Method NAV Scenario based net NAV SoS net NAV Rank 24

25 Enhanced Scenario-Based Method The Enhanced Scenario-Based Method extends the Scenario-Based one by using the additional operation : ˆ NAV φ(p) = M i= M ˆ NAV φ(p)+i 2M + 1 with M small. This is a smoothing operation on the next neighbors. 25

26 Enhanced Scenario-Based Method NAV Enhanced scenario based net NAV SoS net NAV Rank 26

27 Comparison of Methods The next step is the comparison of the LSMC and the Enhanced Scenario-Based Method. 27

28 Comparison of the LSMC (with no interaction terms) and ESB methods NAV Enhanced scenario based net NAV LSMC net NAV Rank => LSMC with no interaction terms is not validated. 28

29 Comparison of the LSMC (with interaction terms) and ESB methods NAV Enhanced scenario based net NAV LSMC net NAV Rank => LSMC with interaction terms is validated. 29

30 Sensitivity Analysis of NAV We now study the sensitivity of NAV 99.5% with respect to the main model risk parameters. 30

31 NAV w.r.t. Equity Volatility net NAV Sigma Equity 31

32 NAV w.r.t. Interest Rate Level 100 TMG = net NAV Risk free rate curve level 32

33 NAV w.r.t. Lapse Rate net NAV Lapse rate parameter alpha 33

34 NAV w.r.t. Mortality Rate net NAV Mortality table multiplier coefficient 34

35 Conclusion The LSMC and ESB methods perform well for computing NAVs and SCRs. Only LSMC gives NAV as a function of the risk-drivers and can be used for sensitivity analyses. Framework applicable to all kinds of life insurance contracts. 35