Market-Consistent Valuation of Long-Term Insurance Contracts

Transcription

1 Market-Consistent Valuation of Long-Term Insurance Contracts Madrid June 2011 Jan-Philipp Schmidt Valuation Framework and Application to German Private Health Insurance

2 Slide 2 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Motivation What is the shareholders value from long-term insurance contracts? How do the special characteristics of German health insurance affect the shareholders value and risk associated with the value? How big is the financial risk of companies private health insurance portfolios?

3 Slide 3 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Literature review Projection of insurance contracts in balance sheet setup: Kling et al. (2007) Gerstner et al. (2008) Market-consistent valuation of single insurance contracts: Bacinello (2003) Market-consistent valuation of portfolios of insurance contracts: Sheldon and Smith (2004) Castellani et al. (2005) Wüthrich et al. (2010) Principles of market-consistent embedded value: CFO Forum (2009)

4 Slide 4 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Agenda Framework Stochastic Environment Insurance Company Valuation Results

5 Slide 5 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Agenda Framework Stochastic Environment Insurance Company Valuation Results

6 Slide 6 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Framework Stochastic Environment: External factors (Ω, F, P) prob. space X 1,..., X n r.v. with X i : Ω R T 1 i n Insurance Company: Cash flows Y = f (X 1,..., X n ) Valuation: Value e.g. E(g(Y ))

7 Slide 7 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Agenda Framework Stochastic Environment Insurance Company Valuation Results

8 Slide 8 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Stochastic Environment Capital market model by Jarrow and Yildirim (2003) (JY-Model) Risk factors: forward real and nominal interest rate, inflation

9 Slide 8 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Stochastic Environment Capital market model by Jarrow and Yildirim (2003) (JY-Model) Risk factors: forward real and nominal interest rate, inflation JY-Model (Heath-Jarrow-Morton framework, under P) df n(t, T ) = α n(t, T )dt + ς n(t, T )dw P n (t) df r (t, T ) = α r (t, T )dt + ς r (t, T )dw P r (t) di(t) = I(t)μ(t)dt + I(t)σ I dw P I (t)

10 Slide 8 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Stochastic Environment Capital market model by Jarrow and Yildirim (2003) (JY-Model) Risk factors: forward real and nominal interest rate, inflation JY-Model (Heath-Jarrow-Morton framework, under P) df n(t, T ) = α n(t, T )dt + ς n(t, T )dw P n (t) df r (t, T ) = α r (t, T )dt + ς r (t, T )dw P r (t) di(t) = I(t)μ(t)dt + I(t)σ I dw P I (t) t [0, T ], I(0) = I 0 > 0, f n(0, T ) = fn M (0, T ) and f r (0, T ) = fr M (0, T ). (Wn P, Wr P, WI P ) Brownian motion with correlations ρ n,r, ρ n,i and ρ r,i.

11 Slide 9 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Stochastic Environment Assume that for t [0, T ] : ς n(t, T ) = σ ne an(t t) and ς r (t, T ) = σ r e ar (T t).

12 Slide 9 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Stochastic Environment Assume that for t [0, T ] : ς n(t, T ) = σ ne an(t t) and ς r (t, T ) = σ r e ar (T t). JY-Model (Short rate framework, under Q n ) dn(t) = (θ n(t) a nn(t))dt + σ ndw n(t) dr(t) = (θ r (t) ρ r,i σ r σ I a r r(t))dt + σ r dw r (t) di(t) = I(t)(n(t) r(t))dt + I(t)σ I dw I (t) (W n, W r, W I ) Brownian motion with correlations ρ n,r, ρ n,i and ρ r,i and fn(0, t) ( θ n(t) = + a nf n(0, t) + σ2 n 1 e 2ant) T 2a n θ r (t) = fr (0, t) T + a r f r (0, t) + σ2 r 2a r ( 1 e 2ar t).

13 Slide 10 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Agenda Framework Stochastic Environment Insurance Company Valuation Results

14 Slide 11 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Private health insurance policies in Germany Life-long guarantee of coverage for predefined medical reimbursement. Insurer renounces the right to cancel the contract. Annual premium calculated by principle of equivalence (similar to life-techniques) assuming future claim development based on current situation. No benefits in case of death or surrender. Policyholder pays additional premium until age 60 (10 percent of actuarial premium). Additional reserve serves to limit excessive premium increase.

15 Slide 12 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Special contract characteristics Increase of average claim size (linked to inflation) leads to check of basis for actuarial calculation. If increase of average claim size is significant then insurance company adjusts expected claim size and interest rate. Premium payment includes surcharge (surcharge factor).

16 Slide 12 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Special contract characteristics Increase of average claim size (linked to inflation) leads to check of basis for actuarial calculation. If increase of average claim size is significant then insurance company adjusts expected claim size and interest rate. Premium payment includes surcharge (surcharge factor). Options and Guarantees Short-term interest rate guarantee. Medical reimbursement. Surrender.

17 Slide 13 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Surplus distribution Surplus resulting from asset development, risk and cost surplus is aggregated at the end of the year. Positive surplus: Shareholders receive at most 20 % (surplus factor). Negative surplus: Up to 100 % financed by shareholders. Asymmetric impact on shareholders value.

18 Interest Bond market Asset return nominal interest real interest price zero bond 1yr to maturity price zero bond 2yr to maturity price zero bond 3yr to maturity asset return interest for premium and reserve time time time Interest surplus Insurance surplus Total surplus 6e+06 4e+06 interest surplus 3e+06 2e+06 risk surrender surcharge 5e+06 4e+06 3e+06 s.total s.sh 2e+06 0e+00 1e+06 0e+00 1e+06 2e+06 1e+06 0e+00 1e+06 2e time time time Claims Equity Cash flow 2500 real claims 5e+06 equity 4e+05 cash flow for valuation calculatory claims 4e+06 required capital 3e e+06 2e+05 2e e+06 1e+05 0e+00 0e age time time

19 Slide 14 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Agenda Framework Stochastic Environment Insurance Company Valuation Results

20 Slide 15 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Valuation Definition: Present Value of Future Profits PVFP := E Qn ( T t=0 exp Y t cash flow between shareholder and company ( t ) ) n udu Y t 0

21 Slide 15 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Valuation Definition: Present Value of Future Profits PVFP := E Qn ( T t=0 exp ( t ) ) n udu Y t 0 Y t cash flow between shareholder and company Generate K scenarios of external factors X 1,..., X T under risk-neutral measure Q n. Present Value of Future Profits Estimation ( ˆPVFP := 1 K T ) B k,t Yt k K k=1 t=0 with B k,t discount factor for scenario k at time t.

22 Slide 16 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Valuation Generate sample of K scenarios of external factors X 1,..., X n under risk-neutral measure Q n. Definition: Average Scenario This scenario yields X i,t := 1 K PVFP of Average Scenario with B t discount factor at time t. PVFP AS := K k=1 T t=0 X k i,t B t Y t

23 Slide 17 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Quantification of Financial Risk Def.: Time Value of Financial Options and Guarantees TVFOG := PVFP AS ˆPVFP Time value is the additional value ascribable to the potential for benefits under the option to increase in value prior to expiry (CFO Forum, 2009).

24 Slide 18 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Agenda Framework Stochastic Environment Insurance Company Valuation Results

25 Slide 19 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Example: one model point Contracts specifics 5, 000 contracts from male insureds age 40, contracts exist since 10 years initial annual premium 1,764 C and average claim size 1,197 C

26 Slide 19 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Example: one model point Contracts specifics 5, 000 contracts from male insureds age 40, contracts exist since 10 years initial annual premium 1,764 C and average claim size 1,197 C Capital market (values from Jarrow and Yildirim (2003)) Nominal: a n = , σ n = , n 0 = 0.04 Real: a r = , σ r = , r 0 = 0.01 Inflation: σ i = , I 0 = 100 Dependency: ρ n,r = , ρ r,i = , ρ n,i =

27 Slide 19 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Example: one model point Contracts specifics 5, 000 contracts from male insureds age 40, contracts exist since 10 years initial annual premium 1,764 C and average claim size 1,197 C Capital market (values from Jarrow and Yildirim (2003)) Nominal: a n = , σ n = , n 0 = 0.04 Real: a r = , σ r = , r 0 = 0.01 Inflation: σ i = , I 0 = 100 Dependency: ρ n,r = , ρ r,i = , ρ n,i = Liablities at t = 0: Actuarial Reserve: 60, 964, 712 C Bonus Account: 5, 000, 000 C Equity: 1, 500, 000 C

28 Slide 20 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Results Present Value of Future Profits 3,000,000 surplus 20 % surplus 15 % surplus 10 % 2,000,000 ˆPVFP 1,000, ,000,000 surcharge in percent of premium

29 Slide 21 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Results Time Value of Options and Guarantees 3,000,000 surplus 20 % surplus 15 % surplus 10 % 2,000,000 TVFOG 1,000, ,000,000 surcharge in percent of premium

30 Slide 22 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Conclusion Surcharge factor has a big impact on present value of future profits. We observe an exposure to financial risk in terms of the time value. High variance in present value of future profits.

31 Slide 23 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Thank you very much for your attention. Jan-Philipp Schmidt University of Ulm Institute of Insurance Science Helmholtzstraße Ulm Germany

32 Slide 24 Market-Consistent Valuation of Long-Term Insurance Contracts AFIR 2011 in Madrid June 2011 Literature A. Bacinello. Pricing Guaranteed Life Insurance Participating Policies with Annual Premiums and Surrender Option. North American Actuarial Journal, 7(3):1 17, G. Castellani, M. D. Felice, F. Moriconi, and C. Pacati. Embedded Value in Life Insurance, CFO Forum. Market Consistent Embedded Value Principles T. Gerstner, M. Griebel, M. Holtz, R. Goschnick, and M. Haep. A general asset-liability management model for the efficient simulation of portfolios of life insurance policies. Insurance: Mathematics and Economics, 42(2): , R. Jarrow and Y. Yildirim. Pricing Treasury Inflation Protected Securities and Related Derivatives using an HJM Model. Journal of Financial and Quantitative Analysis, 38(2): , A. Kling, A. Richter, and J. Ruß. The interaction of guarantees, surplus distribution, and asset allocation in with-profit life insurance policies. Insurance: Mathematics and Economics, 40(1): , T. Sheldon and A. Smith. Market Consistent Valuation of Life Assurance Business. British Actuarial Journal, 10(3): , M. Wüthrich, H. Furrer, and H. Bühlmann. Market-Consistent Actuarial Valuation. Springer, 2010.