PARTIAL SPLITTING OF LONGEVITY AND FINANCIAL RISKS: THE LIFE NOMINAL CHOOSING SWAPTIONS

Transcription

1 Interest rate risk transfer PARTIAL SPLITTING OF LONGEVITY AND FINANCIAL RISKS: THE LIFE NOMINAL CHOOSING SWAPTIONS H. Bensusan (Société Générale) joint work with N. El Karoui, S. Loisel and Y. Salhi Longevity and Pension Funds February 4th, 2011 Thanks to C. Michel and all the R & D team of CACIB

2 OUTLINE OF THE TALK Interest rate risk transfer 1 INTEREST RATE RISK TRANSFER 2 MORTALITY AND LONGEVITY MODELING 3 QUANTATIVE ANALYSIS

3 INTRODUCTION Interest rate risk transfer Improvements in longevity are bringing new issues and challenges at various levels: social, political, economic and regulatory. Need of MORE AND MORE CAPITAL to face this long-term risk Hedging longevity risk is now an important element of risk management for many organisations

4 Interest rate risk transfer LONGEVITY RISK: RISK MANAGEMENT SUBJECT FOR INSURANCE COMPAGNIES The insurance industry is also facing some specific challenges related to longevity risk. Accurate longevity projections are delicate (prospective life tables) Modeling the embedded risk (such long term interest rate risk) remains challenging Important to find a suitable and efficient way to cross-hedge or to transfer part of the longevity risk to reinsurers or to financial markets

5 HETEROGENEITY Interest rate risk transfer Longevity patterns and longevity improvements are very different for different countries, and different geographic area. Factors affecting the mortality socio-economic level (occupation, income, education...) gender marital status living environment (pollution, nutritional standards, hygienic...) wealth

6 BASIS RISK I Interest rate risk transfer Difference between the national mortality data and the one from an insured portfolio. Insurance companies have much more detailed information They know the exact ages at death and not only the year of death (time continuous data) Cause of death are specified Characteristics of the policyholders : socio economic level, living conditions... BUT limited size of their portfolios (in comparison to national populations : individuals from 19 different insurance companies) small range of the observation period Furthermore, insurers are tending to select individuals (given their health and medical history for example). This heterogeneity is very important for longevity risk transfer based on NATIONAL INDICES: for too important basis risk, the hedge would be too imperfect

7 OUTLINE Interest rate risk transfer Hedging of life-insurance products 1 INTEREST RATE RISK TRANSFER 2 MORTALITY AND LONGEVITY MODELING 3 QUANTATIVE ANALYSIS

8 Interest rate risk transfer FINANCIAL RISK TRANSFER Hedging of life-insurance products Insurance companies are exposed to interest rate risk Annuities at rate k Investments in interest rate products (Bonds,...) Looking for a product that transfers interest rate risk while keeping longevity risk: Insurance can manage longevity risk Banks can manage intereste rate risk Product with decorrelation of both risks

9 YES BUT HOW? Interest rate risk transfer Hedging of life-insurance products Difficulties for launching a pure longevity product: No longevity market (No shared reference) Information asymmetry Modelling and pricing (evaluation in historical probability) Pure interest rate products

10 REAL PORTFOLIO Interest rate risk transfer Hedging of life-insurance products FIGURE: Age distribution of policyholders

11 PORTFOLIO EVOLUTION Interest rate risk transfer Hedging of life-insurance products FIGURE: Survival extreme scenarii of policyholders Using our longevity model

12 CHALLENGES Interest rate risk transfer Hedging of life-insurance products Interest rate hedging of life-insurance products : Static hedge of interest rate risk Swaps Dynamic hedge of forward risk Swaptions Longevity Swaptions with variable nominal Stochastic evolution of longevity Choice of the nominal profile

13 Interest rate risk transfer LIFE NOMINAL CHOOSING SWAPTION Hedging of life-insurance products Swaption on a swap with variable nominal N t with strike k : [ P N ] swaption = E Q T (k SV T (T 0, T N, δ, N t)) + B(T, T i )N Ti δb(0, T) i=1 Choice of α T [0, 1] by the insurer at date T (with available information) Hedge on the nominal series N α T t = α TN t + (1 α T)N + t Forward swap rate with variable nominal SV T(T 0, T N, δ, α T, N t, N + t ) for the series N t determined by α T Evaluation of "Life Nominal Choosing Swaption" (LNCS) at strike k : P LNCS δb(0, T) = E Q N T max 0 α T 1 (k SV T (T 0, T N, δ, α T, N, N + )) + B(T, T i )(α T N T + (1 α T )N + i T ) i i=1 E Q T max 0 l n (k SV T (T 0, T N, δ, l N n, N, N + )) + B(T, T i )( l n N T + (1 l i n )N+ T ) i i=1

14 CHOICE OF α T (EXAMPLE) Interest rate risk transfer Hedging of life-insurance products FIGURE: Choice of the parameter α T in 2019 Using our longevity model

15 CHOICE OF α T (EXAMPLE) Interest rate risk transfer Hedging of life-insurance products FIGURE: Choice of the parameter α T in 2029 Using our longevity model

16 OUTLINE Interest rate risk transfer 1 INTEREST RATE RISK TRANSFER 2 MORTALITY AND LONGEVITY MODELING 3 QUANTATIVE ANALYSIS

17 Interest rate risk transfer CAIRNS, BLAKE AND DOWD (CBD) MODEL (2006) The 2-factor CBD model gives the dynamics of the annual mortality rate q t(x) at age x during the year t: logitq t(x) = A 1 (t) + xa 2 (t). ( where the function logit is defined by logit(x) = ln x 1 x ).

18 Interest rate risk transfer INDIVIDUAL MORTALITY MODEL AIM: Reduce the basis risk by estimating the deviation of the "individual mortality" from the average mortality (as given by mortality tables). Find individual characteristics (such as socio-economic level or income) that can explain mortality Take them into account in a stochastic mortality model The mortality model by age and trait (feature) is calibrated on national mortality data and on specific data (i.e. with information on individual characteristics)

19 Interest rate risk transfer MARITAL STATUS INFLUENCE AT INITIAL DATE (MALES) FIGURE: Logit of mortality rate for French males in 2007 with different marital status

20 Interest rate risk transfer MORTALITY PROJECTIONS IN 2017 (MALES) FIGURE: Logit of mortality rate for French males in 2017 with different marital status

21 Interest rate risk transfer MICRO/MACRO MODELLING Macroscopic models: Evolution of population with macroscopic information Whole population Average evolution Microscopic models: Evolution at the level of the individual using accurate information Sample of the population Interaction between individuals Uncertainty Scenarii for population evolution

22 Interest rate risk transfer INDIVIDUAL BASED MODELS IN POPULATION DYNAMICS Microscopic models in ecology: Microscopic model for a trait-structured population (N. Fournier et S. Méléard 2004) Extension to an age-structured population (C. Tran Viet 2006) Suggested extensions for a human population: Rate of trait evolution e (marriage rate, divorce rate,...) Evolution rates depending on time and and beging random : addition of stochastic exogeneous factors Yt b, Yt b and Yt e Immigration modelling

23 Interest rate risk transfer MICROSCOPIC EVOLUTION PROCESS Occurence of an event at time T n: 1 At time Tn, selection of a person in the population 2 age and traits of individuals allow to calculate probabilities: b(x, a, YT b n ) birth intensity d(x, a, YT d n ) death intensity e(x, a, YT b n ) trait evolution intensity 3 A random variable is drawn to select an event Death: the person is removed fromš the population Birth: a person is added in the population with an age a = 0 and a trait x drawn in a distribution k b (x, a, x ) Trait evolution: the person with trait x is replaced by a person with trait x drawn in a distribution k e (x, a, x ) and with the same age a Nothing: an auxiliary event occurs

24 Interest rate risk transfer MICRO/MACRO LINK C. TRAN VIET (2006) McKendrick (1926) and VonFoerster (1959) : Equation in demography ( g t + g ) (a, t) = d(a)g(a, t), g(0, t) = b(a)g(a, t)da a Approximation (a.s) for large populations: ( g t + g ) [ (ω, x, a, t) = d a g(ω, x, 0, t) = b χ g(ω, x, a, 0) = g 0 (ω, x, a), + 0 (x, a, Y d t (ω) ) + e (x, a, Y e t (ω)) ] g(ω, x, a, t) e(x, a, Yt e )ke (x, a, x)g(ω, x, a, t)p(dx ) χ ) (x, a, Yt b (ω) k b (x, a, x)g(ω, x, a, t)p(dx )da

25 Interest rate risk transfer EVOLUTION SCENARII IN 2097 FIGURE: Evolution Scenarii of the size of the population in 2097 with a mortality scenario

26 PROJECTION SCENARII Interest rate risk transfer Double stochasticity (mortality process and evolution process) various scenarii Approximation to large propulation relevant behaviour Average scenario is realistic and close to INSEE projections Identification of the scenarii panel (extreme or not) generated by the model

27 OUTLINE Interest rate risk transfer Basis risk Pricing of LNCS 1 INTEREST RATE RISK TRANSFER 2 MORTALITY AND LONGEVITY MODELING 3 QUANTATIVE ANALYSIS

28 Interest rate risk transfer USE OF THE LONGEVITY MODEL Basis risk Pricing of LNCS Survival scenarii of a specific group (N persons): Standard method (no modelling of population dynamics): Mortality model 1 mortality scenario N survival scenarii (for each person) provides 1 evolution scenario of a group (N persons) Suggested modelling : Individual mortality model 1 mortality scenario Microscopic model for population dynamics provides 1 evolution scenario of a group (N persons) Efficient modelling numerically

29 Interest rate risk transfer Basis risk Pricing of LNCS BASIS RISK: IMPACT OF MARITAL STATUS (MALES) FIGURE: Central scenario of the annuities for a portfolio with 60-years old French men with different marital status

30 Interest rate risk transfer LIFE NOMINAL CHOOSING SWAPTIONS Basis risk Pricing of LNCS Product on maximum: impact of rates correlation 2-factor HJM calibrated on European rate curve and on specific swaptions (maturity and tenor adapted) Assume constant correlation between successive forward rates: ρ = correl(f (., T), f (., T + dt)) Focus on price impact of k and ρ

31 INITIAL RATE CURVE Interest rate risk transfer Basis risk Pricing of LNCS FIGURE: European rate curve as of 8 April 2009

32 Interest rate risk transfer NOTION OF COST ON ANNUITIES Basis risk Pricing of LNCS Annuities: P annuities = k LVL(α), where LVL(α) = N i=1 (αn T i + (1 α)n + T i )B(T, T i) Life Nominal Choosing Swaption: c = P LNCS called cost on annuities LVL(α) P LNCS = c LVL(α) Interpretation : Without product : annuities at rate k without hedging interest rate risk With product : annuities at rate k + c with hedging interest rate risk

33 CORRELATION IMPACT Interest rate risk transfer Basis risk Pricing of LNCS Successive rate Swap correlation Price Price Price Price Cost on correlation 10Y/12Y α = 0 α = 0.5 α = 1 LNCS Annuities ρ = % 3015 bp 2656 bp 2244 bp 3020 bp 0.87% ρ = % 2954 bp 2580 bp 2215 bp 2960 bp 0.853% ρ = % 2886 bp 2529 bp 2183 bp 2897 bp 0.835% ρ = % 2813 bp 2474 bp 2147 bp 2828 bp 0.815% ρ = % 2732 bp 2412 bp 2107 bp 2751 bp 0.793% ρ = % 2641 bp 2342 bp 2061 bp 2667 bp 0.769% ρ = % 2540 bp 2264 bp 2009 bp 2574 bp 0.742% TABLE: Evolution of the price at k = 4.3% depending on successive rate correlation When ρ decreases Price of swaptions with variable nominal decreases Price of the switch option increases from = 5bp to = 34bp exotic nature increases but remains low

34 CONCLUSIONS Interest rate risk transfer Basis risk Pricing of LNCS Estimation of basis risk: portfolio heterogeneity Pure interest rate product suited to expectations of banks/insurance companies Could use more sophisticated model for interest rate

35 Interest rate risk transfer Basis risk Pricing of LNCS Thank you for your attention

36 ARTICLES/PROJECTS Interest rate risk transfer Basis risk Pricing of LNCS Understanding, Modelling and Managing Longevity Risk: Key issues and Main Challenge (Longevity group) Risques de taux et de longévité : modélisation dynamique et applications aux produits dérivés d assurance-vie (PhD thesis, H. Bensusan) Detection of changes in longevity trends (N. El Karoui, S. Loisel, C. Mazza) Microscopic modelling of population dynamic: an analysis of longevity risk (H. Bensusan, N. El Karoui) On inter-age correlations in stochastic mortality models (S. Loisel and Y. Salhi) Pricing longevity securities ( N. El Karoui, S. Loisel, C. Hillairet, Y. Salhi) Growth optimal portfolio and Long term interest rate (I. Camilier, N. El Karoui, C. Hillairet) Variables annuities (S. Loisel, T.L. Nguyen)