BASIS RISK AND SEGREGATED FUNDS

Transcription

1 BASIS RISK AND SEGREGATED FUNDS Capital oversight of financial institutions June 2017 June

2 INTRODUCTION The view expressed in this presentation are those of the author. No responsibility for them should be attributed to the AMF. June

3 INTRODUCTION The AMF is the body mandated by the government to regulate Québec s financial market The AMF has been created in 2004 The AMF distinguish itself by virtue of its integrated regulation of the Québec financial sector, notably in the areas of insurance, securities, derivatives, deposit institutions June

4 INTRODUCTION The segregated funds can introduce a lot of risk for insurers Popular segregated funds guarantees : GMDB, GMMB and GMWB The AMF has to make sure that the risk management related to segregated funds is appropriate The AMF has to make sure that the capital requirements related to segregated funds products are appropriate June

5 MODEL APPROVAL At the AMF, we have developed an optional formula that recognizes dynamic hedging in the capital calculation Qualitative requirements (i. e. governance, model risk, changes to model, etc.) Quantitative requirements (i. e. actuarial assumptions, stress testing, calibration criterias, etc.) June

6 MODEL APPROVAL Approval process of segregated fund model Administrative process EMSFP Phase 1 Phase 2 Qualitative requirements Qualitative and quantitative requirements EMSFP Quantitative requirements Phase 3 Phase 4 Insurer model Capital calculation June

7 CAPITAL CALCULATION Quantification of capital for segregated fund model with dynamic hedging: o Market risk + basis risk (R1) (calibration criteria) o Lapse rate (R2) (shock of 40 % and null lapse rates under certain conditions) o Mortality (R3) (shock of 16 %) o Longevity (R4) (300 % of the ICA table) o Expenses (R5) (shocks: 15 % administrative expenses and 20 % transaction) June

8 CAPITAL CALCULATION Requirements after diversification benefit (RAM): under the following constraint: 5 RAM R R i, j 1 i, j i j R R R 0.3 R i i, j i j i i 1 i, j 1 i 1 June

9 RESEARCH PROJECTS AT AMF Asset model Interest rate model Development of an internal model with and without hedging Research segregated funds Dependence between interest rate and asset model Lapse risk Basis risk Mortality improvement June

10 Asset models: RESEARCH PROJECTS AT AMF o Regime-swithching Log-Normal o Regime-swithching Garch o Stochastic volatility model o etc. Interest rate models: o Model proposed by canadian institute of actuaries in an educational note (i. e. CIR) o Regime-switching model with many factors o etc. June

11 RESEARCH PROJECTS AT AMF Segregated fund models with and without hedging: o The AMF has developped the segregated fund models: one with hedging and one without hedging Lapse assumption: o Comparison between insurers June

12 RESEARCH PROJECTS AT AMF Basis risk modeling: o One of the methods used in the industry : Fund mapping where o What are the methods to model basis risk besides fund mapping? o Should 0? R I I A 1 1 n n o Impact of basis risk on capital 2 ~ N 0, June

13 RESEARCH PROJECTS AT AMF Basis risk modeling: o Research project financed by the Fonds d éducation de la saine gouvernance (i. e. FESG ) o Research project conducted by Frédéric Godin and Denis-Alexandre Trottier o Goal of the research project: quantify the impact of the basis risk for a segregated fund model with dynamic hedging June

14 A simple hedging scheme for variable annuities in the presence of basis risk Frédéric Godin, Ph.D., FSA, ACIA Concordia University, Laval University Emmanuel Hamel, Ph.D. Candidate Laval University Denis-Alexandre Trottier, Ph.D. Candidate Laval University CIA Annual Meeting 2017, Quebec June 22, /32

15 Summary A simple hedging for variable annuities is presented GMMB guarantee. A cross-hedge is performed hedging with a liquid futures, generates basis risk. We find that basis risk materially reduces hedging efficiency, an investment component within the hedging portfolio can improve outcomes. 2/32

16 Outline 1 Assets Dynamics 2 Variable annuities 3 Hedging 4 Simulation study 5 Conclusion 3/32

17 Assets description Assets : Great-West Life Canadian Equity (Bissett) BEL 1 (underlying asset of the variable annuity) TSX60 Futures (hedging instrument) Notation : t : indicates time (in months), F = {F t } t 0 : Great-West Fund Value process, S = {S t } t 0 : TSX60 Futures price process. 1. Returns data publicly available 4/32

18 Monthly returns dynamics h = {h t } t 0 : two-state Markov chain. Great-West Fund return : ( ) R (F ) t+1 log Ft+1 = µ (F ) F h t + σ (F ) h t z (F ) t+1. t TSX60 futures price return : R (S) t+1 log ( St+1 S t Joint distribution : [ ] z (F ) t+1 z (S) t+1 N ) = µ (S) h t + σ (S) h t z (S) t+1. ([ ] [ ]) 0 1 ρht,. 0 ρ ht 1 5/32

19 Monthly returns time series Great-West TSX60 futures /32

20 Maximum likelihood Table 1.2 Model estimation by maximum likelihood. µ (j) 1 (j) 1 µ (j) 2 Great-West Life Canadian Equity (Bissett) BEL fund (j = F ) (0.0023) (0.0020) (0.0085) (0.0060) TSX60 futures (j = S) (0.0025) (0.0022) (0.0119) (0.0090) Correlations (0.0131) (0.0294) Markov chain transition probabilities P 1,1 P 2, (0.0173) (0.0543) (j) 2 Note : Standard errors are presented within parentheses. 7/32

21 Most likely regime path Regime Most probable regime Great-West (monthly returns) /32

22 Outline 1 Assets Dynamics 2 Variable annuities 3 Hedging 4 Simulation study 5 Conclusion 9/32

23 Policy account dynamics The policy account is fully invested F (Great-West Fund), A = {A t } t 0 : policy account value, max(0, K A T ) : guarantee payoff at maturity (GMMB), T : guarantee maturity, K : guaranteed amount, g tot : total monthly fees rate, g opt : hedging monthly fees rate, Dynamics : A t+1 = (1 g tot )A t F t+1 F t, t {0,..., T 1} A t = F t A 0 F 0 (1 g tot ) t, t {0,..., T } 10/32

24 Model assumptions Constant monthly risk-free rate r, Constant monthly lapse rate b, Deterministic mortality (including improvement), No surrender charges, No ratchet neither, reset (constant strike price K). Extensions relaxing those assumptions are feasible. 11/32

25 Variable annuity guarantee value B t = e rt : risk-free asset price at t, CF t : cash outflow for the insurer at time t, Π t : variable annuity guarantee value at t (upcoming cash flows), g opt is determined such that Π 0 = 0, l t : proportion of insured still active at t. Π t = B t E Q A t T j=t+1 CF j B j F t, CF t = g opt l t 1 + max(0, K A T )l T 1 t=t. 1 g }{{ tot }{{}} maturity benefit fees 12/32

26 Risk-neutral measure The risk-neutral measure Q is chosen such that the model dynamics are preserved, but with different parameters : ( µ (F ) j is replaced by r 1 2 ( µ (S) j is replaced by 1 2 σ (S) j σ (F ) j ) 2, j {1, 2}, ) 2, j {1, 2}. The risk-free rate r does not appear in the drift of S a since this asset is futures. 13/32

27 Mortality and lapses Assumption : deterministic mortality and lapse rates. Assumes full diversification of lapse and mortality risk through a large number of insured. Notation : b : monthly lapse rate, tp x : probability the a policyholder aged x survives t months. Proportion of policyholder whose policy is still active at t : l t = (1 b) t tp x, t {0,..., T }. 14/32

28 tp x as a function of t for x = 55 years old policyholder Probabilité de survie Nombre de mois Note : mortality rates are obtained using base rates of the CPM 2014 mortality table (CIA, 2014) and projecting mortality improvement as indicated by CIA (2010). 15/32

29 Outline 1 Assets Dynamics 2 Variable annuities 3 Hedging 4 Simulation study 5 Conclusion 16/32

30 Hedging mechanics Monthly portfolio rebalancing, Non-self-financing portfolio, CF t : cash outflow for the insurer at t V θ t : pre-injection and pre-cash flow hedging portfolio value, V θ t+ = Π t : post-injection and post-cash flow portfolio value. Cash injection required at t : I t = Π t V θ t + CF t. 17/32

31 Delta approximation of the injection An approximation using a Taylor expansion gives where θ (S) t+1 I t+1 = (1 e r )Π t + t δf t θ (S) t+1 δs t (+ neglected terms) is the number of outstanding futures positions between (t, t + 1], t Πt F t δf t F t+1 F t, δs t S t+1 S t. is the guarantee delta, 18/32

32 Hedging problem (general case) At each period t, the risk related to the next injection I t+1 is minimized, R denotes the chosen risk measure, F t is the market information at t, is the number of outstanding futures positions between (t, t + 1]. θ (S) t+1 We solve : θ (S) t+1 = arg min R(I t+1 F t ). θ (S) t+1 19/32

33 Mean-variance R(I t+1 F t ) = Var P (I t+1 F t ) + 2λE P [I t+1 F t ]. This method involves a mean-variance trade-off. This is consistent with the objective of minimizing capital requirements (measured by the CVaR). λ 0 : mean-variance trade-off parameter. Solution (under the delta approximation) : θ (S) t+1 = Cov P (F t+1, S t+1 F t ) t Var P + λ EP [S t+1 F t ] S t (S t+1 F t ) Var P (S t+1 F t ). 20/32

34 Outline 1 Assets Dynamics 2 Variable annuities 3 Hedging 4 Simulation study 5 Conclusion 21/32

35 Simulation study objective We want to compare the following strategies : 1. No hedging : θ (S) t+1 0 for all t, 2. Cross-hedging with the TSX 60 futures, 3. Hedging with shares of the underlying fund. Note : the last strategy is unavailable in practice since shorting shares of the mutual funds is often impossible. 22/32

36 Capital and Reserve The sum and the capital requirement and the reserve [ T ] Reserve + Capital = CVaR P 0.95 e rt I t t=1 where and [ T ] Reserve = CVaR P 0.80 e rt I t [ T t=1 Capital = CVaR P 0.95 t=1 e rt I t ] Reserve. 23/32

37 Prix d exercice du GMMB K 100 Valeur initiale de F F Simulation Valeur initiale Parameters de S (monthly frequency) S Assets Dynamics Variable annuities Hedging Simulation study Conclusion Maturity T 120 Survival probability tp 55 Figure Lapse rate b 0.34% Total fee rate g tot 0.29% Risk-free rate r 0.25% Strike price K 100 Initial value of de F F Initial value of S S /32

38 Results : descriprive statistics of T t=1 e rt I t Mean Standard Deviation CVaR P 0.80 CVaR P 0.95 CVaR P 0.99 No hedging Cross-hedging with futures (with basis risk) Mean-variance = = = = = 23? Hedging with the underlying fund (no basis risk) Mean-variance = = 1.4? = = = /32

39 Results analysis In absence of hedging, we obtain CVaR P 0.95 = When hedging with underlying fund shares, the minimal variance hedging (λ = 0) yields CVaR P 0.95 = 3.5. When hedging with the futures the minimal variance hedge λ = 0 gives CVaR P 0.95 = 17.0, the hedge λ = 5 gives CVaR P 0.95 = 9.5, the hedge λ = 23 gives CVaR P 0.95 = 7.6. Minimizing the conditional variance is not necessarily optimal in the presence of basis risk 26/32

40 Why is minimal variance suboptimal? Continuous-time limit maximal risk reduction : Std P( It+1 vm ) Ft Std P( ) I nh = 1 Cor P( ) δf t, δs t 2, F t Ft t+1 It+1 vm : Minimal variance cash injection It+1 nh : Cash injection without hedging For example, = 0.44 : for a 90% correlation, the minimal variance injection standard deviation is reduced by only 56% with respect to the no hedging case. 27/32

41 ... moreover Cash injection for the λ mean-variance trade-off : I t+1 = I vm t+1 + λg t+1, G t+1 E[ δs t /S t F t ] Var ( δs t /S t F t ) δs t S t. Probability the λ trade-off increases the subsequent injection is : P ( G t+1 > 0 F t ) 37.9%. However, when summing over all time steps : ( T ) P G t e rt > 0 2.0%. t=1 Diversification greatly reduces downside risk. 28/32

42 In summary... In the presence of basis risk, reducing risk is very difficult. Time diversification significantly reduces the risk increase generated by a mean-variance tradeoff. This explains why the minimal variance strategy (which aims purely at minimizing local risk) is sub-optimal. It is reasonable to think that conclusions of this study apply also for mutual funds other than the Great-West fund. 29/32

43 Outline 1 Assets Dynamics 2 Variable annuities 3 Hedging 4 Simulation study 5 Conclusion 30/32

44 Conclusion Despite a very high correlation ( 90%), basis risk can have a significant impact on capital requirements. Minimizing the conditional variance of injections is not necessarily optimal in the presence of basis risk. A mean-variance tradeoff allows reducing capital requirements by more than 50% with respect to a minimal variance strategy. 31/32

45 Conclusion Next research steps : Complexify the mode stochastic risk-free rate dynamic lapses and stochastic mortality non-constant guarantee strike price Include additional hedging assets hedge with multiple indices interest rate swaps 32/32