Pricing and Risk Management of Variable Annuity Guaranteed Benefits by Analytical Methods Longevity 10, September 3, 2014

Transcription

1 Pricing and Risk Management of Variable Annuity Guaranteed Benefits by Analytical Methods Longevity 1, September 3, 214 Runhuan Feng, University of Illinois at Urbana-Champaign Joint work with Hans W. Volkmer, University of Wisconsin-Milwaukee

2 Seite 2 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Variable annuity guaranteed benefits

3 Seite 3 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Variable annuity guaranteed benefits Brief introduction Guaranteed minimum maturity benefit Individual model Guaranteed minimum maturity benefit - Average model Guaranteed minimum withdrawal benefit

5 Seite 5 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Product design Arguably the most complex equity-based guarantee available to individual investors; Policyholders make contributions (called purchase payments) into subaccounts; typically single purchase payment at the inception; The term of the product can be broken down into two parts: Accumulation phase; (resembles mutual funds) Income phrase. (various types of guaranteed benefits)

6 Seite 6 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Product design The value of each account varies with the performance of the particular fund in which it invests; F t = F S t S e mt, t T, where m is the annualized rate of total fees and charges. Ft total value of subaccounts at time t; St value of equity-index at time t; Without any guarantee, equity participation involves no risk to the variable annuity writer, who merely acts as a steward of the policyholders funds.

7 Seite 7 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 In order to compete with mutual funds, nearly all major variable annuity writers start to offer various types of investment guarantees, which transfer certain financial risks to the insurers. (Liabilities) Guaranteed Minimum Maturity Benefit (GMMB) Guaranteed Minimum Death Benefit (GMDB) Guaranteed Minimum Withdrawal Benefit (GMWB) Guaranteed Minimum Accumulation Benefit (GMAB) Guaranteed Minimum Income Benefit (GMIB) Nick-named as the GMxB series. (x=m,d,w,a,i)

11 Seite 11 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Guaranteed Minimum Maturity Benefit(GMMB) Policyholder receives the greater of a minimum guarantee and account value. The GMMB writer is liable for the difference between the guarantee and account value, should the former exceeds the latter.

12 Seite 12 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 M&E charge, rider charge are made on a daily basis as a certain percentage of subaccount values. (Incomes) M t = m x F t, t T, where m x is the charge allocated to fund GMxB rider.(m x < m)

13 Seite 13 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Guaranteed Minimum Maturity Benefit(GMMB) Gross liability: (T is the maturity date) e rt (G F T ) + I(τ x > T ), where τ x is the future lifetime of policyholder aged x at issue. (put-option-like payoff) Net liability= Gross liability - Margin offset income T τx L = e rt (G F T ) + I(τ x > T ) e rs m e F s ds. (exotic option?) Mostly negative and rarely positive. F investment fund value; G guaranteed benefit; T maturity date; r risk-free interest rate; m e GMMB fee rate.

14 Seite 14 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Commonly used risk measures Quantile risk measure (Value-at-Risk) V α := inf{y : P[L y] α}. The minimum capital required to ensure that there is sufficient fund to cover future liability with the probability of at least α. Conditional tail expectation (Expected Shortfall) CTE α := E[L L > V α ]. Capital required to cover the liabilities exceeding the quantile measure with the probability of at most 1 α.

15 Seite 15 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 How do we model the equity-index process? The model depends on the assumptions on the dynamics of the equity-index. F t = F S t S e mt, t < T. Net liability= Gross liability - Margin offset income T τx L = e rt (G F T ) + I(τ x > T ) e rs m e F s ds. There is standard life table model for the future lifetime τ x ; No consensus on equity-index/asset price process S t. F investment fund value; G guaranteed benefit; T maturity date; r risk-free interest rate; M margin offset (fees).

16 Seite 16 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 In principle, insurance companies are allowed to use their own stochastic models for equity returns. However, the stochastic models should meet the calibration criteria set by the American Academy of Actuaries. Tabelle : Calibration Standard for Total Return Wealth Ratios Percentile 1 Year 5 Years 1 Years 2 Years 2.5% n/a 5.% % % %

17 Seite 17 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Common equity-return models listed in the AAA guildeline Independent Lognormal (ILN) (Geometric Brownian motion) Monthly Regime-Switching Lognormal Model with 2 Regimes (RSLN2) Monthly Regime-Switching Lognormal Model with 3 Regimes (RSLN3-M) Daily Regime-Switching Lognormal Model with 3 Regimes (RSLN3-D) Stochastic Log Volatility with Varying Drift (SLV)

18 Seite 18 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Essentially, all models are wrong, but some are useful... George E.P. Box

19 Seite 19 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 For simplicity, we assume the dynamics of equity prices is driven by geometric Brownian motion. S t = S exp(µt + σb t ), t. (If the subaccounts are automatically rebalanced (with fixed porportion) and each account is driven by a GBM, then the total fund is also driven by a GBM.)

20 Seite 2 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Analytical methods Conditioning on that the policyholder s death occurs after maturity T L = e rt (G F T ) e rs m e F s ds ( ) T = e rt G e rt F T + m e e rs F s ds e rt G X T (identity in distribution) where X is determined by ] dx t = [(µ σ2 2 m r)x t + m e dt + σx t db t.

21 Seite 21 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Risk measures P(L > V α ) = P(L > V α τ x > T )P(τ x > T ) for some constant K. Similarly, one can show that = P(e rt G X T > V α )P(τ x > T ) = P(X T < K )P(τ x > T ) CTE α = 1 α E[(e rt G X T )I {XT <K }]P(τ x > T ).

22 Seite 22 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Using spectral expansion, we obtained P(L > V α ) = x ( 2π 2 exp 1 ( ( W ν+1 2, ip 2 1 2wx ) W 1 ν 2, ip 2 4wx ) ( w ν+1 2 exp ) ( 1 Γ ν+ip 2x 2 ) 1 4x e (ν2 +p 2 )t/2 ) 2 sinh(πp)p dp where W is the Whitter function of the second kind and t := σ2 T 4, ν = 2(µ m r) σ 2, x = σ2 4m e, K = x w, w = e rt G V α F. We also obtained similar results for E(L I {L >V α}), which determines the conditional risk measure CTE α.

23 Seite 23 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Comparative study: GMMB (Joint work with Hans W. Volkmer) 1-year GMMB with full fund of initial deposit G/F = 1; Mean and standard deviation of log-returns per annum µ =.9, σ =.3; Risk-free discount rate per annum r =.4; M&E charges and rider charges per annum m =.1; GMMB rider charge 35 basis points of account value m e =.35. Methods Direct integration Inverse Laplace Monte Carlo V 95% /F % % % Initial value 33% (28%, 33%) - Time (mins) CTE 95% /F 4.41% 4.42% 4.29% Time (mins)

24 Seite 24 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Comparative study: GMMB (Joint work with Hans W. Volkmer) The same valuation assumptions. Methods Integration Inverse Lap Spectral Green V 9% % % % % Initial 1% (12%, 14%) (12%, 14%) (12%, 14%) Time 3.674(mins) 5.32(mins) (secs).172(secs) CTE 9% % % % % Time 1.867(mins).285(mins) 2.953(secs) (secs) Tabelle : A comparison of computational methods for the GMMB rider

26 Seite 26 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 In practice, valuation actuaries use spreadsheets to run simulations based on traditional life insurance reserving methods. In each period, they compute Change in Surplus = Fee Income - Guaranteed Benifits - Expenses + Interest on Surplus The spreadsheet calculations are in essence the difference equation from a mathematical point of view. It is easy to show that the net liability used in practice for the GMMB is T L = e rt (G F T ) + T p x sp x e rs m e F s ds.

27 Seite 27 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Connection between individual and average models Individual model: L = e rt (G F T ) + I(τ x > T ) Average model: L = e rt (G F T ) + T p x T T τx e rs m e F s ds. sp x e rs m e F s ds. Connection: Assuming all policies are of equal size and the future lifetimes of policyholders are mutually indepedent, we can show that 1 N N i=1 L (i) L, almost surely. F investment fund value; G guaranteed benefit; T maturity date; r risk-free interest rate; m d GMMB fee rate.

28 Seite 28 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Differences between individual and average models Individual model: (Two sources of randomness) L = e rt (G F T ) + I(τ x > T ) T τx e rs m e F s ds. Average model: (One source of randomness, the mortality risk is fully diversified.) L = E[L F T ] = e rt (G F T ) + T p x T sp x e rs m e F s ds. They make no difference for pricing since E[L ] = E[L ] = E[E[L F T ]], yet they are different in terms of tail probabilities. We can compute the tail probability P(L > V ) using numerical PDE methods but it is much slower than the computation of P(L > V ) through Laplace transform.

29 Seite 29 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Comparison of Individual and Average Models Average Model Individual Model.12.1 Tail Probability Net Liability Abbildung : Survival functions of net liabilities

31 Seite 31 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Guaranteed Minimum Withdrawal Benefit(GMWB) Contains no life insurance component. Provides a minimal payout based on the initial purchase payment. For example: A policyholder is guaranteed the ability to withdraw $7 per annum per $1 of initial investment until the original $1 has been fully exhausted. (The benefit expires after $1/$ years.)

32 Seite 32 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Illustration of a scenario with GMWB payments Pictures taken from Milevsky and Salisbury (26)

33 Seite 33 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Illustration of a scenario without GMWB payments Pictures taken from Milevsky and Salisbury (26)

34 Seite 34 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Mathematical Formulation Assume the dynamics of the equity-index is given by ds t = µs t dt + σs t db t. The accumulated value of VA subaccounts F t = F S t S e mt wt, t. The dynamics of VA investment fund is driven by df t = [(µ m)f t w] dt + σf t dw t, F = G >.

35 Seite 35 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Mathematical Formulation (Joint work with Hans W. Volkmer) Pricing from an investor s point of view (Milevsky and Salisbury (26)) Present value of guaranteed income (Maturity: T = G/w) w T e rt dt = w r (1 e rt ). Investment income e rt F T I(τ > T ), τ := inf{t : F t < }. The fair price mw is determined by F = w r (1 e rt ) + E Q [e rt F T I(τ > T )]. w withdrawal rate; r risk-free interest rate; T guaranteed period; F fund value; τ first time fund is exhausted.

36 Seite 36 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Mathematical Formulation (Joint work with Hans W. Volkmer) Pricing from an insurer s point of view Present value of Income (Assets) τ T m w e rt F t dt. Present value of Outgo (Liabilities) w T τ e rt dt I(τ < T ) = w r (e rτ e rt )I(τ < T ). The fair price m w is determined by [ ] τ T E Q m w e rt F t dt = w [ ] r EQ (e rτ e rt )I(τ < T ). w withdrawal rate; r risk-free interest rate; T guaranteed period; F fund value; m w fees to fund GMWB.

37 Seite 37 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Pricing from an investor s point of view F = w r (1 e rt ) + E Q [e rt F T I(τ > T )]. Pricing from an insurer s point of view [ ] τ E Q T m w e rt F t dt = w [ ] r EQ (e rτ e rt )I(τ < T ). Milevsky and Salisbury (26) used numerical PDE methods to price the GMWB from an investor s perspective. We found closed-form solutions in both cases. One can prove using Dynkin s formula that the two methods of pricing are equivalent when m w = m.

38 Seite 38 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Future Work Computation of risk measures for the GMWB would be rather difficult. (Importance sampling?) w T τ T τ e rt T dt m w e rt F t dt. Guaranteed Lifetime Withdrawal Benefit (GLWB): The guarantee lasts until the policyholder s death. Individual model: (Tx is the future lifetime of the policyholder at age x) w Tx Average model: w τ τ T x e rt dt m w e rt tp x dt m w τ T x τ e rt F t dt. e rt tp x F t dt.

39 Seite 39 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Future Work Computation of risk measures for the GMWB would be rather difficult. (Importance sampling?) w T τ T τ e rt T dt m w e rt F t dt. Guaranteed Lifetime Withdrawal Benefit (GLWB): The guarantee lasts until the policyholder s death. Individual model: (Tx is the future lifetime of the policyholder at age x) w Tx Average model: w τ τ T x e rt dt m w e rt tp x dt m w τ T x τ e rt F t dt. e rt tp x F t dt.

40 Seite 4 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 References GMMB & GMDB: R. Feng, H.W. Volkmer. Analytical calculation of risk measures for variable annuity guaranteed benefits. Insurance: Mathematics and Economics (212), 51(3), R. Feng, H.W. Volkmer. Spectral methods for the calculation of risk measures for variable annuity guaranteed benefits. ASTIN Bulletin, (214), to appear. R. Feng. A comparative study of risk measures for the guaranteed minimum maturity benefit by a PDE method. North American Actuarial Journal (214), to appear. GMWB: M.A. Milevsky, T.S. Salisbury. Financial valuation of guaranteed minimum withdrawal benefits. Insurance: Mathematics and Economics, 38(1), R. Feng, H.W. Volkmer. An identity of hitting times and its application to the pricing of guaranteed minimum withdrawal benefit. Preprint.

41 Seite 41 Pricing & Risk Management of Variable Annuity Benefits by Analytical Methods Longevity 1 September 3, 214 Thank you very much for your attention!