Managing Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives

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Managing Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives Simon Man Chung Fung, Katja Ignatieva and Michael Sherris School of Risk & Actuarial Studies University of New South Wales Australia ASTIN, AFIR/ERM and IACA Colloquia Sydney, 23-27 August, 25 Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 / 26

Introduction Provision of Retirement Income Products Examples of retirement income products: life annuities, deferred annuities, variable annuities with guarantees (financial options) Selling them can be risky - long term nature Duration of payments depend on survival of annuitants Idiosyncratic mortality risk can be handled by diversification Systematic mortality risk (longevity risk) is significant Hedging longevity risk using longevity swaps and caps Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 2 / 26

Outline Pricing and hedging analysis of longevity derivatives on a hypothetical life annuity portfolio subject to longevity risk" Propose and calibrate a two-factor Gaussian mortality model Derive analytical pricing formulas for longevity swaps and caps Investigate hedging features of longevity swaps and caps w.r.t. different assumptions on () the market price of longevity risk (2) term to maturity of the hedging instruments (3) portfolio size Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 3 / 26

Longevity risk modelling How to model survival probability? Intensity-based approach: Cox process with stochastic intensity µ. If µ is known then deal with inhomogeneous Poisson process, where probability of k events in [t, T ] is given by ( T t ) k µ(s) ds µ(s) ds P(N T N t = k F T ) = e T t k! Death time is modelled as the first jump time of a Cox process Given only F t, we set k = in Eq. () and use iterated expectation to obtain the expected survival probability ( S x+t (t, T ) = E P e T ) µ x+s (s)ds t Ft (2) Useful for pricing and the corresponding density function of death time can be obtained () Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 4 / 26

Mortality model The mortality intensity µ x (t) (in full: µ x+t (t)) of a cohort aged x at time t = is modelled by µ x (t) = Y (t) + Y 2 (t), (3) where Y (t) is a general trend that is common to all ages, and Y 2 (t) is an age-specific factor, satisfying the following dynamics where dw (t) dw 2 (t) = ρdt dy (t) = α Y (t) dt + σ dw (t) (4) dy 2 (t) = (αx + β)y 2 (t) dt + σe γx dw 2 (t) (5) Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 5 / 26

Survival probability Proposition Under the two-factor mortality model, the expected (T t)-survival probability of a person aged x + t at time t is given by S x+t (t, T ) = Et (e P T ) µ x (s)ds t = e 2 Γ(t,T ) Θ(t,T ) (6) where Θ(t, T ) and Γ(t, T ) are the mean and the variance of the integral µ x (s) ds respectively. T t We will use the fact that the integral T µ t x (s) ds is normally distributed with known mean and variance to derive analytical pricing formulas for longevity derivatives Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 6 / 26

Parameter estimation Figure: Central death rates m(x, t)..4.3 m (x,t).2. 9 8 Age (x) 7 6 97 98 99 Year (t) 2 2 Australian male ages x = 6,..., 95 and years t = 97,...28 Intensity assumed constant over each integer age and calendar year; approximated by central death rates m(x, t) Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 7 / 26

Parameter estimation Figure: Difference of central death rates m(x, t)..5. m (x,t).5.5. 9 8 Age (x) 7 6 97 98 99 Year (t) 2 2 Sample variance: Var( m x ); Model variance: Var( µ x )=(σ 2 + 2σ σρe γx + σ 2 e 2γx ) t Minimizing 9 x=6,65... (Var( µ x σ, σ, γ, ρ) Var( m x )) 2 with respect to the parameters {σ, σ, γ, ρ}. Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 8 / 26

5 5 2 25 3 35 4 3.5 3 2.5 Age x = 65 5th perc. 25th perc. 5th perc. 75th perc. 95th perc. 4 3.5 3 2.5 Age x = 75 5th perc. 25th perc. 5th perc. 75th perc. 95th perc. µ x (t) 2 µ x (t) 2.5.5.5.5 t 5 5 2 25 3 35 t Survival Probability S x (,T).8.6.4.2 Age x = 65 99% CI Mean Survival Probability S x (,T).8.6.4.2 Age x = 75 99% CI Mean 5 5 2 25 3 35 T 5 5 2 25 3 35 T Other parameters are calibrated to the empirical survival curves aged 65 and 75 in 28 by minimizing T x x=65,75 j= ) 2 (Ŝx (, j) S x (, j) (7) Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 9 / 26

Risk-adjusted measure Assuming, under the risk-adjusted measure Q, we have dy (t) = α Y (t) dt + σ d W (t) (8) dy 2 (t) = (αx + β λσe γx ) Y 2 (t) dt + σe γx d W 2 (t) (9) where λ is the market price of longevity risk; λ is estimated using the proposed price of the BNP/EIB longevity bond (Meyricke & Sherris (24)) Survival Probability S 65 (,T).9.8.7.6.5.4.3 λ = λ = 4.5 λ = 8.5 λ = 2.5.2. 5 5 2 25 3 35 Horizon T Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 / 26

(Index-based) Longevity swaps Consider an annuity provider/hedger who has a future liability that depends on the (stochastic) survival probability of a cohort The provider wants certainty for the estimated survival probability S-forward: At maturity T, pays fixed leg, K(T ) (, ), and receives floating leg - the realized survival probability e T µ x (s) ds. The payoff from the S-forward is, assuming the notional amount is, e T Zero price at inception means that e r T E Q (e T µ x (s) ds K(T ) () ) µ x (s) ds K(T ) = () and hence K(T ) = E Q (e T ) µ x (s) ds (2) A longevity swap is a portfolio of S-forwards Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 / 26

The mark-to-market price process F (t) of an S-forward is F (t) = e r(t t) Et (e Q T ) µ x (s) ds K = e r(t t) Et (e Q t µx (s) ds e T ) µ x (s) ds t K = e r(t t) ( Sx (, t) S x+t (t, T ) K ) (3) S x (, t) := e t µx (s) ds Ft is the realized survival probability, which is observable given F t. Let ˆn denotes the number of survivors in [, t] and the initial population of the cohort is n, then we have S x (, t) ˆn n (4) S x+t (t, T ) is the risk-adjusted survival probability; If an analytical expression for S x+t (t, T ) is available then an explicit formula for F (t) is obtained Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 2 / 26

where S t = S x (, t), S t = S x+t (t, T ), d = Γ(t,T ) ( ln {K/( St St )} + 2 Γ(t, T ) ) and Φ( ) denotes the CDF of the standard normal R.V. Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 3 / 26 (Index-based) Longevity caps A longevity cap is a portfolio of caplets - the payoff of a longevity caplet is max {(e T ) } µ x (s) ds K, (5) Similar to an S-forward but is able to capture the upside potential - when survival probability is overestimated The price of the longevity caplet: Cl(t) = e r(t t) Et ((e Q T )) + µ x (s) ds K Proposition Under the two-factor mortality model, the price at time t of a longevity caplet Cl(t), with maturity T and strike K, is given by ( ) Cl(t) = S t S t e r (T t) Φ Γ(t, T ) d Ke r (T t) Φ ( d) (6)

Idea of proof: since the integral I := T µ t x (s) ds is normally distributed, e I is a log-normal random variable The standard deviation Γ(t, T ) of the integral can be interpreted as the volatility of the (risk-adjusted) aggregated longevity risk for an individual aged x + t at time t, for the period from t to T Figure: Caplet price as a function of (left panel) T and K and (right panel) λ where K =.4 and T = 2..5.5.45.4.4 Cl ( ; T, K).3.2 Caplet Price.35..3 5 2 Time to Maturity (T) 25 3.5.4.3 Strike (K).2..25.2 2 4 6 8 2 4 λ Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 4 / 26

Managing longevity risk in a life annuity portfolio Consider a life annuity portfolio consists of n policyholders aged 65 in year 28; Premium of the annuity ($ per year upon survival) is given by ω x a x = B(, T ) S x (, T ; λ) (7) T = For the annuity provider the present value (P.V.) of asset is A = n a x P.V. of (random) liability for policyholder k: τ k L k = B(, T ) (8) T = P.V. of liability for the whole portfolio: L = n k= L k Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 5 / 26

Interested in the discounted surplus distribution per policy where D no = A L D no n (9) 9 8 7 Discounted Surplus Distribution Per Policy n = 2 n = 4 n = 6 n = 8 6 Density 5 4 3 2.5.5 $ Figure: No longevity risk (σ = σ = ); Idiosyncratic mortality risk is reduced as n increases Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 6 / 26

Swap-hedged annuity portfolio: where F swap = n ˆT T = D swap = A L + F swap (2) B(, T ) (e T ) µ x (s) ds S x (, T ) is the random cash flow from the longevity swap; n here acts as the notional amount; ˆT : term to maturity Cap-hedged annuity portfolio: where F cap = n ˆT T = (2) D cap = A L + F cap C cap (22) B(, T ) max {(e T ) } µ x (s) ds S x (, T ), is the random cash flow from the longevity cap and C cap = n ˆT T = (23) Cl (; T, S x (, T )) (24) is the price of the longevity cap Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 7 / 26

Examining hedge features of longevity swaps and caps with respect to (w.r.t.) different assumptions on () the market price of longevity risk λ (2) term to maturity of the hedging instruments ˆT and (3) the portfolio size n Table: Parameters for the base case. λ ˆT (years) n 8.5 3 4, Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 8 / 26

3.5 3 2.5 λ = no hedge swap cap 3.5 3 2.5 λ = 4.5 no hedge swap cap 2 2.5.5.5.5.5.5.5.5.5.5.5.5 3.5 3 2.5 λ = 8.5 no hedge swap cap 3.5 3 2.5 λ = 2.5 no hedge swap cap 2 2.5.5.5.5.5.5.5.5.5.5.5.5 Figure: Effect of the market price of longevity risk λ on the discounted surplus distribution per policy Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 9 / 26

Table: Hedging features of a longevity swap and a cap w.r.t. market price of longevity risk λ. Mean Std.dev. Skewness VaR.99 ES.99 λ = No hedge -.76.3592 -.284 -.922 -.27 Swap-hedged -.89.78 -.99 -.84 -.223 Cap-hedged -.86.254.855 -.393 -.355 λ = 4.5 No hedge.52.3592 -.284 -.766 -.943 Swap-hedged.48.78 -.99 -.73 -.294 Cap-hedged.682.254.855 -.2425 -.2746 λ = 8.5 No hedge.2978.3592 -.284 -.648 -.7973 Swap-hedged.24.78 -.99 -.547 -.938 Cap-hedged.25.254.855 -.93 -.2224 λ = 2.5 No hedge.4475.3592 -.284 -.465 -.6476 Swap-hedged.398.78 -.99 -.354 -.744 Cap-hedged.69.254.855 -.489 -.8 Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 2 / 26

3.5 3 2.5 Term = years no hedge swap cap 3.5 3 2.5 Term = 2 years no hedge swap cap 2 2.5.5.5.5.5.5.5.5.5.5.5.5 3.5 3 2.5 Term = 3 years no hedge swap cap 3.5 3 2.5 Term = 4 years no hedge swap cap 2 2.5.5.5.5.5.5.5.5.5.5.5.5 Figure: Effect of the term to maturity ˆT of the hedging instruments on the discounted surplus distribution per policy Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 2 / 26

Table: Hedging features of a longevity swap and cap w.r.t. term to maturity ˆT. Mean Std.dev. Skewness VaR.99 ES.99 ˆT = Years No hedge.2978.3592 -.284 -.648 -.7973 Swap-hedged.282.29 -.387 -.577 -.749 Cap-hedged.2893.2989 -.266 -.58 -.7592 ˆT = 2 Years No hedge.2978.3592 -.284 -.648 -.7973 Swap-hedged.74.794 -.757 -.3656 -.56 Cap-hedged.2234.23.26 -.387 -.5259 ˆT = 3 Years No hedge.2978.3592 -.284 -.648 -.7973 Swap-hedged.24.78 -.99 -.547 -.938 Cap-hedged.25.254.855 -.93 -.2224 ˆT = 4 Years No hedge.2978.3592 -.284 -.648 -.7973 Swap-hedged -.9.668.277 -.66 -.869 Cap-hedged.984.999.527 -.99 -.23 Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 22 / 26

Table: Longevity risk reduction R = Var( D ) of a longevity swap and a cap w.r.t. Var( D) different portfolio size (n). n 2, 4, 6, 8, R swap 92.6% 96.% 97.2% 97.7% R cap 64.9% 67.3% 68.% 68.6% Var( D ) variance of discounted surplus for hedged portfolio Var( D) variance of discounted surplus for unhedged portfolio Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 23 / 26

Table: Hedging features of a longevity swap and a cap w.r.t. portfolio size (n). Mean Std.dev. Skewness VaR.99 ES.99 n = 2, No hedge.2973.3646 -.2662 -.636 -.87 Swap-hedged.2.99 -.65 -.22 -.2653 Cap-hedged.2.26.922 -.2432 -.2944 n = 4, No hedge.2978.3592 -.284 -.648 -.7973 Swap-hedged.24.78 -.99 -.547 -.938 Cap-hedged.25.254.855 -.93 -.2224 n = 6, No hedge.2977.3566 -.2786 -.6363 -.8 Swap-hedged.24.594 -.3346 -.259 -.66 Cap-hedged.24.26.59 -.639 -.25 n = 8, No hedge.2982.3554 -.292 -.66 -.7876 Swap-hedged.29.536 -.556 -.9 -.595 Cap-hedged.29.992.66 -.598 -.99 Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 24 / 26

Summary The proposed two-factor Gaussian mortality model is capable of modelling mortality intensities of different ages simultaneously Longevity swaps and caps can be priced analytically under the model; Standard derivation of the integral T µ t x (s) ds plays the role of volatility in longevity derivatives pricing Swap-hedged annuity portfolio is insensitive to the market price of risk λ Longevity swaps and caps are markedly different as hedging instruments when term to maturity ˆT or portfolio size n is large Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 25 / 26

Thank you for your attention Katja Ignatieva ASTIN, AFIR/ERM and IACA 25 Sydney 25 26 / 26

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