Investigation of Dependency between Short Rate and Transition Rate on Pension Buy-outs. Arık, A. 1 Yolcu-Okur, Y. 2 Uğur Ö. 2

Transcription

1 Investigation of Dependency between Short Rate and Transition Rate on Pension Buy-outs Arık, A. 1 Yolcu-Okur, Y. 2 Uğur Ö. 2 1 Hacettepe University Department of Actuarial Sciences 06800, TURKEY 2 Middle East Technical University Institute of Applied Mathematics 06800, TURKEY June 7, 2017

2 The Outline of the Study 1 2 3

3 Aim of the Study The main purpose of the study Introduce a general valuation expression for pension buy-outs 1 as an extension of Arık et al. (2017), 2 under the dependence assumption between short rate and transition rates, 3 in a continuous Markovian setting.

4 What is a Pension Buy-out? Pensioners liability contributions benefit DB pension scheme Buy-out insurer P buyout

5 P buyout (t) = [ ti ] tm t i >t E Q e r(s)ds t max{pa(t + i ) (PA(t i ) N(t i ).C), 0} I t L(t) r( ) : the stochastic short rate for t 0. L( ) is the liability of the pension scheme as defined in Definition 1. a(.) is the fair price of an immediate life annuity deal. B(.) is the fair price of a zero coupon bond. PA(t + i ) is the value of the pension portfolio at the beginning of time t i+1.

6 Definition (Liability Process) The pension liability process L(t) at time t, t [0, T ], is determined as L(t) = N(t) a(t, x), where N(t) : the number of survivors in the model at time t (determined according to the force of mortality rate dynamics). For the valuation of a(t, x), see the next slides.

7 An Illness-Death for a DB Pension Scheme 0 µ 01 t 1 healthy µ 10 t infected µ 02 t µ 12 t 2 dead Figure 1: The illness-death model for a hypothetical DB pension scheme

8 Correlated Transition and Short Rates 1 Suppose η(t) = µ 01 t, µ(t) = µ.2 t and X is a 3-dimensional affine process: (η(t), µ(t), r(t)) = c(t) + Γ(t)X(t), (1) where c : R + R 3 and Γ : R + R Hence ( dη(t) dµ(t) dr(t) ) = ( c1 (t) c 2 (t) c 3 (t) ) dt + ( 0 0 Γ13 Γ 21 0 Γ 23 0 Γ 32 Γ 33 ) ( dx1 (t) dx 2 (t) dx 3 (t) ) 3 The relevant SDEs are as dη(t) = c 1 (t)dt + Γ 13 dx 3 (t), dµ(t) = c 2 (t)dt + Γ 21 dx 1 (t) + Γ 23 dx 3 (t), dr(t) = c 3 (t)dt + Γ 32 dx 2 (t) + Γ 33 dx 3 (t).

9 How to Derive Possible Transition Probabilities in Figure 1? Chapman-Kolmogorov equations are held t+hp ij x = k S hp kj x+t t pik x, where i, j S and S = {0, 1, 2}. Hence, Kolmogorov forward differential equations are d dt (tp00 x ) = tp01 x µ10 t t px 00 t + µ 02 t ] d dt (tp01 x ) = tpx 00 µ 01 t t px 01 [µ 10 t + µ 12 t ] d dt (tp02 x ) = tpx 00 µ 02 t + t px 01 µ 12 t d dt (tp10 x ) = tpx 11 µ 10 t t px 10 [µ 01 t + µ 02 t ] d dt (tp11 x ) = tpx 10 µ 01 t t px 11 [µ 10 t + µ 12 t ] d dt (tp12 x ) = tp10 x µ02 t + t px 11 t. (2)

10 (Continued) The value of pension portfolio right after possible adjustments at the end of time t i where t i = t + i for i = 1, 2,... The value of pension portfolio at time t i P buyout (t) = [ ti ] tm t i >t E Q e r(s)ds t + max{ PA(t i ) ( PA(t i ) N(t i ).C), 0} I t L(t) Here, PA(t + i ) = max{pa(t i ) N(t i ) C, L(t i )}.

11 The value of pension assets under P da k (t) = A k (t)[α k dt + σ k dw k (t)] (3) Cov(A k (t), A l (t)) = ρ kl σ k σ l. Cov(W k (t), W l (t)) = ρ kl t, k = 1, 2, 3; l = 1, 2, 3; k l. 1 ρ kl : the correlation coefficient between assets k and l. 2 α k : the drift term for asset k where k = 1, 2, 3. 3 σ k : the instantaneous volatility

12 (Continued) The value of the pension portfolio under Q d log PA(t) = ( r 1 ) 2 σ2 W dt + 3 π k (t)σ k dw Q k (t) (4) k=1 1 r is the risk free rate (stochastic). 2 π(t) = [π 1 (t), π 2 (t), π 3 (t)] are the weights of the assets in the portfolio. 3 3 σw 2 = π k (t)π l (t)ρ kl σ k σ l. k,l=1 4 Moreover, dw P k (t) = dw Q k (t) ( 3 1 π k(t)α k r 3 1 π k(t)σ k ) dt.

13 Assumptions 1 No annual contributions. No pension gap at inception. 2 No inflation risk and π UK = [0.10, 0.85, 0.05]. 3 Initial state is state 0. Benefits are payable for state 0. 4 No differential mortality; µ 02 t = µ 12 t = µ(t). 5 µ 10 t = 0.1µ 01 t and µ 01 t = η(t). 6 Exponential jump sizes with mean j. 7 x = 65, N(0) = 100, C = 60000, dt = 1/252 Parameter set for the application OU process, X 1 x = 0, γ = , σ 1 = , x 1 (0) = CIR model, X 2 κ = 0.2, θ = 0.04, σ 2 = 0.1, x 2 (0) = 0.04 CP process, X 3 λ = 0, , 0.001, j = 0.1, 0.01

14 (Continued) The plan funds are assumed to be invested in the S&P UK stock total return index A 1 (t), the Merrill Lynch UK Sterling corporate bond index A 2 (t) and the 3-month UK cash total return index A 3 (t). Simulation Approach 1 Generate transition rates for illness-death model based on the continuous Markov process and calculate the liability process, 2 Generate asset processes and pension asset portfolio, 3 Apply the main formula to determine buy-out premiums depending on Monte Carlo simulation under various sample paths.

15 Main Scenario Definition (Short rate and mortality rate dynamics under measure Q) dµ(t) = dx 1 (t) + dx 3 (t) dr(t) = dx 2 (t) dx 3 (t) dη(t) = dx 3 (t), (5) by choosing Γ as ( 0 0 Γ13 Γ 21 0 Γ 23 0 Γ 32 Γ 33 ) = ( ). (6) dx 1 (t) = γ( x t X 1 (t))dt + σ 1 dw 1 (t) dx 2 (t) = κ(θ X 2 (t))dt + σ 2 X 2 (t)dw 2 (t) dx 3 (t) = dj(t), (7) where W 1 (t) and W 2 (t) are independent Wiener processes under measure Q.

16 Buy-out Premiums Table 1: Actuarial fair prices of the buy-out deal under MC iterations based on different levels of λ and j P buyout (0) j = 0.1 j = 0.01 λ = λ = λ =

17 Confidence Interval for Buy-out Premiums Definition (Determination of 95% confidence interval for buy-out premiums P buyout (0)) The confidence interval for P buyout (0) is calculated as follows: P buyout (0) = P buyout (0) = tm t i =1 PV payoff(t i ) tm L(0) t i =1 PV payoff(t i ), (8) L(0) where P buyout (0) and P buyout (0) show the lower and upper [ bounds of the ti ] confidence interval respectively. Here, PV payoff (t i ) = E Q e r(s)ds 0 H(t i ). PV payoff (t i ) = µ payoff (t i ) 1.96[σ payoff (t i )/ N] PV payoff (t i ) = µ payoff (t i ) [σ payoff (t i )/ N]

18 (Continued) Premium Premium Premium Premium Simulations (a) λ = 0 Simulations (b) λ = , j = 0.01 Figure 2: Calculated buy-out premiums with the corresponding confidence intervals when λ = 0 and λ =

19 Summary 1 A different setting based on a continuous time Markov process to obtain buy-out premiums, 2 Affine term structure as a method for modeling dependent short rate and transition rates, 3 Analytical solution is the next step.

20 References Arık, A., Yolcu-Okur, Y., Şahin, Ş and Uğur, Ö., 2017, Pricing Pension Buy-outs under Stochastic Interest and Mortality Rates, accepted for the publication in Scandinavian Actuarial Journal. Biffis, E., 2005, Affine Processes for Dynamic Mortality and Actuarial Valuations. Buchardt, K., 2014, Dependent Interest and Transition Rates in Life Insurance. Cox, J.C., Ingersoll, J.E. and Ross, S.A., 1985, A Theory of the Term Structure of Interest Rates. Deshmukh, S., 2012, Multiple Decrement s in Insurance. Fernholz, E.R., 2002, Stochastic Portfolio Theory. Haberman, S. and Pitacco, E., 1999, Actuarials s for Disability Insurance. Lin,Y., Shi, T. and Arik, A., 2017, Pricing Buy-ins and Buy-outs, Journal of Risk and Insurance, Vol:84, pp Vasicek, O., 1977, An Equilibrium Characterization of the Term Structure.

21 Thanks for your attention.