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doi: 10.3934/jimo.2021188
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Valuation of cliquet-style guarantees with death benefits

1. 

Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong, China

* Corresponding author: Yaodi Yong

Received  June 2021 Revised  September 2021 Early access November 2021

Fund Project: This research was supported by the Research Grants Council of the Hong Kong Special Administrative Region (Project No. HKU 17305018) and a CRGC grant from the University of Hong Kong

In this paper, we consider the problem of valuing an equity-linked insurance product with a cliquet-style payoff. The premium is invested in a reference asset whose dynamic is modeled by a geometric Brownian motion. The policy delivers a payment to the beneficiary at either a fixed maturity or the time upon the insured's death, whichever comes first. The residual lifetime of a policyholder is described by a random variable, assumed to be independent of the asset price process, and its distribution is approximated by a linear sum of exponential distributions. Under such characterization, closed-form valuation formulae are derived for the contract considered. Moreover, a discrete-time setting is briefly discussed. Finally, numerical examples are provided to illustrate our proposed approach.

Citation: Yaodi Yong, Hailiang Yang. Valuation of cliquet-style guarantees with death benefits. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021188
References:
[1]

M. J. Brennan and E. S. Schwartz, The pricing of equity-linked life insurance policies with an asset value guarantee, Journal of Financial Economics, 3 (1976), 195-213.  doi: 10.1016/0304-405X(76)90003-9.

[2]

N. Bowers, H. U. Gerber, J. C. Hickman, D. A. Jones and C. J. Nesbit, Actuarial Mathematics, 2nd edition, The Society of Actuaries, Illinois, 1997.

[3]

P. P. Boyle and E. S. Schwartz, Equilibrium prices of guarantees under equity-linked contracts, Journal of Risk and Insurance, 44 (1977), 639-660.  doi: 10.2307/251725.

[4]

P. P. Boyle and W. Tian, The design of equity-indexed annuities, Insurance Math. Econom., 43 (2008), 303-315.  doi: 10.1016/j.insmatheco.2008.05.006.

[5]

Y. F. ChiuM. H. Hsieh and C. Tsai, Valuation and analysis on complex equity indexed annuities, Pacific-Basin Finance Journal, 57 (2019), 101175.  doi: 10.1016/j.pacfin.2019.101175.

[6]

Z. CuiJ. Kirkby and D. Nguyen, Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps, Insurance Math. Econom., 74 (2017), 46-62.  doi: 10.1016/j.insmatheco.2017.02.010.

[7]

D. Dufresne, Fitting combinations of exponentials to probability distributions, Appl. Stoch. Models Bus. Ind., 23 (2007), 23-48.  doi: 10.1002/asmb.635.

[8]

D. Dufresne, Stochastic life annuities, N. Am. Actuar. J., 11 (2007), 136-157.  doi: 10.1080/10920277.2007.10597441.

[9]

M. Dai and Y. K. Kwok, American options with lookback payoff, SIAM J. Appl. Math., 66 (2005), 206-227.  doi: 10.1137/S0036139903437345.

[10]

L. Feng and V. Linetsky, Computing exponential moments of the discrete maximum of a Lévy process and lookback options, Finance Stoch., 13 (2009), 501-529.  doi: 10.1007/s00780-009-0096-x.

[11]

R. Feng and X. Jing, Analytical valuation and hedging of variable annuity guaranteed lifetime withdrawal benefits, Insurance Math. Econom, 72 (2017), 36-48.  doi: 10.1016/j.insmatheco.2016.10.011.

[12]

H. U. Gerber and E. S. W. Shiu, Pricing lookback options and dynamic guarantees, N. Am. Actuar. J., 7 (2003), 48-67.  doi: 10.1080/10920277.2003.10596076.

[13]

H. U. GerberE. S. W. Shiu and H. Yang, Valuing equity-linked death benefits and other contingent options: A discounted density approach, Insurance Math. Econom., 51 (2012), 73-92.  doi: 10.1016/j.insmatheco.2012.03.001.

[14]

H. U. GerberE. S. W. Shiu and H. Yang, Valuing equity-linked death benefits in jump diffusion models, Insurance Math. Econom., 53 (2013), 615-623.  doi: 10.1016/j.insmatheco.2013.08.010.

[15]

H. U. GerberE. S. W. Shiu and H. Yang, Geometric stopping of a random walk and its applications to valuing equity-linked death benefits, Insurance Math. Econom., 64 (2015), 313-325.  doi: 10.1016/j.insmatheco.2015.06.006.

[16]

J. M. Harrison, Brownian Motion and Stochastic Flow Systems, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1985.

[17]

M. Hardy, Ratchet equity indexed annuities, In Contributions to: Proceedings of the 14th Annual International AFIR Colloquium, Boston, 2004.

[18]

P. Hieber, Cliquet-style return guarantees in a regime switching Lévy model, Insurance Math. Econom., 72 (2017), 138-147.  doi: 10.1016/j.insmatheco.2016.11.009.

[19]

J. L. Kirkby, American and exotic option pricing with jump diffusions and other Lévy processes, Journal of Computational Finance, 22 (2018), 89-148.  doi: 10.21314/JCF.2018.355.

[20]

J. L. Kirkby and D. Nguyen, Equity-linked guaranteed minimum death benefits with dollar cost averaging, Insurance Math. Econom., 100 (2021), 408-428.  doi: 10.1016/j.insmatheco.2021.04.012.

[21]

M. Kijima and T. Wong, Pricing of ratchet equity-indexed annuities under stochastic interest rates, Insurance Math. Econom., 41 (2007), 317-338.  doi: 10.1016/j.insmatheco.2006.11.005.

[22]

H. Lee, Pricing equity-indexed annuities with path-dependent options, Insurance Math. Econom., 33 (2003), 677-690.  doi: 10.1016/j.insmatheco.2003.09.006.

[23]

X. LiangC. C. L. Tsai and Y. Lu, Valuing guaranteed equity-linked contracts under piecewise constant forces of mortality, Insurance Math. Econom., 70 (2016), 150-161.  doi: 10.1016/j.insmatheco.2016.06.004.

[24]

L. QianW. WangR. Wang and Y. Tang, Valuation of equity-indexed annuity under stochastic mortality and interest rate, Insurance Math. Econom., 47 (2010), 123-129.  doi: 10.1016/j.insmatheco.2010.06.005.

[25]

C. C. SiuS. C. P. Yam and H. Yang, Valuing equity-linked death benefits in a regime-switching framework, Astin Bull., 45 (2015), 355-395.  doi: 10.1017/asb.2014.32.

[26]

S. Tiong, Valuing equity-indexed annuities, N. Am. Actuar. J., 4 (2000), 149-170.  doi: 10.1080/10920277.2000.10595945.

[27]

E. R. Ulm, The effect of the real option to transfer on the value of guaranteed minimum death benefits, Journal of Risk and Insurance, 73 (2006), 43-69.  doi: 10.1111/j.1539-6975.2006.00165.x.

[28]

E. R. Ulm, Analytic solution for return of premium and rollup guaranteed minimum death benefit options under some simple mortality laws, Astin Bull., 38 (2008), 543-563.  doi: 10.1017/S0515036100015282.

[29]

E. R. Ulm, Analytic solution for ratchet guaranteed minimum death benefit options under a variety of mortality laws, Insurance Math. Econom., 58 (2014), 14-23.  doi: 10.1016/j.insmatheco.2014.06.003.

[30]

Z. Zhang, Y. Yong and W. Yu, Valuing equity-linked death benefits in general exponential Lévy models, J. Comput. Appl. Math., 365 (2020), 112377, 18pp. doi: 10.1016/j.cam.2019.112377.

show all references

References:
[1]

M. J. Brennan and E. S. Schwartz, The pricing of equity-linked life insurance policies with an asset value guarantee, Journal of Financial Economics, 3 (1976), 195-213.  doi: 10.1016/0304-405X(76)90003-9.

[2]

N. Bowers, H. U. Gerber, J. C. Hickman, D. A. Jones and C. J. Nesbit, Actuarial Mathematics, 2nd edition, The Society of Actuaries, Illinois, 1997.

[3]

P. P. Boyle and E. S. Schwartz, Equilibrium prices of guarantees under equity-linked contracts, Journal of Risk and Insurance, 44 (1977), 639-660.  doi: 10.2307/251725.

[4]

P. P. Boyle and W. Tian, The design of equity-indexed annuities, Insurance Math. Econom., 43 (2008), 303-315.  doi: 10.1016/j.insmatheco.2008.05.006.

[5]

Y. F. ChiuM. H. Hsieh and C. Tsai, Valuation and analysis on complex equity indexed annuities, Pacific-Basin Finance Journal, 57 (2019), 101175.  doi: 10.1016/j.pacfin.2019.101175.

[6]

Z. CuiJ. Kirkby and D. Nguyen, Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps, Insurance Math. Econom., 74 (2017), 46-62.  doi: 10.1016/j.insmatheco.2017.02.010.

[7]

D. Dufresne, Fitting combinations of exponentials to probability distributions, Appl. Stoch. Models Bus. Ind., 23 (2007), 23-48.  doi: 10.1002/asmb.635.

[8]

D. Dufresne, Stochastic life annuities, N. Am. Actuar. J., 11 (2007), 136-157.  doi: 10.1080/10920277.2007.10597441.

[9]

M. Dai and Y. K. Kwok, American options with lookback payoff, SIAM J. Appl. Math., 66 (2005), 206-227.  doi: 10.1137/S0036139903437345.

[10]

L. Feng and V. Linetsky, Computing exponential moments of the discrete maximum of a Lévy process and lookback options, Finance Stoch., 13 (2009), 501-529.  doi: 10.1007/s00780-009-0096-x.

[11]

R. Feng and X. Jing, Analytical valuation and hedging of variable annuity guaranteed lifetime withdrawal benefits, Insurance Math. Econom, 72 (2017), 36-48.  doi: 10.1016/j.insmatheco.2016.10.011.

[12]

H. U. Gerber and E. S. W. Shiu, Pricing lookback options and dynamic guarantees, N. Am. Actuar. J., 7 (2003), 48-67.  doi: 10.1080/10920277.2003.10596076.

[13]

H. U. GerberE. S. W. Shiu and H. Yang, Valuing equity-linked death benefits and other contingent options: A discounted density approach, Insurance Math. Econom., 51 (2012), 73-92.  doi: 10.1016/j.insmatheco.2012.03.001.

[14]

H. U. GerberE. S. W. Shiu and H. Yang, Valuing equity-linked death benefits in jump diffusion models, Insurance Math. Econom., 53 (2013), 615-623.  doi: 10.1016/j.insmatheco.2013.08.010.

[15]

H. U. GerberE. S. W. Shiu and H. Yang, Geometric stopping of a random walk and its applications to valuing equity-linked death benefits, Insurance Math. Econom., 64 (2015), 313-325.  doi: 10.1016/j.insmatheco.2015.06.006.

[16]

J. M. Harrison, Brownian Motion and Stochastic Flow Systems, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1985.

[17]

M. Hardy, Ratchet equity indexed annuities, In Contributions to: Proceedings of the 14th Annual International AFIR Colloquium, Boston, 2004.

[18]

P. Hieber, Cliquet-style return guarantees in a regime switching Lévy model, Insurance Math. Econom., 72 (2017), 138-147.  doi: 10.1016/j.insmatheco.2016.11.009.

[19]

J. L. Kirkby, American and exotic option pricing with jump diffusions and other Lévy processes, Journal of Computational Finance, 22 (2018), 89-148.  doi: 10.21314/JCF.2018.355.

[20]

J. L. Kirkby and D. Nguyen, Equity-linked guaranteed minimum death benefits with dollar cost averaging, Insurance Math. Econom., 100 (2021), 408-428.  doi: 10.1016/j.insmatheco.2021.04.012.

[21]

M. Kijima and T. Wong, Pricing of ratchet equity-indexed annuities under stochastic interest rates, Insurance Math. Econom., 41 (2007), 317-338.  doi: 10.1016/j.insmatheco.2006.11.005.

[22]

H. Lee, Pricing equity-indexed annuities with path-dependent options, Insurance Math. Econom., 33 (2003), 677-690.  doi: 10.1016/j.insmatheco.2003.09.006.

[23]

X. LiangC. C. L. Tsai and Y. Lu, Valuing guaranteed equity-linked contracts under piecewise constant forces of mortality, Insurance Math. Econom., 70 (2016), 150-161.  doi: 10.1016/j.insmatheco.2016.06.004.

[24]

L. QianW. WangR. Wang and Y. Tang, Valuation of equity-indexed annuity under stochastic mortality and interest rate, Insurance Math. Econom., 47 (2010), 123-129.  doi: 10.1016/j.insmatheco.2010.06.005.

[25]

C. C. SiuS. C. P. Yam and H. Yang, Valuing equity-linked death benefits in a regime-switching framework, Astin Bull., 45 (2015), 355-395.  doi: 10.1017/asb.2014.32.

[26]

S. Tiong, Valuing equity-indexed annuities, N. Am. Actuar. J., 4 (2000), 149-170.  doi: 10.1080/10920277.2000.10595945.

[27]

E. R. Ulm, The effect of the real option to transfer on the value of guaranteed minimum death benefits, Journal of Risk and Insurance, 73 (2006), 43-69.  doi: 10.1111/j.1539-6975.2006.00165.x.

[28]

E. R. Ulm, Analytic solution for return of premium and rollup guaranteed minimum death benefit options under some simple mortality laws, Astin Bull., 38 (2008), 543-563.  doi: 10.1017/S0515036100015282.

[29]

E. R. Ulm, Analytic solution for ratchet guaranteed minimum death benefit options under a variety of mortality laws, Insurance Math. Econom., 58 (2014), 14-23.  doi: 10.1016/j.insmatheco.2014.06.003.

[30]

Z. Zhang, Y. Yong and W. Yu, Valuing equity-linked death benefits in general exponential Lévy models, J. Comput. Appl. Math., 365 (2020), 112377, 18pp. doi: 10.1016/j.cam.2019.112377.

Table 1.  Valuation results w.r.t $ g $, $ \alpha = 90\%,\ n = 10,\ \sigma = 0.25 $
$ g(\%)/\mathcal{M} $ $ \mathcal{M}_3 $ $ \mathcal{M}_5 $ $ \mathcal{M}_{10} $
BM RM BM RM BM RM
0.50 1.8154 4.3346 1.8154 4.3342 1.8155 4.3336
1.00 1.8549 4.3390 1.8549 4.3385 1.8550 4.3379
1.50 1.8962 4.3462 1.8962 4.3458 1.8962 4.3451
2.00 1.9392 4.3564 1.9392 4.3560 1.9393 4.3554
2.50 1.9842 4.3697 1.9842 4.3693 1.9842 4.3687
3.00 2.0311 4.3861 2.0311 4.3857 2.0311 4.3850
3.50 2.0802 4.4056 2.0801 4.4052 2.0802 4.4045
4.00 2.1314 4.4284 2.1314 4.4279 2.1314 4.4273
$ g(\%)/\mathcal{M} $ $ \mathcal{M}_3 $ $ \mathcal{M}_5 $ $ \mathcal{M}_{10} $
BM RM BM RM BM RM
0.50 1.8154 4.3346 1.8154 4.3342 1.8155 4.3336
1.00 1.8549 4.3390 1.8549 4.3385 1.8550 4.3379
1.50 1.8962 4.3462 1.8962 4.3458 1.8962 4.3451
2.00 1.9392 4.3564 1.9392 4.3560 1.9393 4.3554
2.50 1.9842 4.3697 1.9842 4.3693 1.9842 4.3687
3.00 2.0311 4.3861 2.0311 4.3857 2.0311 4.3850
3.50 2.0802 4.4056 2.0801 4.4052 2.0802 4.4045
4.00 2.1314 4.4284 2.1314 4.4279 2.1314 4.4273
Table 2.  Valuation results w.r.t $ \alpha $, $ g = 2.5\%,\ n = 10,\ \sigma = 0.25 $
$ \alpha(\%)/\mathcal{M} $ $ \mathcal{M}_3 $ $ \mathcal{M}_5 $ $ \mathcal{M}_{10} $
BM RM BM RM BM RM
75 1.6405 3.1056 1.6405 3.1054 1.6405 3.1053
80 1.7465 3.4771 1.7465 3.4768 1.7465 3.4765
85 1.8608 3.8963 1.8608 3.8960 1.8608 3.8955
90 1.9842 4.3698 1.9842 4.3694 1.9842 4.3687
$ \alpha(\%)/\mathcal{M} $ $ \mathcal{M}_3 $ $ \mathcal{M}_5 $ $ \mathcal{M}_{10} $
BM RM BM RM BM RM
75 1.6405 3.1056 1.6405 3.1054 1.6405 3.1053
80 1.7465 3.4771 1.7465 3.4768 1.7465 3.4765
85 1.8608 3.8963 1.8608 3.8960 1.8608 3.8955
90 1.9842 4.3698 1.9842 4.3694 1.9842 4.3687
Table 3.  Valuation results w.r.t $ \sigma $, $ \alpha = 90\%,\ g = 2.5\%,\ n = 10 $
$ \sigma/\mathcal{M} $ $ \mathcal{M}_3 $ $ \mathcal{M}_5 $ $ \mathcal{M}_{10} $
BM RM BM RM BM RM
0.05 1.0447 1.1639 1.0448 1.1639 1.0447 1.1639
0.10 1.2271 1.6011 1.2271 1.6011 1.2271 1.6012
0.15 1.4440 2.2334 1.4440 2.2334 1.4440 2.2334
0.20 1.6954 3.1246 1.6954 3.1244 1.6954 3.1242
0.25 1.9842 4.3698 1.9842 4.3694 1.9842 4.3687
0.30 2.3140 6.1005 2.3139 6.0997 2.3139 6.0982
$ \sigma/\mathcal{M} $ $ \mathcal{M}_3 $ $ \mathcal{M}_5 $ $ \mathcal{M}_{10} $
BM RM BM RM BM RM
0.05 1.0447 1.1639 1.0448 1.1639 1.0447 1.1639
0.10 1.2271 1.6011 1.2271 1.6011 1.2271 1.6012
0.15 1.4440 2.2334 1.4440 2.2334 1.4440 2.2334
0.20 1.6954 3.1246 1.6954 3.1244 1.6954 3.1242
0.25 1.9842 4.3698 1.9842 4.3694 1.9842 4.3687
0.30 2.3140 6.1005 2.3139 6.0997 2.3139 6.0982
Table 4.  Valuation results w.r.t $ n $, $ \alpha = 90\%,\ g = 2.5\%,\ \sigma = 0.25 $
$ T/\mathcal{M} $ $ \mathcal{M}_3 $ $ \mathcal{M}_5 $ $ \mathcal{M}_{10} $
BM RM BM RM BM RM
1 1.0716 1.1596 1.0717 1.1597 1.0716 1.1597
3 1.2271 1.5508 1.2272 1.5510 1.2275 1.5518
5 1.4046 2.0747 1.4048 2.0750 1.4053 2.0766
7 1.6105 2.7868 1.6106 2.7870 1.6112 2.7887
10 1.9842 4.3698 1.9842 4.3694 1.9842 4.3687
$ T/\mathcal{M} $ $ \mathcal{M}_3 $ $ \mathcal{M}_5 $ $ \mathcal{M}_{10} $
BM RM BM RM BM RM
1 1.0716 1.1596 1.0717 1.1597 1.0716 1.1597
3 1.2271 1.5508 1.2272 1.5510 1.2275 1.5518
5 1.4046 2.0747 1.4048 2.0750 1.4053 2.0766
7 1.6105 2.7868 1.6106 2.7870 1.6112 2.7887
10 1.9842 4.3698 1.9842 4.3694 1.9842 4.3687
Table 5.  Closed-form formula (20) v.s. Monte Carlo, BM case
$ \mathcal{M}_3 $ $ \alpha $(%)
0.75 0.8 0.85 0.9
BM 1.6405 1.7465 1.8608 1.9842
(time) (0.0012) (0.0011) (0.0010) (0.0013)
MC 1.6412 1.7471 1.8613 1.9845
(time) (301.9655) (313.2406) (310.3458) (312.7912)
$ \mathcal{M}_3 $ $ \alpha $(%)
0.75 0.8 0.85 0.9
BM 1.6405 1.7465 1.8608 1.9842
(time) (0.0012) (0.0011) (0.0010) (0.0013)
MC 1.6412 1.7471 1.8613 1.9845
(time) (301.9655) (313.2406) (310.3458) (312.7912)
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