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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 & 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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