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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:

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##### References:
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
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
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
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
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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