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Article Contents

# Valuation of cliquet-style guarantees with death benefits

• * Corresponding author: Yaodi Yong

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.

Mathematics Subject Classification: Primary: 91G05, 91G80; Secondary: 91G15.

 Citation:

• 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

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

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

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

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)

Tables(5)