Valuation of cliquet-style guarantees with death benefits

Yaodi Yong; Hailiang Yang

doi:10.3934/jimo.2021188

Article Contents

2023, Volume 19, Issue 1: 359-375. Doi: 10.3934/jimo.2021188

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Valuation of cliquet-style guarantees with death benefits

Yaodi Yong^1, , and
Hailiang Yang^1,

1.
Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong, China

^* Corresponding author: Yaodi Yong
^* Corresponding author: Yaodi Yong

Received: June 2021

Revised: September 2021

Early access: November 2021

Published: January 2023

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

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Abstract

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.

Keywords:

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

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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} $
$ g(\%)/\mathcal{M} $	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

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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} $
$ \alpha(\%)/\mathcal{M} $	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

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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} $
$ \sigma/\mathcal{M} $	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

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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} $
$ T/\mathcal{M} $	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

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Table 5. Closed-form formula (20) v.s. Monte Carlo, BM case

$ \mathcal{M}_3 $	$ \alpha $(%)
$ \mathcal{M}_3 $	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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