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Implicit options in life insurance: An overview


Proper pricing and risk assessment of implicit options in life insurance contracts has gained substantial attention in recent years, which is reflected in a growing literature in this field. In this article, we first present the different contract designs in Europe and the United States and point out differences in the contract design. Second, a comprehensive overview and description of implicit options contained in these contracts is provided. With focus on participating contracts, we present contract design, valuation methods, and main results of several recent articles in this field. The study indicates that current developments regarding regulation (Solvency II, Swiss Solvency Test), accounting (IFRS), customer needs, and secondary life insurance markets may lead to a trend away from traditional contract design of participating policies and toward new products that are of a more transparent modular form such as variable annuities. These new contracts will contain fewer basic guarantees and a set of additional, adequately priced options.


Die adäquate Bewertung und Risikomessung von impliziten Optionen haben in den letzten Jahren stark an Bedeutung gewonnen. Ziel des vorliegenden Beitrags ist zunächst die Darstellung von Lebensversicherungsverträgen in Europa und den USA sowie das Aufzeigen von zentralen Unterschieden. Darüber hinaus werden ein umfassender Überblick und eine Beschreibung von impliziten Optionen gegeben, die in diesen Verträgen enthalten sind. Des Weiteren werden Vertragsdesign, Bewertungsmethoden und zentrale Ergebnisse von mehreren Artikeln vorgestellt, die sich mit traditionellen gemischten Kapitallebensversicherungsverträgen befassen. Die Untersuchung zeigt, dass die Entwicklungen im Hinblick auf Regulierung (Solvency II, Swiss Solvency Test), Aufsicht (IFRS), Kundenbedürfnisse und dem Zweitmarkt für Lebensversicherungen zu einem Trend weg von traditionellen Lebensversicherungsverträgen und hin zu modularen Produkten führen. Diese haben ein transparenteres Design mit wenigen Basisgarantien und einer Auswahl an zusätzlichen, adäquat bewerteten Optionen und Garantiebausteinen.

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  1. 1.

    The case of Equitable Life is discussed in O'Brien (2006).

  2. 2.

    For an overview and analysis of the Solvency II process, see Eling et al. (2007).

  3. 3.

    Developments in the field of International Financial Reporting Standards and the fair valuation approach are discussed in Jørgensen (2004).

  4. 4.

    Paragraph 5 of SFAS 157 defines the fair value as “the price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date.” In IFRS, fair value is generally defined as “the amount for which an asset could be exchanged, or a liability settled, between knowledgeable, willing parties in an arm's length transaction” (IASB 2006).

  5. 5.

    In 2005, the European share of worldwide life insurance premiums amounted to 37.6%, and the United States had a 26.2% share (Enz 2006).

  6. 6.

    For the following description of life insurance policies, see Bowers et al. (1997) and Trieschmann et al. (2005).

  7. 7.

    A comparison of different smoothing schemes in the United Kingdom can be found in Haberman et al. (2003).

  8. 8.

    See Enz (2006, p. 17) for a description of the development of unit-linked products.

  9. 9.

    An extensive discussion of unit-linked insurance products is provided in Helfenstein and Barnshaw (2003).

  10. 10.

    The four biggest life insurance markets in Europe together represented 70.9% of the total European life insurance premiums in 2004: United Kingdom (27.5%), France (19.1%), Germany (12.4%), and Italy (11.9%) (see European Insurance and Reinsurance Federation CEA 2006, p. 12). According to the European Insurance and Reinsurance Federation CEA (2006, p. 10), the share of unit-linked products in total life insurance premiums in the United Kingdom increased from 70% to 78% in 2004.

  11. 11.

    See Enz (2006, p. 17).

  12. 12.

    See, e. g., Grosen and Jørgensen (2000).

  13. 13.

    See, e. g., Briys and de Varenne (1997); Grosen and Jørgensen (2002).

  14. 14.

    See, e. g., Steffensen (2002); Linnemann (2003, 2004); Gatzert and Schmeiser (2008).

  15. 15.

    See Albizzati and Geman (1994); Bacinello (2001, 2003a, 2003b); Grosen and Jørgensen (1997, 2000); Steffensen (2002).

  16. 16.

    See Gatzert et al. (2008).

  17. 17.

    See Dillmann and Ruß (2001).

  18. 18.

    See Milevsky and Promislow (2001); Boyle and Hardy (2003); Haberman and Ballotta (2003).

  19. 19.

    Participating endowment contracts made up approximately 80% of the total liabilities for contracts with guaranteed annuity options (Boyle and Hardy 2003). The remaining part was mainly composed of unit-linked contracts.

  20. 20.

    See O'Brien (2006).

  21. 21.

    Carson 1994; Carson and Ostaszewski 2004.

  22. 22.

    An example of how minimum interest rate guarantees in participating life insurance contracts can be priced is provided in the Appendix A.

  23. 23.

    See Grosen and Jørgensen (2000); Bacinello (2003a, 2003b); Gatzert and Schmeiser (2008).

  24. 24.

    See results of the Swiss Solvency Test preliminary analysis field test conducted by the Swiss regulators in 2005 (Bundesamt für Privatversicherungen 2005).

  25. 25.

    The Swiss Solvency Test preliminary analysis field test in 2005 showed that particularly the lapsation and longevity scenarios can have substantial effects in terms of the risk-bearing capital.

  26. 26.

    See Gatzert and Kling (2007) for a more detailed analysis of fair pricing and shortfall risk for different contract types.

  27. 27.

    Let \( ( W_{t}), 0 \le t \le T \), be a standard Brownian motion on a probability space \( (\Omega , \mathcal{F} , \mathbb{P} ) \) and \( (\mathcal{F} _{t}), 0 \le \) \( t \le T, \) be the filtration generated by the Brownian motion.

  28. 28.

    For details see, e. g., Björk (2004).

  29. 29.

    For details concerning risk-neutral valuation, see, e. g., Björk (2004).

  30. 30.

    Note that this equation does not consider the limited liability of equityholders. For an example, see, e. g., Grosen and Jørgensen (2002).

  31. 31.

    For Monte Carlo simulation, see, e. g., Glasserman (2004).


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Correspondence to Nadine Gatzert.



Modeling and pricing of a participating life insurance contract with minimum interest rate guarantee

The following example presents a modeling and pricing procedure for participating life insurance contracts. Pricing is conducted using risk-neutral valuation.Footnote 26 We con sider an insurance company, where policyholders make an exogenously given up-front payment of \( P_0 \) at contract inception (at time \( t = 0 \)), and \( E_0 \) is the initial payment made by the equityholders. The total value of initial contributions is then invested in assets.

Modeling the assets

We assume that the total market value of the asset portfolio follows a geometric Brownian motion.Footnote 27 Under the objective (real-world) measure \( \mathbb{P}, \) the asset process is described by

$$ dA ( t ) = \mu ( t ) A ( t ) dt + \sigma ( t ) A ( t ) dW^{\mathbb{P}} ( t )\,, $$

with asset drift \( \mu (t), \) volatility \( \sigma (t) \) that are assumed to be constant over time and a \( \mathbb{P} \)-Brownian motion \( W^{\mathbb{P}}. \) We assume a complete, perfect, and frictionless market. Thus, the solution of the stochastic differential equationFootnote 28 is given by

$$ A(t) = A(t-1) \cdot \exp \left( {\mu -\frac{\sigma ^2}{2} + \sigma \cdot ( {W^{\mathbb{P}} ( t ) - W^{\mathbb{P}} ( {t-1} )} )} \right)\,, $$

with \( A(0)=P_0 +E_0. \) Since pricing is conducted using risk-neutral valuation, the measure needs to be changed to the (risk-neutral) unique equivalent martingale measure \( \mathbb{Q}. \) In this setting, the drift of the geometric Brownian motion changes to the risk-free interest rate r, and development of the assets is given by

$$ dA ( t ) = rA ( t ) dt + \sigma ( t ) A ( t ) dW^{\mathbb{Q}} ( t )\,, $$

where \( W^{\mathbb{Q}} \) is a \( \mathbb{Q} \)-Brownian motion. The solution of this stochastic differential equation under \( \mathbb{Q} \) is given analogously as under \( \mathbb{P}. \)

Modeling the liabilities

Let P denote the policyholder's account, i. e., the book value of the policy reserves. Every year \( t = 1,2, {\ldots}\,, \) the policy reserve P is compounded with the policy interest rate \( r_{\text{P}}(t) \) that depends on the type of guarantee (point-to-point vs. cliquet-style interest rate guarantee) and the type of surplus distribution provided by the insurer. Thus, the development of the policy can in general be described by

$$ P ( t ) = P ( {t-1} ) \cdot ( {1+r_{\text{P}} ( t )} ) = P_0 \cdot \prod\limits_{i=1}^t {( {1+r_{\text{P}} ( i )} )}, t=1,2,{\ldots} $$

Depending on the performance of the insurer's asset base, the customer receives the accumulated book value \( P(T{\kern0.5pt}) \) of the contract at maturity T, and, for some contract designs, a terminal bonus \( S(T{\kern0.5pt}). \) Hence, the payoff \( L(T{\kern0.5pt}) \) to the policyholder is

$$ L ( T{\kern0.5pt}) = P ( T{\kern0.5pt}) + S ( T{\kern0.5pt}) = P_0 \cdot \prod\limits_{t=1}^T {( {1+r_{\text{P}} ( t )} ) + S(T{\kern0.5pt})}\,. $$

The exact form of the terminal bonus \( S(T) \) and the policy interest rate \( r_{\text{P}}(t) \) depends on the contract specifications, with one example being provided below. The amount of surplus credited to the policy reserves (represented by \( r_{\text{P}}(t)) \) as well as the terminal bonus payment typically depends on the insurance company's financial situation and thus on the development of the insurer's asset portfolio, respectively.

Fair contracts

We determine fair contracts using risk-neutral valuation, such that

$$ V_0 ( {L ( T{\kern0.5pt})} ) = e^{-rT} {\mathbb{E}}^{\mathbb{Q}} ( {L ( T{\kern0.5pt})} )\,, $$

where \( {\mathbb{E}}^{\mathbb{Q}} ( \cdot ) \) denotes the expectation under the risk-neutral martingale measureFootnote 29 \( \mathbb{Q}. \) A contract is considered fair if the time zero market value \( V_0 ( \cdot ) \) of the final payoff \( L ( T{\kern0.5pt}) \) under the risk-neutral measure \( \mathbb{Q} \) is equal to the up-front premium \( P_{0} \) paid by the policyholder,Footnote 30

$$ P_0 = V_0 ( {L ( T{\kern0.5pt})} ) = e^{-rT} {\mathbb{E}}^{\mathbb{Q}} ( {L ( T{\kern0.5pt})} )\,. $$

This equation can be used to find parameter combinations of fair contracts. Models that do not allow for explicit analytical expressions can be analyzed using simulation techniques.Footnote 31

Example: Point-to-point guarantee

In the case of a point-to-point guarantee, the single up-front premium \( P_{0} \) is compounded with the guaranteed interest rate g. The policy interest rate is thus given by \( r_{\text{P}} ( t ) = g, \) such that at expiration of the contract, the guaranteed payment is

$$ P ( T{\kern0.5pt}) =P_0 \cdot \prod\limits_{i=1}^T {( {1+g} )} = P_0 \cdot ( {1+g} )^T\,. $$

Additionally, the customer receives a fraction \( \delta \) of the terminal surplus, i. e., if \( A ( T{\kern0.5pt}) - L ( T{\kern0.5pt}) \textgreater 0. \) Hence, the final payoff \( L ( T{\kern0.5pt}) \) (without accounting for the limited liability of equityholders) can be summarized by

$$ L ( T{\kern0.5pt}) = P (T{\kern0.5pt}) + S (T{\kern0.5pt}) = P_0 \cdot \prod\limits_{i=1}^T {( {1+g} )} + \delta \cdot \max ( {A(T{\kern0.5pt})-P(T{\kern0.5pt}),0} )\,. $$

Thus, the payoff can be decomposed into two parts: the first term is a bond with a fixed payoff, whereas the second term is the payoff of a European call option on \( A(T) \) with (stochastic) strike price \( P(T). \) The closed-form solution for the market value \( V_0 ( {L ( T )} ) \) of the payoff using European option pricing theory is

$$ V_0 ( {L ( T )} ) = e^{-rT} {\mathbb{E}}^{\mathbb{Q}} ( {L ( T )} ) = e^{-rT} \cdot P(T) + \delta \cdot \big( {A_0 \cdot \Phi ( {d_1 } ) - P(T) \cdot e^{-rT} \cdot \Phi ( {d_2 } )} \big) $$


$$ d_1 = \frac{\ln ( {A_0 /P ( T )} ) + ( {r+\sigma ^{2} /2} ) \cdot T}{\sigma \cdot \sqrt T } \quad \text{and} \quad d_2 = d_1 - \sigma \cdot \sqrt T\,. $$

Further types of interest rate guarantees are presented and examined in, e. g., Gatzert and Kling (2007).

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Gatzert, N. Implicit options in life insurance: An overview. ZVersWiss 98, 141–164 (2009).

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  • Life Insurance
  • International Financial Reporting Standard
  • Insurance Contract
  • Contract Design
  • Premium Payment