Chiao Da Management Review Vol. 29 No. 1, 2009
pp.79-101
考慮、壽險次級市場下的保單計償新思
維:以變額壽險為例
The Alternative Pricing Approach for Variable Life
Insurance IncorporatingSecondary Life Insurance
Market
羅，~!t 明Lieh-MingLuo
輔仁大學 國際貿易與金融學~
Department ofInternational Trade & Finance, Fu Jen Catholic University 許和鈞 Her-Jiun Sheu
國立暨南國際大學 財務金融學象
Dep缸tmentofBanking and Finance, National Chi Nan University
摘要:變頭壽險係單主要特點為保早到期給付金頌連結於投資標的資產之市 場價格，因係單所附係證收益可視同賣權價值，典型評價模式即結合現代財 務選擇權理論與傳統壽險精算等價原理，文獻上後績的相關研究皆沿用此計 價原則。由於近年來美國壽險係單次級市場的快速發展與成長，使得壽險係 單在市場交易的流動性大為增加，壽險係單已不單是為獲取保障的保險契 約，亦是可交易的資產組合，此市場特徵提供 T 應用財務選擇權計價模式的 重要條件，因此，有別於上述傳統典型的計價方法，本研究以純粹的財務選 擇權定價觀點，特別納入壽險次級市場因素，針對變額壽險係單提出另一種 計價核式，並以此觀點檢視變額壽險保翠的傳統計價方法及其特性。在本研 究的計價祭構下，除証明了傳統計價方法所求得之價格將對應一組特定的風 險中立淚1 J.主外，數值分析結果亦說明了無套利的合理價格與傳統計價結呆的 關餘，結果顯示，保萃的無套利合理價格範囝將因速動資產價值的波動性、 無風險利率及死亡機率型態之改變而呈不同方向的變動，尤其會隨著壽險係 單在次級市場的流動性風險溢酬增加而擴大。
1 Corresponding author: Dep缸師lentof Intemational Trade & Finaoce, Fu Jen Catholic University,
80 The Alternative Pricing Approach for Variable Lij告 Insurance Incorporating Secondl呵崢 Insurance 岫rket
關鍵詞:變額壽險保草;選擇權定價理論;壽險保單次級市場
Abstract: One distinguishing feature of variable life insurance policy is that the benefit payable at expiration depends on the market value of the linked reference portfolio as contrasted with traditional life insurance policies. The conventional pricing approach combines 仕aditional law of large number considerations and financial mathematics. Subsequent relevant studies follow such the valuation approaches. Because recently secondary life insurance markets in America are developing and growing rapidl弘 liquidity of life insurance contracts has significantly improved. So life insurance contracts could not only be guarantees against losses, but also could be seen as tradable portfolio assets. This market characteristic could serve an extra condition for the application of option pricing model to the valuation of variable life insurance. In this article, in comparison with the conventional pricing approaches for variable life insurance, an alternative valuation method is developed with pure option pricing approach especially incorporating the secondary life insurance market. The conventional valuation approach and its properties are reviewed and its derived price is proved as a special one with respect to a specific risk-neutral probability measure in the present valuation 企amework. Numerical analysis illustrates the relationship between no-arbitrage price bounds and the conventional pricing approach as well.
The results indicate that no-arbitrage bounds of the insurance contract would be influenced by asset price volatility, risk -free rate and mortality pa位ern in different directions, and particularly would be augmented with liquidity risk premium in the secondary life insurance market
Keywords: Variable life insurance; Option pricing approach; Secondary life insurance market
1. Introduction
Secondary life insurance markets have been growing rapidly in America. A wide variety of similar products in secondary life insurance market have been developed
,
including viatical settlements,
accelerated death benefits (ADBs) and life settlements. Secondary life insurance markets allow consumers to sell theirChiao Da Management Review Vol. 29 No. 1, 2009 81
policies to independent financial cornpanies or originally-issued insurance cornpany for getting rnoney back (Bhattacharya et. al. 2004). So life insurance contracts could not only serve as guarantees against losses, but also could be seen as tradable portfolio assets. This rnarket characteristic could offer an extra condition for the application of option pricing rnodel to the valuation of life insurance contract. This article focuses on the valu且tion of variable life insurance under a pure option pricing frarnework. The conventional approach for the valuation of variable life insurance cornbines traditional law of large nurnber considerations and financial rnathernatics. It usually assurnes the independence between the stochasticity of a reference fund and rnortality distribution as well as the insurer's risk neu仕ality with respωt to rnortality. The logic behind those assurnptions is that insurers usually suppose the policyholders with the sarne age will have the sarne death distribution (said to be hornogenous) and each
policyholder法 death is independent of other這. Thus only when insurers can obtain a large nurnber of independent hornogenous insurance buyers, the conventional pricing approach could be applied. However, not all of insurance cornpanies can satisfy cornpletely the requirernent for pooling arrangernents. In cornparison with the conventional pricing approach for variable life insurance,
this study develops an alternative valuation rnethod with the pure option pricing approach especially incorporating the secondary life insurance rnarket. Without requiring the independence assurnption in the conventional approach, the price process of a reference fund and 也e death process of an insured are considered jointly to create an underlying stochastic process. A typical option pricing approach usually begins with assurning an underlying asset following a specific stochastic process. Contingent payrnents at each tirne are deterrnined by exercise price. The variable life insurance also could be 甘eatedas a contingent clairn of th巴
rnarket structure we create. This proposed approach will lead to prices that coincide with those deterrnined by the conventional pricing approach (CPA henceforth) suggested by Brennan and Schwartz (1976, 1977, 1979). A different insight into properties of the conventional pricing approach has been explored
The distinguishing feature of variable life insurance is that b巴nefit payable at expiration
82 The Alternative Pricing Approach for •卸的bleL拚 Insurance Incorpora伽rgSeconda.η 排 Insurance 岫rket
consist of stocks, bonds or other financial assets with mutual funds as typical cases. Because policyholders have to bear more risk for this type of insurance product, insurers need to enhance the product attraction by posing additional guarantees. Thus these insurance products 句pically provide policyholder with a minimum guaranteed value on death of the insured or on maturity of the con仕act This kind of insurance product is called the guaranteed variable life insurance policy. The benefit of the insurance contract thus can depend on the performance of the linked reference fund and the guarant的d values. The conventional pricing approach (CPA) initiated by Brennan and Schwartz basically s個rts by calculating the market price of the payoffs which occur at each time point within the contract term and then take account of the expectation on the mortality. In brief, the CPA integrates the option pricing theory and the principle of equivalence. For example,
Brennan and Schwartz (1976, 1977) assumes that the price ofthe reference fund follows a geometric Brownian motion process and then the guarantees are treated as European-type put options which could be solved using Black-Scholes model (Black and Scholes, 1973); hence the fair price can be derived by specifying the market value of contingent payo宜s times the mortality probability using independence assumption between market and mortality risks as well as the traditional law of large numbers. With the CPA, several relevant studies in the literature have been carried out. Some works consider other contract designs with different s個ctures of benefits such as the caps (Eker and Persson, 1996) and the endogenous minimum guarantees (Bacinello and Ortu, 1993). stochastic interest rates are incorporated into pricing models in several studies (Nielsen and Sandmann, 1995; Bacinello and Persson, 2002; Gaillardetz, 2008). Besides,
Bilodeau (1997) and Bacinallo (2003
,
2005,
2008) consider different types of options embedded in the contract under the CPA framework. Following Brennan and Schwartz (1976, 1977), this study is also concentrated on endowment policy with guarantees which is primarily the combination of pure endowment policy and term insurance.As suggested by Embrechts (2000)
,
institutional issues such as the increasing collaboration between insurance companies and banks, and deregulation of insurance markets will be regarded as two further importantChiao Da Management Review Vol. 29 No. J, 2009 83
aspects. To search for combinations and unification of methodologies and traditional principle for the two fields of insurance and finance may deserve as a considerable issue. Obviously the variable life insurance products can involve both financial and insurance risks. F or exampl巴， Melnikov and Romanyuk (2008) highlight the implications of efficient hedging for the management of financial and insurance risks of variable life insurance policies with numerical examples. As to th巴 topic we concem here, one may wonder how the fair price of the variable life insurance with guarantees could be determined if both the independence assumptions and the insurer's risk neu甘ality are violated. Accordingly, we jointly consider the price process of a referen∞ fund and the process mortality risk. We integrate the two risk processes into a new stochastic process. The variable insurance productsωuld be seen as contingent claims of the new underlying process. To calculate the no-arbitrage price of the variable life insurance, life insurance portfolio is viewed as a tradable asset in secondary life insurance market. Investors then can have an additional basis asset to build the portfolio for duplicating the contingent claims. Under the market structure specified here, the complete market property cannot be preserved and insurers cannot replicate perfectly the contingent payment to policyholders at a future date. So the variable life insurance products couldn't be duplicated exactly with the portfolio consisting of the basis assets. Accordingly，位le risk-neutral probability measure in this market s甘ucture is not unique and thus the corresponding
no-arbi仕age pnce compos的 an interval (see, Chl Pliska 1997). Hence this could leave an open pricing problem in the incompletene必. Even liquidity of life insurance portfolio cannot reach as high degree as general financial securities. In fact, the insurance con甘acts could be sold to get money back earlier under special conditions. For example
,
the senior life policies settlement or viatical settlement can be 仕aded in the second life insurance market in USA. Although the assumption about the trading prop巴向rof life insurance con仕acts don't meet fully real world, we can make efforts to reposition the CPA with this treatment in the option pricing framework. As compared with the Black-Scholes model, we won't render this model useless due to that the continuous self-financing strategy cannot be car84 The Alternative Pricing Approach for 均'riable L!J告 Insurance Incorporating Secondary Lij全 InsuranceMarket
hedging problems for variable life insurance in an incomplete market. His main contribution is to obtain the optimal investment s仕ategies that minimize the variance of 也e insurer' s fuωre cost based on the criterion of risk-minimization instituted by FoIImer and Sondermann (1986). Nonetheless life insurance portfolio is not assumed as tradable assets in his seminal articIe.
This paper aims at developing an altemative pricing method and reviewing the CPA for variable life insurance using a pure no-arbitrage viewpoint. For simplicity, this work is restricted to the single premium case. The first task of this study is to present the general form of the no-arbitrage price bound for the insurance contracts. After that, we could verify that CPA wiIl produce a no-arbitrage price with resp的tto a specific risk-neutral probabiIity measure. The relationship between the present pricing approach and the CPA is explored
Through numerical analysis, we investigate and discuss how certain key financial factors can inf1uence the relationship between the no-arbitrage price bound and the pric巴 derived by th巳 CPA. If the reasonable price of the insurance contract would not be determined uniquely and could be affected by several financial parameters, those facts implies that insurers need to specify a pricing practice more sophisticatedly since they have more f1exibility inpricing such insurance product.
The remaining of the paper f10ws as foIIows. The market structure for the variable life insurance contract is built firstly in Section 2. In S的tion 3, the underiying discrete process of the insuτance contract, which is the consequence of combining the reference fund process and the mortality dis仕ibution， is established. We present the general form of the proposed approach incorporating the secondary life insurance market and explore the relationship between the CPA and the proposed approach as weII. In Section 4, the numericaI analysis is employed to iIIustrate the properties of fair price bound of the variable life insurance for various situations. FinaIIy, concIusive remarks are provided in Section 5.
2. The Market Structure for the Insurance Contract
Chiao Da Management Review Vol. 29 No. 1, 2009 85
which can be affected by both the market risk and mortality risk. In this section
,
we first set up the market risk model for the insurance con仕act and then take the mortality risk into account. Here we consider the variable life insurance contract with guaranteed value that issues at the beginning of the contract term andmaωres
T
years later. The market risk associated with the insurance contract comes from a stochastic evolution of the return rates of the reference fund. To demonstrate a discrete-time model, each policy year is divided into n periods of equal length such that the total period is N = TI ð, with ð,= lIn. Hence there are totally N periods during Tyears. The t-th period is denoted 丸， for t= 1,2, ... N-l, N. Following Bacinello (2003) and Moller (2001), this study also uses 也e CRR model proposed by Cox, Ross and Rubinstein (1979) to deal with pricing problems about the variable life insurance properly. This discrete model assumes that the risk-free interest rate r is constant and the financial market consists oftwo basis tradable asset, a reference stock (or 臼nd)S and a risk 企ee 臨叫.TheCRR model may be viewed as an approximation ofthe Black-Scholes model due to its important properties of converging asymptotically to the later.τbe reference fund price follows a stochastic proc臼s: S=
{S, : t E [0, T]} or S=
{丸，丸 ""， SN} . The market price of the reference fund is set up as a binomialla位ice. With a fixed volatility coe宜icientσ> ð，'的 ln( l+r)， we can speci秒 the accompanied upward-moving factor u = 呻(叫0.5)， and downward-moving factor d = lIu. Theunit price ofthe fund at the end ofthe t-th period (品) would be either u~忌。rdS，in
the next period for t = 1,2,..., N. ~耳 is adapted to the filtration It of the binomial
proι，，5S. Let B
,
=
Bo(l + r)的 with the constant annual interest rate r > 0 for t =1,2,..., N. Typically 企ictionless market Ís assumed to simplify the analyses. The
financial 甘eatrnents usually are based on the assumption of no-arbitrage opportunities.
During any trading time period (e.g., the t-th period), to each contingent claim
j(
St, t), a unique self-financing 仕ading s個tegy exists that can duplicate the payoff. With this s仕的egy， a portfolio consisting of a certain number of reference fund and a certain amount of risk-free asset can be formed at any time to exactly meet the claim and there is no need ωmake additional inflow or outflow of capital. Such the financial market is called complete ifthe contingent claim can be86 The Alternative Pricing Approach for Variable Life Insurance Incorporating Seconda.吵 LifeInsurance Market
duplicated perfectly and hence can be priced uniquely. So the no司arbitrage
condition could be satisfied. As is well known, the CRR model is a complete market model. Consequently, the no-arbitrage condition is equivalent to the existence of a risk-neutral probability measure under which all financial prices,
discounted by the risk-free rate, are martingale. The unique risk-neutral probability measure, which is conditional on the information at time t, is
q =[(I+r)企 -d]/(u-d) and I-q=[u 一 (1 +r)叮 /(u -d) ( 1 、、', ••• for {St+I = uSt} and {St+l dSt}, respectively. 百lerefore， the risk-的u個l
probability measure is defined by th巴 sequence
Q
=
{Q',i 11 至 t 歪 N，。三 j 三 t} , (2)for all POSSIblepaths{SrJ=SJ-JdJ|I 計三 N，O 至 j 訂}
with
G
r=t-lqt J(1-q)J. U)
The arbitrage-台'eeprice of a derivative ofthe underlying asset with payo位j{，品，。 在ttime t, for t = 1, 2,. . ., N, , denoted by P[. may be written by
4=EQ[ZL(1+r)raf(丸，叫 (3)
where EQ is the conditional expectation operator with resp的t to the risk-neu虹al probability measure in (2).
甘心 insurance contract involves the risk associated with the 訕訕re
development of the referenc巳如nd as well as the uncertainty about the mortality of insured. Whereas the financial risk affects the amount of benefit for the policyholders, the mortality risk determines 出e times in which the benefit is due. For each period，出ere are two states for the insured's life status, i.e., alive and dead. Hence the death discrete process also could be set up as a binomial 甘ee.1t is
assumed 由at a policyholder makes a single investment amount into the fund at the initial ofthe contract. Let m be the units invested in the fund at the initial time. Without any losses of generali勾心 m is fixed to one. For an insured with age x, let
Chiao Da Management Review Vol. 29 No. J, 2009 87
the mort耳lity of the t-th period be denoted by qx+'-l for t = 1, 2,..., N. The
mortality distribution could be extract巴d from a mort耳lity table. Typically the guarantee asset value of the variable life insurance may be set to a function of time (t) and the market value ofthe fund at the purchase date (So). For simplicity,
we suppose that the guarantee asset value is a constant, denoted by G (typically G is a percentage of So), in this study. That is, j(品， t)= Max (，品， G). 1n other words,
with the specification for the guarantees function, the benefit of this contract would be m似的，的 atthe end ofthe t-th period ifthe insured dies during the t-th period or mω(品，的 at the end of the N-th period if the insured survives to the maturity date. However,j(S" t) could be set to a more complex form. For exam阱，
a guaranteed retum is given by f(Sρ t) = S'_1 max(1 + (S, - S,_I) / SI-1,1 + K) ,
where κis the guaranteed retum.
3.
The Pricing Model3.1
The Review of theCPA
for Variable Life InsuranceBasically the CPA is derived by combining no-arbitrage argument and
甘aditional large number principle from insurance. The contingent payoff at each time can be fairly priced with no-arbitrage argument as described in the precious section. The CPA is justified with the law of large number since insurers 句rpically
hold a portfolio of large number of contracts. The CPA usually assumes that the death process is stochastically independent of the reference fund. Hence the pricing problem can be resolved by specifying the payoff of Max (，品， G) times the probability ofmortality at time t. 1t is implied that the insurer is risk-neutral with respect to mortality in the CPA. Here w巳 only concem about the core work for actuarial valuation, so any problem about expenses or 0值ler transaction cost is ignored. Based on the assumptions, the fair net premium can be derived with the
CPA through first calculating the market values of all payoffs according to (3) and then taking account of the expec個ìion on the mortality. Therefore with the financial set-up in the previous section, the fair price ofthe variable life insurance with guarantees, denoted by pCPA , can be writlen as
88
N
pCPA=
L
ι呻1=1
The Alternative Pricing Approach for Var阻bleLife Insurance Incorpor，αting Secondary LiJ是 InsuranceMarket
where
Q
is defined in (2)，叫 IQx =(1-
Qx)(1-
qx+J)...(1-
qx刊.2) qx叫， is the probability that the insured dies within the t-th period, and Npx = (1 - qx)(1
-qx+J) ...(1 -qx+N.J) is the probability that the insured is still alive at the end ofthe N-th period.
3.2 The Establishment of the
U
nderlying Process for Variable Life InsuranceIn order to establish an underiying process that can include all probabilities,
the price process of the reference fund and the death process of an insured are considered jointly. By integrating the two binomial processes
,
the new underlying discrete process is created as shown in Figure 1. All states occurring possibly have been considered in the new underlying process since the stochastic processes for both mortality and reference fund price have been specified already. Whether the asset price process and the death process are mutually independent or not, Figure 1 actually covers all possible outcomes. So the independence assumption between market risk and mortality risk cannot be required.The special feature for this model setting is to regard the life insurance portfolio as tradable assets such as the viatical settlements and the accelerated death benefits (ADBs). Under taking the term insurance into account, there thus are three basis assets in the market structure. The pure endowment policy is excluded because it may just be duplicated with a portfolio consisting of both the
risk.企'ee asset and the term insurance. It is also assumed that the market price of term insurance for a coverage period is determined by the traditional actuarial method, i.e., the principle of equivalence. This assumption implies that the net premium of term insurance portfolio under the principle of equivalence is regarded as a fair price accepted commonly by participants in the insurance market. In fact， ωtrade ordinary term insurance in the secondary markets is not illegal, e 忌， the senior life policies settlement. Nevertheless, the liquidity of life insurance contracts actually cannot be as good as general financial securities. For this reason, we assume that investors will require a risk premium to compensate
Chiao Da Managemenl Review Vol. 29 No. 1, 2009 89
for bearing more liquidity risk 企om the life insurance contracts. So a premium loading factor, denoted by λ， will exist for each trading period, which can be determined by the secondary life insurance market.
Figure 1
The Underlying Process Describing Jointly Market and Mortality Risks for the Variable Life Insurance Contract
t= to t= tl t= t2 t= t3 (A, S,,) (D, S22)
...--(A, S,,) -唔唔量 (D, S'4) \ \ (D, S\2) (A, S,,) /---.
(A, So) / _____/可D，
S,,) (A, S13) (D, S14)\\ (D， 名，)As illustrated in Figure 1, there are four possible stat俗，namely alive-up 徊，
90 The Alternatiν'e Pricing Approach戶rVariable Life Insurance Incorporating Secondary Life Insurance Mar.ιd
next period conditional on the sta峙，徊 ， S'-l). Thus our pricing model would be incomplete market since it involves four states but only three basis assets in each trading period. Based on the properties of an incomplete market, the risk-neutral probability measure would not be unique and the no-arbitrage price of a derivative asset would be an interval (see
,
e.g.,
Pliska 1997). The general form of the no-arbitrage price bounds will be ca1culated in th巳 followingsubsection.3.3
The Calculation ofthe No-Arbitrage Price BoundsBefore considering the multi-period model, the two-period model could be established firstly as an example with simpler ca1culation. We can refer the stochastic process of 出e two四period model to the first two periods in Figure 1.
Each period in the process involves four states and three basis assets. For example,
just consider the upper part of tl時間ond period in the two-period model. The all
payo臼s in every state for the three basis assets are exhibited in the Figure 2. Based on the no-arbi甘age ∞ndition， the conditional risk-neutral probability measure, q2 = (q21, q22, q23 ， 但4)， corresponding to the probability from (A, Sll) to
(A, S2J), (D, S22)， μ， S23), and (D, S24) respectively, will satis
fY
the following equat\On system: (1+r)也 (u q21 + u q22+ d q23+ d q24) =1, q22/ qX+l( 1+λ)+ q24/ qx+l(I+À.) =1, q21 + q22+ q23+ q24=1, q2I, q22， 但3， q24 逞。 (5)The above first two equations are set respectively according to the risk-neutral property of the fund and the term insurance. The solution of the risk-neutral probability measure for the second period can be expressed as
q2= [q -qx+l( 1+λ) 十的， qx+l( 1+λ) 間的， 1-q -的，的]，
where max(O, qX+l (1+入)- q) < 血2 < min(1-q, qx+l(l+λ)) and q is defined in equation (1). By the same arguments, one can veri砂 that another conditional measure of the second period 企omμ， S13) to (A ， 品5)， (D, S26), (A, S27), and (D,
Chiao Da Management Review Vol. 29 No. 1, 2009 91
S28), is the same as q2, and the risk-neutral probability measure of the first period, q1 = (ql1, q山 q心 qI4)，is
q1 = [q -qx( 1+λ)+α]， qx( 1+λ)- 且1 ， 1-q -u]，且1] ，
where max(O, qx(1 +A.) -q) < 血1 < min(1-q, qxC l+λ)). The risk-neutral probability measure, denoted by
Q;
, can be obtained through making up of ql and q2. Therefore, no-arbitrage price of the variable life insurance with guarantees for the two-period model may be derived as九 = EQ，(乏 (1+ r)一位 max(丸 ， G)) (6)
1t is obvious that
Q
2 would not be unique and P2 could serve as bounds. According to (6), the lower bound of P2 can be ca1culated under the corresponding risk-neutral probability measure Q 2 by settingα1
=
min(l-q,qx(1 + λ)) and α2=
min(1-q,qX+I(1 + λ)) ， whereas th巴 upperbound of P2 can be obtained by setting α1
=
max(O,qx(1 +λ)-q) andα2
=
max(O， qx刊(1 +λ) 一 q). For example,ifqx(l+ λ) 至 q ， qx+tC l+ λ) 三 q ， qx (1 + λ) 至 l-q ， qx+I (1 + λ) 至 1-q, the lower and upper bounds could be derived by setting α1::: qx刊_1(1 +λ) and α， =0 for t = 1,2 respectively, i.e.,
Figure 2
The Pavoffs in Four States for the Three Basis Assets
u
(1 +r)^
。
u (I+r)" 11(1+ λ )qx+l
(1 +r)^
d (I+r)^ 11( 1+ λ )qx+1
92 The Alternative Pricing Approachfor 均riable LiJ告 Insurance Incorporating Secondary Life Insurance Market
p"
L=
(1+
rr2 企 {qV +q(l-q)G +[I-q -qx(1 + λ)]G}+(I+rrA (I+ λ)qxG and p,.U=
(1 +rr
2A {[l-q,(
1 + λ)](qu2 + G - qG) + (1-q)G} + (1 + r) 在 (1+λ)qxuExtending the result to the multi-period case, the form of the no-arbitrage price bounds for the N-period model would be obtained in the same recursive solution. The conditional risk-neutral probability measure ofthe t-th period, q, = (q,], qt2, 駒，
q
,
4), wh岫 is independent of the st泌的 ofprevious period, could be wri位enasq
,=
(q -qx+l.l(1+λ)+ 斜， qx叫(1+λ) 一帥， 1-q 的'的) (7) withmax(O, qX+I.j(1 +λ) -q) < 的 <min(l-q, qx+t.j (1 +A.)), (8)
for 1 三五 t 三至N. Then, the risk-neutral probability measure of the N-period model,
denoted by Q', can be obtained by combining all 兮， for 1 至 t 三三 N. The no-arbitrage price ofthe insurance contract could be expressed by
N
P
=
EQ (L: (1 叫一地 max(丸 ， G)) (9)1=1
Similarly, the lower and upper bounds of P, denoted by
PL
andr
, may be obtained with respect to the risk-neu扯到 probability measures, Q' by setting 的=min(l- q, qx+l.l(1+À.)) and a, = max(O, qx+刊(1+A.) - q) respective勻" for 1 三三 t 三五 N.
Separately, according to (9), the no-arbi仕ageprice bound ofthe contract would be influenced by both mortality and loading factor. This result implies that the range of fair price will be larger for an elder insured than for a younger one. On the other hand, high loading factor then would amplify the range between the no-arbitrage price bounds.
3
.4
The CPA as A Special CaseThe CPA can be reviewed in the proposed valuation framework with the option pricing thinking. It is shown that the price obtained by the CPA is a
Chiao Da Managemenl Review Vol. 29 No. 1. 2009 93
no-arbitrage one with respect to the specific risk-neutral probability measure
Q"
,which is one of the risk-neutral probability measure Q* 吐岱efined w抽且昀i =
(ο1-q)泊qx耐+甘件t←仙.
min(οI-q， q叫1( 1+λ)) is heldand thus the relationship satisfies (8). The price formula of the CPA in (4) could be derived with the risk-neutral probabili句
measure Q's*a吋 presentedas follows:
N
pC叫
=L
余叫州l叫岫|N
= EQ (L (1 +r) 且 max(丸 ， G))
1=1
This means that the price obtained by the CPA (pCPA) is one of the no-arb帥ge
prices under the new underlying process setting. Because pCPA lies within the
r.L ___ T"'tCPA _ ro.U
no-arbitrage price bounds, i.e., r 三三 P 三三 pU， the properties of their relationship become an interesting issue. Thus, numerical analysis is conducted for this purpose in the next section.
Under the market s甘ucture specified in this study, we also can utilize the optimal portfolio pricing approach (see Ch 9, L闊的erger 1997) to get a set of risk-neutral probability measure and determine the fair price for the insurance contract. However we need a位ditionallyto define a utìlity function and specify the optimal portfolio choice criterion for policyholders. As mentioned in the text,
policyholders (or investors) have three basis assets to form the portfolio. The optimal portfolio pricing approach is based on the assumption that policyholders would make decision for allocating optimally their money among the altematives Similar to the discussion in Subse芯tion 3.3, the multiple-period problem has the same solution as th巳 sequence of one-period problem. Thus, N-period problem could also be reduced ωone-period problem here. Denote the corresponding real probabilities for the four possible states of the t-th period conditional on the previous state，抖，斗1)as shown in Figure 1, including alive叩 (A ， 的-1)，的d叩
(D,
USt-l)
, alive-down(A
,dSt_
1) and dead-down(D
,dSt_
1), p;妞， p? ， pfandP ~d respectively for t =1,2,. .., N. We take the first period (t =1) as an illustrative example. A portfolio of these basis assets is represented by a 3-dimentional vector
ß
=(ß
t,ß2
,ß3).
The initial price of each asset is denoted by ki, for i = 1-3.94 The Alternative Pricing Approach for Variable LiJ告 Insurance Incorporating Seconda;吵 LifeInsurance Market
Suppose that a policyholder has an initial wealth wo. The fu仙re wealth would be governed by corresponding random variable. A utiIity function U provides a procedure for ranking random wealth leveIs. If w
,
is the random wealth at the end of the first period, we write w,
> 0 to indicate that the variable is never less than zero and it is strictly positive with some positive probability. The random payoffs for the three asset are represented by dj, i = 1~3. For simplicity, we ignore theliquidity problem about the insurance contract and let λ= O. The investor wishes to form a portfolio to maximize the expected utility of the future weaIth, i.e., w,. Thus the policyholder's problem is:
maxE[U(w
1
月，仰伊ct 的主βA=W2 月1> 0，三βλ 三 Wo
;",,1 ;-1
The problem therefore becomes:
max E[U
(L
ßidi)]' subject to W1
汁，主然失 =Wo
.Î=1 ;=1
By introducing a Lagrange multiplier
r
for the constraint, the necessary conditions are found by differentiating the Lagrangian:L=E[U(Lβ\dJ]-Y(Lβλ 一 W
O
)i=l i=1
W恤 resp叫 to
each ßi.Using 叫=
L:=I
ß: d i for the payoff of the optimalportfolio
,
this giv臼 E[U'(w;)d;]=]1ci for i = 1~3. Since the risk-仕的 asset(i = 1) has the total return of (1+r)'\ it follows that if k,
= 1,
then d,
= (1+r)".
Thus,
we obtainy=
E[U'(w;)](1 + r)" . S的st山tingthis value forr
would yield] ) 1.i ]-v i-B Ju-uvt 、 。一 U T[ 圳、 -E
U-r'
TI-r t 一+ -1 -( 一一 E LκTherefore the risk-neutral probabilities of the first period，記 = CifllJiI2 ，l13' 且4),
could be derived as
~pfuh(叫)pfdUJJw:)PFUA(w:)pFUA(叫)、
Chiao Da Management Review Vol. 29 No. 1, 2009 95
With the same argument, the risk -neutral probabilities for the t-th period, 記=(記l' 且2>記3' 記4), may be written by
~pfu;1(w:)pfuh(w:)p?UA(w:)p尸U:
4
(w;) 、q
,
=~一一一一一一一 一一一一一一一一一一一一一一一-，)t 、 E[U'(w;)]' E[U'(w;)] , E[U'(w;)] , E[U'(w;)] J
By doing so, a set of risk-neutral probability measure 也atdepends on consumer's utility function could be derived as well.
Now the CPA is revisited in views ofthe optimal portfolio pricing approach. According to the ration theory (see Ch16, Luenberger 1997), the relationship
瓦1弘/瓦2弘 zpfvfd/pfhdp戶， for t =1, 2,..., N, would be held if each 甘ading period /:; is enough small. Under the condition where the tw。可pes of risks are independent each other, it follows 互瓦r品4/ 互瓦'2瓦弘3 = pjffh曲ze pf尸d/
p{'"
P瓦ι= 互ι. As a result, the independence w愉 respect to real probabilities is equivalent to the independence with respect to the risk-的U甘alprobabilities. With this condition, the risk-neutral probabilities 革=(扎 ， Qt2 ， qt3 ， qt4)derived from
th巳 optimal portfolio pricing approach would be equal to (7) with u
,
= (1- qω)q弘x+廿t叫.}for 1 三歪三 t 三歪三 N. Consequently, it can be veritied that 出eoptimal portfolio pricing approach could achieve the same result as the CPA does under certain conditions. So this pricing approach also could serve as another applicable valuation method for the variable life insurance using the created underlying stochastic process of this paper.
4. Numerical Results
In this section, the results of some numerical experiments for the comparison between the proposed pricing approach and the CPA 訂e illustrated. It
is attempted to understand how the no-arbi仕age price bounds of the insurance contract is affected by some tinancial parameters. Since ouτnumerical analysis is aimed to catch some comparative properties between the aforementioned two
approach仿， the real mortality is ignored here. Instead, we consider different
pa位ems of the mortality distribution in which different mortality growth rates could be presented. For simplicity, we set T= 1, n =12 (N= 12) and So =100,000. This setting won't get any losses of generality. Note that the choice for n implies a
96 The Alternative Pricing Approach for Variable Life Insurance Incorporating Secondaη Life Insurance Market
monthly change in the unit price of the reference fund. Several numerical experiments are mad巳 with respect to five p訂ameters， i.e., the volatility coefficient (σ)， the interest rate (吟， the guarantee value (G), the pattern of mortality distribution and the loading factor for liquidity risk premiums (À.).叮le results are reported in table 1 through table 4.
Table 1
Insurance Premiums Versus the Volatility Coefficient G
σ 10% 20% 30% 40% 50%
Upper bound pU 100041 101223 103366 106372 109454
Conventiona1 approach pCPA 100039 101193 103316 106314 109392
Lower bound PL 100036 101158 103264 106255 109331
No-arbitrage interva1 p u_ PL 5 65 102 117 123
(pU_ pCPA)/( pu_ PL) 。 400 0.462 0.490 。 496 。.504
Table 2
Insurance Premiums Versus the Interest Rate r
σ 2% 4% 6% 8% 10%
Upper bound pU 104256 103786 103366 1029日9 102652
Conventional approach pCPA 104189 103729 103316 102947 102617
Lower bound PL 104124 103670 103264 102901 102576
No-arbitrage interval p u_ PL 132 116 102 88 76
Chiao Da Management Revi由•Vol. 29 No. !, 2009 97
Table 3
Insurance Premiums Versus the Guarantee Value G (Percentage of SO)
G 75% 80% 85% 90% 95%
Upper bound pU 101555 102381 103366 105037 106742
Conventiona1 approach pCPA 101510 102329 103316 104997 106737 Lower bO\U1d Ý 101463 102275 103264 104954 106729 No-缸bi仕'ageinterva1 pu_ Ý 92 106 102 83 13
(pU_ pCPA)/( pu_ Ý) 。 489 0.491 0.490 0.482 。.3 85
Table4
Insurance Premiums Versus the Pattern ofMortality Distribution
Growth rate of morta1ity -10% -5% 。% 5% 10%
Upper bound pU 103400 103385 103366 103343 103314
Conventiona1 approach pCPA 103358 103339 103316 103289 103256
Lower bound Ý 103314 103291 103264 103231 103193
No-arbitrage interva1 P尺 Ý 86 94 102 112 121
(pU.自 pCPA)/(pu_ Ý) 。 488 。 489 。 490 。 482 0.479
Table 5
Insurance Premiums Versus the Loading factor À
λ 。% 5% 10% 15% 20%
Upper bound pU 103366 103366 103366 103366 103366
Conventiona1 approach pCPA 103316 103316 103316 103316 103316
Lower bound Ý 103264 103257 103253 103248 103243
No-arbitrage interval pu_ Ý 102 109 113 118 123
98 The Alternative Pricing Approach戶'rVariable Life Insurance Incorporating Secondary Life Insurance Market
First, the no-arbitrage price bound and pCPA are calculated when the volatility coefficient (σ) varies between 10% and 50% with a step of 10%. We fix r = 6%， λ= 0 and K = 85% of So. A simple pa位em of mortality distribution is given, where qX+I-1 is fixed to 0.1 % for 1 三三 f 三三 N. The results of this numerical experiment are presented in Table 1. It is obvious that all the premiums obtained,
including the upper bound price pU, the lower bound price
PL
and the pC凹， increase with the volatility coefficient (σ). This result agrees wi由 the general properties of option pricing theory. Moreover, we notice that the no-arbi仕ageprice interval, pU -
PL
, and 由e ratio, (pu _ pCPA)/(r -
PL)
increase withσ. In other words，位le no-arbitrage interval becomes larger and pBs becomes relativelycIoser to
PL
asσbecomes larger. Then, as presented in table 2, the premiums are derived when the interest rate (r),
v訂ies between 2% and 10% with a step of 2% based on the conditions of 0" = 30%， λ= 0, K = 85% of So and the same mortalitydistribution. According to the results in Table 2
,
all the premiums decrease with the interest rate (r). Besides, both of no-arbitrage price interval and the ratio, (pu-pCPA)/(pU _ pL), decrease with r. Furthermore, settingσ= 30%, r= 6%， λ=0 and the same mortality distribution, the premiums are calculated when the guarantee value (G) varies between 75% and 95% of Sowith a step of 5%, as presented in Table 3. We notice that all the premiums increase with the guarantee value (的 This obviously meets the prediction of option pricing theory. Moreover, it is also observed that the larger the guaranteed value, the narrower the no也.bitrage price interval (except in the situations oflower guaranteed values)
In addition, di任erent pa位ems of mortality dis仕ibution are taken into account as well. Fixingσ =30%， r = 6%， λ= 0 and G = 85%，出e premlums are calculated according to different growth rates of mo此ality， including -10%, -5%, 。%， 5%, and 10%. For example, the mortality growth rate of 5 % implies the relationship, q x+1 / q x+I-1 = 1+5%. The results are presented in Table 4. It is found
that all the premiums decrease with the growth rate of mortality. However, the no-arbitrage interval increases with the growth rate of mortality. Finally, we test the effect of the loading factor λon the no-arbi個ge bounds. Using the same setting, i.e.，叮 =30%， r = 6%, K = 85% and zero growth rate of mortality, the premiums are obtained for various loading factors (λ.) with 0 <丸< 8%. As
Chiao Da Management Review Vol. 29 No. 1, 2009 99
exhibited in Table 5, it is obvious tbat the no-arbitrage bound increases with the loading factor. This result implies high liquidity of life insurance contracts in secondary insurance market can decrease possible range ofthe no-arbitrage pri∞.
In summa句" according to the numerical results, all premiums of the variable life insurance policy increase withσand G, but decrease wi也 r and the growth rate of mortality. And 甘le no-arbitrage price interval increases witb σ， λ
and tbe growth rate of mortality, but decrease with r and G. Additionally, almost values of the ratio, (pu _ pC凹)/(pU _
r)
, would be between 0.4 and 0.6. This implies that pCPA usually falls in the middle area of tbe no-arbitrage intervalsAccordingly，企om tbe viewpoint of pure market-value based, the reasonable prices of the contracts couldn't be deterτnined only by the traditional criterion while pCPA could serve as a benchmark for pricing in practice. That is, the reasonable prices could depend on market situations and 曲的 insurance
companies could keep more cushions in making tbe pricing strategy.
5. Conclusive Remarks
In this article, life insurance contracts are seen as tradable portfolio assets since secondary life insurance markets allow consumers to cash out life insurance holding prior to death. Considering this characteristic of secondary life insurance markets
,
we propose altemative valuation methods for variable life insurance under a pure option pricing 企amework. Two proposed approaches, i.e.,no-arbitrage pricing method and optimal portfolio pricing method, could lead to the results that coincide with the price determined by tbe conventional valuation principle. Actually the price obtained by the conventional principle would be verified to be a special case of our pricing framework. The result indicates that fair prices of tbe insurance contract may not be limited to those determined by the conventional pricing approach. It then implies 也泣， from the market-value based perspective, insurers need to speci秒 a more sophisticated pricing practice since they have more flexibility in pricing such insurance product. The optimal pricing policy may depend on market situations and consumer's utility function. The numerical analysis results show the properties of no-arbitrage price of tbe
100 The Alternative Pricing Approach for Variable Lij全 Insurance
Incorporating Secondary Life Insurance Market
insurance contract as weIl. A different insight into prop巳rties of the conventional pricing approach has been explored. Although this research is restricted to the single-premium case, the pricing model we propose could be extended to the annual-premium case for advanced applications. The valuation approach explored in this paper may be applied further to solve the analogous pricing problems related to other insurance contracts types
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