Applications of Central Limit Theorems for Equity-linked Insurance Runhuan Feng Department of Mathematics University of Illinois at Urbana-Champaign
[email protected]
Yasutaka Shimizu Department of Applied Mathematics Waseda University
[email protected]
April 14, 2016
Abstract In both the past literature and industrial practice, it was often implicitly used without any justification that the classical strong law of large numbers applies to the modeling of equitylinked insurance. However, as all policyholders’ benefits are linked to common equity indices or funds, the classical assumption of independent claims is clearly inappropriate for equity-linked insurance. In other words, the strong law of large numbers fails to apply in the classical sense. In this paper, we investigate this fundamental question regarding the validity of strong laws of large numbers for equity-linked insurance. As a result, extensions of classical laws of large numbers and central limit theorem are presented, which are shown to apply to a great variety of equity-linked insurance products. Key Words. Equity-linked insurance; variable annuity guaranteed benefits; risk measures; strong law of large numbers; central limit theorem; individual model; aggregate model.
1
Introduction
One of the most fundamental principles for the insurance business is the pooling of funds from a large number of policyholders to pay for losses that a few policyholders incur. The mathematics behind such a business model is the law of large numbers which dictates that actual average loss would be close to the theoretical mean of loss with a large pool of homogeneous and independent risks. Take for example a traditional pure endowment life insurance that pays a lump sum of B dollars upon survival at the end of T years. Suppose that an insurer sells identical policies to n policyholders all of who are of the same age x. We denote the future lifetime of the i-th policyholder (i) (i) by τx and the survival probability by T px := P(τx > T ). Even though there is uncertainty to each contract with regard to whether the policyholder survives at time T , the strong law of large numbers implies that the percentage of survivorship is almost certain and so is the average benefit payment for each contract, i.e. n
1X BI(τx(i) ≥ T ) −→ E[BI(τx(1) ≥ T )] = B T px , n i=1
1
n → ∞.
Simply put, this is the scientific ground of the insurance practice of pricing and reserving, which mandate a fixed amount of liquid asset on the aggregate level to pay for seemingly uncertain benefits on individual basis. In other words, the mortality risks involved in all individual contracts are diversified through the pooling of a large number of homogeneous contracts. The past few decades has seen the rapid growth of investment-combined insurance products, which allow policyholders to reap the benefits of equity investment on their premiums. Insurers around the world have developed a variety of equity-linked insurance. However, this market innovation brought financial risks into insurance contracts, in conjunction with traditional mortality risk. Since policy benefits are often linked to the same equity-indices or funds, there is no diversification of financial risks amongst each cohort of policyholders. If the equity market performs poorly, there is an erosion of policy values to all contracts at the same time. From the viewpoint of statistics, the individual policy benefits are no longer independent random variables and hence classical laws of large numbers do not apply. Nevertheless, it is fairly common both in practice and in the actuarial literature that mortality risks are implicitly assumed to be diversified in premium and reserve calculations for equity-linked insurance. See examples in Sections 3.2, 4.1 – 4.3. Is there any theoretical basis of such a widely used assumption? In this paper, we intend to address this question and explore various sets of assumptions under which the law of large numbers can be extended to equity-linked insurance. In Section 2, we provide a concrete example to quantify and analyze the interaction of mortality and financial risks. We extend laws of large numbers to a general framework of survival benefits in Section 3 and to that of death benefits in Section 4. We conclude the paper with a numerical example for the central limit theorem result and its application to the calculation of risk measures.
2
Guaranteed Minimum Maturity Benefit
In this section, we take the guaranteed minimum maturity benefit (GMMB) as a model example, although similar results can also be obtained for all other types of equity-linked insurance to be introduced in later sections. In Feng and Volkmer [6], the net liability of the GMMB is defined to be the present value of future outgo less the present value of future income on a standalone contract basis. We shall describe the cost and benefit of the GMMB rider for the sake of completeness. Let L(τ (i) ), i ∈ {1, 2, · · · , n} be the GMMB net liability for the i-th policyholder. Assume all policyholders make the same amount of initial purchase payment at the policy issue (this assumption will be loosen in the next section). Denote the guaranteed maturity benefit by G and the evolution of investment accounts by {Ft : t ≥ 0}, which is modeled by some stochastic process on the probability space (Ω, F, P, {Ft : t ≥ 0}). Throughout the paper, we assume {Ft : t ≥ 0} to be a non-negative process. Rider fees and charges are collected continuously as a fixed percentage me of the account values. Let r be the annual yield rate on bonds backing up the liabilities. The GMMB rider is an equity-linked insurance analog of the traditional pure endowment insurance. It offers a policyholder the greater of a minimum value and account value, should the policyholder survive
2
the maturity T . Since the bulk of the guaranteed amount G comes out of the policyholder’s own investment account, the out-of-pocket cost of the GMMB rider for the insurer is determined by e−rT (G − FT )+ I(τx(i) > T ), where (x)+ = max{x, 0}. To compensate for the GMMB liability, the insurer receives a continuous flow of fee incomes until the earlier of policyholder’s death and the maturity, the present value of which is given by Z T ∧τx(i) e−rs me Fs ds. 0
Then the present value of individual net liability (gross liability less fee income) is given by Z T ∧τx(i) (i) −rT (i) e−rs me Fs ds. L(τ ) := e (G − FT )+ I(τx > T ) −
(1)
0
It was also shown in Feng [8] that the actual model used by practitioners through spreadsheet calculations is the average net liability model Z T ∗ (i) −rT −rs L := E L(τ ) FT = T px e (G − FT )+ − me Fs ds. (2) s px e 0
(i)
Observe that there are two sources of randomness, namely τx and {Ft : t ≥ 0} in the formulation of net liability in (1), whereas only financial risk is present in the formulation of net liability in (2). Before discussing the connection between these two types of models, we digress to investigate the effect on the tail behavior of undiversifiable risks. Let us consider a set of n pairwise symmetric random variables, i.e. (Xi , Xj ) has the same distribution for all i, j = 1, · · · , n. Note that Xi and Xj do not need to be independent. The pairwise symmetry is equivalent to the statement that (X1 , · · · , Xi−1 , Xi+1 , · · · , Xn ) has the same distribution as (X1 , · · · , Xj−1 , Xj+1 , · · · , Xn ) for any i, j = 1, · · · , n. Proposition 2.1. Let {X1 , X2 , · · · } be a set of pairwise symmetric random variables. For any positive integer n, n+1
n
i=1
i=1
1 X 1X Xi ≤cx Xi . n+1 n
(3)
where X ≤cx Y means that Eg(X) ≤ Eg(Y ) for any convex function g. Proof. For any convex function g, observe that " !# n+1 n+1 X 1 n+1 X 1 X 1 E g Xi = E g Xj . n+1 n+1 n i=1
i=1
j=1,j6=i
Applying Jensen’s inequality to a discrete random variable, we can show that for any convex function g and x1 , x2 , · · · , xn ∈ R, ! n n 1X 1X g(xi ) ≥ g xi . n n i=1
i=1
3
Therefore, n+1 n+1 n+1 n+1 n X 1 n+1 X X X X X 1 1 1 1 1 E g Xj ≤ E g Xj = E g Xj , n+1 n n+1 n n+1 n i=1
i=1
j=1,j6=i
i=1
j=1,j6=i
j=1
where the last equality follows from the fact that (X1 , · · · , Xn ) are pairwise symmetric. Therefore, we have proved that for any convex function g " !# n+1 n X X 1 1 E g ≤ E g Xi Xj , n+1 n i=1
j=1
which establishes the convex order in (3). It follows immediately that for p ∈ (0, 1) (c.f. Dhaene et al. [5, Theorem 3.2]), ! ! n+1 n 1 X 1X TVaRp Xi ≤ TVaRp Xi . n+1 n i=1
i=1
The tail-value-at-risk of the average of n losses is a decreasing function of the sample size n. In other words, the tail risk of average loss can always be reduced by diversification through a large pool of policies. When the components of X are independent, then the strong law of large P numbers implies that the limit of TVaRp [(1/n) nj=1 Xj ] is E[Xj ] as n → ∞. However, this is not true in general when the components are dependent, as in the case of equity-linked insurance contracts. A discussion of systematic versus diversifiable risks can be found in Busse, Dacorogna and Kratz [2] with detailed numerical examples. The above provides a theoretical justification of their observations. Denuit et al. [4, Proposition 3.4.23] provides a special case of Proposition 2.1 which requires the assumption of independent and identical distributed risks. As alluded to earlier, individual net liabilities (L(τ (1) , L(τ (2) ), · · · , L(τ (n) )) are not mutually independent. Hence it is critical for us to address the following questions. Is the following convergence used in practice justifiable in some limit sense? n
1X L(τi ) −→ L∗ , n
n → ∞.
i=1
If so, under what conditions can we reach the limit? How realistic are they? It is clear from the Jensen’s inequality that for any n ∈ Z, n
1X L(τi ) ≥cx L∗ , n i=1
which implies that for all p ∈ (0, 1) and n ∈ N, TVaRp
! n 1X L(τi ) ≥ TVaRp (L∗ ) . n i=1
4
(4)
In Corollary 2.1, we shall prove that tail risks of averages decrease with the sample size and eventually converges to that of an average model, i.e. as n → ∞, ! n 1X TVaRp L(τi ) ↓ TVaRp (L∗ ) , n i=1
In other words, the uncertainty of L∗ is entirely attributable to financial risk whereas the mortality risk is fully diversified.
2.1
GMMB - identical and fixed initial payments
Throughout the paper, we assume that all policyholders’ initial purchase payments are deposited in the same equity index or equity fund. Proposition 2.2. Assume that (i) all initial purchase payments are of equal size and their growths are driven by the same fund {Ft : t ≥ 0}; (ii) the future lifetimes of policyholders are independent RT and identically distributed; (iii) 0 |e−rs Fs | ds < ∞ a.s.. Then the convergence in (4) is true almost surely. Proof. Recall that n
1X L(τ (i) ) := e−rT (G − FT )+ I¯nx (T ) − n i=1
Z
T
e−rs me Fs I¯nx (s)ds,
0
where n
1X I¯nx (t) := I(τx(i) > t). n i=1
(i) Since τx (i = 1, . . . , n) are i.i.d., the Glivenko-Cantelli theorem says that supt≥0 |I¯nx (t) − t px | → 0 almost surely. Thus, for a.a. ω ∈ Ω, Z T Z T −rs x −rs ¯ e me Fs In (s) ds − e me Fs s px ds 0
0
≤ sup |I¯nx (s) − s px |me RT 0
2.2
T
|e−rs Fs | ds → 0
a.s.
0
s∈[0,T ]
if
Z
|e−rs Fs | ds < ∞ a.s.
GMMB - identically distributed initial payments
Unlike exchange-traded futures and options which are standardized with unit contract sizes, variable annuities are sold to individual investors with varying purchase payments. The part (i) of Proposition 2.2 does not always hold true in practice. One possible way for the limit in (4) to hold true on policies with different payments is to randomize the initial payments. That is to 5
say, without any a priori knowledge of policyholders, we assume that their contributions all come from a general distribution of a “random generator” determined by social economic development of the general population. Different sizes of purchase payments from individuals are considered observations of the “random generator”. To isolate the impact of random initial payment, we use the assumption that the investment scales with the initial purchase payment. We rewrite the individual liability for i-th policyholder (i = 1, 2, . . . , n): L(τ
(i)
) := e
−rT
(G
(i)
−
(i) FT )+ I(τx(i)
Z
(i)
T ∧τx
> T) −
e−rs me Fs(i) ds,
(5)
0
where (i)
• F0 , the initial purchase payment of the i-th policyholder. (i)
• Ft
(i)
:= e−mt F0 St /S0 . All individual accounts are linked to the same equity index {St , t ≥ 0}. (i)
• G(i) := γF0 , where γ determines the guaranteed amount G(i) as a percentage of the i-th policyholder’s purchase payment. Then the aggregate net liability of all n policies is determined by n X i=1
L(τ
(i)
)=
n X
−rT
e
(G
(i)
−
(i) FT )+ I(τx(i)
> T) −
i=1
n Z X i=1
ST = e−rT (γ − e−mT )+ n Hnx (T ) − S0
T
(i)
e−rt me Ft I(τx(i) > t) dt,
0
T
Z
me e−(r+m)t n Hnx (t)
0
St dt, S0
where n
Hnx (t) :=
1 X (i) F0 I(τx(i) > t) n
(6)
i=1
(i) (i) Proposition 2.3. Suppose that F0 , (i = 1, . . . , n) are i.i.d., independent of τx , and that F¯0 := (i) E[F0 ] < ∞. Then the average of liabilities converges almost surely: n
1X ¯∗ L(τ (i) ) = L n→∞ n lim
a.s.,
i=1
where ¯ ∗ := e−rT (γ F¯0 − F¯T )+ T px − L
Z
T
e−rt t px me F¯t dt
(7)
0
with F¯t := e−mt F¯0 St /S0 . (i) ¯ ∗ = L∗ . In particular, if F0 ≡ F0 (constant) for all i = 1, 2, · · · , n, then F¯0 = F0 and consequently L
6
Proof. Since a path of (St )t∈[0,T ] is almost surely continuous, it follows that Z
T
e−(r+m)t St dt < ∞ a.s.
(8)
0
Hence, from Lemma A.1, it follows that Z
T
−(r+m)t
e
Hnx (t)St dt
T
Z →
0
e−(r+m)t F¯0 t px · St dt
a.s.
0
Therefore, Z T n 1X (i) −rT −mT ¯ me e−(r+m)t F¯0 · t px · St /S0 dt L(τ ) → e (γ − e ST /S0 )+ F0 · T px − n 0 i=1 Z T −rT ¯ ¯ e−rt t px me F¯t dt. =e (γ F0 − FT )+ · T px −
a.s.
0 (i)
The case of F0 ≡ F0 follows trivially from the above result. (i)
Remark 2.1. In practice, F0 are usually different deterministic numbers. In such a case, we can (1) (2) (n) regard the initial purchase payments as independent samples (realizations) of (F0 , F0 , . . . , F0 ) (i) with identical distributions. The distribution of F0 determines the likelihood of the i-th policyholder choosing to make an initial purchase payment of certain size. Corollary 2.1. Under the assumptions of Proposition 2.3, for all p ∈ (0, 1), ! n 1X L(τi ) = TVaRp (L∗ ). lim TVaRp n→∞ n
(9)
i=1
Proof. In [19, Example 7.31(b), p. 163], it was shown that tail-value-at-risk satisfies the “Lebesguecontinuity” (c.f. [19, Definition 7.23, p. 160]). Observe that L(n) ≤ e−rT G. The limit (9) follows immediately from the “Lebesgue-continuity”.
3
General aggregate liability of survival benefits
We now present a general framework in which aggregate liabilities of all kinds of equity-linked insurance with survival benefits can be analyzed. On a probability space (Ω, F, F, P) be a filtered probability space, where F := (Ft )t≥0 , and let D := D[0, ∞) be a space of c` adl` ag functions x = (xt )t≥0 with a uniform norm. Let Bt := σ(x : xs ; s ≤ t) and B := (Bt )t≥0 . The filtration F combines both the financial information (equity returns) G := (Gt )t≥0 and the insurance information (mortality rates) T := (Tt )t≥0 in such a way that F t = Gt ∨ T t . For practical purposes, we assume that Gt and Tt are independent under P. 7
We consider the following individual liability for the policyholder of age x with the initial (i) purchase payment F0 for i = 1, . . . , n, Z ∞ (i) (i) (i) (i) L := F0 At (F )I(τx > t) dt + BT (F )I(τx > T ) , (10) 0
where: • F = (Ft )t≥0 : G-adapted process, which corresponds to a portfolio of financial assets to which all subaccounts are linked. The process should be scaled to unit size so that F0 = 1 under P. • At (·), Bt (·) : D → R, is Bt -measurable for each t ≥ 0 (B-adapted). For example, At (F ), which is Ft -measurable, is some value at time t, depending on a path of F up to time t. • T : G-stopping time with T > 0 a.s.: {T ≤ t} ∈ Gt . (i)
(i)
• τx : T-stopping time for each i = 1, . . . , n: {τx ≤ t} ∈ Tt , with i.i.d. property. (i)
• F0 (i = 1, . . . , n): T0 -measurable, i.i.d. random variables. Then the average aggregate liability can be written as follows n
X ¯ n := 1 L(i) = L n i=1
∞
Z
At (F ) · Hnx (t) dt + BT (F ) · Hnx (T ),
(11)
0
where Hnx (t) is defined in (6). In a manner similar to Proposition 2.3, we first present a result which can be viewed as a strong law of large numbers for the equity-linked insurance model in (10). (i) (i) Proposition 3.1. Suppose that (i) F0 , (i = 1, . . . , n) are i.i.d., independent of τx , (ii) F¯0 := (i) E[F0 ] < ∞ and (iii) Z ∞ |At (F )| dt + |BT (F )| < ∞ a.s. (12) 0
Then it holds that ¯n
¯∗
lim L = L := F¯0
n→∞
Z
∞
At (F ) · t px dt + BT (F ) · T px
a.s.,
(13)
0
Proof. Letting H x (t) := F¯0 · t px , we see from Lemma A.1 that as n → ∞, Z ∞ n ∗ ¯ ¯ |L − L | ≤ |At (F )| dt + |BT (F )| · sup |Hnx (t) − H x (t)| → 0. 0
t∈[0,∞)
We can also obtain the central limit theorem for the equity-linked insurance model in (10).
8
¯ 0 := E|F (i) |2 < Proposition 3.2. Suppose the same assumptions as in Proposition 3.1, and that Q 0 ∞. Then it holds as n → ∞ that Z d ∞ √ n ∗ ¯ ¯ At (F ) · G(t) dt + BT (F ) · G(T ), (14) n L −L → 0
where G is a Gaussian process with mean zero and covariance for any t, s ≥ 0 Cov (G(t), G(s)) =
¯ − t px · s px F¯ 2 . 0
t∨s px Q0
Proof. Note that the functional K : Ω × D → R Z ∞ K[h] := At (F ) · h(t) dt 0
is almost surely continuous in h. Actually, for hn , h ∈ D with hn → h in D, it follows that Z ∞ |At (F )| dt · sup |hn (t) − h(t)| → 0, n → ∞. |K[hn ] − K[h]| ≤ 0
Note that
√
t∈[0,∞)
¯n − L ¯ ∗ = K[Gn ] − BT (F )Gn (T ) n L
√ where Gn (t) := n (Hnx (t) − H x (t)). Since Gn is independent of A, B, T and F , the consequence holds true by Lemma A.2 and the continuous mapping theorem. Remark 3.1. The process G = (G(t))t≥0 in Proposition 3.2 is called “P -Brownian bridge” with the distribution P given by (i)
P := P (dz1 , dz2 ) = P(F0 ∈ dz1 , τx(i) ∈ dz2 ). (i)
See van der Vaart and Wellner [25], Section 2.1. Note that, if F0 bridge on [0, ∞) with G(0) = G(∞) = 0 a.s.
≡ 1, then G is a Brownian
(i)
Remark 3.1. Consider a special case of Theorem 3.2, where F0 ≡ 1. Then we have Z ∞ √ ¯n − L ¯ ∗ →d n L At (F ) · G0 (1 − t px ) dt + BT (F ) · G0 (1 − T px ), 0
where G0 is a standard Brownian bridge on [0, 1] with the covariance function is given by Cov(G0 (t), G0 (s)) = s ∧ t − st. (i)
Indeed, as F0 ≡ 1 and s < t, the covariance of the Gaussian process in Proposition 3.2 has Cov (G(t), G(s)) = t px − t px · s px = (1 − t px ) − (1 − t px )(1 − s px ) = (1 − t px ) ∧ (1 − s px ) − (1 − t px )(1 − s px ) = Cov (G0 (1 − t px ), G0 (1 − s px )) . 9
3.1
An approximate aggregate model
When running stochastic models for reserving and capital requirements, practitioners often group contracts with similar characteristics and run liability projections based on representative policy cells. A possible theoretical justification is provided in Proposition 3.1. However, it should be pointed out that the convergence as predicted in Proposition 2.2 is in general faster than the convergence with random initial purchase payment in Proposition 3.1. A larger pool of policyholders is required to balance out the randomness introduced by initial purchase payments than otherwise the pool of policyholders with similar contract sizes. An empirical evidence of slow convergence ¯ ∗ described in can be seen in the numerical example in Section 5. In this case, the average model L ¯ n described in (11). (13) would not be a very good approximation of the true aggregate liability L Therefore, we provide an approximate model based on the central limit theorem in Proposition 3.1 for a relatively small sample size n. It follows immediately from Proposition 3.2 that Z ∞ 1 n ∗ ¯ ¯ At (F ) · G(t) dt + BT (F ) · G(T ) + op (n−1/2 ), L =L +√ n 0 Therefore, we may use the sum of the average model and the first order term in the asymptotic expansion to approximate the aggregate model. Consider a large number T > 0 such that Z ∞ Z T At (F ) · G(t) dt ≈ At (F ) · G(t) dt a.s. 0
0
Then we can discretize the integral in the first order term to obtain N
X 1 ¯n ≈ L ¯ ∗ + √1 Atk (F ) · G(tk ) · ∆k + √ BtN (F ) · G(tN ), L n n k=1
¯ ∗ + √1 C> G, =L n
(15)
where 0 = t0 < t1 < · · · < tN = T , ∆k = tk − tk−1 for k = 1, · · · , N and C := (∆1 At1 (F ), . . . , ∆N −1 AtN −1 (F ), ∆N ATN (F ) + BtN (F ))> , G := (G(t1 ), . . . , G(tN ))> . ¯ ∗ and C are independent of G, which follows a multivariate Observe that both the average model L ¯ 0 − F¯ 2 t px · t px for normal distribution NN (0, Σ) with the covariance matrix with Σij = ti px Q j 0 i i, j = 1, · · · , N and i > j. In other words, given each sample path, Ln approximately follows a mixed normal distribution.
3.2
Example: Guaranteed lifetime withdrawal benefit (GLWB)
In Feng and Volkmer [7], the net liability for the guaranteed minimum withdrawal benefit (GMWB) is formulated from an insurer’s perspective. The GMWB rider allows a policyholder to withdraw 10
a capped amount every year, which is typically a fixed percentage of initial purchase payment. Therefore, the instantaneous change in the policyholder’s investment account at any instance t is determined by the current balance times the per unit financial return from linked equity less the per unit fee charge, less the amount of withdrawal dSt − m dt − gF0 dt, dFt = Ft St where g is the withdrawal percentage of the initial purchase payment per time unit. The rider guarantees that the policyholder can withdraw at this rate up until the initial purchase payment is fully refunded, regardless of the performance of the investment account. Because the withdrawal rate is fixed, the time until full refund is also determined by a constant T = F0 /w. Observe that as long as the policyholder’s account remains positive, the withdrawal amount is taken out of his/her own account, which is not a liability of the insurer. However, as soon as the account hits zero before T , the GMWB benefit kicks in and the insurer pays w per time unit up until time T . The rider charges are taken continuously at the rate mw per time unit. Therefore, the present value of the insurer’s net liability (benefit payments less fee incomes) is determined by Z T Z T0 ∧T −rs L := gF0 e ds − mw e−rs Fs ds, T0 ∧T
0
where T0 := inf{t|Ft = 0} is the ruin time at which the investment account hits zero. Because T is relatively short, the mortality risk in this product is rather negligible. An extension of this benefit is the guaranteed lifetime withdrawal benefit (GLWB) where the minimum withdrawal amount is guaranteed until the time of death, in which case the mortality diversification becomes an important factor. With a slide abuse of notation, we shall now use the scaled version of {Ft , t ≥ 0} such that F0 = 1. Using similar arguments, we can formulate the present value of the insurer’s net liability as Z τx Z T0 ∧τx (i) (i) −rs −rs L := F0 g e ds − mw e Fs ds . (16) T0 ∧τx
0
We can rewrite (16) as Z ∞ Z (i) (i) −rs (i) L = F0 g e I(τx > s)I(Fs ≤ 0)ds − mw 0
∞
−rs
e
Fs I(τx(i)
> s)I(Fs > 0)ds ,
0
which conforms to the general liability model (11) with At (F ) = g e−rt I(Ft ≤ 0) − mw e−rt Ft I(Ft > 0),
Bt (F ) = 0.
It follows from the strong law of large numbers in Proposition 3.1 that as n → ∞, Z ∞ n ∗ ¯ ¯ L −→ L := F0 g e−rt I(Ft < 0) − mw e−rt Ft I(Ft > 0) t px dt. 0
Similar results can be obtained for the central limit theorem in Proposition 3.2. 11
4
General aggregate liability with death benefits
In this section, we consider a general framework of aggregate liabilities with death benefits. Define the following individual liability for the policyholder of age x with the initial purchase payment (i) F0 for i = 1, . . . , n, Z ∞ (i) (i) (i) (i) L := F0 At (F )I(τx > t) dt + Bτ (i) (F )I(τx < T ) (17) x
0
In practice, death benefits are usually payable at the end of the period of death (e.g. a year or a month, etc.) to allow for time of claim submission, investigation and settlement. For simplicity, we consider a practical modification of (17), where payments are made at the end of each year. Similar results can be written for death benefits payable monthly or quarterly. Z ∞ [T ] X (i) Bk+1 (F )I k < τx(i) ≤ k + 1 , L(i) := F0 At (F )I(τx(i) > t) dt + (18) 0
k=x
where [T ] is the integer part of T , almost surely. Even though the finite time T may seem to be rather restrict, it can be arbitrarily large so that any reasonable human mortality model can be considered by truncation in this framework with any desired level of accuracy. (i) (i) (i) Proposition 4.1. Suppose that {F0 }i=1,...,n is independent of {τx }i=1,...,n with F¯0 := E[F0 ] < ∞. Moreover, suppose that |Bk (F )| < ∞ a.s. for each k ∈ N, and that
Z
∞
|At (F )| dt + 0
∞ X
|Bk+1 (F )|t px < ∞
a.s.
(19)
k=x
Then it holds for any T > 0 that ¯n = L ¯ ∗ := F¯0 lim L
n→∞
∞
Z
At (F ) · t px dt + F¯0
0
where [T ] is the integer part of T and
[T ] X
Bk+1 (F )k| qx
a.s.,
k=x k| qx
= k px − k+1 px .
Proof. Note that n
X (i) ¯n = 1 F0 L n i=1
Z
∞
At (F )I(τx(i)
0
n
[T ]
i=1
k=x
1 X (i) X > t) dt + F0 Bk+1 (F )I k < τx(i) ≤ k + 1 . n
We denote the two terms by Axn and Bnx respectively. Under the assumption, we can apply Lemma A.1 to Hnx . Then there exists a P-null set N (⊂ Ω) such that Z
∞
|At (F )| dt < ∞; 0
∞ X
|Bk+1 (F )|t px < ∞;
sup |Hnx (t) − H x (t)| → 0, t∈[0,∞)
k=x
12
and that each |Bk (F )| (k ∈ N) is finite, for all ω ∈ Ω \ N . Then it follows for all ω ∈ Ω \ N that Z ∞ Z ∞ x x lim An − At (F ) · H (t) dt ≤ |At (F )| dt sup |Hnx (t) − H x (t)| → 0. n→∞
0
0
t∈[0,∞)
(i) (i) (i) Since I k < τx ≤ k + 1 = I τx > k − I τx > k + 1 , we must have Bnx
=
[T ] X
Bk+1 (F ) [Hnx (k) − Hnx (k + 1)] .
k=x
Denote x
B :=
[T ] X
Bk+1 (F ) [H x (k) − H x (k + 1)] .
k=x
Now we consider the following two cases: (i)
(I) There exists some T0 such that P(τx > [T0 ]) = 0; (i)
(II) For any t > 0, P(τx > t) > 0. As case (I), note that Hnx (t), H x (t) ≡ 0 a.s. for any t > [T0 ], and that [T0 ] X
|Bk+1 (F )| < ∞ a.s.
k=x
due to the finite summation. Therefore it follows for all ω ∈ Ω \ N that |Bnx
x
|Hnx (t)
− B | ≤ 2 sup
x
− H (t)|
t∈[0,[T0 ]]
[T0 ] X
|Bk+1 (F )| → 0.
k=x
As case (II), since t px is decreasing in t, it follows for any T > 0 that there exists a constant δ :=
H x (t) = H x ([T + 1]) > 0.
inf k∈[T,∞)∩N
Then |Bnx
x
−B |≤
[T ] X k=x
|Hnx (k) − H x (k)| |Hnx (k + 1) − H x (k + 1)| + |Bk+1 (F )|H (k) H x (k) H x (k) x
[T ]
X 2F¯0 |Bk+1 (F )|k px → 0. ≤ sup |Hnx (t) − H x (t)| δ t∈[0,∞) k=x
This completes the proof.
Here we also present the central limit theorem for the general model of net liability for equitylinked insurance with death benefits. 13
¯ 0 := E|F (i) |2 < Proposition 4.2. Suppose the same assumptions as in Proposition 4.1, and that Q 0 ∞. Then it holds as n → ∞ that Z [T ] X d ∞ √ n ∗ ¯ ¯ At (F ) · G(t) dt + n L −L → Bk+1 (F ) [G(k) − G(k + 1)] , (20) 0
k=x
where G is a Gaussian process with mean zero and covariance for any t, s ≥ 0 Cov (G(t), G(s)) =
¯ − t px · s px F¯ 2 . 0
t∨s px Q0
Proof. Using the notation in Lemma A.2, we can write √
¯n − L ¯∗ = n L
Z
∞
At (F ) · Gn (t) dt + 0
Z
[T ] X
Bk+1 (F ) [Gn (k) − Gn (k + 1)]
k=x ∞
At (F ) · Gn (t) dt
= 0
+ B(F )>
π(x,x+1,...,[T ]) ◦ Gn − π(x+1,...,[T ]+1) ◦ Gn
where B(F ) = (Bx+1 (F ), . . . , B[T ] (F ))> and π(t1 ,...,td ) is a canonical projection from `∞ ([0, ∞)) to Rd such that, for each function g ∈ `∞ ([0, ∞)), π(t1 ,...,td ) ◦ g = (g(t1 ), . . . , g(td ))> . Note that a mapping π(t1 ,...,td ) is continuous with respect to the uniform norm endowed in `∞ ([0, ∞)). (i) Since F = (Ft )t≥0 and τx ’s are independent of each other, by the similar argument as in the proof of Proposition 3.2, we see from the continuous mapping theorem that Z d ∞ √ n ∗ ¯ ¯ At (F ) · G(t) dt n L −L → 0
+ B(F )> π(x,x+1,...,[T ]) ◦ G − π(x+1,...,[T ]+1) ◦ G Z T [T ] X At (F ) · G(t) dt + Bk+1 (F ) [G(k) − G(k + 1)] . = 0
k=x
This completes the proof.
4.1
Example: Guaranteed minimum death benefit (GMDB)
The GMDB rider is the most common among all guaranteed benefits for variable annuity contracts. It provides policyholders with a minimum protection of account values at the time of death. It was first introduced in Milevsky and Posner [15] and its pricing problems were studied extensively by Ulm and coauthors in [22], [21] and [23]. The risk measures of net liability for GMDB were analyzed in Feng [8]. From the perspective of insurers, the gross liability of the GMDB is determined by the amount by which the policyholder’s account value exceeds the minimum guarantee at the time of death. Hence its present value is given by (i) e−rτx G − Fτ (i) I(τx(i) < T ). x
14
+
The present value of accumulated fee income is given by (i)
T ∧τx
Z
e−rs me Fs ds.
0
Therefore, the present value of the net liability for an individual contract is given by L(τx(i) )
(i)
:= e
−rτx
G − Fτ (i)
+
x
I(τx(i)
Z
(i)
T ∧τx
< T) −
e−rs md Fs ds,
0
where md is the rate at which the GMDB rider charges are collected. Note, however, it is usually a component of the M&E fee. It follows immediately that the GMDB net liability is a special case of (17) where At (F ) = e−rt md Ft I(t < T ), Bt (F ) = e−rt (γ − Ft )+ . The framework in (17) can also be used to accommodate other common guarantee features such as ratchet and roll-up options. Under the ratchet option, the guarantee base increases with the account value, should the later exceeds the former. The continuous-time version of the ratchet option is given by Gt = sup Fs . 0≤s≤t
Under the roll-up option, the guarantee base accrues compound interest at a constant rate, i.e. Gt = F0 eδt , where δ is the crediting interest rate. It is also increasingly common that the management fees are taken as a fixed percentage of the guarantee base, i.e. (i)
T ∧τx
Z
e−rs me Gs ds.
0
Therefore, the present value of the net liability for an individual contract is given by (17) where At (F ) = e−rt md Gt I(t < T ),
4.2
Bt (F ) = e−rt (Gt − Ft )+ .
Example: Guaranteed lifetime withdrawal benefit (GLWB)
Now we revisit the GLWB rider introduced in Section 3.2 from a policyholder’s point of view. The present value of GLWB can also be formulated from an investor’s perspective by L
(i)
Z := gF0 0
(i)
τx
(i)
e−rt dt + e−rτx max{Fτ (i) , 0}. x
From an investor’s point of view, the GLWB is a life annuity plus a call option payable upon death. This can be considered in the framework of (17) by taking Ft −rt −rt At (F ) = ge , Bt (F ) = e max ,0 . F0 15
4.3
Example: Equity-linked annuities
Similar to variable annuities, equity-indexed annuities also offer policyholders participation in equity returns. There are many differences in the operation and management of the two types of products. For example, purchase payments of variable annuities are typically invested in subaccounts separate from insurers’ own general accounts and are managed by third party vendors, whereas premiums from equity-indexed annuities are collected and invested within insurers’ own accounts. Therefore, the downside risk of policyholders’ investment is partially transferred to variable annuity writers, while the full responsibility of investment with equity-indexed annuities is vested in the insurers. The pricing and risk management of equity-indexed annuities are extensively studied in the literature. Here we name a few well-known papers on this subject, although the list is by no means comprehensive. For example, Tiong [20], Lee [11], Lin and Tan [13] developed closed-form solutions to a variety of common product designs in the market. Barbarin and Devolder [1] compared two pricing approaches based on risk measures and risk-neutral pricing on equity-linked insurance. Hedging methods were developed for pricing and risk measures in Melnikov and Romanyuk [16], Melnikov and Smirnov [17] for contracts linked to multiple assets. A general framework of pricing equity-linked death benefits using Laplace transform based approaches can be seen in Gerber, Shiu and Yang [10], Lin [12], etc. It is natural to think of equity-linked annuities as enhanced versions of call options. Here we use the framework of Gaillardetz and Lakhmiri [9] to demonstrate the insurers’ liabilities from most equity-indexed annuity can be analyzed in the framework of (11). In Gaillardetz and Lakhmiri [9], the equity-indexed annuities provide both death and survival benefits. Consider the present value of a N -year equity-indexed annuity with the payoff L(i) := e−r(τ
(i) ∧N )
D(τx(i) ∧ N ),
where D(t) represents the product design which varies from company to company. For example, the so-called “point-to-point” with capped and floored interests can be written as St − 1 , (1 + η)t , β(1 + g)t , D(t) = max min 1 + α S0 where α is known as the participation rate and determines the percentage of profit sharing with policyholders, β represents the percentage of the initial premium that is guaranteed to receive the minimum return of g per dollar, η provides a upper bound on the interest return. It follows immediately from Proposition 4.1 that as n → ∞, ¯ n −→ L
Z
N
e−rt D(t) t px dt + e−rT T px D(T ),
almost surely.
0
Observe that the average model on the right-hand side is in fact used in Gaillardetz and Lakhmiri [9, Eq. (12)-(14)] to derive the fair value of equity-indexed annuities.
16
5
Numerical example: Application of Central Limit Theorem
We provide an example to show the effectiveness of approximations based on the central limit theorem in (15). Consider the computation of risk measures for a 10-year variable annuity contract with a GMMB rider. Suppose the identical policy is sold to a cohort of male policyholders all of age 65 at policy issue, except that the policyholders may deposit different amount of initial purchase payment. Equity model: The policy offers a single fund which invest 100% on equity. Hence the policyholders’ investment accounts are all linked to an equity index modeled by a geometric Brownian motion (also known as independent lognormal model in the industry) dFt = (µ − m)Ft dt + σFt dBt ,
t ≥ 0,
(21)
where the mean and standard deviation of log-returns per annum are set to be µ = 0.09 and σ = 0.3 respectively. The yield rate per annum on the assets backing up the guarantee liabilities is r = 0.04. The M&E fee per annum is m = 0.01, which includes the GMMB rider charge of 35 basis points per annum of the investment account, me = 0.0035. Survival model: We assume that the future lifetimes of all policyholders are mutually independent. The survival model is based on an excerpt of a period life table for male and calendar year 2010 published in the actuarial study by the U.S. Social Security Administration in 2005. x
qx
k
k px
x
qx
k
k px
65
0.01753
0
1
71
0.03059
6
0.87275
66
0.01932
1
0.98246
72
0.03343
7
0.84606
67
0.02122
2
0.96348
73
0.03633
8
0.81778
68
0.02323
3
0.94304
74
0.03942
9
0.78807
69
0.02538
4
0.92113
75
0.04299
10
0.75700
70
0.02785
5
0.89775 Table 1: Life table
Random initial payments: Each policyholder makes an independent choice of initial purchase payment. However, the random choices are assumed to follow a common distribution. There are four cases of distributions governing initial purchase payments to be considered. For all i = 1, 2, · · · , (i)
Case A: (mean 1, variance 0) F0 is set to be a fixed number 1; (F 0 = 1, Q0 = 1) (i) Case B: (mean 1, variance 1/12) F0 follows a uniform distribution on [0.5, 1.5]; (F 0 = 1, Q0 = 1.08¯ 3) (i)
Case C: (mean 1, variance 1) F0 follows an exponential distribution with mean 1; (F 0 = 1, Q0 = 2) (i)
Case D: (mean 1, variance 2) F0 (F 0 = 1, Q0 = 3)
follows a Pareto distribution with parameters α = 4 and θ = 3.
17
The guarantee is set to be a full refund of mean initial purchase payment, i.e. γ = 1. Since all investment accounts are linked to the same equity fund/index, then (i)
Ft
(i) F0
= e−mt
St , S0
for all t ≥ 0, i = 1, · · · , n.
In this case, the insurer’s net liability to each individual is given by (5). Therefore, the average aggregate liability, which is the arithmetic average of n individual net liabilities, is determined by (11) with St St −rt −(m+r)t γ− Bt (F ) = e At (F ) = e me I(t < T ), . S0 S0 + In practice, fees and benefits are paid on discrete time points. So we use the discrete version for simulations. Let t0 = 0 < t1 < · · · < tN = T be a partition of [0, T ] and ∆k = tk − tk−1 , L(τ
(i)
−rT
)=e
(G
(i)
−
(i) FT )+ I(τx(i)
> T) −
N X
(i)
e−rtk me Ftk ∆k I(τ (i) > tk ).
k=1
In this example, we compare the risk measures of three types of aggregate liabilities. n
1. The true average aggregate liability of n policies, L = (1/n) ∗
Pn
i=1 L(τ
(i) );
n
2. The limiting case from the average model, L = limn→∞ L almost surely; (discretized version of (7)) " # N X S S ∗ t T L = F 0 e−rT γ − e−rtk me k ∆k tk px . T px − S0 + S0 k=1
∗
3. An approximate average aggregate liability, L + X n , where the random variable X n is the “correction” term from the central limit theorem. Details can be found in 3.1. In each (i) experiment, we simulate the evoluation of a policyholder’s investment account {Ft , t = t0 , t1 · · · , tN }. For each path, we draw a random number from a normal distribution with ¯ 0 − F¯ 2 t px · t px for i > j and mean 0 and variance C> ΣC with Σij = ti px Q j 0 i > C = e−rt1 ∆1 me Ft1 , · · · , e−rtN −1 ∆N −1 me FtN −1 , e−rtN [∆N me FtN + (G − FtN )+ ] . √ The correction term X n is determined by 1/ n times the random number, where n is the number of policies in the pool. For each run of the whole simulation procedure, we make one million projections of equity returns using the geometric Brownian motion model in (21) with ∆k = 1/12 for all k’s. In other words, fees are assumed to be collected on a monthly basis. Along each sample path of equity n ∗ returns, both L and L are computed using the above-mentioned formulas. Note that we consider n the individual model n = 1 and cohorts of 5, 10 and 30 individuals (n = 5, 10, 30) for L . The (random) initial purchase payments in the aggregate model (n > 1) are always equally split among 18
Case A n n
CTE0.9 (L ) Run time ∗
CTE0.9 (L + X n )
Aggregation of individual models 1
5
10
30
∞
0.30284 (0.00067) 90 0.29170 (0.00063)
0.26554 (0.00036) 122 0.26603 (0.00038)
0.26217 (0.00040) 159 0.26287 (0.00038)
0.26037 (0.00051) 311 0.26086 (0.00038)
0.25993 (0.00045)
Run time
166
Case B n n
CTE0.9 (L ) Run time ∗
CTE0.9 (L + X n )
Aggregation of individual models 5
10
30
∞
0.30515 (0.00060) 88 0.30175 (0.00072)
0.26699 (0.00030) 122 0.26832 (0.00045)
0.26289 (0.00054) 159 0.26373 (0.00044)
0.26058 (0.00040) 309 0.26083 (0.00047)
0.25955 (0.00085)
168
Case C
n
CTE0.9 (L ) Run time ∗
CTE0.9 (L + X n )
Aggregation of individual models 5
10
30
∞
0.32111 (0.00110) 95 0.39188 (0.00126)
0.27949 (0.00066) 142 0.29500 (0.00075)
0.27003 (0.00063) 199 0.27786 (0.00051)
0.26302 (0.00051) 430 0.26553 (0.00047)
0.25980 (0.00137)
169
Case D
n
CTE0.9 (L ) Run time ∗
CTE0.9 (L + X n )
Average model
1
Run time n
Average model
1
Run time n
Average model
Aggregation of individual models
Average model
1
5
10
30
∞
0.32151 (0.00149) 89 0.46672 (0.00126)
0.28493 (0.00092) 123 0.31878 (0.00080)
0.27450 (0.00077) 161 0.29181 (0.00057)
0.26500 (0.00055) 317 0.27062 (0.00041)
0.25979 (0.00033)
Run time
168
Table 2: GMMB net liabilities: the average model versus the aggregation of individual models policies all of which are linked to the same equity index. In other words, both the aggregate model and the average model have the same total amount of purchase payments to begin with. Bear in mind that as only the mean and variance of initial purchase payment F0 are required, we use the same set of scenarios from the equity index for all three cases. We sort the one million realizations 19
n
∗
of L and L to form their respective empirical distributions, from which the 0.9 million-th largest order statistics are used to estimate of the 90% value-at-risk measures and arithmetic averages of the 0.1 million largest order statistics to estimate the 90% conditional tail expectation risk measures. Then the whole procedure is repeated 20 times to obtain a random sample of risk measure estimates. In Table 2, we show means and standard deviations (in brackets) of 20 estimates of risk measures under the four assumptions of initial purchase payments (Cases A, B, C, D). The purpose of this example is to verify numerically the prediction from (14) that as purchase payments are equally split among more and more policies, the distribution of the aggregate net liability of all policies eventually converges to that of the average model. All computations in this section are carried out on a personal computer with Intel Core i7-4700MQ CPU at 2.40GHz and an RAM of 8.00 GB. Run times are recorded in minutes. It is not surprising that the risk measures of liabilities with uniformly distributed initial purchase payments are larger than those with fixed initial payments, as the former introduces more uncertainty to account values. Similarly, the liabilities with exponentially distributed purchase payments result in even larger risk measures than those with uniformly distributed payments, as the exponential randomization leads to a larger variance than the uniform randomization. The first row in each case of Table 2 contains the CTE risk measures of aggregate net liabilities as well as the CTE risk measure of the corresponding average net liability. Keep in mind that the average model is the limit of the aggregate net liabilities as the number of i.i.d. policies goes to infinity, as shown in Proposition 3.1. The second row in each case shows the approximated risk measures, which are obtained from adding “correction” terms to L∗ in the average model, as shown in (15). Note that the additional time for the computation of “correction” terms is negligible for the average model. It becomes clear from Table 2 that the algorithm based on the approximation method can be more efficient than simulations based on individual contracts if there is an aggregation of a large number of individual contracts. Comparing vertically the efficiency of approximations for all four cases, we see that the convergence appears to be faster in the cases of fixed and uniformly distributed purchase payment than that in the cases of exponential and Pareto distributed purchase payment. As the variation of initial payments increases from Cases A to D, a larger and larger pool of policies are necessary to balance the influence of large policies and to exhibit the effect of the central limit theorm in the average model.
Acknowledgement The authors would like to thank Xiaochen Jing for working out numerical examples in this paper.
20
A
Auxiliary lemmas
(i) (i) Lemma A.1. Suppose that F0 , (i = 1, . . . , n) are i.i.d., independent of τx , and that F¯0 := (i) E[F0 ] < ∞. Then it holds that
lim
sup |Hnx (t) − H x (t)| = 0
n→∞ t∈[0,∞)
a.s.,
(22)
where H x (t) := F¯0 · t px . (i)
(i)
Proof. Note that Z (i) = (F0 , τx ) are i.i.d., and that n
Hnx (t) =
1X ft (Z (i) ), n i=1
where ft (z) = z1 I(z2 > t) with z = (z1 , z2 ), and we set f∞ (z) ≡ 0. Letting F = {ft |t ∈ [0, ∞)};
(i)
P (dz1 , dz2 ) = P(F0 ∈ dz1 , τx(i) ∈ dz2 ),
we shall show that F is P -Glivenko-Cantelli; see, e.g., van der Vaart [24], Section 19.2. From Theorem 19.4 in [24], the proof ends if -bracketing number of F is finite: N[ ] (, F, L1 (P )) < ∞
for every > 0.
First, for every > 0, we shall take some T > 0 such that Z |ftk−1 (z1 , z2 )| P (dz1 , dz2 ) = F¯0 P(τx(i) > T ) <
(23)
and fix this and T . Take a sequence (tk )k=1,...,m such that 0 = t0 < t1 < · · · < tm = T , and put tm+1 = ∞. Then, for each t ∈ (0, T ), there exists some tk−1 and tk with t ∈ [tk−1 , tk ] such that ftk ≤ ft ≤ ftk−1 . Hence a bracket [ftk , ftk−1 ] is an -bracket in L1 (P ) if and only if Z |ftk−1 (z1 , z2 ) − ftk (z1 , z2 )| P (dz1 , dz2 ) = F¯0 P(tk−1 < τx(i) < tk ) < ,
(24)
which is possible by taking (tk − tk−1 ) as being small enough (m large enough). Moreover, from the definition of T in (23), [f∞ , fT ] is also the -bracket. Therefore, for any > 0, there exists m large enough such that m+1 [ F⊂ [ftk , ftk−1 ] k=1
that is, N[ ] (, F, L1 (P )) = m + 1 < ∞. This completes the proof. 21
As usual, let `∞ ([0, ∞)) be a space of all bounded function z : [0, ∞) → R with uniform norm kzk := supt∈[0,∞) |z(t)|. For example, the family F = {ft (z)|t ∈ [0, ∞)} ⊂ `∞ ([0, ∞)) since kf· (z)k < z1 < ∞ for every z = (z1 , z2 ). ¯ 0 := E|F (i) |2 < ∞. Lemma A.2. Suppose the same assumptions as in Lemma A.1, and that Q 0 Then, for √ Gn (t) := n (Hnx (t) − H x (t)) , t ∈ [0, ∞). it holds that Gn →d G
in `∞ ([0, ∞)),
(25)
as n → ∞, where G = (G(t))t∈[0,∞) is a Gaussian process with zero mean and covariance function given by ¯ 0 − t px · s px F¯ 2 . E [G(t)G(s)] = t∨s px Q 0 Proof. Using the same notation as in the proof of Lemma A.1, we can check the bracketing integral condition given in Theorem 19.5 by van der Vaart [24]: Z 1q J[ ] (1, F, L2 (P)) := log N[ ] (, F, L2 (P)) d < ∞. (26) 0
¯ 0 < ∞, it is easy to see by the same argument as in the proof of Under the assumption that Q Lemma A.1; see especially the argument in (24), a bracket [ftk , ftk−1 ] can be an -bracket in L2 (P) because Z ¯ 0 P(tk−1 < τ (i) < tk ) < {ftk−1 (z1 , z2 ) − ftk (z1 , z2 )}2 P (dz1 , dz2 ) = Q x if m is large enough. Hence the total number of brackets, m + 1, can be chosen smaller than C/ for a constant C > 0, which implies that N[ ] (, F, L2 (P)) < C/. Therefore, we have that J[ ] (1, F, L2 (P)) < ∞. The covariance function is given by P (ft fs ) − P (ft )P (fs ) with (i)
P := P (dz1 , dz2 ) = P(F0 ∈ dz1 , τx(i) ∈ dz2 ). Then the last result is obvious. The following corollary is an immediate consequence from the continuous mapping theorem. Corollary A.1. Under the same assumptions as in Lemma A.2, it holds that sup |Gn (t)| →d sup |G(t)| t∈[0,∞)
in `∞ ([0, ∞)),
t∈[0,∞)
as n → ∞, where G is a Gaussian process given in Lemma A.2. 22
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