Insurance: Mathematics and Economics 37 (2005) 599–614
Optimal stopping behavior of equity-linked investment products with regime switching Ka Chun Cheunga,1 , Hailiang Yangb,∗ a
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive, Calgary, Alberta, Canada b Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong Received March 2005 ; received in revised form June 2005; accepted 9 June 2005
Abstract In recent years, there is a growing interest in equity-linked investment products. The return credited to such product depends on the return of some underlying reference index. A prominent example is the equity-indexed annuities (EIAs). A special feature of many of the equity-linked products is that the holders are entitled the right to surrender the product prior to maturity. In this paper, we will study the optimal surrender time for a equity-linked product in a discrete-time setting. We assume that the market environment will switch among different regimes in a Markovian way, and the return of the reference index will have different distributions in different regimes. Assuming a CRRA preference, we have obtained the optimal surrender policy. Properties of the optimal surrender behavior, in particular the effect of regime switching, are examined. © 2005 Elsevier B.V. All rights reserved. Keywords: Equity-linked products; Markov regime switching model; Optimal surrender time; Stochastic orders; Utility function
1. Introduction Since Black and Scholes (1973) and Merton (1973) introduced their path-breaking work on option-pricing, there has been an explosive growth in the trading activities on derivative products in the worldwide financial markets. Influenced by the pace of innovation in financial markets, many insurance products have some kind of derivative features nowadays. In particular, equity-linked products are getting more popular in recent years. Equity-indexed annuity (EIAs) is such a product. Essentially, EIA is an equity-linked deferred annuity whose returns are based on the performance of an equity mutual fund or a stock index, e.g. S&P 500. For detailed discussion on EIAs and other ∗ 1
Corresponding author. Tel.: +852 28578322; fax: +852 28589041. Tel.: +1 403 2108697; fax: +1 403 282 5150. E-mail addresses:
[email protected] (K.C. Cheung),
[email protected] (H. Yang).
0167-6687/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2005.06.005
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related products, see Aase and Persson (1994), Brennan and Schwartz (1976), Gerber and Pafuni (2001), Gerber and Shiu (2003), Hardy (2003), Tiong (2000) and the references therein. Equity-indexed annuity is one of the most successful innovations in the financial markets in the last decade. Since Keyport Life launched the “Key Index” in 1995, the notional amount of EIAs sold has increased dramatically in recent years. The design of EIAs is very flexible. This flexibility makes EIA a useful investment and risk management tool. A special feature of EIA is that the holders are entitled the right to surrender the product prior to maturity. This leads to the problem of when the EIA account holder should surrender the product. In a recent paper, Moore and Young (2005) considered the problem of optimal surrender time under a Black–Scholes type model with infinite horizon. Currently, the research on EIAs is mainly in the actuarial circle. As EIA is in fact a derivative product, EIA should be an interesting topic in mathematical finance as well. Many people are now interested in the interplay between actuarial science and finance (see, for example, Embrechts, 2000; Embrechts and Samorodnitsky, 2003). We believe that many interesting problems which lie in the interplay between finance and insurance need to be investigated and some important results could be expected. When we deal with financial problems, it is crucially important to use an appropriate model. Due to the complexity of the financial world, it is almost impossible to model the dynamic of the stock price or index price perfectly. However, there are some works showing that the regime switching model is indeed a good model. Hardy (2001) used monthly data from the Standard and Poor’s 500, and the Toronto Stock Exchange 300 indices to fit a regimeswitching lognormal model. The fit of the regime-switching model to the data was compared with other econometric models. Regime switching models are a class stochastic models used in almost every area of application. The use of regime switching model in financial modelling can be traced back at least to Hamilton (1989). Di Masi et al. (1994) considered the European options under the Black–Scholes formulation of the market in which the underlying economy switches among a finite number of states. Buffington and Elliott (2002) discussed the American options under this set-up. Zariphopoulou (1992) considered an investment–consumption model with regime switching. Zhang (2001) derived an optimal stock selling rule for a Markov-modulated Black–Scholes model. Yin and Zhou (2003) studied a discrete-time version of Markowitz’s mean–variance portfolio selection problem, where the market parameters depend on a finite-state Markov chain. Zhou and Yin (2003) considered a continuous-time version of the Markowitz mean–variance portfolio selection problem for a market consisting of one bank account and multiple stocks. The market parameters depend on the market mode that switches among a finite number of states. Noarbitrage pricing problem in a financial market driven by continuous time homogeneous Markov chain was studied in Norberg (2003). In Cheung and Yang (2004), optimal asset allocation problem under a discrete regime switching model was considered. With short-selling and leveraging constraints, the existence and uniqueness of the optimal trading strategy were obtained. Some natural properties of the optimal strategy were also obtained. The objective of this paper is to study the optimal surrender policy for an equity-linked product like an EIA with a finite-horizon. The methodology and ideas used in this paper are related to those in the papers dealing with optimal portfolio selection. Optimal portfolio selection has been investigated by many authors under different models. Samuelson (1969) considered a discrete time consumption investment model with the objective of maximizing the overall expected consumption. He advocated a dynamic stochastic programming approach and succeeded in obtaining the optimal decision for a consumption investment model. Merton (1969) first used the stochastic optimal control method in continuous finance. He was able to obtain closed form solution to the problem of optimal portfolio strategy under specific assumptions about asset returns and investor preferences. He showed that, under the assumptions of geometric Brownian motion for the stock returns and HARA utility, the optimal proportion invested in the risky asset portfolio is constant through time. For recent developments on this subject, we refer the readers to the books Campbell and Viceira (2001) and Korn (1997). In this paper, we will assume that the market environment will switch among different regimes in a Markovian way. The return of the underlying reference index, and hence the growth rate of the EIA1 will have different 1
In the following discussion, we will use EIA as a representative of equity-linked product.
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distributions in different regimes. Since our focus is on the effect of the regime-switching on the optimal surrender time, the effect of mortality and other product features, like the various embedded guarantees, will be ignored. In particular, we will compare the surrender behavior in different regimes if the distribution of the growth rate of the EIA in different regimes can be ordered in some suitable sense.
2. The model Let {ξn ; n = 0, 1, . . . , N} be a time-homogeneous Markov chain with state space S = {1, 2, . . . , M} and transition probability matrix P = (pij ). We assume that every entry of P is strictly positive. This implies that the Markov chain is irreducible. Suppose {Sn ; n = 0, 1, . . . , N} is the price process of the reference index. This process will satisfy the following Markov-switching model: Sn+1 = Sn
M i=1
Rin 1{ξn =i}
(1)
where 1{...} is the indicator function and Rin represents the return of the reference index in the period [n, n + 1] if the Markov chain is in regime i at time n. This dynamic is interpreted as follows: at time n, if the Markov chain ξ is at regime i, the return in the coming period [n, n + 1] would be Rin . We assume that ξn is observable at time n, but the investor cannot predict what the regime for the next period is. For simplicity, we will often write (1) as: Sn+1 = Sn Rξnn . For empirical evidence and more detailed discussion on the regime-switching model, see, for example, Hardy (2001). As in Cheung and Yang (2004), we assume that 1. for each i ∈ S, the random returns Ri0 , Ri1 , . . . , RiN−1 are strictly positive, integrable and are identically distributed with common distribution function Fi ; j 2. in different time periods, the random returns are independent, i.e. ∀i, j ∈ S, Rin is independent of Rm for m = n; 3. the Markov chain {ξ} is independent of the random returns in the following sense: P(ξn+1 = j, Rinn ∈ B | ξ0 = i0 , . . . , ξn = in ) = pin j P(Rinn ∈ B) for all i0 , . . . , in , j ∈ S, B ∈ B(R) and n = 0, 1, . . . , N − 1. All the random variables considered in this paper are defined on a common probability space (, F, P), and we ξ use {Fn } to denote the natural filtration generated by the process {(ξn , Rnn )}n=0,1,...,N−1 . This filtration represents the flow of information available to the investor. In particular, ξn is observable at time n, but ξn+1 is not. In general, the interest earned by the EIA is based on the return of the reference index. If we denote the value process of the EIA by {Wn ; n = 0, 1, . . . , N}, then we will have: ξ
Wn+1 = W0 fn (R00 , . . . , Rξnn ),
n = 0, 1, . . . , N − 1,
where for each n, fn : Rn+1 + −→ R+ is a measurable function that represents the rule employed by the insurance company to calculate return credited to the EIA account. See Tiong (2000) for some examples of such rule.
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In this paper, we will restrict our attention to the so-called “cliquet” design, which is one of the most common EIA designs. Under the cliquet design, fn takes the following form: fn (r0 , r1 , . . . , rn ) =
n
f (rk )
k=0
where f : R+ −→ R+ is a measurable function such that f (Ri ) is integrable for all i. For the details about the cliquet design, one may consult Tiong (2000). In this case, the process {W} will follow the dynamic Wn+1 = W0
n k=0
ξ
f (Rkk ) = Wn f (Rξnn ).
From the assumptions of the model, it is not difficult to see that the two-dimensional process {ηn = (ξn , Wn ); n = 0, 1, . . . , N} is a Markov process with respect to filtration {F}. We can define the transition operator T by: TG(i, w) = E[G(ξ1 , W1 ) | ξ0 = i, W0 = w] for all measurable G : S × (0, ∞) −→ R such that the expectation on the right exists. An investor can decide the time to surrender the EIA. If it is surrendered at time τ, the surrender value is simply Wτ . Here, we assume that there is no early surrender charge for simplicity. The investor make the surrender decision based on the information carried by the filtration {F}, hence τ has to be an {F}- stopping time. The objective of the investor is to maximize the discounted expected utility of the surrender value over all the {F}- stopping time which is bounded by N. Denote by Tn the set of all {F}-stopping time τ with n ≤ τ ≤ N. If the initial regime is i and the initial value of the EIA is w > 0, then the problem is to solve: U(Wτ ) max E0 = max J(τ; i, w) = J ∗ (i, w), (2) τ∈T0 τ∈T0 (1 + r)τ where r is a discount rate and U is a utility function which is increasing and concave. Maximizing the expected discounted utility as the optimization criterion, first adopted in Samuelson (1969), is quite common in the financial economics literature, especially in problems related to optimal control. Henceforth, we will assume that all the expectations concerned exist and are finite. 3. Optimal surrender time for general utility Optimal stopping problem has been studied by many authors, see, for example, Altieri and Vargiolu (2001) and Shiryaev (1973). In Altieri and Vargiolu (2001), an optimal stopping problem with constraint was solved through the dynamic programming method. The dynamic programming method in some sense is more transparent then the classical martingale method using Snell Envelope (c.f. Neveu (1975)). It can reveal clearly the underlying financial argument. Hence, the dynamic programming method will be used to solve our optimal stopping problem. Actually, one can observe that this method is essentially re-deriving the argument of Snell Envelope.
For (i, w) ∈ D = S × (0, ∞) and n = 0, 1, . . . , N − 1, we define recursively the following: ¯ N (i, w) = U(w), W
(3)
if hn (i, w) > 0 U(w), ¯ n (i, w) = , W 1 ¯ n+1 (i, w), if hn (i, w) = 0 TW 1+r
(4)
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hN (i, w) = 1, hn (i, w) = U(w) −
+
1 ¯ n+1 (i, w) TW 1+r
603
(5) .
(6)
Theorem 1. The optimal surrender time is given by: τˆ0 (i, w) = inf{n ≥ 0;
hn (ξn , Wn ) > 0 | ξ0 = w, W0 = w},
(7)
and the maximum expected discounted utility of the surrender value is: ¯ 0 (i, w). J ∗ (i, w) = J(ˆτ0 ; i, w) = W
(8)
Proof. For n = 0, 1, . . . , N, (i, w) ∈ D and τ ∈ Tn , define: U(Wτ ) U(Wτ ) ξ Jn (τ; i, w) = E = i, W = w = E n n n (1 + r)τ−n (1 + r)τ−n and Jn∗ (i, w) = max Jn (τ; i, w) = Jn (ˆτn (i, w); i, w) τ∈Tn
(9)
and τn∗ (i, w) = inf{l ≥ n;
hl (ξl , Wl ) > 0 | ξn = i, Wn = w}.
Then our original problem (2) is embedded in the class of problems (9). The J and J ∗ in (2) are just the J0 and J0∗ in (9). When n = N, which is the date of maturity, the investor is forced to surrender the EIA, hence: ¯ N (i, w) JN∗ (i, w) = U(w) = W
and
∗ τˆN (i, w) = N = τN (i, w).
We now prove by induction that for n = 0, 1, . . . , N − 1, ¯ n (i, w) Jn∗ (i, w) = W
(10)
and τˆn (i, w) = τn∗ (i, w).
(11)
Suppose that the above equations are true for some n = k + 1, . . . , N. At time k, given that ξk = i and Wk = w, the investor has two choices: either surrender the EIA immediately or wait for another period. If it is surrendered immediately at time k, the investor can get the amount U(w); if one more period is waited, the investor may get: 1 ∗ E[Jk+1 (ξk+1 , Wk+1 ) | ξk = i, Wk = w] 1+r because of the dynamic programming principle, and (s)he will choose to surrender the EIA at time ∗ (ξ τˆk+1 (ξk+1 , Wk+1 ) = τk+1 k+1 , Wk+1 ) by the induction hypothesis. Using the induction hypothesis again, we may rewrite the above expression as: 1 1 ∗ ¯ k+1 (ξk+1 , Wk+1 ) | ξk = i, Wk = w] (ξk+1 , Wk+1 ) | ξk = i, Wk = w] = E[Jk+1 E[W 1+r 1+r 1 ¯ k+1 (i, w). = TW 1+r
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Therefore, the investor will surrender the EIA at time k if 2 U(w) >
1 ¯ k+1 (i, w), TW 1+r
which is equivalent to hk (i, w) > 0. If the above condition does not hold, then (s)he will wait for another period and surrender the EIA at time ∗ (ξ τˆk+1 (ξk+1 , Wk+1 ) = τk+1 k+1 , Wk+1 ). Hence, we have: τˆk (i, w) =
k, ∗ (ξ τk+1 k+1 , Wk+1 ),
if hk (i, w) > 0 if hk (i, w) = 0
= τk∗ (i, w).
Therefore, (10) and (11) are also true for n = k. This completes the induction step and finishes the proof.
It should be remarked that we have defined hN to be identically equal to 1. In fact, it is equally possible for hN to take any strictly positive values. The reason is that we want to ensure that the stopping time τˆ0 is bounded by N, hence belongs to T0 .
4. The case of power utility In order to gain more insights about the optimal surrender strategy and to obtain some explicit formulae, we will henceforth restrict our attention to the case where U(w) =
wγ γ
(12)
where 0 < γ < 1. The results for γ < 0 can be established similarly and are not done here. ¯ n . For n = 0, 1, . . . , T − 1 and each In the case of power utility, we have the explicit form of the function W i ∈ S, define B(i) = E[(f (Ri ))γ ], (i)
CN = 1,
M (i)
B Cn(i) = max 1, 1+r
j=1
pij Cn+1 . (j)
Proposition 1. When the utility function is given by (12), then for (i, w) ∈ D, ¯ n (i, w) = 1 wγ Cn(i) . W γ 2
The strictly inequality sign “>” could well be replaced by “≥”.
(13)
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Moreover, the optimal surrender time to problem (2) is M B(ξn ) (j) τˆ0 = inf 0 ≤ n ≤ N − 1; 1 > pξn j Cn+1 ∧ N. 1+r
605
(14)
j=1
Here, we assume that inf φ = +∞. From this proposition, we can see that the optimal surrender time does not depend on the value of the EIA. At any time instant, once we know which regime the Markov chain is in, we can decide whether we have to stop or not. Proof. Fix (i, w) ∈ D. We first prove (13) by induction. When n = N, γ (i) ¯ N (i, w) = U(w) = w = 1 wγ CN W γ γ (i)
as CN = 1. Now suppose that (13) is true for n = k + 1, then we have 1 ¯ k+1 (i, w) = 1 E[W ¯ k+1 (ξk+1 , Wk+1 ) | ξk = i, Wk = w] TW 1+r 1+r 1 1 (ξk+1 ) γ = | ξk = i, Wk = w] E[Wk+1 Ck+1 1+rγ =
M M 1 wγ wγ B(i) (j) (j) pij Ck+1 E[(f (Rik ))γ ] = pij Ck+1 . 1+r γ γ 1+r j=1
j=1
Together with (4), we know that 1 γ 1 1 γ wγ 1 ¯ ¯ ¯ w , T Wk+1 (i, w) = max w , T Wk+1 (i, w) Wk (i, w) = max γ 1+r γ γ 1+r M (i) 1 γ 1 B (j) (i) = w max 1, pij Ck+1 = wγ Ck . γ 1+r γ j=1
This proves that (13) is true for all possible n. To prove (14), we first note that for n = 0, 1, . . . , N − 1 and (i, w) ∈ D, + + M γ γ B(i) w 1 w (j) ¯ n+1 (i, w) = hn (i, w) = U(w) − TW − pij Cn+1 1+r γ γ 1+r +
=
M wγ B(i) (j) pij Cn+1 1− γ 1+r j=1
and hence hn (i, w) > 0 ⇐⇒ 1 >
M B(i) (j) pij Cn+1 . 1+r j=1
Now (14) follows.
j=1
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From (13), the maximum expected discounted utility of the surrender value, given that the initial value of the (i) EIA is w, the initial regime is i and the number of time period to maturity is T, is proportional to CT and the optimal surrender time also depends on Cn(i) s. We may want to explore some properties of the Cn(i) , with a hope of gaining more insights about the optimal surrender strategy. The next proposition, which describes how Cn(i) changes with n for each fixed i, tells us that the longer the time to maturity, the higher the expected discounted utility of the surrender value the investor can achieve. Proposition 2. For every i ∈ S, we have (i)
(i)
(i)
C0 ≥ C1 ≥ · · · ≥ CN .
(15)
Proof. For each i ∈ S, M (i) B (j) (i) = max 1, pij CN ≥ 1 = CN . 1+r
(i)
CN−1
j=1
(i)
(i)
Now suppose that Ck ≥ Ck+1 for all i ∈ S, then (i)
Ck−1 =
M M B(i) B(i) (j) (j) (i) pij Ck ≥ pij Ck+1 ≥ Ck . 1+r 1+r j=1
This completes the proof by induction.
j=1
The previous proposition concerns with the sizes of Cn(i) , n = 0, 1, . . . , T , for fixed i. We then want to compare the sizes of Cn(i) , i ∈ S, for fixed n. In order to do so, the concept of stochastic ordering between random variables and the stochastic monotonicity of the transition matrix become very useful. Definition 1. Suppose that X and Y are two random variables with distribution functions FX and FY , respectively. If for any increasing and concave function h, we always have E[h(X)] ≤ E[h(Y )]
(16)
whenever the expectations exist, then we say X is dominated by Y in the sense of second-order stochastic dominance and is denoted by X ≤SSD Y or FX ≤SSD FY . The concept of stochastic dominance is a commonly used concept in finance, see, for example, Huang and Litzenberger (1988). For a detailed mathematical treatment, we refer to the books by M¨uller and Stoyan (2002) and Shaked and Shanthikumar (1994). Definition 2. Suppose that P = (pij ) is an m × m stochastic matrix. It is called stochastically monotone if m l=k
pil ≤
m
pjl ,
l=k
for all 1 ≤ i < j ≤ m and l = 1, 2, . . . , m.
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The main property of a stochastically monotone matrix that makes it useful is the following. See, for example, Rolski et al. (1999). Lemma 1. Suppose that P is an m × m stochastically monotone matrix. If A = (A1 , . . . , Am )T is an increasing (respectively, decreasing) column vector, then PA is also an increasing (respectively, decreasing) column vector. Now we are ready for the next proposition. Proposition 3. Assume that the transition matrix P of the Markov chain {ξn } is stochastically monotone and f (RM ) ≤SSD · · · ≤SSD f (R1 ),
(17)
then for n = 0, 1, . . . , N, Cn(M) ≤ · · · ≤ Cn(1) .
(18) (i)
Proof. We will again prove the proposition by induction. When n = N, Eq. (18) is true obviously since CN = 1 for all i ∈ S by definition. Suppose that Eq. (18) is also true for n = k + 1, then for 1 ≤ i < j ≤ M, M M M B(i) B(j) B(i) (i) (l) (l) (l) pil Ck+1 ≥ max 1, pjl Ck+1 ≥ max 1, pjl Ck+1 Ck = max 1, 1+r 1+r 1+r l=1
l=1
l=1
(j)
= Ck
where the first inequality follows from Lemma 1, while the second inequality follows from that fact that assumption (17) implies B(M) ≤ · · · ≤ B(1) . In order to study some qualitative properties of the optimal surrender policy, it is reasonable to assume that there is a certain criterion to compare two different regimes. In other words, we should be able to tell which is better among any two regimes. Since each regime is characterized by its corresponding return distribution, it is quite natural to employ the concept of stochastic dominance orders. In fact, stochastic dominance orders have been used extensively in the finance and actuarial science literatures to compare risks and compare returns. See, for example, Cheung and Yang (2004, 2005), Huang and Litzenberger (1988), Kaas et al. (1994, 2001), M¨uller and Stoyan (2002). Condition (17) captures the idea that the returns in all regimes are ordered sequentially: it is the worst in regime M but the best in regime 1. For further discussion, see Cheung and Yang (2005). It was demonstrated in Cheung and Yang (2005) through some empirical figures that assuming the transition matrix P to be stochastically monotone is not that unrealistic. Together with condition (17), this means that if the current market is at a better regime, then it will have a smaller probability of switching to the worst l regimes in the next time period, for any l. Such a financial market is deemed as regular, and is consistent with our intuition.
5. Optimal stopping behavior when N is large Next, we want to study how an investor should behave when the time to maturity is long. To make the discussion easier, we first introduce the time-reversed version of the function Cn(i) . For i ∈ S, define ˜ (i) = 1, C 0
(19)
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608
M (i) B ˜ n(j) , = max 1, pij C 1+r
˜ (i) C n+1
n = 0, 1, 2 . . . .
(20)
j=1
By Proposition 1, we know that if there are n(≥ 1) more time periods till the end of the horizon and the current regime is at state i, then we should surrender the EIA if 1>
M B(i) ˜ (j) , pij C n−1 1+r j=1
or continue holding the EIA otherwise. Let Dn(i) be the expression on the right-hand side of the above inequality, (n) then our surrender decision is closely related to size of Di ; in particular, whether it is greater than 1 or not. Based (i) ˜ n , we can obtain the corresponding recursive formula for Dn(i) : on the recursive formula for C (i)
D1 = (i)
B(i) , 1+r
Dn+1 =
(21)
M B(i) pij max(1, Dn(j) ), 1+r
n = 1, 2, . . . .
(22)
j=1
The monotonic properties of Dn(i) are summarized in the next lemma. Lemma 2. 1. For each i ∈ S, (i)
(i)
(i)
D1 ≤ D2 ≤ D3 ≤ · · · .
(23)
2. Assume that the transition matrix P of the Markov chain {ξn } is stochastically monotone and f (RM ) ≤SSD · · · ≤SSD f (R1 ),
(24)
then for any n ≥ 1 and i = 2, 3, . . . , M, Dn(M) ≤ · · · ≤ Dn(1) .
(25)
and (i−1)
(i)
Dn(i−1) − Dn(i) ≤ Dn+1 − Dn+1 .
(26)
Proof. The proofs of (23) and (25) are similar to that of Propositions 2 and 3, respectively, and are omitted. To (M) (1) prove (26), we first show that it is true when n = 1. From (25), we have D1 ≤ · · · ≤ D1 . Because the function max(1, ·) is increasing (and convex), we have (M)
(1)
max(1, D1 ) ≤ · · · ≤ max(1, D1 ). Since the transition matrix P is stochastically monotone, for i = 2, 3, . . . , M,
K.C. Cheung, H. Yang / Insurance: Mathematics and Economics 37 (2005) 599–614
(i−1)
D2
(i)
− D2 =
M M M B(i−1) B(i) B(i−1) (j) (j) (j) pi−1j max(1, D1 ) − pij max(1, D1 ) ≥ pij max(1, D1 ) 1+r 1+r 1+r j=1
−
j=1
j=1
M M M B(i) (j) (j) (i−1) (i) (i−1) (i) pij max(1, D1 ) = [D1 −D1 ] pij max(1, D1 ) ≥ [D1 −D1 ] pij 1+r j=1
(i−1)
= D1
j=1
(i) (i−1)
(j−1)
j=1
− D1
where the last inequality follows from the fact that D1 holds for some n = k − 1 ≥ 1. We first note that if max(1, Dk
609
(j−1)
(i)
− D1 ≥ 0 by assumption (24). Now suppose that (26)
(j)
(j)
) − max(1, Dk−1 ) ≥ max(1, Dk ) − max(1, Dk−1 )
(27)
is true for j = 2, 3, . . . , M, then (26) also holds for n = k because (i−1)
(i)
(i−1)
[Dk+1 − Dk+1 ] − [Dk
(i)
(i−1)
(i−1)
− Dk ] = [Dk+1 − Dk =
(i)
(i)
] − [Dk+1 − Dk ]
M B(i−1) (j) (j) pi−1j [max(1, Dk ) − max(1, Dk−1 )] 1+r j=1
−
M B(i) (j) (j) pij [max(1, Dk ) − max(1, Dk−1 )] 1+r j=1
≥
M B(i−1) (j) (j) pij [max(1, Dk ) − max(1, Dk−1 )] 1+r j=1
−
M B(i) (j) (j) pij [max(1, Dk ) − max(1, Dk−1 )] 1+r j=1
(i−1)
= (D1
(i)
− D1 )
M j=1
(j)
(j)
pij [max(1, Dk ) − max(1, Dk−1 )] ≥ 0.
Eq. (27) can be shown as follows: (j−1)
[max(1, Dk
(j−1)
(j)
(j)
) − max(1, Dk−1 )] − [max(1, Dk ) − max(1, Dk−1 )] (j−1)
≥ max(1, Dk
(j−1)
(j)
(j−1)
+ (Dk−1 − Dk−1 ) − (Dk
(j−1)
(j)
(j)
− Dk )) (j)
− max(1, Dk−1 ) − max(1, Dk ) + max(1, Dk−1 ) (j−1)
(j)
(j)
(j)
(j−1)
(j)
= [max(1, Dk−1 − Dk−1 + Dk ) − max(1, Dk )] − [max(1, Dk−1 ) − max(1, Dk−1 )] ≥ 0 where the first inequality follows from the induction hypothesis and that the map max(1, ·) is increasing, while the last inequality follows from the convexity of such map. From now on, we will always assume that the transition probability matrix P is stochastically monotone and that (17) is true. Such assumptions are made to ensure that the market possesses a certain amount of regularity so that further analysis can be done. In this case, the return credited to the EIA is the best at regime 1 but the worst at
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regime M. Since (17) is true, we have B(1) ≥ · · · ≥ B(M) , or equivalently (1)
(M)
D1 ≥ · · · ≥ D1 .
(28) (i)
Recall that if there is only one time period left, then the investor should surrender the EIA at regime i if D1 is strictly less than 1. Roughly speaking, we can interpret D(i) as a measure of the relative return in regime i. If it is large, the investor will be inclined to continue holding the EIA so as to enjoy the potentially high return. Without loss of generality, we will assume that all the inequalities in (28) are strict. As the only information (j) (j+1) about the random variable f (Ri ) that is relevant here is its moment B(i) , if B(j) = B(j+1) (and hence D1 = D1 ) for some j, then we can group the two regimes together and consider them as a single regime. It is easy to see that the augmented transition probability matrix, whose dimension is (M − 1) × (M − 1) now, is again stochastically monotone. To analyze the optimal behavior of the investor when the time to maturity is long, we need to investigate the asymptotic behavior of the function Dn(i) when n → ∞ for each i ∈ S. Three different scenarios have to be considered, according to whether all the terms in (28) are less than 1, greater than 1 or some of them are less than 1 but some of them are greater than 1. (1)
(M)
Proposition 4. If 1 > D1 > · · · > D1 , then for each i ∈ S, (i)
(i)
(i)
D1 = D2 = D3 = · · · . (1)
(M)
Proof. If 1 > D1 > · · · > D1 , then for any i ∈ S, (i)
D2 =
M M B(i) B(i) (j) (i) pij max(1, D1 ) = pij = D1 < 1 1+r 1+r j=1
j=1
and if 1 > Dn(1) > · · · > Dn(M) , then for any i ∈ S, (i)
Dn+1 =
M M B(i) B(i) (i) pij max(1, Dn(j) ) = pij = D1 < 1. 1+r 1+r j=1
This finishes the proof.
j=1
In particular, this proposition says that 1 > Dn(i) for all i ∈ S and all n ≥ 1. This means that if it is optimal to surrender the EIA when there is 1 period left regardless of which regime the Markov chain ξ is staying at, then the EIA holder should surrender the EIA immediately at time zero regardless of which regime ξ is staying in. In other words, the EIA is not worth buying. The reason is that under the stated hypothesis, the relative return of the EIA in all the regimes are not attractive enough. (1)
(M)
Proposition 5. If D1 > · · · > D1 Dn(i) ≥ 1.
≥ 1, then for i ∈ S and n ≥ 1,
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Proof. The proof follows directly from part 1 of Lemma 2, which states that Dn(i) is monotonically increasing in n for each i ∈ S. This proposition means that if it is optimal to continue holding the EIA when there is 1 period left regardless of what the current regime is, then it is also optimal to continue holding the EIA at any time, at any regime. This implies that it is not optimal to surrender the EIA prior to maturity and we should wait until the end of the horizon. The reason is that under the hypothesis of the proposition, the return credited to the EIA, even in the worst state, is good enough to attract us to at least hold the EIA for one more period. The scenarios described in the previous two propositions are two extreme cases. The third scenario is an intermediate one: (1)
(l)
(l+1)
D1 > · · · > D1 ≥ 1 > D1
(M)
> · · · > D1 .
(29)
This means that at regimes 1, 2, . . . , l, the return credited to the EIA in the coming period would be “high”, but at regimes l + 1, . . . , M, the return would be “low”. When there is only one period left, we should continue holding the EIA if the Markov chain {ξ} is at regimes 1, 2, . . . , l; otherwise, we should surrender the EIA. As in the previous the cases, we would like to know whether the limits limn→∞ Dn(i) , i ∈ S, are greater than one or not. To make our discussion easier and more intuitive, we will use the following terminology: a regime is said to be BAD, (respectively, GOOD) at a certain time if it is optimal to, (respectively, not to) surrender the EIA at that instant due to the low (respectively, high) return in the coming period. Hence, regime i is BAD (respectively, GOOD) when there are n periods left if Dn(i) < 1 (respectively ≥ 1). Using this terminology and under condition (29), we may classify regimes 1, . . . , l as “GOOD” and regimes l + 1, . . . , M as “BAD”, when there is only one time period left. From part 1 of Lemma 2, we know that for i = 1, 2, . . . , l, Dn(i) ≥ 1,
∀n ≥ 1.
This means that no matter how many time periods are left, it is always optimal to continue holding the EIA as long as we are at one of the initially3 GOOD regimes. In other words, GOOD regimes remain GOOD forever. From part 2 of Lemma 2, it is clear that at any time, if it is optimal to continue holding the EIA at a certain regime, then it is also optimal to continue holding the EIA at that time if the Markov chain is at a better regime. On the other hand, it may appear intuitive that if the time to maturity is sufficiently long and the Markov chain is irreducible, then we should continue holding the EIA even if we are at an initially BAD regime. The argument is that if we have sufficient time, then the Markov chain ξ will eventually switch to one of the GOOD regimes by the irreducibility of the Markov chain. Hence it may be worth holding the EIA for a while. To verify whether this intuition is correct, it is equivalent to determining whether the limits
L(i) = lim Dn(i) , n→∞
i∈S
are greater than 1 or not. As {Dn(i) }n≥1 is an increasing sequence, limit L(i) exists for each i, though it may be infinite. From the recursive equation that defines Dn(i) , the limits L(·) are either all finite or all equal to +∞. If they are all finite, then we also have L(1) ≥ · · · ≥ L(M) by (25). From (21) and (22), we know that the limits L(i) depend only on the transition probability matrix P and the values (i) of D1 , i ∈ S. The following proposition provides a simple sufficient condition that guarantees the infiniteness of all the limits L(i) . 3 Since, we are working in a time-reversed system, “initially” here means when there is only one time period left. Similarly, the word “eventually” will be used to signify that the number of time periods left is tending to infinity.
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Proposition 6. Assume that (29) is true. If there exists an j ∈ {1, 2, . . . , l} such that pjj ≥
1 (j)
D1
,
(30)
then L(1) = · · · = L(M) = +∞. Proof. Assume that L(i) < ∞ for all i and there is a j ∈ {1, 2, . . . , l} such that pjj ≥ on both sides gives: (j)
L(j) = D1
M
1 (j) . D1
pjn max(1, L(n) ).
In (22), letting n → ∞
(31)
n=1
for each j ∈ S. As ∞ > L(1) ≥ · · · ≥ L(M) , there exists k ∈ {0, 1, . . . , M − l} such that ∞ > L(1) ≥ · · · ≥ L(l+k) ≥ 1 ≥ L(l+k+1) ≥ · · · ≥ L(M) . Then (31) could be rewritten as l+k
pjn L(n) −
n=1
L(j) (j)
D1
M
=−
pjn ,
n=l+k+1
which is impossible because the right-hand side is negative while the left-hand side is positive since all the coefficients of the L(n) s are positive by assumption (30). Roughly speaking, Proposition 6 says that when the product of the probability of the Markov chain staying at a GOOD state, i, and the expected return credited to the EIA account when Markov chain is at regime i is large, all states will eventually become GOOD states. In general, it is very difficult to obtain sufficient and necessary conditions for L(i) to be greater than 1 or not. In what follows, we will demonstrate numerically that the intuition discussed above is not always true. That means it may happen that all the limits L(l+1) , . . . , L(M) are less than 1, or the limits L(l+1) , . . . , L(l+k) are greater than one but L(l+k+1) , . . . , L(M) are less than 1. We will demonstrate various possibilities about the values of the limits L(1) , . . . , L(M) . As noted before, we will only focus on that case (1) (l) (l+1) (M) > · · · > D1 . that D1 > · · · > D1 ≥ 1 > D1 Suppose that M = 3, r = 0.1 and the transition probability matrix P is given by
0.6
0.3
0.1
P = 0.3 0.5 0.2 . 0.3 0.3 0.4 One can easily check that this matrix is stochastically monotone. The following three cases illustrate three different (1) possibilities about the sizes of the limits L(i) s. In all three cases, D1 is greater than 1 and hence regime 1 is initially (2) (3) a GOOD regime; on the other hand, both D1 and D1 are less than 1 and hence regimes 2 and 3 are BAD regimes initially.
K.C. Cheung, H. Yang / Insurance: Mathematics and Economics 37 (2005) 599–614 (1)
(2)
613
(3)
1. (D1 , D1 , D1 ) = (1.3, 0.9, 0.8) =⇒ (L(1) , L(2) , L(3) ) = (+∞, +∞, +∞). This means that regimes 2 and 3 become GOOD eventually. (1) (2) (3) 2. (D1 , D1 , D1 ) = (1.2, 0.8, 0.7) =⇒ (L(1) , L(2) , L(3) ) = (1.71, 0.97, 0.85). As anticipated, L(1) is greater than 1 and hence regime 1 remains GOOD forever. However, regimes 2 and 3 remain BAD forever as both L(2) and L(3) are less than 1. (1) (2) (3) 3. (D1 , D1 , D1 ) = (1.2, 0.85, 0.7) =⇒ (L(1) , L(2) , L(3) ) = (1.88, 1.13, 0.91). The limit L(3) is less than 1, meaning that regime 3 remains BAD forever. However, the limit L(2) is greater than 1, meaning that although regime 2 initially is BAD, it becomes GOOD eventually. 6. Conclusion In this paper, we have studied the optimal surrender time for equity-linked products, like the equity-indexed annuity, in a discrete-time model with regime switching. Under the power utility, we have obtained the closed form solution for the optimal surrender time, and have investigated its properties. In order to do so, we introduced the concepts of stochastic dominance and stochastic monotonicity which turn out carry some nature financial interpretations. We also studied the properties of the optimal stopping policy when the investment time period is long. Some interesting results have been obtained. In practice, the investor can surrender part of the investment. In this case, the investor has to decide how much to surrender at any time instant. The objective becomes maximizing the sum of expected discounted utility of the surrender values. This problem essentially becomes a dynamic intertemporal consumption problem. A similar model, together with the presence of default risk, was studied in Cheung and Yang (2005). In the current study, it is assumed that the investment horizon N is deterministic. However, N could be random in reality. For example, N may represents the remaining lifetime of the investor. Let Tx be the future lifetime of the investor who is at age x at time 0. Assuming that the utility is zero if the investor dies before surrendering the EIA, we may modified the dynamics of {W} as Wn+1 = Wn f (Rξnn )1{Tx ≥n+1} . The random behavior of N can be specified by the conditional survival probabilities, see, for example, Bowers et al. (1997). Optimal surrender policy can be obtained similarly. However, some of the monotonic properties about the optimal policy revealed in Section 5 will be lost. One possible direction of further investigation is to explore necessary and sufficient conditions for L(i) to be greater than 1. This is a hard problem and meaningful, as indicated in our example, it is not always true that the BAD regimes will become GOOD eventually. Another future research problem is to investigate the continuous time model. Some similar conclusions can be expected. Acknowledgments This work was supported by Research Grants Council of HKSAR (Project No.: HKU 7239/04H). The authors wish to thank the anonymous referee for helpful comments and suggestions. References Aase, K.K., Persson, S.A., 1994. Pricing of unit-linked life insurance policies. Scandinavian Actuarial Journal 1, 26–52. Altieri, A., Vargiolu, T., 2001. Optimal default boundary in a discrete time setting. In: Kohlmann, M., Tang, S. (Eds.), Mathematical Financ, Trends in Mathematics. Birkh¨auser.
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