optimal stopping behavior of equity-linked investment products with regime switching

Insurance: Mathematics and Economics 37 (2005) 599–614

Optimal stopping behavior of equity-linked investmentproducts 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 dependson the return of some underlying reference index. A prominent example is the equity-indexed annuities (EIAs). A special featureof many of the equity-linked products is that the holders are entitled the right to surrender the product prior to maturity. In thispaper, we will study the optimal surrender time for a equity-linked product in a discrete-time setting. We assume that the marketenvironment will switch among different regimes in a Markovian way, and the return of the reference index will have differentdistributions in different regimes. Assuming a CRRA preference, we have obtained the optimal surrender policy. Properties ofthe 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

SinceBlack and Scholes (1973)andMerton (1973)introduced their path-breaking work on option-pricing, therehas 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 derivativefeatures nowadays. In particular, equity-linked products are getting more popular in recent years. Equity-indexedannuity (EIAs) is such a product. Essentially, EIA is an equity-linked deferred annuity whose returns are based onthe performance of an equity mutual fund or a stock index, e.g. S&P 500. For detailed discussion on EIAs and other

∗ Corresponding author. Tel.: +852 28578322; fax: +852 28589041.1 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

600 K.C. Cheung, H. Yang / Insurance: Mathematics and Economics 37 (2005) 599–614

related products, seeAase and Persson (1994), Brennan and Schwartz (1976), Gerber and Pafuni (2001), Gerberand 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. SinceKeyport Life launched the “Key Index” in 1995, the notional amount of EIAs sold has increased dramatically inrecent years. The design of EIAs is very flexible. This flexibility makes EIA a useful investment and risk managementtool. A special feature of EIA is that the holders are entitled the right to surrender the product prior to maturity. Thisleads to the problem of when the EIA account holder should surrender the product. In a recent paper,Moore andYoung (2005)considered the problem of optimal surrender time under a Black–Scholes type model with infinitehorizon.

Currently, the research on EIAs is mainly in the actuarial circle. As EIA is in fact a derivative product, EIA shouldbe an interesting topic in mathematical finance as well. Many people are now interested in the interplay betweenactuarial science and finance (see, for example,Embrechts, 2000; Embrechts and Samorodnitsky, 2003). We believethat many interesting problems which lie in the interplay between finance and insurance need to be investigated andsome important results could be expected.

When we deal with financial problems, it is crucially important to use an appropriate model. Due to the complexityof 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 regime-switching lognormal model. The fit of the regime-switching model to the data was compared with other econometricmodels. Regime switching models are a class stochastic models used in almost every area of application. The useof regime switching model in financial modelling can be traced back at least toHamilton (1989). Di Masi et al.(1994)considered the European options under the Black–Scholes formulation of the market in which the underlyingeconomy switches among a finite number of states.Buffington and Elliott (2002)discussed the American optionsunder 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 marketparameters depend on a finite-state Markov chain.Zhou and Yin (2003)considered a continuous-time version ofthe Markowitz mean–variance portfolio selection problem for a market consisting of one bank account and multiplestocks. The market parameters depend on the market mode that switches among a finite number of states. No-arbitrage pricing problem in a financial market driven by continuous time homogeneous Markov chain was studiedin Norberg (2003). In Cheung and Yang (2004), optimal asset allocation problem under a discrete regime switchingmodel was considered. With short-selling and leveraging constraints, the existence and uniqueness of the optimaltrading 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 afinite-horizon. The methodology and ideas used in this paper are related to those in the papers dealing with optimalportfolio 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 maximizingthe overall expected consumption. He advocated a dynamic stochastic programming approach and succeeded inobtaining the optimal decision for a consumption investment model.Merton (1969)first used the stochastic optimalcontrol method in continuous finance. He was able to obtain closed form solution to the problem of optimalportfolio strategy under specific assumptions about asset returns and investor preferences. He showed that, underthe assumptions of geometric Brownian motion for the stock returns and HARA utility, the optimal proportioninvested in the risky asset portfolio is constant through time. For recent developments on this subject, we refer thereaders to the booksCampbell and Viceira (2001)andKorn (1997).

In this paper, we will assume that the market environment will switch among different regimes in a Markovianway. 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.

K.C. Cheung, H. Yang / Insurance: Mathematics and Economics 37 (2005) 599–614 601

distributions in different regimes. Since our focus is on the effect of the regime-switching on the optimal surrendertime, the effect of mortality and other product features, like the various embedded guarantees, will be ignored. Inparticular, we will compare the surrender behavior in different regimes if the distribution of the growth rate of theEIA 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 spaceS = {1, 2, . . . , M} and transi-tion probability matrixP = (pij). We assume that every entry ofP is strictly positive. This implies that the Markovchain is irreducible. Suppose{Sn; n = 0, 1, . . . , N} is the price process of the reference index. This process willsatisfy the following Markov-switching model:

Sn+1 = Sn

M∑i=1

Rin1{ξn=i} (1)

where1{...} is the indicator function andRin represents the return of the reference index in the period [n, n + 1] if

the Markov chain is in regimei at timen. This dynamic is interpreted as follows: at timen, if the Markov chainξis at regimei, the return in the coming period [n, n + 1] would beRi

n. We assume thatξn is observable at timen,but the investor cannot predict what the regime for the next period is. For simplicity, we will often write(1) as:

Sn+1 = SnRξnn .

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 eachi ∈ S, the random returnsRi0, R

i1, . . . , R

iN−1 are strictly positive, integrable and are identically distributed

with common distribution functionFi;2. in different time periods, the random returns are independent, i.e.∀i, j ∈ S, Ri

n is independent ofRjm 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) = pinjP(Rin

n ∈ B)

for all i0, . . . , in, j ∈ S, B ∈ B(R) andn = 0, 1, . . . , N − 1.

All the random variables considered in this paper are defined on a common probability space (�,F, P), and weuse{Fn} to denote the natural filtration generated by the process{(ξn, R

ξnn )}n=0,1,...,N−1. This filtration represents

the flow of information available to the investor. In particular,ξn is observable at timen, butξn+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:

Wn+1 = W0fn(Rξ00 , . . . , Rξn

n ), n = 0, 1, . . . , N − 1,

where for eachn, 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. SeeTiong (2000)for some examples of such rule.


In this paper, we will restrict our attention to the so-called “cliquet” design, which is one of the most commonEIA designs. Under the cliquet design,fn takes the following form:

fn(r0, r1, . . . , rn) =n∏

k=0

f (rk)

wheref : R+ −→ R+ is a measurable function such thatf (Ri) is integrable for alli. For the details about thecliquet design, one may consultTiong (2000). In this case, the process{W} will follow the dynamic

Wn+1 = W0

n∏k=0

f (Rξk

k ) = Wnf (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 operatorT by:

TG(i, w) = E[G(ξ1, W1) | ξ0 = i, W0 = w]

for all measurableG : 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 timeτ, the surrender value is simply

Wτ . Here, we assume that there is no early surrender charge for simplicity. The investor make the surrender decisionbased on the information carried by the filtration{F}, henceτ has to be an{F}- stopping time. The objective of theinvestor is to maximize the discounted expected utility of the surrender value over all the{F}- stopping time whichis bounded byN. Denote byTn the set of all{F}-stopping timeτ with n ≤ τ ≤ N. If the initial regime isi and theinitial value of the EIA isw > 0, then the problem is to solve:

maxτ∈T0

E0

[U(Wτ)

(1 + r)τ

]= max

τ∈T0J(τ; i, w) = J∗(i, w), (2)

wherer is a discount rate andU is a utility function which is increasing and concave. Maximizing the expecteddiscounted utility as the optimization criterion, first adopted inSamuelson (1969), is quite common in the financialeconomics literature, especially in problems related to optimal control. Henceforth, we will assume that all theexpectations 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)andShiryaev (1973). In Altieri and Vargiolu (2001), an optimal stopping problem with constraint was solved throughthe dynamic programming method. The dynamic programming method in some sense is more transparent then theclassical martingale method using Snell Envelope (c.f.Neveu (1975)). It can reveal clearly the underlying financialargument. 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, ∞) andn = 0, 1, . . . , N − 1, we define recursively the following:

WN (i, w) = U(w), (3)

Wn(i, w) =

U(w), if hn(i, w) > 01

1 + rT Wn+1(i, w), if hn(i, w) = 0

, (4)


hN (i, w) = 1, (5)

hn(i, w) =[U(w) − 1

1 + rT Wn+1(i, w)

]+. (6)

Theorem 1. The optimal surrender time is given by:

τ0(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:

J∗(i, w) = J(τ0; i, w) = W0(i, w). (8)

Proof. Forn = 0, 1, . . . , N, (i, w) ∈ D andτ ∈ Tn, define:

Jn(τ; i, w) = E

[U(Wτ)

(1 + r)τ−n

∣∣∣∣ ξn = i, Wn = w

]= En

[U(Wτ)

(1 + r)τ−n

]

and

J∗n (i, w) = max

τ∈TnJn(τ; i, w) = Jn(τn(i, w); i, w) (9)

and

τ∗n(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). TheJ andJ∗ in (2) are just theJ0 andJ∗0

in (9).Whenn = N, which is the date of maturity, the investor is forced to surrender the EIA, hence:

J∗N (i, w) = U(w) = WN (i, w) and τN (i, w) = N = τ∗

N (i, w).

We now prove by induction that forn = 0, 1, . . . , N − 1,

J∗n (i, w) = Wn(i, w) (10)

and

τn(i, w) = τ∗n(i, w). (11)

Suppose that the above equations are true for somen = k + 1, . . . , N. At time k, given thatξk = i andWk = w,the investor has two choices: either surrender the EIA immediately or wait for another period. If it is surrenderedimmediately at timek, the investor can get the amountU(w); if one more period is waited, the investor may get:

1

1 + rE[J∗

k+1(ξk+1, Wk+1) | ξk = i, Wk = w]

because of the dynamic programming principle, and (s)he will choose to surrender the EIA at timeτk+1(ξk+1, Wk+1) = τ∗

k+1(ξk+1, Wk+1) by the induction hypothesis. Using the induction hypothesis again, we mayrewrite the above expression as:

1

1 + rE[J∗

k+1(ξk+1, Wk+1) | ξk = i, Wk = w] = 1

1 + rE[Wk+1(ξk+1, Wk+1) | ξk = i, Wk = w]

= 1

1 + rT Wk+1(i, w).


Therefore, the investor will surrender the EIA at timek if 2

U(w) >1

1 + rT Wk+1(i, w),

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 timeτk+1(ξk+1, Wk+1) = τ∗

k+1(ξk+1, Wk+1). Hence, we have:

τk(i, w) ={

k, if hk(i, w) > 0

τ∗k+1(ξk+1, Wk+1), if hk(i, w) = 0

= τ∗k (i, w).

Therefore,(10) and (11)are also true forn = k. This completes the induction step and finishes the proof.�

It should be remarked that we have definedhN to be identically equal to 1. In fact, it is equally possible forhN

to takeany strictly positive values. The reason is that we want to ensure that the stopping timeτ0 is bounded byN,hence belongs toT0.

4. The case of power utility

In order to gain more insights about the optimal surrender strategy and to obtain some explicit formulae, we willhenceforth restrict our attention to the case where

U(w) = wγ

γ(12)

where 0< γ < 1. The results forγ < 0 can be established similarly and are not done here.In the case of power utility, we have the explicit form of the functionWn. For n = 0, 1, . . . , T − 1 and each

i ∈ S, define

B(i) = E[(f (Ri))γ ],

C(i)N = 1,

C(i)n = max

1, B(i)

1+r

M∑j=1

pijC(j)n+1

.

Proposition 1. When the utility function is given by (12), then for (i, w) ∈ D,

Wn(i, w) = 1

γwγC(i)

n . (13)

2 The strictly inequality sign “>” could well be replaced by “≥”.


Moreover, the optimal surrender time to problem (2) is

τ0 = inf

0 ≤ n ≤ N − 1; 1 >

B(ξn)

1 + r

M∑j=1

pξnjC(j)n+1

∧ N. (14)

Here, we assume that infφ = +∞. From this proposition, we can see that the optimal surrender time does notdepend on the value of the EIA. At any time instant, once we know which regime the Markov chain is in, we candecide whether we have to stop or not.

Proof. Fix (i, w) ∈ D. We first prove(13)by induction. Whenn = N,

WN (i, w) = U(w) = wγ

γ= 1

γwγC

(i)N

asC(i)N = 1. Now suppose that(13) is true forn = k + 1, then we have

1

1 + rT Wk+1(i, w) = 1

1 + rE[Wk+1(ξk+1, Wk+1) | ξk = i, Wk = w]

= 1

1 + r

1

γE[Wγ

k+1C(ξk+1)k+1 | ξk = i, Wk = w]

= 1

1 + r

wγ

γ

M∑j=1

pijC(j)k+1E[(f (Ri

k))γ ] = wγ

γ

B(i)

1 + r

M∑j=1

pijC(j)k+1.

Together with(4), we know that

Wk(i, w) = max

(1

γwγ,

1

1 + rT Wk+1(i, w)

)= max

(1

γwγ,

wγ

γ

1

1 + rT Wk+1(i, w)

)

= 1

γwγ max

1,

B(i)

1 + r

M∑j=1

pijC(j)k+1

= 1

γwγC

(i)k .

This proves that(13) is true for all possiblen.To prove(14), we first note that forn = 0, 1, . . . , N − 1 and (i, w) ∈ D,

hn(i, w) =[U(w) − 1

1 + rT Wn+1(i, w)

]+=wγ

γ− wγ

γ

B(i)

1 + r

M∑j=1

pijC(j)n+1

+

= wγ

γ

1 − B(i)

1 + r

M∑j=1

pijC(j)n+1

+

and hence

hn(i, w) > 0 ⇐⇒ 1 >B(i)

1 + r

M∑j=1

pijC(j)n+1.

Now (14) follows. �


From(13), the maximum expected discounted utility of the surrender value, given that the initial value of theEIA is w, the initial regime isi and the number of time period to maturity isT, is proportional toC(i)

T and the optimalsurrender time also depends onC(i)

n s. We may want to explore some properties of theC(i)n , with a hope of gaining

more insights about the optimal surrender strategy. The next proposition, which describes howC(i)n changes with

n for each fixedi, tells us that the longer the time to maturity, the higher the expected discounted utility of thesurrender value the investor can achieve.

Proposition 2. For every i ∈ S, we have

C(i)0 ≥ C

(i)1 ≥ · · · ≥ C

(i)N . (15)

Proof. For eachi ∈ S,

C(i)N−1 = max

1,

B(i)

1 + r

M∑j=1

pijC(j)N

≥ 1 = C

(i)N .

Now suppose thatC(i)k ≥ C

(i)k+1 for all i ∈ S, then

C(i)k−1 = B(i)

1 + r

M∑j=1

pijC(j)k ≥ B(i)

1 + r

M∑j=1

pijC(j)k+1 ≥ C

(i)k .

This completes the proof by induction.�

The previous proposition concerns with the sizes ofC(i)n , n = 0, 1, . . . , T , for fixed i. We then want to compare

the sizes ofC(i)n , i ∈ S, for fixedn. 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 thatX andY are two random variables with distribution functionsFX andFY , respectively.If for any increasing and concave functionh, we always have

E[h(X)] ≤ E[h(Y )] (16)

whenever the expectations exist, then we sayX is dominated byY in the sense of second-order stochastic dominanceand is denoted byX ≤SSD Y or FX ≤SSD FY .

The concept of stochastic dominance is a commonly used concept in finance, see, for example,Huang andLitzenberger (1988). For a detailed mathematical treatment, we refer to the books byMuller and Stoyan (2002)andShaked and Shanthikumar (1994).

Definition 2. Suppose thatP = (pij) is anm × m stochastic matrix. It is called stochastically monotone if

m∑l=k

pil ≤m∑

l=k

pjl,

for all 1 ≤ i < j ≤ m andl = 1, 2, . . . , m.


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,

C(M)n ≤ · · · ≤ C(1)

n . (18)

Proof. We will again prove the proposition by induction. Whenn = N, Eq.(18) is true obviously sinceC(i)N = 1

for all i ∈ S by definition. Suppose that Eq.(18) is also true forn = k + 1, then for 1≤ i < j ≤ M,

C(i)k = max

(1,

B(i)

1 + r

M∑l=1

pilC(l)k+1

)≥ max

(1,

B(i)

1 + r

M∑l=1

pjlC(l)k+1

)≥ max

(1,

B(j)

1 + r

M∑l=1

pjlC(l)k+1

)

= C(j)k

where the first inequality follows fromLemma 1, while the second inequality follows from that fact that assumption(17) impliesB(M) ≤ · · · ≤ B(1). �

In order to study some qualitative properties of the optimal surrender policy, it is reasonable to assumethat there is a certain criterion to compare two different regimes. In other words, we should be able to tellwhich is better among any two regimes. Since each regime is characterized by its corresponding return dis-tribution, it is quite natural to employ the concept ofstochastic dominance orders. In fact, stochastic domi-nance orders have been used extensively in the finance and actuarial science literatures to compare risks andcompare returns. See, for example,Cheung and Yang (2004, 2005), Huang and Litzenberger (1988), Kaas etal. (1994, 2001), Muller and Stoyan (2002). Condition (17) captures the idea that the returns in all regimesare ordered sequentially: it is the worst in regimeM but the best in regime 1. For further discussion, seeCheung and Yang (2005).

It was demonstrated inCheung and Yang (2005)through some empirical figures that assuming the transi-tion matrix P to be stochastically monotone is not that unrealistic. Together with condition(17), this means thatif the current market is at a better regime, then it will have a smaller probability of switching to the worstlregimes in the next time period, for anyl. Such a financial market is deemed as regular, and is consistent with ourintuition.

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 discussioneasier, we first introduce the time-reversed version of the functionC(i)

n . For i ∈ S, define

C(i)0 = 1, (19)


C(i)n+1 = max

1,

B(i)

1 + r

M∑j=1

pijC(j)n

, n = 0, 1, 2 . . . . (20)

By Proposition 1, we know that if there aren(≥ 1) more time periods till the end of the horizon and the currentregime is at statei, then we should surrender the EIA if

1 >B(i)

1 + r

M∑j=1

pijC(j)n−1,

or continue holding the EIA otherwise. LetD(i)n be the expression on the right-hand side of the above inequality,

then our surrender decision is closely related to size ofD(n)i ; in particular, whether it is greater than 1 or not. Based

on the recursive formula forC(i)n , we can obtain the corresponding recursive formula forD(i)

n :

D(i)1 = B(i)

1 + r, (21)

D(i)n+1 = B(i)

1 + r

M∑j=1

pij max(1, D(j)n ), n = 1, 2, . . . . (22)

The monotonic properties ofD(i)n are summarized in the next lemma.

Lemma 2.

1. For each i ∈ S,

D(i)1 ≤ D

(i)2 ≤ D

(i)3 ≤ · · · . (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 andi = 2, 3, . . . , M,

D(M)n ≤ · · · ≤ D(1)

n . (25)

and

D(i−1)n − D(i)

n ≤ D(i−1)n+1 − D

(i)n+1. (26)

Proof. The proofs of(23) and (25)are similar to that ofPropositions 2 and 3, respectively, and are omitted. Toprove(26), we first show that it is true whenn = 1. From(25), we haveD(M)

1 ≤ · · · ≤ D(1)1 . Because the function

max(1, ·) is increasing (and convex), we have

max(1, D(M)1 ) ≤ · · · ≤ max(1, D(1)

1 ).

Since the transition matrixP is stochastically monotone, fori = 2, 3, . . . , M,


D(i−1)2 − D

(i)2 = B(i−1)

1 + r

M∑j=1

pi−1j max(1, D(j)1 ) − B(i)

1 + r

M∑j=1

pij max(1, D(j)1 ) ≥ B(i−1)

1 + r

M∑j=1

pij max(1, D(j)1 )

− B(i)

1 + r

M∑j=1

pij max(1, D(j)1 ) = [D(i−1)

1 −D(i)1 ]

M∑j=1

pij max(1, D(j)1 ) ≥ [D(i−1)

1 −D(i)1 ]

M∑j=1

pij

= D(i−1)1 − D

(i)1

where the last inequality follows from the fact thatD(i−1)1 − D

(i)1 ≥ 0 by assumption(24). Now suppose that(26)

holds for somen = k − 1 ≥ 1. We first note that if

max(1, D(j−1)k ) − max(1, D(j−1)

k−1 ) ≥ max(1, D(j)k ) − max(1, D(j)

k−1) (27)

is true forj = 2, 3, . . . , M, then(26)also holds forn = k because

[D(i−1)k+1 − D

(i)k+1] − [D(i−1)

k − D(i)k ] = [D(i−1)

k+1 − D(i−1)k ] − [D(i)

k+1 − D(i)k ]

= B(i−1)

1 + r

M∑j=1

pi−1j[max(1, D(j)k ) − max(1, D(j)

k−1)]

− B(i)

1 + r

M∑j=1

pij[max(1, D(j)k ) − max(1, D(j)

k−1)]

≥ B(i−1)

1 + r

M∑j=1


k−1)]

− B(i)

1 + r

M∑j=1


k−1)]

= (D(i−1)1 − D

(i)1 )

M∑j=1


k−1)] ≥ 0.

Eq.(27)can be shown as follows:

[max(1, D(j−1)k ) − max(1, D(j−1)

k−1 )] − [max(1, D(j)k ) − max(1, D(j)

k−1)]

≥ max(1, D(j−1)k + (D(j−1)

k−1 − D(j)k−1) − (D(j−1)

k − D(j)k ))

− max(1, D(j−1)k−1 ) − max(1, D(j)

k ) + max(1, D(j)k−1)

= [max(1, D(j−1)k−1 − D

(j)k−1 + D

(j)k ) − max(1, D(j)

k )] − [max(1, D(j−1)k−1 ) − max(1, D(j)

k−1)] ≥ 0

where the first inequality follows from the induction hypothesis and that the map max(1, ·) is increasing, while thelast inequality follows from the convexity of such map.�

From now on, we will always assume that the transition probability matrixP is stochastically monotone andthat(17) is true. Such assumptions are made to ensure that the market possesses a certain amount of regularity sothat further analysis can be done. In this case, the return credited to the EIA is the best at regime 1 but the worst at


regimeM. Since(17) is true, we have

B(1) ≥ · · · ≥ B(M),

or equivalently

D(1)1 ≥ · · · ≥ D

(M)1 . (28)

Recall that if there is only one time period left, then the investor should surrender the EIA at regimei if D(i)1 is

strictly less than 1. Roughly speaking, we can interpretD(i) as a measure of the relative return in regimei. If it islarge, 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 informationabout the random variablef (Ri) that is relevant here is its momentB(i), if B(j) = B(j+1) (and henceD(j)

1 = D(j+1)1 )

for somej, then we can group the two regimes together and consider them as a single regime. It is easy to see thatthe augmented transition probability matrix, whose dimension is (M − 1) × (M − 1) now, is again stochasticallymonotone.

To analyze the optimal behavior of the investor when the time to maturity is long, we need to investigatethe asymptotic behavior of the functionD(i)

n whenn → ∞ for eachi ∈ S. Three different scenarios have to beconsidered, according to whether all the terms in(28) are less than 1, greater than 1 or some of them are less than1 but some of them are greater than 1.

Proposition 4. If 1 > D(1)1 > · · · > D

(M)1 , then for each i ∈ S,

D(i)1 = D

(i)2 = D

(i)3 = · · · .

Proof. If 1 > D(1)1 > · · · > D

(M)1 , then for anyi ∈ S,

D(i)2 = B(i)

1 + r

M∑j=1

pij max(1, D(j)1 ) = B(i)

1 + r

M∑j=1

pij = D(i)1 < 1

and if 1> D(1)n > · · · > D(M)

n , then for anyi ∈ S,

D(i)n+1 = B(i)

1 + r

M∑j=1

pij max(1, D(j)n ) = B(i)

1 + r

M∑j=1

pij = D(i)1 < 1.

This finishes the proof. �In particular, this proposition says that 1> D(i)

n for all i ∈ S and alln ≥ 1. This means that if it is optimal tosurrender the EIA when there is 1 period left regardless of which regime the Markov chainξ is staying at, then theEIA holder should surrender the EIA immediately at time zero regardless of which regimeξ is staying in. In otherwords, the EIA is not worth buying. The reason is that under the stated hypothesis, the relative return of the EIA inall the regimes are not attractive enough.

Proposition 5. If D(1)1 > · · · > D

(M)1 ≥ 1, then for i ∈ S and n ≥ 1,

D(i)n ≥ 1.


Proof. The proof follows directly from part 1 ofLemma 2, which states thatD(i)n is monotonically increasing inn

for eachi ∈ S. �

This proposition means that if it is optimal to continue holding the EIA when there is 1 period left regardlessof what the current regime is, then it is also optimal to continue holding the EIA at any time, at any regime. Thisimplies 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, isgood 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 inter-mediate one:

D(1)1 > · · · > D

(l)1 ≥ 1 > D

(l+1)1 > · · · > D

(M)1 . (29)

This means that at regimes 1, 2, . . . , l, the return credited to the EIA in the coming period would be “high”, but atregimesl + 1, . . . , M, the return would be “low”. When there is only one period left, we should continue holdingthe EIA if the Markov chain{ξ} is at regimes 1, 2, . . . , l; otherwise, we should surrender the EIA. As in the previousthe cases, we would like to know whether the limits limn→∞ D(i)

n , i ∈ S, are greater than one or not. To make ourdiscussion easier and more intuitive, we will use the following terminology: a regime is said to beBAD, (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, regimei is BAD (respectively, GOOD) when there aren periods left ifD(i)

n < 1 (respectively ≥ 1). Using this terminology and under condition(29), we may classifyregimes 1, . . . , l as “GOOD” and regimesl + 1, . . . , M as “BAD”, when there is only one time period left.

From part 1 ofLemma 2, we know that fori = 1, 2, . . . , l,

D(i)n ≥ 1, ∀n ≥ 1.

This means that no matter how many time periods are left, it is always optimal to continue holding the EIA as longas we are at one of the initially3 GOOD regimes. In other words,GOOD regimes remainGOOD forever. Frompart 2 ofLemma 2, it is clear that at any time, if it is optimal to continue holding the EIA at a certain regime, thenit is also optimal to continue holding the EIA at that time if the Markov chain is at a better regime. On the otherhand, 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 initiallyBAD regime. The argument is that if we havesufficient time, then the Markov chainξ will eventually switch to one of theGOOD regimes by the irreducibility ofthe Markov chain. Hence it may be worth holding the EIA for a while. To verify whether this intuition is correct, itis equivalent to determining whether the limits

L(i) �= limn→∞ D(i)

n , i ∈ S

are greater than 1 or not. As{D(i)n }n≥1 is an increasing sequence, limitL(i) exists for eachi, though it may be infinite.

From the recursive equation that definesD(i)n , the limitsL(·) are either all finite or all equal to+∞. If they are all

finite, then we also haveL(1) ≥ · · · ≥ L(M) by (25).From(21) and (22), we know that the limitsL(i) depend only on the transition probability matrixP and the values

of D(i)1 , i ∈ S. The following proposition provides a simple sufficient condition that guarantees the infiniteness of

all the limitsL(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.


Proposition 6. Assume that (29) is true. If there exists an j ∈ {1, 2, . . . , l} such that

pjj ≥ 1

D(j)1

, (30)

then L(1) = · · · = L(M) = +∞.

Proof. Assume thatL(i) < ∞ for all i and there is aj ∈ {1, 2, . . . , l} such thatpjj ≥ 1D

(j)1

. In (22), lettingn → ∞on both sides gives:

L(j) = D(j)1

M∑n=1

pjn max(1, L(n)). (31)

for eachj ∈ S. As ∞ > L(1) ≥ · · · ≥ L(M), there existsk ∈ {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∑n=1

pjnL(n) − L(j)

D(j)1

= −M∑

n=l+k+1

pjn,

which is impossible because the right-hand side is negative while the left-hand side is positive since all the coefficientsof theL(n)s are positive by assumption(30). �

Roughly speaking,Proposition 6says that when the product of the probability of the Markov chain staying ata GOOD state,i, and the expected return credited to the EIA account when Markov chain is at regimei is large,all states will eventually becomeGOOD states. In general, it is very difficult to obtain sufficient and necessaryconditions forL(i) to be greater than 1 or not. In what follows, we will demonstrate numerically that the intuitiondiscussed above is not always true. That means it may happen that all the limitsL(l+1), . . . , L(M) are less than 1,or the limitsL(l+1), . . . , L(l+k) are greater than one butL(l+k+1), . . . , L(M) are less than 1. We will demonstratevarious possibilities about the values of the limitsL(1), . . . , L(M). As noted before, we will only focus on that casethatD(1)

1 > · · · > D(l)1 ≥ 1 > D

(l+1)1 > · · · > D

(M)1 .

Suppose thatM = 3, r = 0.1 and the transition probability matrixP is given by

P =

0.6 0.3 0.1

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 differentpossibilities about the sizes of the limitsL(i)s. In all three cases,D(1)

1 is greater than 1 and hence regime 1 is initially

aGOOD regime; on the other hand, bothD(2)1 andD

(3)1 are less than 1 and hence regimes 2 and 3 areBAD regimes

initially.


1. (D(1)1 , D

(2)1 , D

(3)1 ) = (1.3, 0.9, 0.8) =⇒ (L(1), L(2), L(3)) = (+∞, +∞, +∞). This means that regimes 2 and 3

becomeGOOD eventually.2. (D(1)

1 , D(2)1 , D

(3)1 ) = (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 remainsGOOD forever. However, regimes 2 and 3 remainBAD forever as bothL(2) andL(3) are less than 1.

3. (D(1)1 , D

(2)1 , D

(3)1 ) = (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 remainsBAD forever. However, the limitL(2) is greater than 1, meaning that althoughregime 2 initially isBAD, it becomesGOOD eventually.

6. Conclusion

In this paper, we have studied the optimal surrender time for equity-linked products, like the equity-indexedannuity, in a discrete-time model with regime switching. Under the power utility, we have obtained the closedform solution for the optimal surrender time, and have investigated its properties. In order to do so, we introducedthe concepts of stochastic dominance and stochastic monotonicity which turn out carry some nature financialinterpretations. We also studied the properties of the optimal stopping policy when the investment time period islong. 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 muchto surrender at any time instant. The objective becomes maximizing the sum of expected discounted utility ofthe surrender values. This problem essentially becomes a dynamic intertemporal consumption problem. A similarmodel, together with the presence of default risk, was studied inCheung and Yang (2005).

In the current study, it is assumed that the investment horizonN is deterministic. However,N could be randomin reality. For example,N may represents the remaining lifetime of the investor. LetTx be the future lifetime of theinvestor who is at agex 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 = Wnf (Rξnn )1{Tx≥n+1}.

The random behavior ofN can be specified by the conditional survival probabilities, see, for example,Bowers etal. (1997). Optimal surrender policy can be obtained similarly. However, some of the monotonic properties aboutthe optimal policy revealed in Section5 will be lost.

One possible direction of further investigation is to explore necessary and sufficient conditions forL(i) to begreater than 1. This is a hard problem and meaningful, as indicated in our example, it is not always true that theBAD regimes will becomeGOOD eventually. Another future research problem is to investigate the continuous timemodel. Some similar conclusions can be expected.

Acknowledgments

This work was supported by Research Grants Council of HKSAR (Project No.: HKU 7239/04H). The authorswish to thank the anonymous referee for helpful comments and suggestions.

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