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Possibilistic individual multi-period consumption–investment models
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ScienceDirect Fuzzy Sets and Systems 274 (2015) 47–61 www.elsevier.com/locate/fss
Possibilistic individual multi-period consumption–investment models Yonggang Li, Peijun Guo ∗ Faculty of Business Administration, Yokohama National University, 79-4 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan Received 29 September 2013; received in revised form 6 January 2015; accepted 10 January 2015 Available online 14 January 2015
Abstract In this paper, we consider a multi-period consumption–investment problem with partially known information for an individual. We emphasize the fact that for an individual there is one and only one result will appear for his/her consumption–investment decision in each period. Different from the existing multi-period consumption–investment models which basically maximize the expected utility, we utilize the one-shot decision theory for individual multi-period consumption–investment problems. In each period, the individual chooses one state (scenario) for each strategy with considering the satisfaction of the outcome and its possibility. The selected state is called the focus point. The multi-period consumption–investment decision models with twelve types of focus points are built to maximize the sum of discounted consumption over the whole process (lifetime) for twelve types of individuals and economic insights are gained by the theoretical analysis. The proposed models are scenario-based decision models which provide a fundamental alternative to analyze individual multi-period consumption–investment behavior. © 2015 Elsevier B.V. All rights reserved. Keywords: Possibility distribution; Decision making; One-shot decision theory; Scenario-based decision theory; Focus point; Multistage decision problem; Dynamic programming; Consumption–investment problem
1. Introduction As an important research topic in economic literature, multi-period consumption–investment problems have received increasing attention since 1960s. A multi-period consumption–investment problem can be simply described as follows: there is an investor with an initial wealth which should be allocated to consumption and a portfolio for investment in the first period. The portfolio will yield uncertain new wealth in the next period and the probability distribution of returns of securities is known. In the next period, the investor should allocate the new wealth to consumption and investment again. The consumption–investment decisions are repeated in each period until the end of life. The multi-period consumption–investment problems were initially studied by Mossin [18] and Samuelson [22], in which the objective of decision making was to maximize the expected utility of consumption. Merton [15,17] in* Corresponding author.
E-mail addresses: [email protected] (Y. Li), [email protected] (P. Guo). http://dx.doi.org/10.1016/j.fss.2015.01.005 0165-0114/© 2015 Elsevier B.V. All rights reserved.
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troduced the models under a continuous-time framework and the stochastic optimal control approach was used in his models. General equilibrium theory for the pricing of risky assets has been used as an extremely powerful tool in the existing researches [3,14,16]. Pliska [20] introduced the discrete time model. Cadenillas [1] gave the detailed survey for consumption–investment problems. Recently, Karatzas and Wang [11] studied an optimal dynamic investment problem in the case where the uncertain time horizon was a stopping time of asset price filtration. Cheung and Yang [2], Sotomayor and Cadenillas [23] investigated dynamic investment–consumption problems in a regime-switching environment. Pliska and Ye [21] considered optimal insurance and consumption rules for a wage earner whose lifetime was random and explicit solutions were found for the family of CRRA utilities. Li and Tan [13] studied the optimality of the consumption–investment problem that reflected the mortality uncertainty. Nielsen and Steffensen [19] investigated the optimal investment, and life insurance strategies under the minimum and maximum consumption. Steffensen [24] examined the optimal consumption and investment rules, particularly in the case where the relative risk aversion with respect to consumption was increasing with age. Kraft et al. [12] studied the optimal consumption-portfolio decision of an investor with recursive preferences of Epstein–Zin type in an incomplete market. Su and Guan [25] utilized multi-objective programming to solve the single period consumption–investment problem with fuzzy coefficients. Until now, researchers considered and solved multi-period consumption–investment problems within the probabilistic framework where the uncertainty of the portfolios’ return was characterized by the probability distribution, and different kinds of utility functions were examined. In fact, an individual’s decision on consumption and investment is a one-time and never-return decision. It means a person cannot return to the past and his/her decision on investment in each period can achieve one and only one result. Guo [7] initially established the one-shot decision theory for dealing with one-shot decision problems. The one-shot decision making problems have been researched in the papers [4–6,8,10]. Instead of the lottery-based choices as in other existing decision theories under uncertainty, the one-shot decision theory provides a scenario-based choice which fits the characteristic of one-shot decision problems. In the one-shot decision theory, the decision process encompasses two steps. The first step is to identify which state should be taken into account for each alternative with considering the possibility of the state and the satisfaction of an outcome. The states selected are called focus points. The second step is to evaluate the alternatives based on the focus points to obtain the optimal alternative. Based on the one-shot decision theory, Guo and Li studied multistage one-shot decision problems in the research [9] where a general approach to multistage one-shot decision making is given and the one-shot optimal stopping problem is analyzed. In this paper, we apply multistage one-shot decision approaches to consumption–investment problems with considering that an individual has one and only one chance to allocate his/her wealth to consumption and investment, and choose a suitable investment strategy in each period. In the proposed model, an individual takes into account how to allocate the initial wealth and determine a consumption–investment plan to maximize the sum of possible discounted consumption over the whole process (lifetime). This paper extends the existing literature in three important dimensions. First, in the existing multistage consumption–investment models, the uncertainty is characterized by the probability distribution whereas we utilize the possibility distribution to describe the uncertainty. There might be an acute lack of information for obtaining the probability distribution in multistage consumption–investment problems. The possibility distribution seems more feasible for such decision problems. To the best of our knowledge, our research is the first which studies multi-period consumption–investment problems with possibilistic information, which is an innovative study on solving such an important economic problem. Second, different from the existing multistage consumption–investment models which utilize the subjective expected utility theory (SEU), our models is based on the one-shot decision theory (OSDT). The SEU-based models use different utility functions to reflect the attitude of a person about risk whereas in the OSDTbased models choosing which type of focus point (scenario) to make a decision reflects the attitude of this person about uncertainty. Since the individual consumption–investment is a typical one-time and never-return decision, the proposed models fit the one-time feature of the individual behavior and economic insights are gained by the theoretical analysis. Third, in possibilistic multistage consumption–investment models, we consider multidimensional states and decisions. It extends the focus point and the decision in the classic one-shot decision theory from the one-dimensional to the multidimensional case. The remainder of the paper is organized as follows. In Section 2, the one-shot decision theory is briefly introduced with considering multidimensional states and alternatives. In Section 3, the approach to multi-period consumption– investment problems with the one-shot decision theory is proposed. In Section 4, the theoretical results are given. Finally, concluding remarks are made in Section 5.
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2. Introduction to the one-shot decision theory Guo [7] initially proposed the one-shot decision theory to solve one-shot decision problems which are for the situations where a decision maker has only one chance to make a decision under uncertainty. In this section we consider the situation that the state and the alternative are multi-dimensional vectors. Suppose the set of an alternative a is A and the set of a state x is S. a and x are vectors in R n and R m , respectively. π(x) is a function defined on S, which is used to characterize the degree to which a state x is to occur. Definition 1. Given a function π : S → [0, 1] if maxx∈S π(x) = 1, then π(x) is called a possibility distribution, where S is the sample space, π(x) is the possibility degree of x. A payoff is a result for the combination of an alternative a and a state x, denoted as v(x, a). The satisfaction level of a decision maker for a payoff can be expressed by a satisfaction function, as defined below. Definition 2. Denote the set of payoffs as V . The function u : V → [0, 1] is called a satisfaction function if it satisfies u(v1 ) > u(v2 ) for v1 > v2 where v1 , v2 ∈ V . Satisfaction function is used to represent the relative position of the payoff obtained from an alternative by the normalizing operator. Because the payoff is the function of x and a, we also can write the satisfaction function as u(x, a). Different from the existing decision theories which are lottery-based, the one-shot decision theory argues that a person makes a one-shot decision based on some particular state which is the most appropriate scenario for him/her while considering the satisfaction level incurred by it and its possibility degree. The one-shot decision process encompasses two steps. The first step is to identify which state should be taken into account for each alternative with considering the possibility of the state and the satisfaction of an outcome. The states focused are called focus points. The second step is to evaluate the alternatives based on the focus points to obtain the optimal alternative. Considering all combination of satisfaction and possibility, twelve types of focus points are provided in the following to help a decision maker in finding out states (scenarios) which are adequate for him/her. ∗ x1α (a) ∈ arg max u(x, a) where X≥α = x|π(x) ≥ α , (1) x∈X≥α
where the given parameter α is a level used to distinguish whether the possibility degree is evaluated as ‘high’ by a decision maker. X≥α = {x|π(x) ≥ α} is the set of states with high possibility degrees and maxx∈X≥α u(x, a) is ∗ ∗ the highest satisfaction level amongst all x ∈ X≥α which can be attained when x = x1α (a) so that x1α (a) is a state ∗ (scenario) with the highest satisfaction and a high possibility to occur. x1α (a) is called as Type I focus point of an alternative a. ∗ x2α (a) ∈ arg min u(x, a) where X≥α = x|π(x) ≥ α . (2) x∈X≥α ∗
∗
Eq. (2) shows that x2α (a) is a state (scenario) with the lowest satisfaction and a high possibility to occur. x2α (a) is called as Type II focus point of an alternative a. ∗ x3α (a) ∈ arg max u(x, a) where X≤α = x|π(x) ≤ α . (3) x∈X≤α ∗
∗
Eq. (3) implies that x3α (a) is a state (scenario) with the highest satisfaction and a low possibility to occur. x3α (a) is called as Type III focus point of an alternative a. ∗ x4α (a) ∈ arg min u(x, a) where X≤α = x|π(x) ≤ α . (4) x∈X≤α
∗
∗
It is clear from (4) that x4α (a) is a state (scenario) with the lowest satisfaction and a low possibility to occur. x4α (a) is called as Type IV focus point of an alternative a. ∗ x5β (a) ∈ arg max π(x) where X≥β (a) = x|u(x, a) ≥ β . (5) x∈X≥β (a)
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The given parameter β is the level to distinguish whether the satisfaction level is evaluated as ‘high’ by a decision ∗ maker. It follows from (5) that x5β (a) is a state (scenario) with the highest possibility to occur and a high satisfaction. ∗
x5β (a) is called as Type V focus point of an alternative a. ∗ x6β (a) ∈ arg min π(x) where X≥β (a) = x|u(x, a) ≥ β .
(6)
x∈X≥β (a)
∗
∗
From (6), we know that x6β (a) is a state (scenario) with the lowest possibility to occur and a high satisfaction. x6β (a) is called as Type VI focus point of an alternative a. ∗ x7β (a) ∈ arg max π(x) where X≤β (a) = x|u(x, a) ≤ β . (7) x∈X≤β (a)
∗
Eq. (7) makes it clear that x7β (a) is a state (scenario) with the low satisfaction and the highest possibility to occur. ∗
x7β (a) is called as Type VII focus point of an alternative a. ∗ x8β (a) ∈ arg min π(x) where X≤β (a) = x|u(x, a) ≤ β .
(8)
x∈X≤β (a)
∗
∗
Obviously, x8β (a) represents a state (scenario) with the low satisfaction and the lowest possibility to occur. x8β (a) is called as Type VIII focus point of an alternative a. ∗ x9 (a) ∈ arg max min π(x), u(x, a) . (9) x∈S
∗
It follows from (9) that x = x9 (a) maximizes g(x, a) = min[π(x), u(x, a)]. Because min[π(x), u(x, a)] represents the lower bound of [π(x), u(x, a)], increasing min[π(x), u(x, a)](maxx∈S min[π(x), u(x, a)]) will increase the possibility degree and the satisfaction level simultaneously. Therefore, arg maxx∈S min[π(x), u(x, a)] represents the state that has ∗ the higher possibility degree and the higher satisfaction level. x9 (a) is called as Type IX focus point of an alternative a. ∗ x10 (a) ∈ arg min max π(x), u(x, a) . (10) x∈S
∗
Eq. (10) shows that x = x10 (a) minimizes h(x, a) = max[π(x), u(x, a)]. Because max[π(x), u(x, a)] represents the upper bound of [π(x), u(x, a)], decreasing max[π(x), u(x, a)](minx∈S max[π(x), u(x, a)]) will decrease the possibility degree and the satisfaction level simultaneously. Therefore, arg minx∈S max[π(x), u(x, a)] is for seeking the state ∗ that has the lower possibility degree and the lower satisfaction level. x10 (a) is called as Type X focus point of an alternative a. ∗ x11 (a) ∈ arg min max 1 − π(x), u(x, a) . (11) x∈S
∗
Similarly, we know x11 (a) represents a state (scenario) with the higher possibility degree and the lower satisfaction ∗ level. x11 (a) is called as Type XI focus point of an alternative a. ∗ x12 (a) ∈ arg min max π(x), 1 − u(x, a) . (12) x∈S
∗
It is easy to understand that x12 (a) is a state (scenario) with the lower possibility degree and the higher satisfaction ∗ level. x12 (a) is called as Type XII focus point of an alternative a. In summary, (1)–(12) are used to seek the focused states featured by the highest satisfaction and a high possibility, the lowest satisfaction and a high possibility, the highest satisfaction and a low possibility, the lowest satisfaction and a low possibility, the highest possibility and a high satisfaction, the lowest possibility and a high satisfaction, the highest possibility and a low satisfaction, the lowest possibility and a low satisfaction, the higher possibility degree and the higher satisfaction level, the lower possibility degree and the lower satisfaction level, the higher possibility degree and the lower satisfaction level, the lower possibility degree and the higher satisfaction level (see Tables 1–3). Choosing which kind of focus point to make a decision is dependent on the attitude of the decision maker on uncertainty and benefit. The relationship amongst twelve types of focus points is given in the paper [7]. In real-world investing, Type IX focus point is used for characterizing an active investor whereas Type X focus point is used for describing
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Table 1 The features of focus points (Types I–IV).
Possibility degree Satisfaction level
Type I
Type II
Type III
Type IV
High The highest
High The lowest
Low The highest
Low The lowest
Table 2 The features of focus points (Types V–VIII).
Possibility degree Satisfaction level
Type V
Type VI
Type VII
Type VIII
The highest High
The lowest High
The highest Low
The lowest Low
Table 3 The features of focus points (Types IX–XII).
Possibility degree Satisfaction level
Type IX
Type X
Type XI
Type XII
Higher Higher
Lower Lower
Higher Lower
Lower Higher
an apprehensive investor because although some scenario occurs with a lower possibility degree, it is still emphasized by an investor as it can lead to a larger loss (lower satisfaction level). Type XI focus point is used for a passive investor whereas Type XII focus point is used for a daring investor because even if some scenario occurs with a lower possibility degree, a higher gain (higher satisfaction level) may lure an investor to consider such a situation. For one alternative, more than one states might exist as one type of focus point. We denote the sets of twelve types of focus points of the alternative a as X1α (a), X2α (a), X3α (a), X4α (a), X5β (a), X6β (a), X7β (a), X8β (a), X9 (a), X10 (a), X11 (a), and X12 (a), respectively. It should be noted that X3α (a) and X4α (a) are empty sets when X≤α = ∅; X7β (a) and X8β (a) are empty sets when X≤β (a) = ∅. For the sake of simplification, in this paper we assume that X3α (a), X4α (a), X7β (a) and X8β (a) are nonempty sets for any alternative a. In one-shot decision, the second step is to obtain the optimal solution based on the focus points of all alternatives. The decision maker contemplates that the focus points are the most appropriate states (scenarios) for him/her and the decision maker chooses one alternative which can bring about the best consequence once the focus point (scenario) comes true. Since there are twelve types of focus points, there are twelve types of optimal alternatives, as shown below. ∗ ∗ Type I optimal alternative: a1 (α) ∈ arg max v x1α (a), a . (13) a∈A
∗ ∗ Type II optimal alternative: a2 (α) ∈ arg max v x2α (a), a .
(14)
∗ ∗ Type III optimal alternative: a3 (α) ∈ arg max v x3α (a), a .
(15)
∗ ∗ Type IV optimal alternative: a4 (α) ∈ arg max v x4α (a), a .
(16)
a∈A
a∈A
a∈A
Clearly, focus points in X1α (a), X2α (a), X3α (a) or X4α (a) have the same payoff. ∗ ∗ Type V optimal alternative: a5 (β) ∈ arg max ∗ min v x5β (a), a . a∈A x5 (a)∈X5 (a) β β
(17)
The maximin operator is needed for the cases where multiple focus points of an alternative a exist. It reflects the conservative attitude of a decision maker.
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Y. Li, P. Guo / Fuzzy Sets and Systems 274 (2015) 47–61 ∗
Type VI optimal alternative: a6 (β) ∈ arg max
min
a∈A x6∗ (a)∈X6 (a) β β
∗
Type VII optimal alternative: a7 (β) ∈ arg max
min
∗ v x6β (a), a .
a∈A x7∗ (a)∈X7 (a) β β
∗
Type VIII optimal alternative: a8 (β) ∈ arg max a∈A
∗
Type IX optimal alternative: a9 ∈ arg max
min
∗ x8β (a)∈X8β (a)
max
9∗
∗ v x7β (a), a .
a∈A x (a)∈X9 (a)
∗ v x8β (a), a .
∗ v x9 (a), a .
(18)
(19)
(20)
(21)
The maximax operator is used for the cases where multiple Type IX focus points of an alternative a exist. It reflects the aggressive attitude of a decision maker. ∗ ∗ Type X optimal alternative: a10 ∈ arg max ∗ min v x10 (a), a . (22) a∈A x10 (a)∈X10 (a)
∗
Type XI optimal alternative: a11 ∈ arg max
min
a∈A x11∗ (a)∈X11 (a)
∗
Type XII optimal alternative: a12 ∈ arg max
12∗
a∈A x
∗ v x11 (a), a .
max
∗ v x12 (a), a .
(23) (24)
(a)∈X12 (a)
It should be noted that the payoff v(x, a) is used for obtaining the optimal alternative instead of the satisfaction u(x, a) in (13)–(24). 3. Multi-period consumption–investment problems with the one-shot decision theory Suppose the initial wealth of an individual is w0 and the length of the process (lifetime) is L. Let us consider how he/she allocates the wealth to make him/her have a largest amount of discounted consumption during the whole process. In each period k, there are a savings account S0 and N securities Sj (1 ≤ j ≤ N ) available for investment. The return rate vector of them at period k + 1 is rk+1 = [rk+1,0 , rk+1,1 , . . . , rk+1,N ]t where rk+1,0 > 0 is the interest rate of the savings account and rk+1,j (1 ≤ j ≤ N ) is the return rate of the j th security. For an individual there is one and only one chance for him/her to allocate the wealth and choose a suitable investment strategy in each period. The individual multi-period consumption–investment decision-making problem is to determine his/her consumption and investment strategy to maximize the sum of the possible discounted consumption over the whole process (lifetime). Clearly, in each period there is a trade-off between consumption and investment, the investor needs to choose a suitable consumption–investment plan. For any period k, wk is the wealth and ck ∈ [0, 1] is the consumption ratio, that is, the proportion of the total wealth devoted to the consumption. Therefore, the amount of wealth consumed by the investor is ck wk and the amount of wealth for investment is wk (1 − ck ). An investment strategy at period k is expressed as ak = [ak,0 , ak,1 , · · · , ak,N ]t where ak,0 is the proportion of the total wealth devoted to the savings account and ak,j (1 ≤ j ≤ N ) is the proportion of the total wealth devoted to the j th security. By the definition of ck and ak , we have ck + N j =0 ak,j = 1, that N is, when the strategy ak is determined, ck can be expressed as ck = ck (ak ) = 1 − j =0 ak,j . Thus, the problem for determining the consumption and investment strategy is reduced to choose a suitable investment strategy which can maximize the total discounted consumption over the whole process (lifetime). We assume that in the final period the individual will consume all the wealth, that is cLwL = wL , cL = 1, where wL is the wealth at period L. The set of all consumption ratios available at period k is expressed as Ck = {ck }. The set of the investment strategies available at period k is Ak = {ak }. In period k, there is one and only one chance for the investor to choose a consumption ratio from Ck and an investment strategy from Ak . By the definition of Ck and Ak , c we know akk is a N + 2 dimensional vector. The rates of returns on the securities at period k + 1 are uncertain, suppose the set of the possible rates of returns is Rk+1 = {rk+1 }. The possibility distribution of Rk+1 is defined as follows.
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Definition 3. The possibility distribution of the return ratio vectors is a function πk+1 : Rk+1 → [0, 1] satisfying maxrk+1 πk+1 (rk+1 ) = 1 and πk+1 (rk+1 ) is the possibility degree of rk+1 in period k + 1. The initial wealth of the individual is w0 . Clearly, there is one and only one chance for this person to allocate his wealth into consumption and investment in each period. By using multi-stage one-shot decision approaches, we can obtain the consumption–investment plan according to his/her preference of the state. First, let us consider the last decision in period L − 1. Giving aL−1 and rL and regarding the wealth wL−1 as a unit, the total amount of consumption in the last two periods is cL−1 + λL aL−1 · (1 + rL ) where λL is the discount factor, 1 = [1, 1, . . . , 1]t and aL−1 · (1 + rL ) is the inner product of aL−1 and 1 + rL . The set of all possible total amounts of consumption in the last two periods is given as GL = gL cL−1 (aL−1 ), aL−1 , rL |aL−1 ∈ AL−1 , rL ∈ RL = cL−1 + λL aL−1 · (1 + rL )|aL−1 ∈ AL−1 , rL ∈ RL . (25) Since the objective is to choose the consumption–investment strategy for maximizing the discounted consumption over periods L − 1 and L, the satisfaction level of the strategy in L − 1 is determined by the total amount of consumption in periods L − 1 and L. We give the satisfaction function for periods L − 1 and L as follows: Definition 4. The function uL : GL → [0, 1] is called a satisfaction function for the period L − 1 and L if it satisfies uL (gL1 ) > uL (gL2 ), for any gL1 > gL2 , gL1 , gL2 ∈ GL . From the definition of uL we can see the higher the total discounted consumption, the higher the satisfaction level. For the same value of cL−1 , the satisfaction function is increasing with the amount consumed in period L. Therefore, the investor consumes all the wealth in the final period L is rational. Using the multi-stage one-shot decision approach, we have the following steps for the last decision period L − 1. c . Since cL−1 = 1 − N Step 1: Seek the focus points for each aL−1 j =0 aL−1,j , cL−1 is the function of aL−1 , denoted L−1 c as cL−1 (aL−1 ). Seeking the focus points of aL−1 is reduced into finding out the focus points for each investment L−1 strategy aL−1 ∈ AL−1 . The focus point of aL−1 , denoted as r∗L (aL−1 ) can be obtained as r∗L (aL−1 ) ∈ arg F πL (rL ), uL gL (cL−1 , aL−1 , rL ) , (26) where F (·) is a general form representing the approaches for seeking one of twelve types’ focus points. For example, F (·) = maxr min[πL , uL ] corresponds to the Type IX focus point of aL−1 . c∗ , that is, the optimal investment strategy a∗L−1 , Step 2: Find out the optimal consumption–investment strategy aL−1 ∗ L−1
which is determined as follows: a∗L−1 ∈ arg
max
aL−1 ∈AL−1
cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) ,
where λL is a discount factor. Denote f1 = max cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) , aL−1 ∈AL−1
(27)
(28)
which is the optimality equation in period L − 1. f1 represents the maximum ratio of the wealth wL−1 which is available for consumption in the last two periods. If the wealth wL−1 and the possibility distribution of the rates of return in period L are given, we can obtain the total discounted consumption in periods L − 1 and L, which equals f1 times wL−1 . Next, we consider the decision in period L − n (n ≥ 2). Set GL−n+1 = gL−n+1 cL−n (aL−n ), aL−n , rL−n+1 |aL−n ∈ AL−n , rL−n+1 ∈ RL−n+1 = cL−n + λL−n+1 aL−n · (1 + rL−n+1 ) fn−1 |aL−n ∈ AL−n , rL−n+1 ∈ RL−n+1 , (29) where λL−n+1 is a discount factor, fn−1 is the optimality equation in period L − n + 1.
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Table 4 The possibility distribution of the return rate of the stock in period 1. Return rate r1,1
0.8r1,0 − 0.2
r1,0
1.3r1,0 + 0.3
1.5r1,0 + 0.5
Possibility degree π1
0.4
0.5
1
0.6
Table 5 The possibility distribution of the return rate of the stock in period 2. Return rate r2,1
0.7r2,0 − 0.3
0.9r2,0 − 0.1
1.2r2,0 + 0.2
1.5r2,0 + 0.5
Possibility degree π2
0.3
0.6
1
0.75
Definition 5. The satisfaction function in period L − n + 1 is given as uL−n+1 : GL−n+1 → [0, 1] satisfying 1 2 1 2 1 2 uL−n+1 (gL−n+1 ) > uL−n+1 (gL−n+1 ) for any gL−n+1 > gL−n+1 , gL−n+1 , gL−n+1 ∈ GL−n+1 . From the definition of uL−n+1 we can see the higher the total discounted consumption, the higher the satisfaction level. We have the following steps for thedecision in period L − n. c Step 1: Find out the focus points of aL−n , that is, the focus points of aL−n ∈ AL−n , denoted as r∗L−n+1 (aL−n ). L−n ∗ rL−n+1 (aL−n ) can be obtained as follows: (30) r∗L−n+1 (aL−n ) ∈ arg F πL−n+1 (rL−n+1 ), uL−n+1 cL−n (aL−n ), aL−n , rL−n+1 , where the same type of focus point as the one obtained in the previous period should be sought due to the logic consistency. Step 2: Obtain the optimal investment strategy in period L − n as follows: a∗L−n ∈ arg max cL−n + λL−n+1 aL−n · 1 + r∗L−n+1 (aL−n ) fn−1 . (31) aL−n ∈AL−n
Denote fn =
max
aL−n ∈AL−n
cL−n + λL−n+1 aL−n · 1 + r∗L−n+1 (aL−n ) fn−1 ,
(32)
which is the optimality equation of period L − n. fn represents the maximum ratio of the wealth wL−n which is available for consuming from period L − n to period L. If we know the wealth wL−n , the possibility distribution of the rates of return in period L − n + 1 and fn−1 , we can obtain the total discounted consumption from period L − n to L, that is equal to fn times wL−n . c∗ (a∗ ) c∗ (a∗ ) c∗ (a∗ ) Repeating (29), (30), (31) and (32), we can obtain the decision sequence 0a∗ 0 , 1a∗ 1 , . . . , L−1a∗ L−1 . For 0
1
L−1
easily understanding the decision procedure introduced above, let us have a look at the following example. Example 1. Given L = 2, there are a savings account S0 and one stock S1 . In period 1, the rate of return of the savings account is r1,0 where r1,0 is a positive constant, the first subscript 1 represents period 1, the second subscript 0 represents the savings account S0 . The rate of return of the stock in period 1 is r1,1 . The possibility distribution of the return ratio in period 1 is shown in Table 4. In period 2, the return rate of the savings account is r2,0 . The return rate of the stock in period 2 is r2,1 whose possibility distribution is shown in Table 5. Let’s take into account the decision in period 1 with Type IX focus point. Assume λ2 = 1+r12,0 where r2,0 is the interest rate of the savings account at period 2, then the problem is reduced to consumption or investment on the stock. The satisfaction function is set as u2 (r2 , a1 ) = (c1 + a1,1 (1 + r2,1 )/(1 + r2,0 ))/1.5 where c1 is the proportion of the total wealth devoted to consumption in period 1 and c1 + a1,1 = 1. Then we can obtain the optimal strategy c9∗ 0.375 r2,0 ∗ ∗ for Type IX focus point is 19∗ = 0 , the focus points of the optimal strategy are r92 (a91 ) = 1.2r2,0 +0.2 and a1
0.625
Y. Li, P. Guo / Fuzzy Sets and Systems 274 (2015) 47–61
55
r2,0 ∗ ∗ r92 (a91 ) = 1.5r2,0 +0.5 . The optimality equation is f1 = 0.375 + 0.625 × 1.5 = 1.3125. Due to λ2 = 1/(1 + r2,0 ) we know the strategies in set {(c1 , a1,0 , 0.625)t |c1 + a1,0 = 0.375, c1 ≥ 0, a1,0 ≥ 0} are all optimal strategies for Type IX focus point. Now let us consider the decision in period 0. Similarly, we assume λ1 = 1/(1 + r1,0 ) where r1,0 is the interest rate of the savings account at period 1. The decision problem is reduced to how to consume or invest on the stock. Set the satisfaction function as u1 (r1 , a0 ) = (c0 + a0,1 f1 (1 + r1,1 )/(1 + r1,0 ))/1.5f1 where c0 is the proportion of the total wealth devoted to consumption in period 0, c1 + a1,1 = 1 and c0 + a0,1 = 1. We can obtain the optimal strategy c9∗ 0 r1,0 ∗ ∗ is 09∗ = 0 . The focus point of the optimal investment strategy is r91 (a90 ) = 1.3r1,0 +0.3 and f2 = 1.3 × f1 = a0
1.70625.
1
4. Theoretical analysis of the models In this section we examine which factors influence consumption and investment and analyze the relationship between different types of focus points. Lemma 1. (1) In any period L − n (1 ≤ n ≤ L), fn ≥ 1 holds. (2) Suppose the optimal investment strategy in period L − n is a∗L−n , then a∗L−n · (1 + r∗L−n+1 (a∗L−n )) ≥ 0, where ∗ rL−n+1 (a∗L−n ) is the focus point of a∗L−n . Proof. (1) In any period L − n, if the investor consumes all, that is cL−n = 1, the ratio of wL−n is equal to 1. By the definition of the optimality equation, fn ≥ 1 holds. (2) In the decision period L − 1, if a∗L−1 · (1 + r∗L (a∗L−1 )) < 0, then ∗ (33) f1 = cL−1 + λL a∗L−1 · 1 + r∗L a∗L−1 < 1. Eq. (33) contradicts with f1 ≥ 1, so we have a∗L−1 · (1 + r∗L (a∗L−1 )) ≥ 0. Similarly, in the decision period L − n (n ≥ 2), if a∗L−n · (1 + r∗L−n+1 (a∗L−n )) < 0, we have ∗ fn = cL−n + λL−n+1 a∗L−n · 1 + r∗L−n+1 a∗L−n fn−1 < 1. Eq. (34) contradicts with fn ≥ 1. So we can say in any period L − n, a∗L−n · (1 + r∗L−n+1 (a∗L−n )) ≥ 0.
(34) 2
Now let us consider the relationship between the optimality equation and the discount factor. Proposition 2. λ1L−n+1 and λ2L−n+1 are discount factors satisfying λ2L−n+1 > λ1L−n+1 , fn1 and fn2 are the optimality equations with λ1L−n+1 and λ2L−n+1 . In any period L − n (1 ≤ n ≤ L), the followings hold. (1) (2) (3) (4)
For Type I focus points fn2 ≥ fn1 , For Type II focus points fn2 ≥ fn1 , For Type III focus points fn2 ≥ fn1 , For Type IV focus points fn2 ≥ fn1 .
Proof. (1) In the last decision period L − 1, the optimality equation with λ2L and λ1L are 2 ∗ λL + λ2L a∗L−1 λ2L · 1 + r∗L a∗L−1 λ2L , λ2L , f12 = cL−1
(35)
and 1 ∗ f11 = cL−1 λL + λ1L a∗L−1 λ1L · 1 + r∗L a∗L−1 λ1L , λ1L . ∗ cL−1 (λ2L )
a∗L−1 (λ2L )
(36)
is the optimal consumption–investment strategy with λ2L and
investment strategy with λ1L when we choose Type I focus point.
∗ cL−1 (λ1L )
is the optimal consumption–
a∗L−1 (λ1L ) ∗ ∗ rL (aL−1 (λ2L ), λ2L ))
and r∗L (a∗L−1 (λ1L ), λ1L )) are
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Y. Li, P. Guo / Fuzzy Sets and Systems 274 (2015) 47–61
the Type I focus points of a∗L−1 (λ2L ) with λ2L and a∗L−1 (λ1L ) with λ1L , respectively. For any investment strategy aL−1 satisfying aL−1 · (1 + r∗L (aL−1 , λ1L )) ≥ 0, we have (37) cL−1 + λ1L aL−1 · 1 + r∗L aL−1 , λ1L ≤ cL−1 + λ2L aL−1 · 1 + r∗L aL−1 , λ1L . For rL ∈ {rL |πL (rL ) ≥ α}, by the definition of the Type I focus point, we obtain cL−1 + λ2L aL−1 · (1 + rL ) ≤ cL−1 + λ2L aL−1 · 1 + r∗L aL−1 , λ2L .
(38)
We know a∗L−1 (λ2L ), a∗L−1 (λ1L ) are the optimal investment strategies with λ2L and λ1L respectively, by the Lemma 1 a∗L−1 λ2L · 1 + r∗L a∗L−1 λ2L , λ2L ≥ 0, (39) and a∗L−1 λ1L · 1 + r∗L a∗L−1 λ1L , λ1L ≥ 0.
(40)
Then by (37) and (38), we know 1 ∗ λL + λ1L a∗L−1 λ1L · 1 + r∗L a∗L−1 λ1L , λ1L f11 = cL−1 1 ∗ λL + λ2L a∗L−1 λ1L · 1 + r∗L a∗L−1 λ1L , λ1L ≤ cL−1 1 ∗ λL + λ2L a∗L−1 λ1L · 1 + r∗L a∗L−1 λ1L , λ2L . ≤ cL−1
(41)
By the definition of the optimality equation, we have 1 ∗ λL + λ2L a∗L−1 λ1L · 1 + r∗L a∗L−1 λ1L , λ2L cL−1 2 ∗ λL + λ2L a∗L−1 λ2L · 1 + r∗L a∗L−1 λ2L , λ2L = f12 . ≤ cL−1
(42)
According to (41) and (42), we obtain f12 ≥ f11 . In period L − n (n ≥ 2), we have 2 2 ∗ λL−n+1 + λ2L−n+1 a∗L−n λ2L−n+1 · 1 + r∗L−n+1 a∗L−n λ2L−n+1 , λ2L−n+1 fn−1 fn2 = cL−n ,
(43)
1 1 ∗ λL−n+1 + λ1L−n+1 a∗L−n λ1L−n+1 · 1 + r∗L−n+1 a∗L−n λ1L−n+1 , λ1L−n+1 fn−1 , fn1 = cL−n
(44)
where
∗ cL−n (λ2L−n+1 )
a∗L−n (λ2L−n+1 )
is the optimal consumption–investment strategy with λ2L−n+1 and
∗ cL−n (λ1L−n+1 )
is the opti-
a∗L−n (λ1L−n+1 ) 1 mal consumption–investment strategy with λL−n+1 when we choose Type I focus point. r∗L−n+1 (a∗L−n (λ2L−n+1 ), λ2L−n+1 ) and r∗L−n+1 (a∗L−n (λ1L−n+1 ), λ1L−n+1 ) are the Type I focus point of a∗L−n (λ2L−n+1 ) with λ2L−n+1 and the Type I focus point of a∗L−n (λ1L−n+1 ) with λ1L−n+1 , respectively. Similar to proving f12 ≥ f11 , for any strategy aL−n 1 ≥ 0, we have satisfying aL−n · (1 + r∗L−n+1 (aL−n , λ1L ))fn−1
1 cL−n + λ1L−n+1 aL−n · 1 + r∗L−n+1 aL−n , λ1L−n+1 fn−1 1 ≤ cL−n + λ2L−n+1 aL−n · 1 + r∗L−n+1 aL−n , λ1L−n+1 fn−1 2 ≤ cL−n + λ2L−n+1 aL−n · 1 + r∗L−n+1 aL−n , λ1L−n+1 fn−1 2 ≤ cL−n + λ2L−n+1 aL−n · 1 + r∗L−n+1 aL−n , λ2L−n+1 fn−1 .
(45)
Then 1 1 ∗ fn1 = cL−n λL−n+1 + λ1L−n+1 a∗L−n λ1L−n+1 · 1 + r∗L−n+1 a∗L−n λ1L−n+1 , λ1L−n+1 fn−1 1 1 ∗ λL−n+1 + λ2L−n+1 a∗L−n λ1L−n+1 · 1 + r∗L−n+1 a∗L−n λ1L−n+1 , λ1L−n+1 fn−1 ≤ cL−n 1 2 ∗ λL−n+1 + λ2L−n+1 a∗L−n λ1L−n+1 · 1 + r∗L−n+1 a∗L−n λ1L−n+1 , λ2L−n+1 fn−1 ≤ cL−n 2 ∗ λL−n+1 + λ2L−n+1 a∗L−n λ2L−n+1 · (1 + r∗L−n+1 a∗L−n λ2L−n+1 , λ2L−n+1 = fn2 . ≤ (cL−n Likewise, we can prove (2), (3) and (4).
2
(46)
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57
From Proposition 2 we know fL2 ≥ fL1 holds with λ2L−n+1 > λ1L−n+1 for Types I, II, III, IV focus points. Since and fL1 w0 are the two maximum total consumption amount, it means if the individual enjoys more consumption at the present time than in the future (decreasing the discount factor), the total consumption amount will decrease. Such conclusions are quite in agreement with the situations encountered in the real world. fL2 w0
c at period L − n (1 ≤ n ≤ L), Type I, Type II, Type Theorem 3. For any consumption–investment strategy aL−n L−n ∗ ∗ III, Type IV, Type IX, Type X, Type XI and Type XII focus points are denoted as r1L−n+1 (aL−n ), r2L−n+1 (aL−n ), ∗ ∗ ∗ ∗ ∗ ∗ 11 12 1∗ r3L−n+1 (aL−n ), r4L−n+1 (aL−n ), r9L−n+1 (aL−n ), r10 L−n+1 (aL−n ), rL−n+1 (aL−n ) and rL−n+1 (aL−n ), respectively. fn , ∗ ∗ ∗ ∗ ∗ ∗ ∗ fn2 , fn3 , fn4 , fn9 , fn10 , fn11 and fn12 are the optimality equations with Type I, Type II, Type III, Type IV, Type IX, Type X, Type XI and Type XII focus points, respectively. We have (1) (2) (3) (4)
∗
∗
fn2 ≤ fn1 , ∗ ∗ fn4 ≤ fn3 , ∗ ∗ 11 fn ≤ fn9 , ∗ ∗ 10 12 fn ≤ f n .
Proof. (1) Let us examine the last decision period L − 1. For any investment strategy aL−1 , by the definitions of Types I and II focus points we have ∗ ∗ uL cL−1 (aL−1 ), aL−1 , r2L (aL−1 ) ≤ uL cL−1 (aL−1 ), aL−1 , r1L (aL−1 ) , (47) that is
∗ ∗ cL−1 + λL aL−1 · 1 + r2L (aL−1 ) ≤ cL−1 + λL aL−1 · 1 + r1L (aL−1 ) .
(48)
2∗
For aL−1 , which is the optimal investment strategy with the Type II focus point, we have ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2∗ 2∗ f12 = cL−1 + λL a2L−1 · 1 + r2L a2L−1 ≤ cL−1 + λL a2L−1 1 + r1L a2L−1 .
(49)
1∗
By the definition of f1 , we have ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2∗ 1∗ cL−1 + λL a2L−1 1 + r1L a2L−1 ≤ cL−1 + λL a1L−1 1 + r1L a1L−1 = f11 . 2∗
(50)
1∗
It follows from (49) and (50) that f1 ≤ f1 holds. In period L − n (n ≥ 2), the optimality functions for Type I and Type II focus points are ∗ 1∗ ∗ ∗ ∗ 1∗ fn1 = cL−n + λL−n+1 a1L−n · 1 + r1L−n+1 a1L−n fn−1 , (51) and
∗ 2∗ ∗ ∗ ∗ 2∗ fn2 = cL−n + λL−n+1 a2L−n · 1 + r2L−n+1 a2L−n fn−1 . 2∗
(52)
1∗
2∗
Supposing fn−1 ≤ fn−1 , for any investment strategy aL−n satisfying aL−n · (1 + rL−n+1 (aL−n )) ≥ 0, we have 2∗ 1∗ ∗ ∗ cL−n + λL−n+1 aL−n · 1 + r2L−n+1 (aL−n ) fn−1 ≤ cL−n + λL−n+1 aL−n · 1 + r2L−n+1 (aL−n ) fn−1 .
(53)
By the definition of Type I focus point, we know 1∗ 1∗ ∗ ∗ cL−n + λL−n+1 aL−n · 1 + r2L−n+1 (aL−n ) fn−1 ≤ cL−n + λL−n+1 aL−n · 1 + r1L−n+1 (aL−n ) fn−1 .
(54)
2∗
2∗
2∗
1∗
2∗
1∗
1∗
Considering (53) and (54), we have fn ≤ cL−n + λL−n+1 aL−n (1 + rL−n+1 (aL−n ))fn−1 ≤ fn . In a similar way, we can prove (2). (3) In the last decision period L − 1, for any investment strategy aL−1 , according to Theorem 2 [7] we have ∗ 9∗ uL cL−1 (aL−1 ), aL−1 , r11 (55) L (aL−1 ) ≤ uL cL−1 (aL−1 ), aL−1 , rL (aL−1 ) , that is
∗ 9∗ cL−1 + λL aL−1 · 1 + r11 L (aL−1 ) ≤ cL−1 + λL aL−1 · 1 + rL (aL−1 ) .
(56)
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Y. Li, P. Guo / Fuzzy Sets and Systems 274 (2015) 47–61 ∗
For the optimal investment strategy a11 L−1 with the Type XI focus point, we have ∗ ∗ 11∗ 11∗ 11∗ 11∗ 11∗ 9∗ 11∗ f111 = cL−1 + λL a11 L−1 · 1 + rL aL−1 ≤ cL−1 + λL aL−1 1 + rL aL−1 . By the definition of ∗
∗
(57)
∗ f19 ,
9 cL−1 + λL a9L−1
we have ∗ ∗ 9∗ 9∗ 9∗ 9∗ 9∗ 1 + r9L a11 L−1 ≤ cL−1 + λL aL−1 1 + rL aL−1 = f1 .
∗
∗
∗
(58) ∗
11 ≤ f 9 . For any investment strategy From (57) and (58) we know f111 ≤ f19 . In period L − n (n ≥ 2), suppose fn−1 n−1 aL−n , by Theorem 3 [9] we have 11∗ 9∗ ∗ 9∗ cL−n + λL−n+1 aL−n · 1 + r11 (59) L−n+1 (aL−n ) fn−1 ≤ cL−n + λL−n+1 aL−n · 1 + rL−n+1 (aL−n ) fn−1 . ∗
For the optimal investment strategy a11 L−n with the Type XI focus point, we have 11∗ 11∗ ∗ ∗ ∗ 11 11∗ f111 = cL−n + λL−n+1 a11 L−n · 1 + rL−n+1 aL−n fn−1 11∗ 9∗ ∗ 11∗ 9∗ ≤ cL−n + λL−n+1 a11 L−n · 1 + rL−n+1 aL−n fn−1 .
(60)
9∗
By the definition of fn , we have 11∗ 9∗ ∗ 11∗ 9∗ cL−n + λL−n+1 a11 L−n · 1 + rL−n+1 aL−n fn−1 ∗ 9∗ ∗ ∗ ∗ 9∗ ≤ cL−n + λL−n+1 a9L−n · 1 + r9L−n+1 a9L−n fn−1 = fn9 . 11∗
It follows (60) and (61) that fn
9∗
≤ fn holds. Similarly, we can prove (4) by using Theorem 2 and Theorem 4 [9].
(61) 2
From Theorem 3 we know that in the same investment environment, a person who is concerned with different Type of focus point will have different imagination about how the initial wealth will be changed and how he/she should consume and invest in each stage. This theorem gives an explanation about why there is a variety of ways of the individual consumption. 1 2 Proposition 4. Suppose in any period L − n (1 ≤ n ≤ L), there are two possibility functions πL−n+1 and πL−n+1 . π1
π2
1 2 and πL−n+1 respectively. If ∀rL−n+1 ∈ RL−n+1 , fn L−n+1 and fn L−n+1 are the optimality equations with πL−n+1 1 2 πL−n+1 (rL−n+1 ) ≥ πL−n+1 (rL−n+1 ) holds, then π1
π2
(1) For Type I focus points fn L−n+1 ≥ fn L−n+1 , π1
π2
(2) For Type II focus points fn L−n+1 ≤ fn L−n+1 , π1
π2
π1
π2
(3) For Type III focus points fn L−n+1 ≤ fn L−n+1 , (4) For Type IV focus points fn L−n+1 ≥ fn L−n+1 . 1 (r ) ≥ π 2 (r ) (∀r ∈ R ), for any investment strategy Proof. (1) In the last decision period L − 1, since πL−1 L L L L−1 L aL−1 , considering the definition of the Type I focus point we have (62) uL cL−1 (aL−1 ), aL−1 , r∗L aL−1 , πL1 ≥ uL cL−1 (aL−1 ), aL−1 , r∗L aL−1 , πL2 ,
where r∗L (aL−1 , πL1 ) and r∗L (aL−1 , πL2 ) are the focus points of aL−1 with πL1 and πL2 , respectively. That is, cL−1 + λL aL−1 · 1 + r∗L aL−1 , πL1 ≥ cL−1 + λL aL−1 · 1 + r∗L aL−1 , πL2 .
(63)
The optimality equations for πL1 and πL2 are 1 π1 ∗ πL + λL a∗L−1 πL1 · 1 + r∗L a∗L−1 πL1 , πL1 , f1 L = cL−1
(64)
2 π2 ∗ πL + λL a∗L−1 πL2 · 1 + r∗L a∗L−1 πL2 , πL2 , f1 L = cL−1
(65)
and
Y. Li, P. Guo / Fuzzy Sets and Systems 274 (2015) 47–61
59
where a∗L−1 (πL1 ), a∗L−1 (πL2 ) are the optimal investment strategies for πL1 and πL2 , respectively. r∗L (a∗L−1 (πL1 ), πL1 ) and r∗L (a∗L−1 (πL2 ), πL2 ) are the Type I focus point of a∗L−1 (πL1 ) with πL1 and the Type I focus point of a∗L−1 (πL2 ) with πL2 , π1
respectively. By the definition of f1 L we have 2 π1 ∗ f1 L ≥ cL−1 πL + λL a∗L−1 πL2 · 1 + r∗L a∗L−1 πL2 , πL1 .
(66)
From (63) we have 2 π2 ∗ πL + λL a∗L−1 πL2 · 1 + r∗L a∗L−1 πL2 , πL1 ≥ f1 L . cL−1 π1
(67)
π2
From (66) and (67) we know f1 L ≥ f1 L . In the period L − n (n ≥ 2), we have 1 1 1 1 1 πL−n+2 π1 ∗ πL−n+1 + λL−n+1 a∗L−n πL−n+1 · 1 + r∗L−n+1 a∗L−n πL−n+1 , πL−n+1 fn−1 fn L−n+1 = cL−n , 2 2 πL−n+1 π L−n+2 ∗ 2 2 2 2 fn πL−n+1 + λL−n+1 a∗L−n πL−n+1 · 1 + r∗L−n+1 a∗L−n πL−n+1 , πL−n+1 fn−1 = cL−n .
(68) (69)
1 1 2 a∗L−n (πL−n+1 ) is the optimal investment strategy with πL−n+1 and a∗L−n (πL−n+1 ) is the optimal investment strat2 ∗ ∗ 1 1 ∗ ∗ 2 2 ) are the focus points of egy with πL−n+1 . rL−n+1 (aL−n (πL−n+1 ), πL−n+1 ) and rL−n+1 (aL−n (πL−n+1 ), πL−n+1 ∗ 1 1 ∗ 2 2 aL−n (πL−n+1 ) with πL−n+1 and the focus point of aL−n (πL−n+1 ) with πL−n+1 , respectively. For any investment π1
2 strategy aL−n satisfying aL−n · (1 + r∗L−n+1 (aL−n , πL−n+1 )) ≥ 0, considering the definition of fn L−n+1 we have 2 πL−n+2 2 fn−1 cL−n + λL−n+1 aL−n · 1 + r∗L−n+1 aL−n , πL−n+1 1 πL−n+2 2 ≤ cL−n + λL−n+1 aL−n · 1 + r∗L−n+1 aL−n , πL−n+1 fn−1 1 πL−n+2 1 fn−1 . ≤ cL−n + λL−n+1 aL−n · 1 + r∗L−n+1 aL−n , πL−n+1
(70)
Eq. (70) implies 1 2 2 2 1 πL−n+2 π1 ∗ πL−n+1 + λL−n+1 a∗L−n πL−n+1 · 1 + r∗L−n+1 a∗L−n πL−n+1 , πL−n+1 fn−1 fn L−n+1 ≥ cL−n
π2
≥ fn L−n+1 .
(71)
In a similar way, we can prove (2), (3), (4). 2 Proposition 4 shows that increasing the uncertainty will make the individual who chooses Type I or Type IV focus point feel that the total amount of consumption available becomes lager, the individual who chooses Type II or Type III focus point believe that the total amount of consumption available becomes smaller. The different feelings about the total amount of consumption available may lead them to make different consumption–investment plans. 1 2 1 In period L − n (1 ≤ n ≤ L), if πL−n+1 and πL−n+1 are isotonically transformable, that is, (πL−n+1 (rL−n+1 ) − 1 2 2 1 1 πL−n+1 (rL−n+1 ))(πL−n+1 (rL−n+1 ) − πL−n+1 (rL−n+1 )) > 0 or |πL−n+1 (rL−n+1 ) − πL−n+1 (rL−n+1 )| 2 2 (rL−n+1 ) − πL−n+1 (rL−n+1 )| = 0 holds for any rL−n+1 , rL−n+1 ∈ RL−n+1 [9], we have the following + |πL−n+1 lemma. 1 2 Lemma 5. In any period L − n (1 ≤ n ≤ L), if πL−n+1 and πL−n+1 are isotonically transformable, for Types V, VI, π1
π2
VII, VIII focus points, we have fn L−n+1 = fn L−n+1 . 1 2 It is trivial to prove Lemma 5 because if πL−n+1 and πL−n+1 are isotonically transformable, for any investment 2 will remain the same as the focus points with strategy aL−n , the Types V, VI, VII, VIII focus points with πL−n+1 1 πL−n+1 . Now let us consider how the rates of return change influence the individual consumption and investment.
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Proposition 6. Suppose in any period L − n (1 ≤ n ≤ L), there are two sets of return ratios RL−n+1 = {rL−n+1 } and RδL−n+1 = {rδL−n+1 |rδL−n+1 = rL−n+1 + δ, rL−n+1 ∈ RL−n+1 , δ ≥ 0}. fn and fnδ are the optimality equations with RL−n+1 and RδL−n+1 , respectively. The followings hold (1) (2) (3) (4)
For Type I focus points fn ≤ fnδ , For Type II focus points fn ≤ fnδ , For Type III focus points fn ≤ fnδ , For Type IV focus points fn ≤ fnδ .
Proof. (1) In the last decision period L − 1, for any investment strategy aL−1 , we have cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) ≤ cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) + δ ,
(72)
where r∗L (aL−1 ) is the Type I focus point of aL−1 with RL . Considering the definitions of Type I focus point and RδL−n+1 we obtain cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) + δ ≤ cL−1 + λL aL−1 · 1 + r∗L (aL−1 , δ) ,
(73)
where r∗L (aL−1 , δ) is the Type I focus point of aL−1 with RδL . For the optimal investment strategy a∗L−1 with RL , from (72) and (73), we have ∗ ∗ f1 = cL−1 + λL a∗L−1 · 1 + r∗L a∗L−1 ≤ cL−1 + λL a∗L−1 · 1 + r∗L a∗L−1 , δ ∗ ≤ cL−1 (δ) + λL a∗L−1 (δ) · 1 + r∗L a∗L−1 (δ), δ = f1δ .
(74)
In any period L − n (n ≥ 2), we have ∗ fn = cL−n + λL−n+1 a∗L−n · 1 + r∗L−n+1 a∗L−n fn−1 ,
(75)
δ ∗ fnδ = cL−n (δ) + λL−n+1 a∗L−n (δ) · 1 + r∗L−n+1 a∗L−n (δ), δ fn−1 ,
(76)
where a∗L−n is the optimal investment strategy with RL−n+1 and a∗L−n (δ) is the optimal investment strategy with RδL−n+1 . r∗L−n+1 (a∗L−n ) and r∗L−n+1 (a∗L−n (δ), δ) are the focus points of a∗L−n with RL−n+1 and the focus point of a∗L−n (δ) with RδL−n+1 , respectively. Similarly, we can prove fn ≤ fnδ . (2) In the last decision period L − 1, for any investment strategy aL−1 , if cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) > cL−1 + λL aL−1 · 1 + r∗L (aL−1 , δ) ,
(77)
where r∗L (aL−1 ) is the Type II focus point of aL−1 with RL and r∗L (aL−1 , δ) is the Type II focus point of aL−1 with RδL , we have cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) > cL−1 + λL aL−1 · 1 + r∗L (aL−1 , δ) − δ .
(78)
Since (78) contradicts with the definition of Type II focus point, we know for any investment strategy cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) ≤ cL−1 + λL aL−1 · 1 + r∗L (aL−1 , δ) .
(79)
By (79), f1 ≤ f1δ holds. Similarly, we can prove fn ≤ fnδ . Likewise, we can prove (3) and (4). 2 Proposition 6 shows that when the economic situation changes for the better, a person believes the total amount of consumption in the future will increase in any period. Such a conclusion is in harmony with the common sense in the real world.
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5. Conclusions In this paper, with considering an individual has one and only one chance to make a decision on how to consume and how to invest in each period with possibilisitc information, the scenario-based multi-period consumption–investment models with the one-shot decision theory are proposed. In the proposed models, the securities’ return ratio vector is governed by the possibility distribution. Since in each period, one and only one return ratio vector will come, an individual contemplate one vector amongst all possible vectors based on the satisfaction level caused by the appearance of this vector and the possibility degree to which this vector appears. The selected return ratio vector is called focus point. There are twelve types of focus points. Which type of focus point should be selected depends on the decision maker’s attitude about satisfaction and possibility at each period. Based on the selected vectors (focus points), the sequence of optimal consumption–investment plan is determined by dynamic programming to maximize the sum of possible discounted consumption over the whole process (lifetime). Several theoretical results are obtained and economic insights are gained. Since the proposed models fit the characteristics of the individual multi-period consumption–investment behavior, possibilistic individual multi-period consumption–investment models can provide a useful vehicle to deal with such an important economic problem. References [1] A. Cadenillas, Consumption–investment problems with transaction costs: survey and open problems, Math. Methods Oper. Res. 51 (1) (2000) 43–68. [2] K.C. Cheung, H. Yang, Optimal investment–consumption strategy in a discrete-time model with regime switching, Discrete Contin. Dyn. Syst., Ser. B 8 (2) (2007) 315–332. [3] E.F. Fama, Risk, return and equilibrium, J. Polit. Econ. 79 (1) (1971) 30–55. [4] P. Guo, One-shot decision approach and its application to duopoly market, Int. J. Inf. Decis. Sci. 2 (3) (2010) 213–232. [5] P. Guo, Private real estate investment analysis within a one-shot decision framework, Int. Real. Estate Rev. 13 (3) (2010) 238–260. [6] P. Guo, R. Yan, J. Wang, Duopoly market analysis within one-shot decision framework with asymmetric possibilistic information, Int. J. Comput. Intell. Syst. 3 (6) (2010) 786–796. [7] P. Guo, One-shot decision theory, IEEE Trans. Syst. Man Cybern., Part A, Syst. Hum. 41 (5) (2011) 917–926. [8] P. Guo, One-shot decision theory: a fundamental alternative for decision under uncertainty, in: P. Guo, W. Pedrycz (Eds.), Human-Centric Decision-Making Models for Social Sciences, in: Studies in Computational Intelligence, vol. 502, Springer-Verlag, Berlin, Heidelberg, 2014, pp. 33–55. [9] P. Guo, Y. Li, Approaches to multistage one-shot decision making, Eur. J. Oper. Res. 236 (2) (2014) 612–623. [10] P. Guo, X. Ma, Newsvendor models for innovative products with one-shot decision theory, Eur. J. Oper. Res. 239 (2) (2014) 523–536. [11] I. Karatzas, H. Wang, Utility maximization with discretionary stopping, SIAM J. Control Optim. 39 (1) (2000) 306–329. [12] H. Kraft, F.T. Seifried, M. Steffensen, Consumption-portfolio optimization with recursive utility in incomplete markets, Finance Stoch. 17 (1) (2013) 161–196. [13] Zh. Li, K.S. Tan, H. Yang, Multiperiod optimal investment–consumption strategies with mortality risk and environment uncertainty, N. Am. Actuar. J. 12 (1) (2008) 47–64. [14] J. Lintner, Security prices, risk and maximal gains from diversification, J. Finance 20 (5) (1965) 587–616. [15] R.C. Merton, Lifetime portfolio selection under uncertainty: the continuous-time case, Rev. Econ. Stat. 51 (3) (1969) 247–257. [16] R.C. Merton, Optimal consumption and portfolio rules in a continuous-time model, J. Econ. Theory 3 (4) (1971) 373–413. [17] R.C. Merton, Continuous-Time Finance, Basil Blackwell, Oxford, 1990. [18] J. Mossin, Optimal multi-period portfolio policies, J. Bus. 41 (2) (1968) 215–229. [19] P.H. Nielsen, M. Steffensen, Optimal investment and life insurance strategies under minimum and maximum constraints, Insur. Math. Econ. 43 (1) (2008) 15–28. [20] S.R. Pliska, Introduction to Mathematical Finance: Discrete Time Models, Blackwell, 1997. [21] S.R. Pliska, J. Ye, Optimal life insurance purchase and consumption/investment under uncertain lifetime, J. Bank. Finance 31 (5) (2007) 1307–1319. [22] P.A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Rev. Econ. Stat. 51 (3) (1969) 239–246. [23] L.R. Sotomayor, A. Cadenillas, Explicit solutions of consumption–investment problems in financial markets with regime switching, Math. Finance 19 (2) (2009) 251–279. [24] M. Steffensen, Optimal consumption and investment under time-varying relative risk aversion, J. Econ. Dyn. Control 35 (5) (2011) 659–667. [25] J. Su, X. Guan, Multi-objective Optimal Strategy for Individual Consumption–Investment with Fuzzy Coefficients, Lecture Notes in Computer Science, vol. 4113, 2006, pp. 919–924.
ScienceDirect Fuzzy Sets and Systems 274 (2015) 47–61 www.elsevier.com/locate/fss
Possibilistic individual multi-period consumption–investment models Yonggang Li, Peijun Guo ∗ Faculty of Business Administration, Yokohama National University, 79-4 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan Received 29 September 2013; received in revised form 6 January 2015; accepted 10 January 2015 Available online 14 January 2015
Abstract In this paper, we consider a multi-period consumption–investment problem with partially known information for an individual. We emphasize the fact that for an individual there is one and only one result will appear for his/her consumption–investment decision in each period. Different from the existing multi-period consumption–investment models which basically maximize the expected utility, we utilize the one-shot decision theory for individual multi-period consumption–investment problems. In each period, the individual chooses one state (scenario) for each strategy with considering the satisfaction of the outcome and its possibility. The selected state is called the focus point. The multi-period consumption–investment decision models with twelve types of focus points are built to maximize the sum of discounted consumption over the whole process (lifetime) for twelve types of individuals and economic insights are gained by the theoretical analysis. The proposed models are scenario-based decision models which provide a fundamental alternative to analyze individual multi-period consumption–investment behavior. © 2015 Elsevier B.V. All rights reserved. Keywords: Possibility distribution; Decision making; One-shot decision theory; Scenario-based decision theory; Focus point; Multistage decision problem; Dynamic programming; Consumption–investment problem
1. Introduction As an important research topic in economic literature, multi-period consumption–investment problems have received increasing attention since 1960s. A multi-period consumption–investment problem can be simply described as follows: there is an investor with an initial wealth which should be allocated to consumption and a portfolio for investment in the first period. The portfolio will yield uncertain new wealth in the next period and the probability distribution of returns of securities is known. In the next period, the investor should allocate the new wealth to consumption and investment again. The consumption–investment decisions are repeated in each period until the end of life. The multi-period consumption–investment problems were initially studied by Mossin [18] and Samuelson [22], in which the objective of decision making was to maximize the expected utility of consumption. Merton [15,17] in* Corresponding author.
E-mail addresses: [email protected] (Y. Li), [email protected] (P. Guo). http://dx.doi.org/10.1016/j.fss.2015.01.005 0165-0114/© 2015 Elsevier B.V. All rights reserved.
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troduced the models under a continuous-time framework and the stochastic optimal control approach was used in his models. General equilibrium theory for the pricing of risky assets has been used as an extremely powerful tool in the existing researches [3,14,16]. Pliska [20] introduced the discrete time model. Cadenillas [1] gave the detailed survey for consumption–investment problems. Recently, Karatzas and Wang [11] studied an optimal dynamic investment problem in the case where the uncertain time horizon was a stopping time of asset price filtration. Cheung and Yang [2], Sotomayor and Cadenillas [23] investigated dynamic investment–consumption problems in a regime-switching environment. Pliska and Ye [21] considered optimal insurance and consumption rules for a wage earner whose lifetime was random and explicit solutions were found for the family of CRRA utilities. Li and Tan [13] studied the optimality of the consumption–investment problem that reflected the mortality uncertainty. Nielsen and Steffensen [19] investigated the optimal investment, and life insurance strategies under the minimum and maximum consumption. Steffensen [24] examined the optimal consumption and investment rules, particularly in the case where the relative risk aversion with respect to consumption was increasing with age. Kraft et al. [12] studied the optimal consumption-portfolio decision of an investor with recursive preferences of Epstein–Zin type in an incomplete market. Su and Guan [25] utilized multi-objective programming to solve the single period consumption–investment problem with fuzzy coefficients. Until now, researchers considered and solved multi-period consumption–investment problems within the probabilistic framework where the uncertainty of the portfolios’ return was characterized by the probability distribution, and different kinds of utility functions were examined. In fact, an individual’s decision on consumption and investment is a one-time and never-return decision. It means a person cannot return to the past and his/her decision on investment in each period can achieve one and only one result. Guo [7] initially established the one-shot decision theory for dealing with one-shot decision problems. The one-shot decision making problems have been researched in the papers [4–6,8,10]. Instead of the lottery-based choices as in other existing decision theories under uncertainty, the one-shot decision theory provides a scenario-based choice which fits the characteristic of one-shot decision problems. In the one-shot decision theory, the decision process encompasses two steps. The first step is to identify which state should be taken into account for each alternative with considering the possibility of the state and the satisfaction of an outcome. The states selected are called focus points. The second step is to evaluate the alternatives based on the focus points to obtain the optimal alternative. Based on the one-shot decision theory, Guo and Li studied multistage one-shot decision problems in the research [9] where a general approach to multistage one-shot decision making is given and the one-shot optimal stopping problem is analyzed. In this paper, we apply multistage one-shot decision approaches to consumption–investment problems with considering that an individual has one and only one chance to allocate his/her wealth to consumption and investment, and choose a suitable investment strategy in each period. In the proposed model, an individual takes into account how to allocate the initial wealth and determine a consumption–investment plan to maximize the sum of possible discounted consumption over the whole process (lifetime). This paper extends the existing literature in three important dimensions. First, in the existing multistage consumption–investment models, the uncertainty is characterized by the probability distribution whereas we utilize the possibility distribution to describe the uncertainty. There might be an acute lack of information for obtaining the probability distribution in multistage consumption–investment problems. The possibility distribution seems more feasible for such decision problems. To the best of our knowledge, our research is the first which studies multi-period consumption–investment problems with possibilistic information, which is an innovative study on solving such an important economic problem. Second, different from the existing multistage consumption–investment models which utilize the subjective expected utility theory (SEU), our models is based on the one-shot decision theory (OSDT). The SEU-based models use different utility functions to reflect the attitude of a person about risk whereas in the OSDTbased models choosing which type of focus point (scenario) to make a decision reflects the attitude of this person about uncertainty. Since the individual consumption–investment is a typical one-time and never-return decision, the proposed models fit the one-time feature of the individual behavior and economic insights are gained by the theoretical analysis. Third, in possibilistic multistage consumption–investment models, we consider multidimensional states and decisions. It extends the focus point and the decision in the classic one-shot decision theory from the one-dimensional to the multidimensional case. The remainder of the paper is organized as follows. In Section 2, the one-shot decision theory is briefly introduced with considering multidimensional states and alternatives. In Section 3, the approach to multi-period consumption– investment problems with the one-shot decision theory is proposed. In Section 4, the theoretical results are given. Finally, concluding remarks are made in Section 5.
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2. Introduction to the one-shot decision theory Guo [7] initially proposed the one-shot decision theory to solve one-shot decision problems which are for the situations where a decision maker has only one chance to make a decision under uncertainty. In this section we consider the situation that the state and the alternative are multi-dimensional vectors. Suppose the set of an alternative a is A and the set of a state x is S. a and x are vectors in R n and R m , respectively. π(x) is a function defined on S, which is used to characterize the degree to which a state x is to occur. Definition 1. Given a function π : S → [0, 1] if maxx∈S π(x) = 1, then π(x) is called a possibility distribution, where S is the sample space, π(x) is the possibility degree of x. A payoff is a result for the combination of an alternative a and a state x, denoted as v(x, a). The satisfaction level of a decision maker for a payoff can be expressed by a satisfaction function, as defined below. Definition 2. Denote the set of payoffs as V . The function u : V → [0, 1] is called a satisfaction function if it satisfies u(v1 ) > u(v2 ) for v1 > v2 where v1 , v2 ∈ V . Satisfaction function is used to represent the relative position of the payoff obtained from an alternative by the normalizing operator. Because the payoff is the function of x and a, we also can write the satisfaction function as u(x, a). Different from the existing decision theories which are lottery-based, the one-shot decision theory argues that a person makes a one-shot decision based on some particular state which is the most appropriate scenario for him/her while considering the satisfaction level incurred by it and its possibility degree. The one-shot decision process encompasses two steps. The first step is to identify which state should be taken into account for each alternative with considering the possibility of the state and the satisfaction of an outcome. The states focused are called focus points. The second step is to evaluate the alternatives based on the focus points to obtain the optimal alternative. Considering all combination of satisfaction and possibility, twelve types of focus points are provided in the following to help a decision maker in finding out states (scenarios) which are adequate for him/her. ∗ x1α (a) ∈ arg max u(x, a) where X≥α = x|π(x) ≥ α , (1) x∈X≥α
where the given parameter α is a level used to distinguish whether the possibility degree is evaluated as ‘high’ by a decision maker. X≥α = {x|π(x) ≥ α} is the set of states with high possibility degrees and maxx∈X≥α u(x, a) is ∗ ∗ the highest satisfaction level amongst all x ∈ X≥α which can be attained when x = x1α (a) so that x1α (a) is a state ∗ (scenario) with the highest satisfaction and a high possibility to occur. x1α (a) is called as Type I focus point of an alternative a. ∗ x2α (a) ∈ arg min u(x, a) where X≥α = x|π(x) ≥ α . (2) x∈X≥α ∗
∗
Eq. (2) shows that x2α (a) is a state (scenario) with the lowest satisfaction and a high possibility to occur. x2α (a) is called as Type II focus point of an alternative a. ∗ x3α (a) ∈ arg max u(x, a) where X≤α = x|π(x) ≤ α . (3) x∈X≤α ∗
∗
Eq. (3) implies that x3α (a) is a state (scenario) with the highest satisfaction and a low possibility to occur. x3α (a) is called as Type III focus point of an alternative a. ∗ x4α (a) ∈ arg min u(x, a) where X≤α = x|π(x) ≤ α . (4) x∈X≤α
∗
∗
It is clear from (4) that x4α (a) is a state (scenario) with the lowest satisfaction and a low possibility to occur. x4α (a) is called as Type IV focus point of an alternative a. ∗ x5β (a) ∈ arg max π(x) where X≥β (a) = x|u(x, a) ≥ β . (5) x∈X≥β (a)
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The given parameter β is the level to distinguish whether the satisfaction level is evaluated as ‘high’ by a decision ∗ maker. It follows from (5) that x5β (a) is a state (scenario) with the highest possibility to occur and a high satisfaction. ∗
x5β (a) is called as Type V focus point of an alternative a. ∗ x6β (a) ∈ arg min π(x) where X≥β (a) = x|u(x, a) ≥ β .
(6)
x∈X≥β (a)
∗
∗
From (6), we know that x6β (a) is a state (scenario) with the lowest possibility to occur and a high satisfaction. x6β (a) is called as Type VI focus point of an alternative a. ∗ x7β (a) ∈ arg max π(x) where X≤β (a) = x|u(x, a) ≤ β . (7) x∈X≤β (a)
∗
Eq. (7) makes it clear that x7β (a) is a state (scenario) with the low satisfaction and the highest possibility to occur. ∗
x7β (a) is called as Type VII focus point of an alternative a. ∗ x8β (a) ∈ arg min π(x) where X≤β (a) = x|u(x, a) ≤ β .
(8)
x∈X≤β (a)
∗
∗
Obviously, x8β (a) represents a state (scenario) with the low satisfaction and the lowest possibility to occur. x8β (a) is called as Type VIII focus point of an alternative a. ∗ x9 (a) ∈ arg max min π(x), u(x, a) . (9) x∈S
∗
It follows from (9) that x = x9 (a) maximizes g(x, a) = min[π(x), u(x, a)]. Because min[π(x), u(x, a)] represents the lower bound of [π(x), u(x, a)], increasing min[π(x), u(x, a)](maxx∈S min[π(x), u(x, a)]) will increase the possibility degree and the satisfaction level simultaneously. Therefore, arg maxx∈S min[π(x), u(x, a)] represents the state that has ∗ the higher possibility degree and the higher satisfaction level. x9 (a) is called as Type IX focus point of an alternative a. ∗ x10 (a) ∈ arg min max π(x), u(x, a) . (10) x∈S
∗
Eq. (10) shows that x = x10 (a) minimizes h(x, a) = max[π(x), u(x, a)]. Because max[π(x), u(x, a)] represents the upper bound of [π(x), u(x, a)], decreasing max[π(x), u(x, a)](minx∈S max[π(x), u(x, a)]) will decrease the possibility degree and the satisfaction level simultaneously. Therefore, arg minx∈S max[π(x), u(x, a)] is for seeking the state ∗ that has the lower possibility degree and the lower satisfaction level. x10 (a) is called as Type X focus point of an alternative a. ∗ x11 (a) ∈ arg min max 1 − π(x), u(x, a) . (11) x∈S
∗
Similarly, we know x11 (a) represents a state (scenario) with the higher possibility degree and the lower satisfaction ∗ level. x11 (a) is called as Type XI focus point of an alternative a. ∗ x12 (a) ∈ arg min max π(x), 1 − u(x, a) . (12) x∈S
∗
It is easy to understand that x12 (a) is a state (scenario) with the lower possibility degree and the higher satisfaction ∗ level. x12 (a) is called as Type XII focus point of an alternative a. In summary, (1)–(12) are used to seek the focused states featured by the highest satisfaction and a high possibility, the lowest satisfaction and a high possibility, the highest satisfaction and a low possibility, the lowest satisfaction and a low possibility, the highest possibility and a high satisfaction, the lowest possibility and a high satisfaction, the highest possibility and a low satisfaction, the lowest possibility and a low satisfaction, the higher possibility degree and the higher satisfaction level, the lower possibility degree and the lower satisfaction level, the higher possibility degree and the lower satisfaction level, the lower possibility degree and the higher satisfaction level (see Tables 1–3). Choosing which kind of focus point to make a decision is dependent on the attitude of the decision maker on uncertainty and benefit. The relationship amongst twelve types of focus points is given in the paper [7]. In real-world investing, Type IX focus point is used for characterizing an active investor whereas Type X focus point is used for describing
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Table 1 The features of focus points (Types I–IV).
Possibility degree Satisfaction level
Type I
Type II
Type III
Type IV
High The highest
High The lowest
Low The highest
Low The lowest
Table 2 The features of focus points (Types V–VIII).
Possibility degree Satisfaction level
Type V
Type VI
Type VII
Type VIII
The highest High
The lowest High
The highest Low
The lowest Low
Table 3 The features of focus points (Types IX–XII).
Possibility degree Satisfaction level
Type IX
Type X
Type XI
Type XII
Higher Higher
Lower Lower
Higher Lower
Lower Higher
an apprehensive investor because although some scenario occurs with a lower possibility degree, it is still emphasized by an investor as it can lead to a larger loss (lower satisfaction level). Type XI focus point is used for a passive investor whereas Type XII focus point is used for a daring investor because even if some scenario occurs with a lower possibility degree, a higher gain (higher satisfaction level) may lure an investor to consider such a situation. For one alternative, more than one states might exist as one type of focus point. We denote the sets of twelve types of focus points of the alternative a as X1α (a), X2α (a), X3α (a), X4α (a), X5β (a), X6β (a), X7β (a), X8β (a), X9 (a), X10 (a), X11 (a), and X12 (a), respectively. It should be noted that X3α (a) and X4α (a) are empty sets when X≤α = ∅; X7β (a) and X8β (a) are empty sets when X≤β (a) = ∅. For the sake of simplification, in this paper we assume that X3α (a), X4α (a), X7β (a) and X8β (a) are nonempty sets for any alternative a. In one-shot decision, the second step is to obtain the optimal solution based on the focus points of all alternatives. The decision maker contemplates that the focus points are the most appropriate states (scenarios) for him/her and the decision maker chooses one alternative which can bring about the best consequence once the focus point (scenario) comes true. Since there are twelve types of focus points, there are twelve types of optimal alternatives, as shown below. ∗ ∗ Type I optimal alternative: a1 (α) ∈ arg max v x1α (a), a . (13) a∈A
∗ ∗ Type II optimal alternative: a2 (α) ∈ arg max v x2α (a), a .
(14)
∗ ∗ Type III optimal alternative: a3 (α) ∈ arg max v x3α (a), a .
(15)
∗ ∗ Type IV optimal alternative: a4 (α) ∈ arg max v x4α (a), a .
(16)
a∈A
a∈A
a∈A
Clearly, focus points in X1α (a), X2α (a), X3α (a) or X4α (a) have the same payoff. ∗ ∗ Type V optimal alternative: a5 (β) ∈ arg max ∗ min v x5β (a), a . a∈A x5 (a)∈X5 (a) β β
(17)
The maximin operator is needed for the cases where multiple focus points of an alternative a exist. It reflects the conservative attitude of a decision maker.
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Type VI optimal alternative: a6 (β) ∈ arg max
min
a∈A x6∗ (a)∈X6 (a) β β
∗
Type VII optimal alternative: a7 (β) ∈ arg max
min
∗ v x6β (a), a .
a∈A x7∗ (a)∈X7 (a) β β
∗
Type VIII optimal alternative: a8 (β) ∈ arg max a∈A
∗
Type IX optimal alternative: a9 ∈ arg max
min
∗ x8β (a)∈X8β (a)
max
9∗
∗ v x7β (a), a .
a∈A x (a)∈X9 (a)
∗ v x8β (a), a .
∗ v x9 (a), a .
(18)
(19)
(20)
(21)
The maximax operator is used for the cases where multiple Type IX focus points of an alternative a exist. It reflects the aggressive attitude of a decision maker. ∗ ∗ Type X optimal alternative: a10 ∈ arg max ∗ min v x10 (a), a . (22) a∈A x10 (a)∈X10 (a)
∗
Type XI optimal alternative: a11 ∈ arg max
min
a∈A x11∗ (a)∈X11 (a)
∗
Type XII optimal alternative: a12 ∈ arg max
12∗
a∈A x
∗ v x11 (a), a .
max
∗ v x12 (a), a .
(23) (24)
(a)∈X12 (a)
It should be noted that the payoff v(x, a) is used for obtaining the optimal alternative instead of the satisfaction u(x, a) in (13)–(24). 3. Multi-period consumption–investment problems with the one-shot decision theory Suppose the initial wealth of an individual is w0 and the length of the process (lifetime) is L. Let us consider how he/she allocates the wealth to make him/her have a largest amount of discounted consumption during the whole process. In each period k, there are a savings account S0 and N securities Sj (1 ≤ j ≤ N ) available for investment. The return rate vector of them at period k + 1 is rk+1 = [rk+1,0 , rk+1,1 , . . . , rk+1,N ]t where rk+1,0 > 0 is the interest rate of the savings account and rk+1,j (1 ≤ j ≤ N ) is the return rate of the j th security. For an individual there is one and only one chance for him/her to allocate the wealth and choose a suitable investment strategy in each period. The individual multi-period consumption–investment decision-making problem is to determine his/her consumption and investment strategy to maximize the sum of the possible discounted consumption over the whole process (lifetime). Clearly, in each period there is a trade-off between consumption and investment, the investor needs to choose a suitable consumption–investment plan. For any period k, wk is the wealth and ck ∈ [0, 1] is the consumption ratio, that is, the proportion of the total wealth devoted to the consumption. Therefore, the amount of wealth consumed by the investor is ck wk and the amount of wealth for investment is wk (1 − ck ). An investment strategy at period k is expressed as ak = [ak,0 , ak,1 , · · · , ak,N ]t where ak,0 is the proportion of the total wealth devoted to the savings account and ak,j (1 ≤ j ≤ N ) is the proportion of the total wealth devoted to the j th security. By the definition of ck and ak , we have ck + N j =0 ak,j = 1, that N is, when the strategy ak is determined, ck can be expressed as ck = ck (ak ) = 1 − j =0 ak,j . Thus, the problem for determining the consumption and investment strategy is reduced to choose a suitable investment strategy which can maximize the total discounted consumption over the whole process (lifetime). We assume that in the final period the individual will consume all the wealth, that is cLwL = wL , cL = 1, where wL is the wealth at period L. The set of all consumption ratios available at period k is expressed as Ck = {ck }. The set of the investment strategies available at period k is Ak = {ak }. In period k, there is one and only one chance for the investor to choose a consumption ratio from Ck and an investment strategy from Ak . By the definition of Ck and Ak , c we know akk is a N + 2 dimensional vector. The rates of returns on the securities at period k + 1 are uncertain, suppose the set of the possible rates of returns is Rk+1 = {rk+1 }. The possibility distribution of Rk+1 is defined as follows.
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Definition 3. The possibility distribution of the return ratio vectors is a function πk+1 : Rk+1 → [0, 1] satisfying maxrk+1 πk+1 (rk+1 ) = 1 and πk+1 (rk+1 ) is the possibility degree of rk+1 in period k + 1. The initial wealth of the individual is w0 . Clearly, there is one and only one chance for this person to allocate his wealth into consumption and investment in each period. By using multi-stage one-shot decision approaches, we can obtain the consumption–investment plan according to his/her preference of the state. First, let us consider the last decision in period L − 1. Giving aL−1 and rL and regarding the wealth wL−1 as a unit, the total amount of consumption in the last two periods is cL−1 + λL aL−1 · (1 + rL ) where λL is the discount factor, 1 = [1, 1, . . . , 1]t and aL−1 · (1 + rL ) is the inner product of aL−1 and 1 + rL . The set of all possible total amounts of consumption in the last two periods is given as GL = gL cL−1 (aL−1 ), aL−1 , rL |aL−1 ∈ AL−1 , rL ∈ RL = cL−1 + λL aL−1 · (1 + rL )|aL−1 ∈ AL−1 , rL ∈ RL . (25) Since the objective is to choose the consumption–investment strategy for maximizing the discounted consumption over periods L − 1 and L, the satisfaction level of the strategy in L − 1 is determined by the total amount of consumption in periods L − 1 and L. We give the satisfaction function for periods L − 1 and L as follows: Definition 4. The function uL : GL → [0, 1] is called a satisfaction function for the period L − 1 and L if it satisfies uL (gL1 ) > uL (gL2 ), for any gL1 > gL2 , gL1 , gL2 ∈ GL . From the definition of uL we can see the higher the total discounted consumption, the higher the satisfaction level. For the same value of cL−1 , the satisfaction function is increasing with the amount consumed in period L. Therefore, the investor consumes all the wealth in the final period L is rational. Using the multi-stage one-shot decision approach, we have the following steps for the last decision period L − 1. c . Since cL−1 = 1 − N Step 1: Seek the focus points for each aL−1 j =0 aL−1,j , cL−1 is the function of aL−1 , denoted L−1 c as cL−1 (aL−1 ). Seeking the focus points of aL−1 is reduced into finding out the focus points for each investment L−1 strategy aL−1 ∈ AL−1 . The focus point of aL−1 , denoted as r∗L (aL−1 ) can be obtained as r∗L (aL−1 ) ∈ arg F πL (rL ), uL gL (cL−1 , aL−1 , rL ) , (26) where F (·) is a general form representing the approaches for seeking one of twelve types’ focus points. For example, F (·) = maxr min[πL , uL ] corresponds to the Type IX focus point of aL−1 . c∗ , that is, the optimal investment strategy a∗L−1 , Step 2: Find out the optimal consumption–investment strategy aL−1 ∗ L−1
which is determined as follows: a∗L−1 ∈ arg
max
aL−1 ∈AL−1
cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) ,
where λL is a discount factor. Denote f1 = max cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) , aL−1 ∈AL−1
(27)
(28)
which is the optimality equation in period L − 1. f1 represents the maximum ratio of the wealth wL−1 which is available for consumption in the last two periods. If the wealth wL−1 and the possibility distribution of the rates of return in period L are given, we can obtain the total discounted consumption in periods L − 1 and L, which equals f1 times wL−1 . Next, we consider the decision in period L − n (n ≥ 2). Set GL−n+1 = gL−n+1 cL−n (aL−n ), aL−n , rL−n+1 |aL−n ∈ AL−n , rL−n+1 ∈ RL−n+1 = cL−n + λL−n+1 aL−n · (1 + rL−n+1 ) fn−1 |aL−n ∈ AL−n , rL−n+1 ∈ RL−n+1 , (29) where λL−n+1 is a discount factor, fn−1 is the optimality equation in period L − n + 1.
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Table 4 The possibility distribution of the return rate of the stock in period 1. Return rate r1,1
0.8r1,0 − 0.2
r1,0
1.3r1,0 + 0.3
1.5r1,0 + 0.5
Possibility degree π1
0.4
0.5
1
0.6
Table 5 The possibility distribution of the return rate of the stock in period 2. Return rate r2,1
0.7r2,0 − 0.3
0.9r2,0 − 0.1
1.2r2,0 + 0.2
1.5r2,0 + 0.5
Possibility degree π2
0.3
0.6
1
0.75
Definition 5. The satisfaction function in period L − n + 1 is given as uL−n+1 : GL−n+1 → [0, 1] satisfying 1 2 1 2 1 2 uL−n+1 (gL−n+1 ) > uL−n+1 (gL−n+1 ) for any gL−n+1 > gL−n+1 , gL−n+1 , gL−n+1 ∈ GL−n+1 . From the definition of uL−n+1 we can see the higher the total discounted consumption, the higher the satisfaction level. We have the following steps for thedecision in period L − n. c Step 1: Find out the focus points of aL−n , that is, the focus points of aL−n ∈ AL−n , denoted as r∗L−n+1 (aL−n ). L−n ∗ rL−n+1 (aL−n ) can be obtained as follows: (30) r∗L−n+1 (aL−n ) ∈ arg F πL−n+1 (rL−n+1 ), uL−n+1 cL−n (aL−n ), aL−n , rL−n+1 , where the same type of focus point as the one obtained in the previous period should be sought due to the logic consistency. Step 2: Obtain the optimal investment strategy in period L − n as follows: a∗L−n ∈ arg max cL−n + λL−n+1 aL−n · 1 + r∗L−n+1 (aL−n ) fn−1 . (31) aL−n ∈AL−n
Denote fn =
max
aL−n ∈AL−n
cL−n + λL−n+1 aL−n · 1 + r∗L−n+1 (aL−n ) fn−1 ,
(32)
which is the optimality equation of period L − n. fn represents the maximum ratio of the wealth wL−n which is available for consuming from period L − n to period L. If we know the wealth wL−n , the possibility distribution of the rates of return in period L − n + 1 and fn−1 , we can obtain the total discounted consumption from period L − n to L, that is equal to fn times wL−n . c∗ (a∗ ) c∗ (a∗ ) c∗ (a∗ ) Repeating (29), (30), (31) and (32), we can obtain the decision sequence 0a∗ 0 , 1a∗ 1 , . . . , L−1a∗ L−1 . For 0
1
L−1
easily understanding the decision procedure introduced above, let us have a look at the following example. Example 1. Given L = 2, there are a savings account S0 and one stock S1 . In period 1, the rate of return of the savings account is r1,0 where r1,0 is a positive constant, the first subscript 1 represents period 1, the second subscript 0 represents the savings account S0 . The rate of return of the stock in period 1 is r1,1 . The possibility distribution of the return ratio in period 1 is shown in Table 4. In period 2, the return rate of the savings account is r2,0 . The return rate of the stock in period 2 is r2,1 whose possibility distribution is shown in Table 5. Let’s take into account the decision in period 1 with Type IX focus point. Assume λ2 = 1+r12,0 where r2,0 is the interest rate of the savings account at period 2, then the problem is reduced to consumption or investment on the stock. The satisfaction function is set as u2 (r2 , a1 ) = (c1 + a1,1 (1 + r2,1 )/(1 + r2,0 ))/1.5 where c1 is the proportion of the total wealth devoted to consumption in period 1 and c1 + a1,1 = 1. Then we can obtain the optimal strategy c9∗ 0.375 r2,0 ∗ ∗ for Type IX focus point is 19∗ = 0 , the focus points of the optimal strategy are r92 (a91 ) = 1.2r2,0 +0.2 and a1
0.625
Y. Li, P. Guo / Fuzzy Sets and Systems 274 (2015) 47–61
55
r2,0 ∗ ∗ r92 (a91 ) = 1.5r2,0 +0.5 . The optimality equation is f1 = 0.375 + 0.625 × 1.5 = 1.3125. Due to λ2 = 1/(1 + r2,0 ) we know the strategies in set {(c1 , a1,0 , 0.625)t |c1 + a1,0 = 0.375, c1 ≥ 0, a1,0 ≥ 0} are all optimal strategies for Type IX focus point. Now let us consider the decision in period 0. Similarly, we assume λ1 = 1/(1 + r1,0 ) where r1,0 is the interest rate of the savings account at period 1. The decision problem is reduced to how to consume or invest on the stock. Set the satisfaction function as u1 (r1 , a0 ) = (c0 + a0,1 f1 (1 + r1,1 )/(1 + r1,0 ))/1.5f1 where c0 is the proportion of the total wealth devoted to consumption in period 0, c1 + a1,1 = 1 and c0 + a0,1 = 1. We can obtain the optimal strategy c9∗ 0 r1,0 ∗ ∗ is 09∗ = 0 . The focus point of the optimal investment strategy is r91 (a90 ) = 1.3r1,0 +0.3 and f2 = 1.3 × f1 = a0
1.70625.
1
4. Theoretical analysis of the models In this section we examine which factors influence consumption and investment and analyze the relationship between different types of focus points. Lemma 1. (1) In any period L − n (1 ≤ n ≤ L), fn ≥ 1 holds. (2) Suppose the optimal investment strategy in period L − n is a∗L−n , then a∗L−n · (1 + r∗L−n+1 (a∗L−n )) ≥ 0, where ∗ rL−n+1 (a∗L−n ) is the focus point of a∗L−n . Proof. (1) In any period L − n, if the investor consumes all, that is cL−n = 1, the ratio of wL−n is equal to 1. By the definition of the optimality equation, fn ≥ 1 holds. (2) In the decision period L − 1, if a∗L−1 · (1 + r∗L (a∗L−1 )) < 0, then ∗ (33) f1 = cL−1 + λL a∗L−1 · 1 + r∗L a∗L−1 < 1. Eq. (33) contradicts with f1 ≥ 1, so we have a∗L−1 · (1 + r∗L (a∗L−1 )) ≥ 0. Similarly, in the decision period L − n (n ≥ 2), if a∗L−n · (1 + r∗L−n+1 (a∗L−n )) < 0, we have ∗ fn = cL−n + λL−n+1 a∗L−n · 1 + r∗L−n+1 a∗L−n fn−1 < 1. Eq. (34) contradicts with fn ≥ 1. So we can say in any period L − n, a∗L−n · (1 + r∗L−n+1 (a∗L−n )) ≥ 0.
(34) 2
Now let us consider the relationship between the optimality equation and the discount factor. Proposition 2. λ1L−n+1 and λ2L−n+1 are discount factors satisfying λ2L−n+1 > λ1L−n+1 , fn1 and fn2 are the optimality equations with λ1L−n+1 and λ2L−n+1 . In any period L − n (1 ≤ n ≤ L), the followings hold. (1) (2) (3) (4)
For Type I focus points fn2 ≥ fn1 , For Type II focus points fn2 ≥ fn1 , For Type III focus points fn2 ≥ fn1 , For Type IV focus points fn2 ≥ fn1 .
Proof. (1) In the last decision period L − 1, the optimality equation with λ2L and λ1L are 2 ∗ λL + λ2L a∗L−1 λ2L · 1 + r∗L a∗L−1 λ2L , λ2L , f12 = cL−1
(35)
and 1 ∗ f11 = cL−1 λL + λ1L a∗L−1 λ1L · 1 + r∗L a∗L−1 λ1L , λ1L . ∗ cL−1 (λ2L )
a∗L−1 (λ2L )
(36)
is the optimal consumption–investment strategy with λ2L and
investment strategy with λ1L when we choose Type I focus point.
∗ cL−1 (λ1L )
is the optimal consumption–
a∗L−1 (λ1L ) ∗ ∗ rL (aL−1 (λ2L ), λ2L ))
and r∗L (a∗L−1 (λ1L ), λ1L )) are
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the Type I focus points of a∗L−1 (λ2L ) with λ2L and a∗L−1 (λ1L ) with λ1L , respectively. For any investment strategy aL−1 satisfying aL−1 · (1 + r∗L (aL−1 , λ1L )) ≥ 0, we have (37) cL−1 + λ1L aL−1 · 1 + r∗L aL−1 , λ1L ≤ cL−1 + λ2L aL−1 · 1 + r∗L aL−1 , λ1L . For rL ∈ {rL |πL (rL ) ≥ α}, by the definition of the Type I focus point, we obtain cL−1 + λ2L aL−1 · (1 + rL ) ≤ cL−1 + λ2L aL−1 · 1 + r∗L aL−1 , λ2L .
(38)
We know a∗L−1 (λ2L ), a∗L−1 (λ1L ) are the optimal investment strategies with λ2L and λ1L respectively, by the Lemma 1 a∗L−1 λ2L · 1 + r∗L a∗L−1 λ2L , λ2L ≥ 0, (39) and a∗L−1 λ1L · 1 + r∗L a∗L−1 λ1L , λ1L ≥ 0.
(40)
Then by (37) and (38), we know 1 ∗ λL + λ1L a∗L−1 λ1L · 1 + r∗L a∗L−1 λ1L , λ1L f11 = cL−1 1 ∗ λL + λ2L a∗L−1 λ1L · 1 + r∗L a∗L−1 λ1L , λ1L ≤ cL−1 1 ∗ λL + λ2L a∗L−1 λ1L · 1 + r∗L a∗L−1 λ1L , λ2L . ≤ cL−1
(41)
By the definition of the optimality equation, we have 1 ∗ λL + λ2L a∗L−1 λ1L · 1 + r∗L a∗L−1 λ1L , λ2L cL−1 2 ∗ λL + λ2L a∗L−1 λ2L · 1 + r∗L a∗L−1 λ2L , λ2L = f12 . ≤ cL−1
(42)
According to (41) and (42), we obtain f12 ≥ f11 . In period L − n (n ≥ 2), we have 2 2 ∗ λL−n+1 + λ2L−n+1 a∗L−n λ2L−n+1 · 1 + r∗L−n+1 a∗L−n λ2L−n+1 , λ2L−n+1 fn−1 fn2 = cL−n ,
(43)
1 1 ∗ λL−n+1 + λ1L−n+1 a∗L−n λ1L−n+1 · 1 + r∗L−n+1 a∗L−n λ1L−n+1 , λ1L−n+1 fn−1 , fn1 = cL−n
(44)
where
∗ cL−n (λ2L−n+1 )
a∗L−n (λ2L−n+1 )
is the optimal consumption–investment strategy with λ2L−n+1 and
∗ cL−n (λ1L−n+1 )
is the opti-
a∗L−n (λ1L−n+1 ) 1 mal consumption–investment strategy with λL−n+1 when we choose Type I focus point. r∗L−n+1 (a∗L−n (λ2L−n+1 ), λ2L−n+1 ) and r∗L−n+1 (a∗L−n (λ1L−n+1 ), λ1L−n+1 ) are the Type I focus point of a∗L−n (λ2L−n+1 ) with λ2L−n+1 and the Type I focus point of a∗L−n (λ1L−n+1 ) with λ1L−n+1 , respectively. Similar to proving f12 ≥ f11 , for any strategy aL−n 1 ≥ 0, we have satisfying aL−n · (1 + r∗L−n+1 (aL−n , λ1L ))fn−1
1 cL−n + λ1L−n+1 aL−n · 1 + r∗L−n+1 aL−n , λ1L−n+1 fn−1 1 ≤ cL−n + λ2L−n+1 aL−n · 1 + r∗L−n+1 aL−n , λ1L−n+1 fn−1 2 ≤ cL−n + λ2L−n+1 aL−n · 1 + r∗L−n+1 aL−n , λ1L−n+1 fn−1 2 ≤ cL−n + λ2L−n+1 aL−n · 1 + r∗L−n+1 aL−n , λ2L−n+1 fn−1 .
(45)
Then 1 1 ∗ fn1 = cL−n λL−n+1 + λ1L−n+1 a∗L−n λ1L−n+1 · 1 + r∗L−n+1 a∗L−n λ1L−n+1 , λ1L−n+1 fn−1 1 1 ∗ λL−n+1 + λ2L−n+1 a∗L−n λ1L−n+1 · 1 + r∗L−n+1 a∗L−n λ1L−n+1 , λ1L−n+1 fn−1 ≤ cL−n 1 2 ∗ λL−n+1 + λ2L−n+1 a∗L−n λ1L−n+1 · 1 + r∗L−n+1 a∗L−n λ1L−n+1 , λ2L−n+1 fn−1 ≤ cL−n 2 ∗ λL−n+1 + λ2L−n+1 a∗L−n λ2L−n+1 · (1 + r∗L−n+1 a∗L−n λ2L−n+1 , λ2L−n+1 = fn2 . ≤ (cL−n Likewise, we can prove (2), (3) and (4).
2
(46)
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57
From Proposition 2 we know fL2 ≥ fL1 holds with λ2L−n+1 > λ1L−n+1 for Types I, II, III, IV focus points. Since and fL1 w0 are the two maximum total consumption amount, it means if the individual enjoys more consumption at the present time than in the future (decreasing the discount factor), the total consumption amount will decrease. Such conclusions are quite in agreement with the situations encountered in the real world. fL2 w0
c at period L − n (1 ≤ n ≤ L), Type I, Type II, Type Theorem 3. For any consumption–investment strategy aL−n L−n ∗ ∗ III, Type IV, Type IX, Type X, Type XI and Type XII focus points are denoted as r1L−n+1 (aL−n ), r2L−n+1 (aL−n ), ∗ ∗ ∗ ∗ ∗ ∗ 11 12 1∗ r3L−n+1 (aL−n ), r4L−n+1 (aL−n ), r9L−n+1 (aL−n ), r10 L−n+1 (aL−n ), rL−n+1 (aL−n ) and rL−n+1 (aL−n ), respectively. fn , ∗ ∗ ∗ ∗ ∗ ∗ ∗ fn2 , fn3 , fn4 , fn9 , fn10 , fn11 and fn12 are the optimality equations with Type I, Type II, Type III, Type IV, Type IX, Type X, Type XI and Type XII focus points, respectively. We have (1) (2) (3) (4)
∗
∗
fn2 ≤ fn1 , ∗ ∗ fn4 ≤ fn3 , ∗ ∗ 11 fn ≤ fn9 , ∗ ∗ 10 12 fn ≤ f n .
Proof. (1) Let us examine the last decision period L − 1. For any investment strategy aL−1 , by the definitions of Types I and II focus points we have ∗ ∗ uL cL−1 (aL−1 ), aL−1 , r2L (aL−1 ) ≤ uL cL−1 (aL−1 ), aL−1 , r1L (aL−1 ) , (47) that is
∗ ∗ cL−1 + λL aL−1 · 1 + r2L (aL−1 ) ≤ cL−1 + λL aL−1 · 1 + r1L (aL−1 ) .
(48)
2∗
For aL−1 , which is the optimal investment strategy with the Type II focus point, we have ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2∗ 2∗ f12 = cL−1 + λL a2L−1 · 1 + r2L a2L−1 ≤ cL−1 + λL a2L−1 1 + r1L a2L−1 .
(49)
1∗
By the definition of f1 , we have ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2∗ 1∗ cL−1 + λL a2L−1 1 + r1L a2L−1 ≤ cL−1 + λL a1L−1 1 + r1L a1L−1 = f11 . 2∗
(50)
1∗
It follows from (49) and (50) that f1 ≤ f1 holds. In period L − n (n ≥ 2), the optimality functions for Type I and Type II focus points are ∗ 1∗ ∗ ∗ ∗ 1∗ fn1 = cL−n + λL−n+1 a1L−n · 1 + r1L−n+1 a1L−n fn−1 , (51) and
∗ 2∗ ∗ ∗ ∗ 2∗ fn2 = cL−n + λL−n+1 a2L−n · 1 + r2L−n+1 a2L−n fn−1 . 2∗
(52)
1∗
2∗
Supposing fn−1 ≤ fn−1 , for any investment strategy aL−n satisfying aL−n · (1 + rL−n+1 (aL−n )) ≥ 0, we have 2∗ 1∗ ∗ ∗ cL−n + λL−n+1 aL−n · 1 + r2L−n+1 (aL−n ) fn−1 ≤ cL−n + λL−n+1 aL−n · 1 + r2L−n+1 (aL−n ) fn−1 .
(53)
By the definition of Type I focus point, we know 1∗ 1∗ ∗ ∗ cL−n + λL−n+1 aL−n · 1 + r2L−n+1 (aL−n ) fn−1 ≤ cL−n + λL−n+1 aL−n · 1 + r1L−n+1 (aL−n ) fn−1 .
(54)
2∗
2∗
2∗
1∗
2∗
1∗
1∗
Considering (53) and (54), we have fn ≤ cL−n + λL−n+1 aL−n (1 + rL−n+1 (aL−n ))fn−1 ≤ fn . In a similar way, we can prove (2). (3) In the last decision period L − 1, for any investment strategy aL−1 , according to Theorem 2 [7] we have ∗ 9∗ uL cL−1 (aL−1 ), aL−1 , r11 (55) L (aL−1 ) ≤ uL cL−1 (aL−1 ), aL−1 , rL (aL−1 ) , that is
∗ 9∗ cL−1 + λL aL−1 · 1 + r11 L (aL−1 ) ≤ cL−1 + λL aL−1 · 1 + rL (aL−1 ) .
(56)
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Y. Li, P. Guo / Fuzzy Sets and Systems 274 (2015) 47–61 ∗
For the optimal investment strategy a11 L−1 with the Type XI focus point, we have ∗ ∗ 11∗ 11∗ 11∗ 11∗ 11∗ 9∗ 11∗ f111 = cL−1 + λL a11 L−1 · 1 + rL aL−1 ≤ cL−1 + λL aL−1 1 + rL aL−1 . By the definition of ∗
∗
(57)
∗ f19 ,
9 cL−1 + λL a9L−1
we have ∗ ∗ 9∗ 9∗ 9∗ 9∗ 9∗ 1 + r9L a11 L−1 ≤ cL−1 + λL aL−1 1 + rL aL−1 = f1 .
∗
∗
∗
(58) ∗
11 ≤ f 9 . For any investment strategy From (57) and (58) we know f111 ≤ f19 . In period L − n (n ≥ 2), suppose fn−1 n−1 aL−n , by Theorem 3 [9] we have 11∗ 9∗ ∗ 9∗ cL−n + λL−n+1 aL−n · 1 + r11 (59) L−n+1 (aL−n ) fn−1 ≤ cL−n + λL−n+1 aL−n · 1 + rL−n+1 (aL−n ) fn−1 . ∗
For the optimal investment strategy a11 L−n with the Type XI focus point, we have 11∗ 11∗ ∗ ∗ ∗ 11 11∗ f111 = cL−n + λL−n+1 a11 L−n · 1 + rL−n+1 aL−n fn−1 11∗ 9∗ ∗ 11∗ 9∗ ≤ cL−n + λL−n+1 a11 L−n · 1 + rL−n+1 aL−n fn−1 .
(60)
9∗
By the definition of fn , we have 11∗ 9∗ ∗ 11∗ 9∗ cL−n + λL−n+1 a11 L−n · 1 + rL−n+1 aL−n fn−1 ∗ 9∗ ∗ ∗ ∗ 9∗ ≤ cL−n + λL−n+1 a9L−n · 1 + r9L−n+1 a9L−n fn−1 = fn9 . 11∗
It follows (60) and (61) that fn
9∗
≤ fn holds. Similarly, we can prove (4) by using Theorem 2 and Theorem 4 [9].
(61) 2
From Theorem 3 we know that in the same investment environment, a person who is concerned with different Type of focus point will have different imagination about how the initial wealth will be changed and how he/she should consume and invest in each stage. This theorem gives an explanation about why there is a variety of ways of the individual consumption. 1 2 Proposition 4. Suppose in any period L − n (1 ≤ n ≤ L), there are two possibility functions πL−n+1 and πL−n+1 . π1
π2
1 2 and πL−n+1 respectively. If ∀rL−n+1 ∈ RL−n+1 , fn L−n+1 and fn L−n+1 are the optimality equations with πL−n+1 1 2 πL−n+1 (rL−n+1 ) ≥ πL−n+1 (rL−n+1 ) holds, then π1
π2
(1) For Type I focus points fn L−n+1 ≥ fn L−n+1 , π1
π2
(2) For Type II focus points fn L−n+1 ≤ fn L−n+1 , π1
π2
π1
π2
(3) For Type III focus points fn L−n+1 ≤ fn L−n+1 , (4) For Type IV focus points fn L−n+1 ≥ fn L−n+1 . 1 (r ) ≥ π 2 (r ) (∀r ∈ R ), for any investment strategy Proof. (1) In the last decision period L − 1, since πL−1 L L L L−1 L aL−1 , considering the definition of the Type I focus point we have (62) uL cL−1 (aL−1 ), aL−1 , r∗L aL−1 , πL1 ≥ uL cL−1 (aL−1 ), aL−1 , r∗L aL−1 , πL2 ,
where r∗L (aL−1 , πL1 ) and r∗L (aL−1 , πL2 ) are the focus points of aL−1 with πL1 and πL2 , respectively. That is, cL−1 + λL aL−1 · 1 + r∗L aL−1 , πL1 ≥ cL−1 + λL aL−1 · 1 + r∗L aL−1 , πL2 .
(63)
The optimality equations for πL1 and πL2 are 1 π1 ∗ πL + λL a∗L−1 πL1 · 1 + r∗L a∗L−1 πL1 , πL1 , f1 L = cL−1
(64)
2 π2 ∗ πL + λL a∗L−1 πL2 · 1 + r∗L a∗L−1 πL2 , πL2 , f1 L = cL−1
(65)
and
Y. Li, P. Guo / Fuzzy Sets and Systems 274 (2015) 47–61
59
where a∗L−1 (πL1 ), a∗L−1 (πL2 ) are the optimal investment strategies for πL1 and πL2 , respectively. r∗L (a∗L−1 (πL1 ), πL1 ) and r∗L (a∗L−1 (πL2 ), πL2 ) are the Type I focus point of a∗L−1 (πL1 ) with πL1 and the Type I focus point of a∗L−1 (πL2 ) with πL2 , π1
respectively. By the definition of f1 L we have 2 π1 ∗ f1 L ≥ cL−1 πL + λL a∗L−1 πL2 · 1 + r∗L a∗L−1 πL2 , πL1 .
(66)
From (63) we have 2 π2 ∗ πL + λL a∗L−1 πL2 · 1 + r∗L a∗L−1 πL2 , πL1 ≥ f1 L . cL−1 π1
(67)
π2
From (66) and (67) we know f1 L ≥ f1 L . In the period L − n (n ≥ 2), we have 1 1 1 1 1 πL−n+2 π1 ∗ πL−n+1 + λL−n+1 a∗L−n πL−n+1 · 1 + r∗L−n+1 a∗L−n πL−n+1 , πL−n+1 fn−1 fn L−n+1 = cL−n , 2 2 πL−n+1 π L−n+2 ∗ 2 2 2 2 fn πL−n+1 + λL−n+1 a∗L−n πL−n+1 · 1 + r∗L−n+1 a∗L−n πL−n+1 , πL−n+1 fn−1 = cL−n .
(68) (69)
1 1 2 a∗L−n (πL−n+1 ) is the optimal investment strategy with πL−n+1 and a∗L−n (πL−n+1 ) is the optimal investment strat2 ∗ ∗ 1 1 ∗ ∗ 2 2 ) are the focus points of egy with πL−n+1 . rL−n+1 (aL−n (πL−n+1 ), πL−n+1 ) and rL−n+1 (aL−n (πL−n+1 ), πL−n+1 ∗ 1 1 ∗ 2 2 aL−n (πL−n+1 ) with πL−n+1 and the focus point of aL−n (πL−n+1 ) with πL−n+1 , respectively. For any investment π1
2 strategy aL−n satisfying aL−n · (1 + r∗L−n+1 (aL−n , πL−n+1 )) ≥ 0, considering the definition of fn L−n+1 we have 2 πL−n+2 2 fn−1 cL−n + λL−n+1 aL−n · 1 + r∗L−n+1 aL−n , πL−n+1 1 πL−n+2 2 ≤ cL−n + λL−n+1 aL−n · 1 + r∗L−n+1 aL−n , πL−n+1 fn−1 1 πL−n+2 1 fn−1 . ≤ cL−n + λL−n+1 aL−n · 1 + r∗L−n+1 aL−n , πL−n+1
(70)
Eq. (70) implies 1 2 2 2 1 πL−n+2 π1 ∗ πL−n+1 + λL−n+1 a∗L−n πL−n+1 · 1 + r∗L−n+1 a∗L−n πL−n+1 , πL−n+1 fn−1 fn L−n+1 ≥ cL−n
π2
≥ fn L−n+1 .
(71)
In a similar way, we can prove (2), (3), (4). 2 Proposition 4 shows that increasing the uncertainty will make the individual who chooses Type I or Type IV focus point feel that the total amount of consumption available becomes lager, the individual who chooses Type II or Type III focus point believe that the total amount of consumption available becomes smaller. The different feelings about the total amount of consumption available may lead them to make different consumption–investment plans. 1 2 1 In period L − n (1 ≤ n ≤ L), if πL−n+1 and πL−n+1 are isotonically transformable, that is, (πL−n+1 (rL−n+1 ) − 1 2 2 1 1 πL−n+1 (rL−n+1 ))(πL−n+1 (rL−n+1 ) − πL−n+1 (rL−n+1 )) > 0 or |πL−n+1 (rL−n+1 ) − πL−n+1 (rL−n+1 )| 2 2 (rL−n+1 ) − πL−n+1 (rL−n+1 )| = 0 holds for any rL−n+1 , rL−n+1 ∈ RL−n+1 [9], we have the following + |πL−n+1 lemma. 1 2 Lemma 5. In any period L − n (1 ≤ n ≤ L), if πL−n+1 and πL−n+1 are isotonically transformable, for Types V, VI, π1
π2
VII, VIII focus points, we have fn L−n+1 = fn L−n+1 . 1 2 It is trivial to prove Lemma 5 because if πL−n+1 and πL−n+1 are isotonically transformable, for any investment 2 will remain the same as the focus points with strategy aL−n , the Types V, VI, VII, VIII focus points with πL−n+1 1 πL−n+1 . Now let us consider how the rates of return change influence the individual consumption and investment.
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Proposition 6. Suppose in any period L − n (1 ≤ n ≤ L), there are two sets of return ratios RL−n+1 = {rL−n+1 } and RδL−n+1 = {rδL−n+1 |rδL−n+1 = rL−n+1 + δ, rL−n+1 ∈ RL−n+1 , δ ≥ 0}. fn and fnδ are the optimality equations with RL−n+1 and RδL−n+1 , respectively. The followings hold (1) (2) (3) (4)
For Type I focus points fn ≤ fnδ , For Type II focus points fn ≤ fnδ , For Type III focus points fn ≤ fnδ , For Type IV focus points fn ≤ fnδ .
Proof. (1) In the last decision period L − 1, for any investment strategy aL−1 , we have cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) ≤ cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) + δ ,
(72)
where r∗L (aL−1 ) is the Type I focus point of aL−1 with RL . Considering the definitions of Type I focus point and RδL−n+1 we obtain cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) + δ ≤ cL−1 + λL aL−1 · 1 + r∗L (aL−1 , δ) ,
(73)
where r∗L (aL−1 , δ) is the Type I focus point of aL−1 with RδL . For the optimal investment strategy a∗L−1 with RL , from (72) and (73), we have ∗ ∗ f1 = cL−1 + λL a∗L−1 · 1 + r∗L a∗L−1 ≤ cL−1 + λL a∗L−1 · 1 + r∗L a∗L−1 , δ ∗ ≤ cL−1 (δ) + λL a∗L−1 (δ) · 1 + r∗L a∗L−1 (δ), δ = f1δ .
(74)
In any period L − n (n ≥ 2), we have ∗ fn = cL−n + λL−n+1 a∗L−n · 1 + r∗L−n+1 a∗L−n fn−1 ,
(75)
δ ∗ fnδ = cL−n (δ) + λL−n+1 a∗L−n (δ) · 1 + r∗L−n+1 a∗L−n (δ), δ fn−1 ,
(76)
where a∗L−n is the optimal investment strategy with RL−n+1 and a∗L−n (δ) is the optimal investment strategy with RδL−n+1 . r∗L−n+1 (a∗L−n ) and r∗L−n+1 (a∗L−n (δ), δ) are the focus points of a∗L−n with RL−n+1 and the focus point of a∗L−n (δ) with RδL−n+1 , respectively. Similarly, we can prove fn ≤ fnδ . (2) In the last decision period L − 1, for any investment strategy aL−1 , if cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) > cL−1 + λL aL−1 · 1 + r∗L (aL−1 , δ) ,
(77)
where r∗L (aL−1 ) is the Type II focus point of aL−1 with RL and r∗L (aL−1 , δ) is the Type II focus point of aL−1 with RδL , we have cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) > cL−1 + λL aL−1 · 1 + r∗L (aL−1 , δ) − δ .
(78)
Since (78) contradicts with the definition of Type II focus point, we know for any investment strategy cL−1 + λL aL−1 · 1 + r∗L (aL−1 ) ≤ cL−1 + λL aL−1 · 1 + r∗L (aL−1 , δ) .
(79)
By (79), f1 ≤ f1δ holds. Similarly, we can prove fn ≤ fnδ . Likewise, we can prove (3) and (4). 2 Proposition 6 shows that when the economic situation changes for the better, a person believes the total amount of consumption in the future will increase in any period. Such a conclusion is in harmony with the common sense in the real world.
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5. Conclusions In this paper, with considering an individual has one and only one chance to make a decision on how to consume and how to invest in each period with possibilisitc information, the scenario-based multi-period consumption–investment models with the one-shot decision theory are proposed. In the proposed models, the securities’ return ratio vector is governed by the possibility distribution. Since in each period, one and only one return ratio vector will come, an individual contemplate one vector amongst all possible vectors based on the satisfaction level caused by the appearance of this vector and the possibility degree to which this vector appears. The selected return ratio vector is called focus point. There are twelve types of focus points. Which type of focus point should be selected depends on the decision maker’s attitude about satisfaction and possibility at each period. Based on the selected vectors (focus points), the sequence of optimal consumption–investment plan is determined by dynamic programming to maximize the sum of possible discounted consumption over the whole process (lifetime). Several theoretical results are obtained and economic insights are gained. Since the proposed models fit the characteristics of the individual multi-period consumption–investment behavior, possibilistic individual multi-period consumption–investment models can provide a useful vehicle to deal with such an important economic problem. References [1] A. Cadenillas, Consumption–investment problems with transaction costs: survey and open problems, Math. Methods Oper. Res. 51 (1) (2000) 43–68. [2] K.C. Cheung, H. Yang, Optimal investment–consumption strategy in a discrete-time model with regime switching, Discrete Contin. Dyn. Syst., Ser. B 8 (2) (2007) 315–332. [3] E.F. Fama, Risk, return and equilibrium, J. Polit. Econ. 79 (1) (1971) 30–55. [4] P. Guo, One-shot decision approach and its application to duopoly market, Int. J. Inf. Decis. Sci. 2 (3) (2010) 213–232. [5] P. Guo, Private real estate investment analysis within a one-shot decision framework, Int. Real. Estate Rev. 13 (3) (2010) 238–260. [6] P. Guo, R. Yan, J. Wang, Duopoly market analysis within one-shot decision framework with asymmetric possibilistic information, Int. J. Comput. Intell. Syst. 3 (6) (2010) 786–796. [7] P. Guo, One-shot decision theory, IEEE Trans. Syst. Man Cybern., Part A, Syst. Hum. 41 (5) (2011) 917–926. [8] P. Guo, One-shot decision theory: a fundamental alternative for decision under uncertainty, in: P. Guo, W. Pedrycz (Eds.), Human-Centric Decision-Making Models for Social Sciences, in: Studies in Computational Intelligence, vol. 502, Springer-Verlag, Berlin, Heidelberg, 2014, pp. 33–55. [9] P. Guo, Y. Li, Approaches to multistage one-shot decision making, Eur. J. Oper. Res. 236 (2) (2014) 612–623. [10] P. Guo, X. Ma, Newsvendor models for innovative products with one-shot decision theory, Eur. J. Oper. Res. 239 (2) (2014) 523–536. [11] I. Karatzas, H. Wang, Utility maximization with discretionary stopping, SIAM J. Control Optim. 39 (1) (2000) 306–329. [12] H. Kraft, F.T. Seifried, M. Steffensen, Consumption-portfolio optimization with recursive utility in incomplete markets, Finance Stoch. 17 (1) (2013) 161–196. [13] Zh. Li, K.S. Tan, H. Yang, Multiperiod optimal investment–consumption strategies with mortality risk and environment uncertainty, N. Am. Actuar. J. 12 (1) (2008) 47–64. [14] J. Lintner, Security prices, risk and maximal gains from diversification, J. Finance 20 (5) (1965) 587–616. [15] R.C. Merton, Lifetime portfolio selection under uncertainty: the continuous-time case, Rev. Econ. Stat. 51 (3) (1969) 247–257. [16] R.C. Merton, Optimal consumption and portfolio rules in a continuous-time model, J. Econ. Theory 3 (4) (1971) 373–413. [17] R.C. Merton, Continuous-Time Finance, Basil Blackwell, Oxford, 1990. [18] J. Mossin, Optimal multi-period portfolio policies, J. Bus. 41 (2) (1968) 215–229. [19] P.H. Nielsen, M. Steffensen, Optimal investment and life insurance strategies under minimum and maximum constraints, Insur. Math. Econ. 43 (1) (2008) 15–28. [20] S.R. Pliska, Introduction to Mathematical Finance: Discrete Time Models, Blackwell, 1997. [21] S.R. Pliska, J. Ye, Optimal life insurance purchase and consumption/investment under uncertain lifetime, J. Bank. Finance 31 (5) (2007) 1307–1319. [22] P.A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Rev. Econ. Stat. 51 (3) (1969) 239–246. [23] L.R. Sotomayor, A. Cadenillas, Explicit solutions of consumption–investment problems in financial markets with regime switching, Math. Finance 19 (2) (2009) 251–279. [24] M. Steffensen, Optimal consumption and investment under time-varying relative risk aversion, J. Econ. Dyn. Control 35 (5) (2011) 659–667. [25] J. Su, X. Guan, Multi-objective Optimal Strategy for Individual Consumption–Investment with Fuzzy Coefficients, Lecture Notes in Computer Science, vol. 4113, 2006, pp. 919–924.