Discrete time multi-period mean-variance model: Bellman type strategy and Empirical analysis
DDiscrete time multi-period mean-variance model: Bellman typestrategy and Empirical analysis *Shuzhen Yang †‡ Abstract : In this paper, we attempt to introduce the Bellman principle for a discrete time multi-period mean-variance model. Based on this new take on the Bellman principle, we obtain a dynamictime-consistent optimal strategy and related e ffi cient frontier. Furthermore, we develop a varyinginvestment period discrete time multi-period mean-variance model and obtain a related dynamic opti-mal strategy and an optimal investment period. This paper compares the highlighted dynamic optimalstrategies of this study with the 1 / n equality strategy, and shows that we can secure a higher returnwith a smaller risk based on the dynamic optimal strategies. Since the foundational work of Markowitz (1952, 1959) on the portfolio selection problem, themean-variance investment model has been used to balance the return and risk of the wealth of portfo-lios (Also see Markowitz (2014)). The single-period mean-variance model, which is used presently,was developed to include an explicit solution of the optimal strategy and its application in real mar-kets. (Further see Merton (1972)). Apart from the single-period framework, many authors considerstudying the multi-period mean-variance model, which optimizes the multi-period objectives witha dynamic strategy. A lot of literature discusses the multi-period mean-variance model, includingboth the discrete and continuous time cases. For a given investment period T , the investor wants tominimize variance in the portfolio’s wealth at T with conditions regarding the return on investment.Because the variance does not satisfy the iterated-expectation property, it is di ffi cult to find a dynamic,time-consistent, optimal strategy for the multi-period mean-variance model. There are two kinds of * Keywords : Dynamic programming; Mean-variance; Multi-period.
Journal of Economic Literature classification Numbers : C61; D81; G11.
MSC2010 subject classification : 90C39; 93E20; 49L10. † Shandong University-Zhong Tai Securities Institute for Financial Studies, Shandong University, PR China,([email protected]). ‡ This work was supported by the National Natural Science Foundation of China (Grant No.11701330) and YoungScholars Program of Shandong University. a r X i v : . [ q -f i n . M F ] N ov ptimal strategies for solving the multi-period mean-variance model: one is called the pre-committedstrategy which is derived by optimizing the variance of wealth at T , and the second is called thegame-theoretic strategy which is derived by defining a local maximum principle.For the pre-committed strategy, a continuous time mean-variance model with one risky assetstock and bond was considered in Richardson (1989) and subsequently the optimal strategy wasgiven. When combining initial time and terminal time, Bajeux-Besnainou and Portait (1998) de-veloped the portfolio strategies for the related mean-variance model. By embedding the multi-periodmean-variance problem into a multi-objective optimization framework, Li and Ng (2000) found a re-lated dynamic optimal strategy. Later, many authors began to study the mean-variance model in con-tinuous time. Based on a similar technique in Li and Ng (2000), the continuous-time mean-varianceproblem was studied in Zhou and Li (2000) and an optimal strategy and e ffi cient frontier were given.In a straightforward manner, a cost-e ffi cient approach to the optimal portfolio selection for the mean-variance problem was proposed in Dybvig (1988). Employing the cost-e ffi cient approach, Bernardand Vandu ff el (2014) solved the problem of a mean-variance in the presence of a benchmark. Wecan see that the optimal strategy was suggested by the cost-e ffi cient approach, which is consistentwith the results of Zhou and Li (2000). Furthermore, we refer the reader to (Bielecki et al., 2005;Dai et al., 2010; Lim and Zhou, 2002; Lim, 2004; Xia, 2005; Bi et al., 2018) for more details of themean-variance problem in continuous time.Based on the optimal control problem of a stochastic di ff erential equation of mean-field type,the stochastic maximum principle to the mean-variance portfolio selection problem was investigatedin Andersson and Djehiche (2011), and the related optimal strategy coexists with that in Zhou andLi (2000). Furthermore, an integral form stochastic maximum principle for general mean-field opti-mal control systems was established in Li (2012), and was applied to solve a mean-field type linearquadratic stochastic control problem, also see Buckdahn et al. (2011). Based on the mean field ap-proach, the continuous time mean-variance portfolio optimization problem was studied in Fischerand Livieri (2016), which obtained a related pre-committed strategy. The optimal control of a generalstochastic McKean-Vlasov equation was studied in Pham and Wei (2017), and the dynamic pro-gramming principle for the value function was established in the Wasserstein space of probabilitymeasures. Furthermore, Pham and Wei (2017) solved the linear-quadratic stochastic McKean-Vlasovcontrol problem and an interbank systemic risk model with common noise were investigated, also seePham and Wei (2018). The explicit solution for the optimal robust portfolio strategies in the case ofuncertain volatilities was given in Ismail and Pham (2019), which coincides with those in Zhou andLi (2000) and Fischer and Livieri (2016).For the game-theoretic strategy, a dynamic method is given in Basak and Chabakauri (2010),which is used to study the mean-variance model by introducing an adjustment term in the objective.2 general time-inconsistent stochastic linear-quadratic control problem was established in Hu et al.(2012), and an equilibrium, instead of optimal control was defined. (Further see Yong (2012)). Inaddition to this, Hu et al. (2012) considered a pre-committed strategy for the mean-variance model.The large population stochastic dynamic games and the Nash Certainty Equivalence based controllaws was investigated in Huang et al. (2007). We refer the readers to (Bensoussan et al., 2016, 2013;Bj¨ork et al., 2014, 2017; Dai et al., 2019) for more details of game-theoretic approach.Recently, Yang (2020) investigated a new method to solve the multi-period mean-variance prob-lem in continuous time. Let X π ( · ) denote the wealth of the investor in the investment time interval[ t , T ] with the initial time t , where π ( · ) is the related strategy. Note that, the variance Var[ X π ( T )]does not satisfy the iterated-expectation property, which deduces that one cannot use the dynamicprogramming principle to solve the mean-variance problem in multi-period mean-variance model. Tosolve the above problem, Yang (2020) introduced a deterministic process Y π ( · ) to represent the meanprocess E [ X π ( · )] such that the variance satisfies Bellman dynamic programming principle. Basedon the idea developed in Yang (2020), we want to investigate the discrete time multi-period mean-variance problem and consider the related empirical analysis in this study. Note that, we cannot usethe Itˆo formula and partial di ff erential equation tool which were used in Yang (2020) to solve themulti-period mean-variance problem in discrete time case. By introducing a deterministic process Y π ( · ), which is equal to the expectation of X π ( · ) with initial value y ∈ R , the related cost functional isgiven as follows: ˜ J ( t , x , y , µ ; π ( · )) = µ E [ (cid:0) X π ( T ) − Y π ( T ) (cid:1) ] − E [ X π ( T )] , (1.1)where µ is the risk aversion coe ffi cient and X π ( T ) with initial value x ∈ R . Hence, we can separatethe process Y π ( T ) from the cost functional (1.1), by defining the value function: V µ ( t , x , y ) = inf π ∈A T − t ˜ J ( t , x , y , µ ; π ( · )) , where A T − t is the set of all adapted strategies. First, we can establish the Bellman principle for valuefunction V µ ( t , x , y ), based on which we can obtain the Bellman type dynamic time-consistent optimalstrategy and dynamic e ffi cient frontier for the value function V µ ( t , x , y ). We denote the related optimalstrategy as STRATEGY I. In particular, when the number of investment period N → ∞ , the Bellmantype optimal strategy converges to the optimal strategy given in Yang (2020).To reduce the variance of the classical mean-variance model in continuous time, Yang (2019)proposed a varying investment period mean-variance model with a constraint on the mean value ofthe portfolio’s wealth, which moves with the varying investment period. In this study, we introduce avarying investment period mean-variance model in discrete time. For a given deterministic investmentperiod τ ∈ N + , we consider a varying investment period structure for the classical discrete mean-3ariance model: τ π = inf { s : E [ X π ( s )] ≥ g ( τ ) , < s ≤ τ } (cid:94) τ, where a (cid:86) a , a , a ∈ R means a (cid:86) a = min( a , a ), and g ( · ) is the varying expected returnon investment X π ( · ). Based on this varying investment period τ π , we obtain the related Bellmantype time-consistent optimal strategy and optimal investment period. We denote the related optimalstrategy as STRATEGY II.Based on the out-of-sample performance of the sample-based mean-variance model, DeMiguelet al. (2009) suggests that the 1 / n equality strategy should serve as a first obvious benchmark. Tocompare the e ffi cient of STRATEGY I, STRATEGY II and 1 / n equality strategy, we use the dailydata of NASDAQ and Dow Jones from the period of Aug. 03, 2009 to Aug. 02, 2019 to constructthe portfolio investments of STRATEGY I, STRATEGY II, and the 1 / n equality strategy. Here, weshow that we can obtain a yearly return of 270 .
78% with Sharpe ratio 0.8077 based on STRATEGYI, a yearly return of 249 .
12% with Sharpe ratio 0.6287 based on STRATEGY II, and a yearly returnof 11 .
65% with Sharpe ratio 0.7370 based on the 1 / n equality strategy.The remainder of this paper is organized as follows: in Section 2, we formulate the discrete timemulti-period mean-variance model. In Section 3, based on the dynamic programming principle of thevalue function, we establish a Bellman type dynamic time-consistent optimal strategy and a dynamictime-consistent relationship between the mean and variance. Furthermore, a varying investment pe-riod discrete mean-variance model is investigated in Section 3. To compare the e ffi ciency of theBellman type dynamic optimal strategy, as deduced in Section 3, with the 1 / n equality strategy, weconstruct the portfolio investment for the index NASDAQ and Dow Jones by the strategies given inSection 4. Finally, we conclude the paper in Section 5. Given a complete filtered probability space ( Ω , F , P ; {F ( s ) } s ≥ t ), and W ( · ), which is a d -dimensionalstandard Brownian motion defined on which with W ( t ) =
0, where F ( s ) is the P -augmentation of thenatural filtration generated by ( W ( t ) , W ( t + , · · · , W ( s )) , t ≤ s ≤ T , where T is the given investmentperiod. We consider that one risk-free bond asset and n risky stock assets are traded in the market,where the bond satisfies: P ( s ) = P ( s − r ( s − , P ( t ) = p , t < s ≤ T , i ’th (1 ≤ i ≤ n ) stock asset is described by P i ( s ) = P i ( s − (cid:20) b i ( s − + d (cid:88) j = σ i j ( s − ∆ W j ( s − (cid:21) , P i ( t ) = p i , t < s ≤ T , where ∆ W j ( s − = W j ( s ) − W j ( s − r ( · ) ∈ R is the risk-free return of the bond, b ( · ) = ( b ( · ) , · · · , b n ( · )) ∈ R n is the expected return of the risky assets. Given initial capital x > γ ( · ) = ( γ ( · ) , · · · , γ n ( · )) ∈ R n ,where γ i ( · ) = b i ( · ) − r ( · ) , ≤ i ≤ n . The investor’s wealth X π ( · ) satisfies X π ( s ) = r ( s − X π ( s − + γ ( s − π ( s − (cid:62) + π ( s − σ ( s − ∆ W ( s − , X π ( t ) = x , t < s ≤ T , (2.1)where σ ( · ) = ( σ ( · ) , · · · , σ d ( · )) ∈ R n × d , σ i ( · ) = ( σ i ( · ) , · · · , σ in ( · )) (cid:62) , π ( · ) = ( π ( · ) , · · · , π n ( · )) ∈ R n isthe capital invested in the risky asset S ( · ) = ( S ( · ) , · · · , S n ( · )) ∈ R n and π ( · ) is the capital invested inthe bond. Thus, we have X π ( · ) = n (cid:88) i = π i ( · ).In this study, we consider the following mean-variance model: J ( t , x ; π ( · )) = Var[ X π ( T )] = E [ (cid:0) X π ( T ) − E [ X π ( T )] (cid:1) ] , (2.2)with the following constraint on the mean, E [ X π ( T )] = L . (2.3)The set of admissible strategies π ( · ) is defined as: A T − t = (cid:26) π ( · ) : π ( s ) ∈ L [ F s , R n ] , t ≤ s < T (cid:27) . The following assumptions are used to obtain the optimal strategy for the proposed model (2.2): H : r ( · ) , b ( · ) and σ ( · ) are deterministic functions. H : r ( · ) > γ ( · ) (cid:44) σ ( · ) σ ( · ) (cid:62) > δ I , where δ > I is the identity matrixof S n , and S n is the set of symmetric matrices. To obtain the Bellman type optimal strategy for the discrete time mean-variance model (2.1), weset the term E [ X π ( · )] as a deterministic process Y π ( · ), which can separate the term E [ X π ( · )] from thevariance Var[ X π ( T )]. In the following, we consider the cost functional: J ( t , x , µ ; π ( · )) = µ Var[ X π ( T )] − E [ X π ( T )] , (2.4)5here µ > ffi cient and can be determined by the mean constraint L in (2.3).Notice that, Var[ X π ( T )] = E [ (cid:0) X π ( T ) − E [ X π ( T )] (cid:1) ] . We cannot obtain the Bellman principle for the term [ E X π ( T )] because [ E ( · )] is a nonlinear functionof E ( · ). To separate the expectation term E [ X π ( T )] from the variance Var[ X π ( T )], we introduce thefollowing auxiliary process Y π ( · ), where Y π ( · ) satisfies Y π ( s ) = r ( s − Y π ( s − + γ ( s − E [ π ( s − (cid:62) ] , Y π ( t ) = y , t < s ≤ T , (2.5)Comparing equations (2.1) and (2.5), it follows that Y π ( s ) = E [ X π ( s )] when x = y , t ≤ s ≤ T .Now, we consider the following general cost functional (2.4):˜ J ( t , x , y , µ ; π ( · )) = µ E [ (cid:0) X π ( T ) − Y π ( T ) (cid:1) ] − E [ X π ( T )] . The value function is defined as V µ ( t , x , y ) = inf π ( · ) ∈A T − t ˜ J ( t , x , y , µ ; π ( · )) . (2.6)We have the following Bellman principle for the value function V µ ( t , x , y ). The proofs of Theorem2.1 and Theorem 2.2 are given in Appendix A. Theorem 2.1.
Let Assumptions H and H hold. For any given ≤ t ≤ s < T , x , y ∈ R , we have,V µ ( t , x , y ) = inf π ( · ) ∈A s − t E [ V µ ( s , X π ( s ) , Y π ( s ))] . (2.7) Theorem 2.2.
Let Assumptions H and H hold. For any given ≤ t < T , x (cid:44) y ∈ R ,V µ ( t , x , y ) = µ ( x − y ) (cid:18) T − (cid:89) s = t r ( s ) (cid:19) − x T − (cid:89) s = t r ( s ) − µ T − (cid:88) s = t β ( s ) , (2.8) is the solution of equation (2.7), where β ( t ) = γ ( t )[ σ ( t ) σ ( t ) (cid:62) ] − γ ( t ) (cid:62) , and the related optimal strategyis π ∗ ( t , x , y ) = µ γ ( t ) (cid:2) σ ( t ) σ ( t ) (cid:62) (cid:3) − (cid:18) T − (cid:89) s = t r ( s ) (cid:19) − , ≤ t < T . The notation (cid:81) T − s = t r ( s ) means r ( t ) r ( t + · · · r ( T − T − < t , we set (cid:81) T − s = t r ( s ) = Remark 2.1.
For any given initial time and states ( t , x , y ) , from Theorem 2.2, we can obtain theoptimal strategy π ∗ ( t , x , y ) = µ γ ( t ) (cid:2) σ ( t ) σ ( t ) (cid:62) (cid:3) − (cid:18) T − (cid:89) h = t r ( h ) (cid:19) − , hich deduces that π ∗ ( s , X π ∗ ( s ) , Y π ∗ ( s )) = µ γ ( s ) (cid:2) σ ( s ) σ ( s ) (cid:62) (cid:3) − (cid:18) T − (cid:89) h = s r ( h ) (cid:19) − , t ≤ s < T . Thus, π ∗ ( s , X π ∗ ( s ) , Y π ∗ ( s )) is independent from the states ( X π ∗ ( s ) , Y π ∗ ( s )) . The Bellman type dynamicoptimal strategy is given as follows: π ∗ ( s ) = µ γ ( s ) (cid:2) σ ( s ) σ ( s ) (cid:62) (cid:3) − (cid:18) T − (cid:89) h = s r ( h ) (cid:19) − , t ≤ s < T . Furthermore, we denote the length of single period as T − tN , and let N → ∞ . It follows that π ∗ ( s ) = µ γ ( s ) (cid:2) σ ( s ) σ ( s ) (cid:62) (cid:3) − e (cid:82) Ts [1 − r ( h )]d h , which is consistent with Theorem 3.2 in Yang (2020). Notice that, V µ ( t , x , y ) and π ∗ ( t , x , y ) are continuous functions of ( x , y ). Letting y → x , we denote V µ ( t , x , x ) = lim y → x V µ ( t , x , y ) = − x T − (cid:89) s = t r ( s ) − µ T − (cid:88) s = t β ( s ) , and related optimal strategy π ∗ ( s ) = µ γ ( s ) (cid:2) σ ( s ) σ ( s ) (cid:62) (cid:3) − (cid:18) T − (cid:89) h = s r ( h ) (cid:19) − , t ≤ s < T . In the following, π ∗ ( · ) is called the Bellman type dynamic optimal strategy of mean variance model(2.2) under constraints (2.3). ffi cient frontier In this section, we derive the dynamic e ffi cient frontier for E [ X π ∗ ( s )] and Var[ X π ∗ ( s )] , t ≤ s < T .Plugging the optimal strategy π ∗ ( s ) = µ γ ( s ) (cid:2) σ ( s ) σ ( s ) (cid:62) (cid:3) − (cid:18) T − (cid:89) h = s r ( h ) (cid:19) − , t ≤ s < T , into equation (2.1), one can obtain that E [ X π ∗ ( · )] and E [ (cid:0) X π ∗ ( · ) (cid:1) ] satisfy the following linear di ff er-ence equations. E [ X π ∗ ( s )] = r ( s − E [ X π ∗ ( s − + β ( s − µ (cid:18) T − (cid:89) h = s − r ( h ) (cid:19) − , E [ X π ∗ ( t )] = x , t < s ≤ T , (2.9)7nd E [ (cid:0) X π ∗ ( s ) (cid:1) ] = E (cid:20)(cid:18) r ( s − X π ∗ ( s − + β ( s − µ (cid:18) T − (cid:89) h = s − r ( h ) (cid:19) − (cid:19) (cid:21) + β ( s − µ (cid:18) T − (cid:89) h = s − r ( h ) (cid:19) − , E [ (cid:0) X π ∗ ( t ) (cid:1) ] = x , t < s ≤ T . (2.10)By equation (2.9), we have (cid:0) E [ X π ∗ ( s )] (cid:1) = (cid:20) r ( s − E [ X π ∗ ( s − + β ( s − µ (cid:18) T − (cid:89) h = s − r ( h ) (cid:19) − (cid:21) , E [ X π ∗ ( t )] = x , t < s ≤ T . (2.11)Note that, Var[ X π ∗ ( s )] = E [ (cid:0) X π ∗ ( s ) (cid:1) ] − (cid:0) E [ X π ∗ ( s )] (cid:1) , t ≤ s ≤ T , combining equations (2.10) and(2.11), it follows that, Var[ X π ∗ ( s )] = ( r ( s − Var[ X π ∗ ( s − + β ( s − µ (cid:18) T − (cid:89) h = s − r ( h ) (cid:19) − , Var[ X π ∗ ( t )] = , t < s ≤ T . (2.12)From equations (2.9) and (2.12), for t ≤ s ≤ T , we can obtain E [ X π ∗ ( s )] and Var[ X π ∗ ( s )] as follows: E [ X π ∗ ( s )] = x s − (cid:89) h = t r ( h ) + (cid:18) T − (cid:89) h = s r ( h ) (cid:19) − s − (cid:88) h = t β ( h )2 µ , Var[ X π ∗ ( s )] = (cid:18) T − (cid:89) h = s r ( h ) (cid:19) − s − (cid:88) h = t β ( h )4 µ . (2.13) Remark 2.2.
Notice that, we introduce the risk aversion coe ffi cient µ in cost functional (2.4). Byequation (2.13), we can obtain µ by constrained condition (2.3) as follows: µ = (cid:80) T − h = t β ( h )2 (cid:0) L − x (cid:81) T − h = t r ( h ) (cid:1) . From equation (2.13), for t < s ≤ T , the relationship between E [ X π ∗ ( s )] and Var[ X π ∗ ( s )] is givenas follows: Theorem 2.3.
Let Assumptions H and H hold. We have Var[ X π ∗ ( s )] = (cid:18) E [ X π ∗ ( s )] − x (cid:81) s − h = t r ( h ) (cid:19) (cid:80) s − h = t β ( h ) , t < s ≤ T , (2.14) where β ( h ) = γ ( h )[ σ ( h ) σ ( h ) (cid:62) ] − γ ( h ) (cid:62) , t ≤ h < T . .4 Comparison with pre-committed strategy In this part of this paper, we compare our Bellman type dynamic optimal strategy and dynamice ffi cient frontier with those in Li and Ng (2000). Using the same setting and notation of this study,we review the main results of Li and Ng (2000), also see Zhou and Li (2000) for the continuous timecase. For the given initial time t and state x , the optimal pre-committed strategy is given as follows: π ∗ ( s ) = γ ( s )[ σ ( s ) σ ( s ) (cid:62) ] − [ λ (cid:18) T − (cid:89) h = s r ( h ) (cid:19) − − X π ∗ ( s )] , t ≤ s < T , (2.15)where λ = (cid:81) T − h = t [ β ( h ) + µ + x T − (cid:89) h = t r ( h ). The related e ffi cient frontier is given as follows:Var[ X π ∗ ( T )] = (cid:18) E [ X π ∗ ( T )] − x (cid:81) T − h = t r ( h ) (cid:19) (cid:81) T − h = t [ β ( h ) + − , (2.16)where E [ X π ∗ ( s )] = x s − (cid:89) h = t r ( h ) β ( h ) + + λ (cid:18) T − (cid:89) h = s r ( h ) (cid:19) − [1 − (cid:18) s − (cid:89) h = t [ β ( h ) + (cid:19) − ] , t ≤ s ≤ T , and E [ X π ∗ ( T )] = x T − (cid:89) h = t r ( h ) + µ (cid:18) T − (cid:89) h = t [ β ( h ) + − (cid:19) . Based on our model, by equality (2.13), we have E [ X π ∗ ( s )] = x s − (cid:89) h = t r ( h ) + (cid:18) T − (cid:89) h = s r ( h ) (cid:19) − s − (cid:88) h = t β ( h )2 µ , with the dynamic optimal strategy π ∗ ( s ) = µ γ ( s ) (cid:2) σ ( s ) σ ( s ) (cid:62) (cid:3) − (cid:18) T − (cid:89) h = s r ( h ) (cid:19) − , t ≤ s < T , By formula (2.15), the optimal pre-committed strategy π ∗ ( · ) at initial time t is given as follows: π ∗ ( t ) = µ γ ( t )[ σ ( t ) σ ( t ) (cid:62) ] − (cid:18) T − (cid:89) h = t r ( h ) β ( h ) + (cid:19) − . Note that β ( · ) >
0, we have that π ∗ ( t ) < π ∗ ( t ), where π ∗ ( t ) < π ∗ ( t ) means that the absolute valueof each element of π ∗ ( t ) is smaller than that of π ∗ ( t ). This is because the optimal pre-committedstrategy cares about the mean and variance of the wealth at investment period T , but not the entireinvestment period { t + , t + , · · · , T } . Therefore, the optimal pre-committed strategy changes alongwith the initial time t . In contrast to this, our dynamic optimal strategy π ∗ ( · ) is derived based onminimizing the cost functional along the lines of investment periods { t + , t + , · · · , T } . Thus, when9e provide the dynamic optimal strategy π ∗ ( · ) at initial time t , it will not change in the followingperiods s ∈ { t + , t + , · · · , T } . In the following, we show the properties of mean and variance underthe pre-committed strategy π ∗ ( · ) and the dynamic strategy π ∗ ( · ). The proof of Proposition 2.1 is givenin Appendix A. Proposition 2.1.
For a given mean level L > x (cid:81) T − h = t r ( h ) at the initial time t under the constrainedcondition (2.3), we have Var[ X π ∗ ( T )] > Var[ X π ∗ ( T )] . (2.17) For a given risk aversion parameter µ > , we have Var[ X π ∗ ( T )] < Var[ X π ∗ ( T )] , E [ X π ∗ ( T )] < E [ X π ∗ ( T )] . (2.18) Remark 2.3.
Note that L > x (cid:81) T − h = t r ( h ) at initial time t, E [ X π ∗ ( T )] = E [ X π ∗ ( T )] = L, based onthe dynamic optimal strategy π ∗ ( · ) and the optimal pre-committed strategy π ∗ ( · ) , the variance of thewealth X π ∗ ( T ) is larger than that of the wealth X π ∗ ( T ) . In contrast to this, for a given risk aversionparameter µ > , the investor can obtain a smaller mean and variance of the wealth X π ∗ ( T ) atinvestment period T within the strategy π ∗ ( · ) , relative to the mean and variance of the wealth X π ∗ ( T ) with the strategy π ∗ ( · ) . Furthermore, for the given investment period T , based on the formulations of E [ X π ∗ ( T )] and E [ X π ∗ ( T )] , we can see that the larger risk aversion parameter µ along with a largermean level L in constrained condition (2.3). Note here, we consider the Bellman type dynamic optimal strategy for the discrete mean-variancemodel with a given investment period T in Section 2. The question is how to determine the investmentperiod T . To answer this, we introduce a varying investment period discrete mean-variance model inthis section. In the following, we use the notation which is given in Section 2. In this section, we set the initial time t = x >
0. For a given deterministictime τ ∈ N + , we first introduce a varying investment period for the classical discrete mean-variancemodel: τ π = inf { s : E [ X π ( s )] ≥ g ( τ ) , < s ≤ τ } (cid:94) τ, (3.1)where a (cid:86) a , a , a ∈ R means a (cid:86) a = min( a , a ), and g ( · ) is the varying expected return oninvestment X π ( · ). 10 emark 3.1. Note that, g ( τ ) is the expected return on investment X π ( · ) before time τ . In this study,we set g ( τ ) = x τ − (cid:89) h = r ( h ) + α x τ − (cid:89) h = θ ( h ) , τ > , (3.2) where x (cid:81) τ − h = r ( h ) is the return when investing all the money into risk-free asset P ( · ) , α x (cid:81) τ − h = θ ( h ) is the excess return, and α > , θ ( · ) > . The objective is to minimize the variance at time τ π , J ( τ π , π ( · )) = E [( X π ( τ π ) − E [ X π ( τ π )]) ] . (3.3)If there exists ( ¯ π ∗ ( · ) , τ ∗ ) minimizing the cost functional (3.3) in the sense of Bellman type time-consistent, we call ¯ π ∗ ( · ) the Bellman type dynamic optimal strategy, τ ∗ the optimal investment period,and ( ¯ π ∗ ( · ) , τ ∗ ) the optimal pair.Note that, inf τ ∈ N + ,π ( · ) ∈A τ − J ( τ π , π ( · )) = inf τ ∈ N + inf π ( · ) ∈A τ − J ( τ π , π ( · )) , (3.4)to obtain the Bellman type dynamic optimal strategy and investment period for the cost functional(3.3), we give the following steps: Step 1:
For the given τ ∈ N + , we solve the first part J ( τ ¯ π , ¯ π τ ( · )) = inf π ( · ) ∈A τ − J ( τ π , π ( · )), andobtain the Bellman type optimal strategy ¯ π τ ( · ) and τ ¯ π = τ . Step 2:
We then solve the second part as J ( τ ∗ ) = inf τ ∈ N + J ( τ, ¯ π τ ( · )), and find the optimal invest-ment period τ ∗ and related optimal strategy ¯ π ∗ ( · ). We first consider the
Step 1 . For a given τ ∈ N + , we suppose that there exists an optimal Bellmantype strategy ¯ π τ ( · ) and investment period τ ¯ π ≤ τ such that J ( τ ¯ π , ¯ π τ ( · )) = inf π ( · ) ∈A τ − J ( τ, π ( · )) , and E [ X ¯ π τ ( τ ¯ π )] = g ( τ ) , E [ X ¯ π τ ( s )] < g ( τ ) , s < τ ¯ π . In the following, we want to show that the optimal strategy which is given in Theorem 2.2 is thestrategy ¯ π τ ( · ). Lemma 3.1.
Let Assumptions H and H hold. For a given τ ∈ N + , the optimal investment period τ ¯ π = τ , and the Bellman type dynamic optimal strategy is ¯ π τ ( s ) = µ γ ( s ) (cid:2) σ ( s ) σ ( s ) (cid:62) (cid:3) − (cid:18) τ − (cid:89) h = s r ( h ) (cid:19) − , ≤ s < τ. roof: Based on the results of Theorem 2.2 and Remark 2.1, for the given investment period τ ¯ π , wecan obtain a Bellman type dynamic strategy for the cost functional J ( τ ¯ π , π ( · )) = E [ (cid:0) X π ( τ ¯ π ) − E [ X π ( τ ¯ π )] (cid:1) ]under the mean constrained E [ X π ( τ ¯ π )] = g ( τ ) . The optimal strategy is given as follows: π ∗ ( s ) = µ γ ( s ) (cid:2) σ ( s ) σ ( s ) (cid:62) (cid:3) − (cid:18) τ ¯ π − (cid:89) h = s r ( h ) (cid:19) − , ≤ s < τ ¯ π . By Theorem 2.3, one can obtainVar[ X π ∗ ( s )] = (cid:18) g ( τ ) − x (cid:81) s − h = r ( h ) (cid:19) (cid:80) s − h = β ( h ) , (3.5)and E [ X π ∗ ( s )] = x s − (cid:89) h = r ( h ) + (cid:18) τ ¯ π − (cid:89) h = s r ( h ) (cid:19) − s − (cid:88) h = β ( h )2 µ , s ≤ τ ¯ π . Notice that, E [ X π ∗ ( s )] is increasing with s ≤ τ ¯ π . Letting µ = (cid:80) τ ¯ π − h = β ( h )2 (cid:0) g ( τ ) − x (cid:81) τ ¯ π − h = r ( h ) (cid:1) , it follows that E [ X π ∗ ( τ ¯ π )] = g ( τ ) , E [ X π ∗ ( s )] < g ( τ ) , s < τ ¯ π , and τ ¯ π = inf { s : E [ X π ∗ ( s )] ≥ g ( τ ) , < s ≤ τ } . Therefore, we have ¯ π τ ( · ) = π ∗ ( · ).Now, we determine the varying investment period τ ¯ π ≤ τ . Note that g ( τ ) = x τ − (cid:89) h = r ( h ) + α x τ − (cid:89) h = θ ( h ) > x τ − (cid:89) h = r ( h ) . By equation (3.5), we can see that Var[ X π ∗ ( s )] is decreasing with s ≤ τ . Thus, Var[ X π ∗ ( · )] takes theminimize value at time τ , which deduces the varying investment period τ ¯ π = τ . This completes theproof. (cid:3) In the following, we consider the
Step 2 to find the optimal period τ ∗ . From the Step 1 , we canobtain the Bellman type optimal strategy,¯ π τ ( s ) = µ γ ( s ) (cid:2) σ ( s ) σ ( s ) (cid:62) (cid:3) − (cid:18) τ − (cid:89) h = s r ( h ) (cid:19) − , ≤ s < τ. τ ¯ π = τ such that J ( τ, ¯ π τ ( · )) = inf π ( · ) ∈A τ − J ( τ π , π ( · )) = (cid:18) g ( τ ) − x (cid:81) τ − h = r ( h ) (cid:19) (cid:80) τ − h = β ( h ) . Based on equation (3.4), we now solve the partinf τ ∈ N + J ( τ, ¯ π τ ( · )) = inf τ ∈ N + (cid:18) g ( τ ) − x (cid:81) τ − h = r ( h ) (cid:19) (cid:80) τ − h = β ( h ) . By Remark 3.1, we have g ( τ ) = x τ − (cid:89) h = r ( h ) + α x τ − (cid:89) h = θ ( h ) , thus, J ( τ, ¯ π τ ( · )) = α x (cid:18) (cid:81) τ − h = θ ( h ) (cid:19) (cid:80) τ − h = β ( h ) . Theorem 3.1.
Let Assumptions H and H hold, and there exists ˆ τ such that for s ≥ ˆ τ , ( θ ( s ) − s − (cid:88) h = β ( h ) − β ( s ) ≥ . We can find the optimal investment period < τ ∗ ≤ ˆ τ such thatJ ( τ ∗ ) = inf τ ∈ N + J ( τ, ¯ π τ ( · )) . The Bellman type dynamic optimal strategy is ¯ π τ ∗ ( s ) = µ γ ( s ) (cid:2) σ ( s ) σ ( s ) (cid:62) (cid:3) − (cid:18) τ ∗ − (cid:89) h = s r ( h ) (cid:19) − , ≤ s < τ ∗ . Proof:
Based on the formula of J ( τ, ¯ π τ ( · )), one obtains J ( τ + , ¯ π τ + ( · )) − J ( τ, ¯ π τ ( · )) = α x (cid:18) (cid:81) τ h = θ ( h ) (cid:19) (cid:80) τ h = β ( h ) − α x (cid:18) (cid:81) τ − h = θ ( h ) (cid:19) (cid:80) τ − h = β ( h ) = α x (cid:18) τ − (cid:89) h = θ ( h ) (cid:19) ( θ ( τ ) − (cid:80) τ − h = β ( h ) − β ( τ ) (cid:80) τ h = β ( h ) (cid:80) τ − h = β ( h ) . (3.6)Note that, there exists ˆ τ such that for s ≥ ˆ τ ,( θ ( s ) − s − (cid:88) h = β ( h ) − β ( s ) ≥ , J ( s , ¯ π s ( · )) − J (ˆ τ, ¯ π ˆ τ ( · )) ≥ . Thus, J ( τ, ¯ π τ ( · )) takes the minimize value at τ ∗ which satisfies τ ∗ ≤ ˆ τ , and the related dynamic optimalstrategy is ¯ π τ ∗ ( s ) = µ γ ( s ) (cid:2) σ ( s ) σ ( s ) (cid:62) (cid:3) − (cid:18) τ ∗ − (cid:89) h = s r ( h ) (cid:19) − , ≤ s < τ ∗ . This completes the proof. (cid:3)
Example 1.
In this example, we consider the Black-Sholes setting. Let r , b , σ, θ be independent fromtime t ∈ N + , γ = ( b − r , · · · , b n − r ) , and β = γ [ σσ (cid:62) ] − γ (cid:62) . For a given τ > , the expect return ofthe wealth X π ( · ) is g ( τ ) = xr τ + α x θ τ . The cost functional in the
Step 1 is given as follows:J ( τ, ¯ π τ ( · )) = α x β θ τ τ , which deduces J ( τ + , ¯ π τ + ( · )) − J ( τ, ¯ π τ ( · )) = α x β θ τ + τ + − α x β θ τ τ = α x θ τ β (cid:20) θ τ + − τ (cid:21) = α x θ τ β (cid:20) ( θ − τ − τ + τ (cid:21) . (3.7) Note that, θ > . Let ˆ τ = (cid:100) θ − (cid:101) , we have ( θ − s − ≥ , s ≥ ˆ τ, and ( θ − s − < , s < ˆ τ. Thus, J ( τ, ¯ π τ ( · )) takes the minimize value at ˆ τ , and τ ∗ = ˆ τ . The related dynamic optimal strategy is ¯ π τ ∗ ( s ) = µ γ (cid:2) σσ (cid:62) (cid:3) − r s − τ ∗ , ≤ s < τ ∗ . In this section, we want to compare three di ff erent strategies in Black-Sholes setting. The expectreturn for period τ is given as follows: g ( τ ) = xr τ + α x θ τ , τ ∈ N + . TRATEGY I:
The Bellman type dynamic time-consistent optimal strategy is given in Theorem2.2 as follows: π ( s ) = µ γ (cid:2) σσ (cid:62) (cid:3) − r s − τ , ≤ s < τ, where τ is the given investment period, and µ = τβ (cid:0) g ( τ ) − xr τ (cid:1) = τβ α x θ τ . Thus, π ( s ) = x αθ τ τβ γ (cid:2) σσ (cid:62) (cid:3) − r s − τ , ≤ s < τ, which means that π ( · ) is the proportional investment of the initial wealth x . The e ffi cient frontier isVar[ X π ( s )] = (cid:18) E [ X π ( s )] − xr s (cid:19) β s , < s ≤ τ. (4.1) STRATEGY II:
The Bellman type dynamic optimal strategy for varying investment period mean-variance model is given in Theorem 3.1 as follows: π ( s ) = µ γ (cid:2) σσ (cid:62) (cid:3) − r s − τ ∗ , ≤ s < τ ∗ , where τ ∗ = (cid:100) θ − (cid:101) , and µ = τ ∗ β (cid:0) g ( τ ∗ ) − xr τ ∗ (cid:1) . We can obtain π ( s ) = x αθ τ ∗ τ ∗ β γ (cid:2) σσ (cid:62) (cid:3) − r s − τ ∗ , ≤ s < τ ∗ . Thus, π ( · ) is also a proportional investment of the initial wealth x . The e ffi cient frontier isVar[ X π ( s )] = (cid:18) E [ X π ( s )] − xr s (cid:19) β s , < s ≤ τ ∗ . (4.2) STRATEGY III: / n equality strategy. We assume that there are n kinds of risky assets, andinvest 1 / n of the wealth into each risky asset. π ( s ) = ( X π ( s ) n , · · · , X π ( s ) n ) , ≤ s < τ. In this part, we set the following values of parameters: r = . , b = . , θ = . , d = n = , α = . , x = , p = p i = , ≤ i ≤ n , σ i j = , ≤ i (cid:44) j ≤ n , . + . i , ≤ i = j ≤ n . (4.3)Based on the above value of parameters, we simulate the risk-free asset and risky assets, andcompare these three strategies STRATEGY I, II, and III with the return and variance. Let the lengthof a single period be one day. For the given initial wealth x =
1, and investment periods τ = ,
90 forSTRATEGY I and III, we simulate M times the path of { X m ,π ( τ ) } Mm = and { X m ,π ( τ ) } Mm = , and define R ( τ ) = (cid:80) Mm = X m ,π ( τ ) M , V ( τ ) = (cid:80) Mm = (cid:2) X m ,π ( τ ) − R ( τ ) (cid:3) M and R ( τ ) = (cid:80) Mm = X m ,π ( τ ) M , V ( τ ) = (cid:80) Mm = (cid:2) X m ,π ( τ ) − R ( τ ) (cid:3) M . For the given initial wealth x =
1, and optimal investment period τ ∗ = (cid:100) θ − (cid:101) =
63 for STRATEGYII, we simulate M times the path of { X m ,π ( τ ∗ ) } Mm = , and define R ( τ ∗ ) = (cid:80) Mm = X m ,π ( τ ∗ ) M , V ( τ ∗ ) = (cid:80) Mm = (cid:2) X m ,π ( τ ∗ ) − R ( τ ∗ ) (cid:3) M . The simulation results are concluded in Table 1.Table 1: Simulation results of STRATEGY I, II, and III with di ff erent investment period and simula-tion times M = x = τ = R (30) = . V (30) = . x = τ = R (90) = . V (90) = . x = τ ∗ = R (63) = . V (63) = . x = τ = R (30) = . V (30) = . x = τ = R (60) = . V (63) = . x = τ = R (90) = . V (90) = . x = τ = g (30) = . X π (30)] = . x = τ = g (90) = . X π (90)] = . x = τ ∗ = g (63) = . X π (63)] = . | R (30) − g (30) | < . , | V (30) − Var[ X π (30)] | < . , | R (90) − g (90) | < . , | V (90) − Var[ X π (90)] | < . , | R (63) − g (63) | < . , | V (63) − Var[ X π (63)] | < . , (4.4)which verifies that the simulation results coincide with the theory results. We take the daily data of NASDAQ and Dow Jones from Aug. 03, 2009 to Aug. 02, 2019.Table 3: Daily data of NASDAQ and Dow JonesThe index Initial time The length (days) Average value Standard deviationNASDAQ Aug. 03, 2009 2518 4488 1798Dow Jones Aug. 03, 2009 2518 16890 4969Employing STRATEGY I, II, and III, we consider investing in risky assets in NASDAQ, DowJones and risk-free asset P ( · ). We set the values of the parameters as follows: the daily return of P ( · ), r = . x =
1, the daily excess expected return θ = . α = . L days. For each given L , we use the following steps to construct the investment portfolios for STRATEGY I, II, and III: wesuppose the prices of index NASDAQ, Dow Jones satisfy the following equation: P i ( s ) = P i ( s − (cid:2) b i + σ i ∆ W ( s ) (cid:3) , ≤ s , i = , , { P N ( s ) } s = and { P D ( s ) } s = , respectively. Step 1:
For a single period’s length L and initial time t , we firstly give the window of history data w = m L +
1. We use the dataset { P N ( s ) } ts = t − w + and { P D ( s ) } ts = t − w + to estimate the return b = ( b , b )and volatility σ = ( σ , σ ) for STRATEGY I and II at time t ≥ w . Denoting, R N ( s ) = P N ( s ) − P N ( s − P N ( s − , R D ( s ) = P D ( s ) − P D ( s − P D ( s − , t − w + ≤ s ≤ t , and I N ( t , s ) = L (cid:88) i = R N (( s − L + i + t − w + , I D ( t , s ) = L (cid:88) i = R D (( s − L + i + t − w + , ≤ s ≤ m . Step 2:
The parameters b = ( b , b ) and σ = ( σ , σ ) are estimated at time t asˆ b ( t ) = (ˆ b ( t ) , ˆ b ( t )) = (cid:18) + (cid:80) m s = I N ( t , s ) m , + (cid:80) m s = I D ( t , s ) m (cid:19) , and ˆ σ ( t ) = ( ˆ σ ( t ) , ˆ σ ( t )) = (cid:18) (cid:80) m s = ( I N ( t , s ) − ˆ b ( t ) + m − , (cid:80) m s = ( I D ( t , s ) − ˆ b ( t ) + m − (cid:19) . Step 3:
For the given investment period τ , STRATEGY I is given as follows:ˆ π ( t , s ) = (cid:18) r s − τ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) , r s − τ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) (cid:19) , ≤ s < τ, where r L = + ( r − L , andˆ µ ( t ) = τ ˆ β ( t )2 (cid:0) g ( τ ) − xr τ L (cid:1) , ˆ β ( t ) = (ˆ b ( t ) − r L ) ˆ σ ( t ) + (ˆ b ( t ) − r L ) ˆ σ ( t ) . Applying STRATEGY I, the wealth is ¯ X ˆ π ( t , s ) = r s − τ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) P N ( t + sL ) P N ( t + ( s − L ) + r s − τ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) P D ( t + sL ) P D ( t + ( s − L ) + (cid:18) ¯ X ˆ π ( t , s − − r s − τ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) − r s − τ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) (cid:19) r L , ¯ X ˆ π ( t , = , ≤ s ≤ τ. (4.5)The optimal investment period of STRATEGY II is τ ∗ = (cid:100) (cid:0) ( θ − L + (cid:1) − (cid:101) , and STRATEGYII is given as follows: ˆ π ( s ) = (cid:18) r s − τ ∗ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) , r s − τ ∗ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) (cid:19) , ≤ s < τ ∗ , and ˆ µ ( t ) = τ ˆ β ( t )2 (cid:0) g ( τ ∗ ) − xr τ ∗ L (cid:1) , ˆ β ( t ) = (ˆ b ( t ) − r L ) ˆ σ ( t ) + (ˆ b ( t ) − r L ) ˆ σ ( t ) . ¯ X ˆ π ( t , s ) = r s − τ ∗ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) P N ( t + sL ) P N ( t + ( s − L ) + r s − τ ∗ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) P D ( t + sL ) P D ( t + ( s − L ) + (cid:18) ¯ X ˆ π ( t , s − − r s − τ ∗ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) − r s − τ ∗ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) (cid:19) r L , ¯ X ˆ π ( t , = , ≤ s ≤ τ ∗ . (4.6)For the investment period τ , the STRATEGY III is given as follows:ˆ π ( s ) = (cid:18) ¯ X ˆ π ( t , s )2 , ¯ X ˆ π ( t , s )2 (cid:19) , ≤ s < τ ∗ , and the wealth as of STRATEGY III is ¯ X ˆ π ( t , s ) = ¯ X ˆ π ( t , s − P N ( t + sL ) P N ( t + ( s − L ) + ¯ X ˆ π ( t , s − P D ( t + sL ) P D ( t + ( s − L ) , ¯ X ˆ π ( t , = , ≤ s ≤ τ. (4.7) Step 4:
For the given initial time t , the length of a single period L and the investment period τ, τ ∗ .We repeat the Steps 1, 2, 3 for w ≤ t ≤ w + K −
1, and denotes the average yearly return and Sharperatio for STRATEGY I, II and III at investment period τ, τ ∗ as follows:Return ( L , τ ) = τ LK K (cid:88) i = ( ¯ X ˆ π ( w + i − , τ ) − , Sharpe ( L , τ ) = (cid:114) τ LK (cid:80) Ki = ¯ X ˆ π ( w + i − , τ ) − K − . τ LK (cid:113)(cid:80) Ki = (cid:2) ¯ X ˆ π ( w + i − , τ ) − K (cid:80) Ki = ¯ X ˆ π ( w + i − , τ ) (cid:3) , Return ( L , τ ∗ ) = τ ∗ LK K (cid:88) i = ( ¯ X ˆ π ( w + i − , τ ∗ ) − , Sharpe ( L , τ ∗ ) = (cid:114) τ ∗ LK (cid:80) Ki = ¯ X ˆ π ( w + i − , τ ∗ ) − K − . τ ∗ LK (cid:113)(cid:80) Ki = (cid:2) ¯ X ˆ π ( w + i − , τ ∗ ) − K (cid:80) Ki = ¯ X ˆ π ( w + i − , τ ∗ ) (cid:3) , Return ( L , τ ) = τ LK K (cid:88) i = ( ¯ X ˆ π ( w + i − , τ ) − , Sharpe ( L , τ ) = (cid:114) τ LK (cid:80) Ki = ¯ X ˆ π ( w + i − , τ ) − K − . τ LK (cid:113)(cid:80) Ki = (cid:2) ¯ X ˆ π ( w + i − , τ ) − K (cid:80) Ki = ¯ X ˆ π ( w + i − , τ ) (cid:3) . Based on the above steps, we now show how to choose the length of single period L . In thefollowing, we set m = K = τ = (cid:100) L (cid:101) , τ ∗ = (cid:100) (cid:0) ( θ − L + (cid:1) − (cid:101) and plot the value ofreturn Return ( L , τ ) , Return ( L , τ ∗ ) , Return ( L , τ ) and Sharpe ( L , τ ) , Sharpe ( L , τ ∗ ), Sharpe ( L , τ )for 1 ≤ L ≤
60: 19igure 1: Average return and Sharpe ratio of STRATEGY I, II, and III along with the single period L T he a v e r age r e t u r n o f S T R A T E G Y Return (L, τ )Return (L, τ * )Return (L, τ ) T he S ha r pe r a t i o o f S T R A T E G Y Sharpe (L, τ )Sharpe (L, τ * )Sharpe (L, τ ) From Figure 1, we can see that the return of STRATEGY I is almost equal to that of STRATEGYII when the length of single period L ≥
10, and the return of STRATEGY I is almost larger thanthat of STRATEGY III for L ≥
1. Furthermore, when L ≥
30, the returns of STRATEGY I andSTRATEGY II are stable. However, the return of STRATEGY III barely changes along with thesingle period L ≥
1. Thus, we plot the Sharpe ratio of STRATEGY I, II and III in Figure 1. Wecan see that the Sharpe ratio of STRATEGY I is almost same as that of STRATEGY II when thelength of single-period L ≥
10, and the Sharpe ratio of STRATEGY III is almost equal with that ofSTRATEGY I and II when the length of single-period L ≥
30. Combining the results in Figure 1, weshow the results in details for the single-period L =
30, and investment periods τ = , τ ∗ = T he v a l ue o f w ea l t h The wealth of STRATEGY I at τ =9 T he v a l ue o f w ea l t h The wealth of STRATEGY II at τ * =2 T he v a l ue o f w ea l t h The wealth of STRATEGY III at τ =9 0 100 200 300 400 500 600 700 800 900 10000.40.60.811.21.41.61.82 The investment times T he v a l ue o f w ea l t h The wealth of STRATEGY III at τ =2 In Figure 2, we repeat the multi-period portfolio investment of STRATEGY I, II, and III 1000times along with w ≤ t ≤ w + i ( τ, L ) , i = , , i ( τ, L ) , i = , , = τ = = τ ∗ = = τ = = τ = L =
30. Note that, theSTRATEGY I and II have same formula but with di ff erent investment periods, τ = > τ ∗ =
2. Theseresults indicate that the multi-period investment portfolio strategy maybe better than the single-periodinvestment portfolio strategy.Based on the results of theory, the expected yearly return Return i ( τ, L ) , i = , i ( τ, L ) , i = , ( τ, L ) = L τ ( g ( τ ) − x ) , Sharpe ( τ, L ) = (cid:114) L τ K ( g ( τ ) − x − . τ L ) (cid:80) w + K − t = w α x (1 + ( θ − L ) τ √ τ ˆ β ( t ) , Return ( τ ∗ , L ) = L τ ∗ ( g ( τ ∗ ) − x ) , Sharpe ( τ ∗ , L ) = (cid:114) L τ ∗ K ( g ( τ ) − x − . τ ∗ L ) (cid:80) w + K − t = w α x (1 + ( θ − L ) τ ∗ √ τ ∗ ˆ β ( t ) , where ˆ β ( t ) is calculated at time t ,ˆ β ( t ) = (ˆ b ( t ) − r L ) ˆ σ ( t ) + (ˆ b ( t ) − r L ) ˆ σ ( t ) , w ≤ t ≤ w + K − . Table 5: Theory results of STRATEGY I and IIInvestment period Expected yearly return Sharpe ratioSTRATEGY I τ = . . τ ∗ = . . (cid:12)(cid:12)(cid:12) Return (9 , − Return (9 , (cid:12)(cid:12)(cid:12) < . , (cid:12)(cid:12)(cid:12)(cid:12) Sharpe (9 , − Sharpe (9 , (cid:12)(cid:12)(cid:12)(cid:12) < . , (cid:12)(cid:12)(cid:12) Return (2 , − Return (2 , (cid:12)(cid:12)(cid:12) < . , (cid:12)(cid:12)(cid:12)(cid:12) Sharpe (2 , − Sharpe (2 , (cid:12)(cid:12)(cid:12)(cid:12) < . , (4.8)which indicates that the experiment results of STRATEGY I and II are nearly with the theory results.22 .3 STRATEGY I with transaction fee and loan interest rate In this part, we introduce transaction fee rate r of risky assets and daily loan interest rate r forthe discrete time multi-period mean-variance model. We show the details of STRATEGY I with thelength of single period L and investment period τ . Using the same setting in Subsection 4.2, weconsider to invest into risky assets NASDAQ, Dow Jones and risk-free asset P ( · ). We set the valuesof the parameters as follows: the daily return of risk-free asset, r = . x = θ = . α = .
5, the transaction fee rate r = .
001 and the loaninterest rate r = . τ , STRATEGY I is given as follows:ˆ π ( t , s ) = (cid:18) r s − τ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) , r s − τ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) (cid:19) , ≤ s < τ, where r L = + ( r − L and τ is the given investment period, andˆ µ ( t ) = τ ˆ β ( t )2 (cid:0) g ( τ ) − xr τ L (cid:1) , ˆ β ( t ) = (ˆ b ( t ) − r L ) ˆ σ ( t ) + (ˆ b ( t ) − r L ) ˆ σ ( t ) . Denoting ω ( t , s ) = r s − τ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) , ≤ s ≤ τ,ω ( t , s ) = r s − τ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) ) , ≤ s ≤ τ,ω ( t , s ) = ¯ X ˆ π ( t , s − − r s − τ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) − r s − τ L (ˆ b ( t ) − r L )2 ˆ µ ( t ) ˆ σ ( t ) , ≤ s ≤ τ. (4.9)Based on STRATEGY I, the related wealth is ¯ X ˆ π ( t , s ) = ω ( t , s ) P N ( t + sL ) P N ( t + ( s − L ) + ω ( t , s ) P D ( t + sL ) P D ( t + ( s − L ) + max( ω ( t , s ) , r L + min( ω ( t , s ) , r L − ( | ω ( t , s ) | + | ω ( t , s ) | ) r ¯ X ˆ π ( t , = , ≤ s ≤ τ, (4.10)where r L = + ( r − L .Based on the introduction of the transaction fee rate of stocks and loan interest rate, we show therelation between the length of single period L and the return and Sharpe ratio of STRATEGY I. In thefollowing, we set m = K = τ = (cid:100) L (cid:101) , and plot the value of average return Return ( L , τ )and Sharpe ratio Sharpe ( L , τ ) for 1 ≤ L ≤
60: whereReturn ( L , τ ) = τ LK K (cid:88) i = ( ¯ X ˆ π ( w + i − , τ ) − , Sharpe ( L , τ ) = (cid:114) τ LK (cid:80) Ki = ¯ X ˆ π ( w + i − , τ ) − K − . τ LK (cid:113)(cid:80) Ki = (cid:2) ¯ X ˆ π ( w + i − , τ ) − K (cid:80) Ki = ¯ X ˆ π ( w + i − , τ ) (cid:3) . L T he a v e r age r e t u r n o f S T R A T E G Y Return (L, τ ) T he S ha r pe r a t i o o f S T R A T E G Y Sharpe (L, τ ) In Figure 3, we build the investment portfolio for NASDAQ, Dow Jones, and risk-free asset usingSTRATEGY I. After considering the transaction fee rate r for risky assets and loan interest rate r L ,we find that the average return of the portfolio of STRATEGY I is larger than 0 when the length ofsingle period L >
10 and becomes stable when L ≥
30. Furthermore, we can see that the Sharperatio of STRATEGY I is nearly 0 . L ≥
30. Therefore, we consider the details of investmentportfolio of STRATEGY I when L = T he v a l ue o f w ea l t h The wealth of STRATEGY I at τ =9 In Figure 4, we repeat the multi-period investment portfolio of STRATEGY I 1500 times alongwith w ≤ t ≤ w + x =
1. Thus we conclude the averageyearly return Return ( τ, L ) and Sharpe ratio Sharpe ( τ, L ) of STRATEGY I from Jan. 2012 to Jan.2018 as follows:Table 6: Average yearly return and Sharpe ratio of STRATEGY I with L =
30, transaction fee rate r = .
001 and daily loan interest rate r = . x = τ = . . x = τ = . . x = τ = . . x = τ = − . − . x = τ = . . x = τ = . . x = . . r = . r ∈{ . , . , · · · , . } a ff ects the average yearly return and Sharpe ratio of portfolio wealth ofSTRATEGY I: 25able 7: Yearly return and Sharpe ratio of STRATEGY I with L =
30 and investment period τ = x = r = .
001 3 . . x = r = .
002 3 . . x = r = .
003 2 . . x = r = .
004 2 . . x = r = .
005 1 . . x = r = .
006 1 . . x = r = .
007 1 . . x = r = .
008 0 . . x = r = .
009 0 . . x = r = . − . − . r . Note here that, we have not introduced the transactionfee rate when deriving STRATEGY I, and considering the transaction fee in the empirical analysis. Itis interesting to find an optimal strategy for the discrete time multi-period mean-variance model withtransaction fee rate. However, we can see that the yearly return is larger than 1 . . r ≤ .
005 under the initial wealth x =
1. Addi-tionally, we want to highlight that STRATEGY I is the proportional investment of the initial wealth x ,which means that the average yearly return and Sharpe ratio of per unit wealth will not change withthe value of initial wealth x . By introducing a deterministic process Y π ( · ) = E [ X π ( · )] with a initial value y , we consider thefollowing value function: V µ ( t , x , y ) = inf π ∈A T − t ˜ J ( t , x , y , µ ; π ( · )) = inf π ∈A T − t E [ µ (cid:0) X π ( T ) − Y π ( T ) (cid:1) − X π ( T )] , (5.1)26here X π ( T ) with the initial value x . From the cost functional (5.1), we can distinguish the wealthprocess X π ( · ) and process Y π ( · ). Based on these setting, we can derive the related dynamic program-ming principle for the value function V µ ( t , x , y ). The main results of this study are given as follows: • Similar with the idea in Yang (2020), we solve a variance type cost functional that contains anonlinear part of the mean process E [ X π ( · )] in discrete time case. This new method can helpus to separate the nonlinear part of the mean process from the variance in cost functional, andthen we can obtain the related dynamic optimal strategy which is time consistent. • Furthermore, we develop a varying investment period discrete time multi-period mean-variancemodel, for which we obtain the related dynamic time-consistent optimal strategy and optimalinvestment period. • To compare our dynamic optimal strategy with the 1 / n equality strategy, we use the daily dataof NASDAQ and Dow Jones to construct portfolio investment for our dynamic optimal strategyand the 1 / n equality, which demonstrates that our optimal strategy is better than 1 / n equalitystrategy at least for index data NASDAQ and Dow Jones in empirical analysis. A Supplementary of the main proofs
The proof of Theorem 2.1 : Based on the method in the proof of Theorem 3.3, Chapter 4 in Yongand Zhou (1999), we can prove this result, also see Yang (2020). For reader’s convenience, we showthe details of the proof. For any given 0 ≤ t ≤ s < T , x , y ∈ R , the value function is given as follows: V µ ( t , x , y ) = inf π ( · ) ∈A T − t ˜ J ( t , x , y , µ ; π ( · )) , (A.1)where ˜ J ( t , x , y , µ ; π ( · )) = µ E [ (cid:0) X π ( T ) − Y π ( T ) (cid:1) ] − E [ X π ( T )] . Note that Y π ( · ) is a deterministic process, for the cost functional ˜ J ( t , x , y , µ ; π ( · )), we can obtain˜ J ( s , X π ( s ) , Y π ( s ) , µ ; π ( · )) = E (cid:2) µ (cid:0) X π ( T ) − Y π ( T ) (cid:1) − X π ( T ) (cid:12)(cid:12)(cid:12) F s (cid:3) . (A.2)We first set, ˜ V µ ( t , x , y ) : = inf π ( · ) ∈A s − t E [ V µ ( s , X π ( s ) , Y π ( s ))] . ε >
0, there exists π ε ( · ) such that V µ ( t , x , y ) + ε > ˜ J ( t , x , y , µ ; π ε ( · )) = E [ µ (cid:0) X π ε ( T ) − Y π ε ( T ) (cid:1) − X π ε ( T )] = E (cid:2) E (cid:2) µ (cid:0) X π ε ( T ) − Y π ε ( T ) (cid:1) − X π ε ( T ) (cid:12)(cid:12)(cid:12) F s (cid:3)(cid:3) = E (cid:2) ˜ J ( s , X π ε ( s ) , Y π ε ( s ) , µ ; π ε ( · )) (cid:3) ≥ E (cid:2) V µ ( s , X π ε ( s ) , Y π ε ( s )) (cid:3) ≥ ˜ V µ ( t , x , y ) . (A.3)Thus V µ ( t , x , y ) + ε > ˜ V µ ( t , x , y ). In contrast, we prove V µ ( t , x , y ) < ˜ V µ ( t , x , y ) + ε in the following.Note that, for any given π ( · ) ∈ A T − t , X π ( s ) = r ( s − X π ( s − + γ ( s − π ( s − (cid:62) + π ( s − σ ( s − ∆ W ( s − , X π ( t ) = x , t < s ≤ T . (A.4)There exists δ > | x − x | + | y − y | < δ, it follows, (cid:12)(cid:12)(cid:12) ˜ J ( t , x , y , µ ; π ( · )) − ˜ J ( t , x , y , µ ; π ( · )) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) V µ ( t , x , y ) − V µ ( t , x , y ) (cid:12)(cid:12)(cid:12) < ε . Thus, we can find a strategy ˆ π ( h ) = π ( h ) , t ≤ h < s , ˜ π ( h ) , s ≤ h < T , where π ( · ) ∈ A s − t is a any given strategy, such that˜ J ( s , X π ( s ) , Y π ( s ) , µ ; ˜ π ( · )) < V µ ( s , X π ( s ) , Y π ( s )) + ε. Thus, for the strategy ˆ π ( · ), we have V µ ( t , x , y ) ≤ E [ µ (cid:0) X ˆ π ( T ) − Y ˆ π ( T ) (cid:1) − X ˆ π ( T )] = E (cid:2) E (cid:2) µ (cid:0) X ˆ π ( T ) − Y ˆ π ( T ) (cid:1) − X ˆ π ( T ) | F s (cid:3)(cid:3) = E (cid:2) ˜ J ( s , X π ( s ) , Y π ( s ) , µ ; ˆ π ( · )) (cid:3) < E (cid:2) V µ ( s , X π ( s ) , Y π ( s )) (cid:3) + ε, (A.5)28or π ( · ) ∈ A s − t is a any given strategy, we have V µ ( t , x , y ) ≤ ˜ V µ ( t , x , y ) + ε, (A.6)which completes the proof. (cid:3) A Proofs
The proof of Theorem 2.2 : For any given x , y ∈ R , we have the following result for time T − V µ ( T − , x , y ) = inf π ( · ) ∈A T − T − E [ µ (cid:0) X π ( T ) − Y π ( T ) (cid:1) − X π ( T )] = inf π ( · ) ∈A T − T − E (cid:20) µπ ( T − σ ( T − σ ( T − (cid:62) ] π ( T − (cid:62) − r ( T − x − γ ( T − π ( T − (cid:62) + µ (cid:2) r ( T − x − y ) + γ ( T − π ( T − (cid:62) − E [ π ( T − (cid:62) ]) (cid:3) (cid:21) . Thus, we can obtain the optimal strategy, π ∗ ( T − , x , y ) = γ ( T − µ [ σ ( T − σ ( T − (cid:62) ] − , and the related value function, V µ ( T − , x , y ) = r ( T − (cid:2) µ r ( T − x − y ) − x (cid:3) − β ( T − µ . Thus, it follows that V µ ( T − , X π ( T − , Y π ( T − = µ r ( T − X π ( T − − Y π ( T − − r ( T − X π ( T − − β ( T − µ . For time T −
2, we have V µ ( T − , x , y ) = inf π ( · ) ∈A T − T − E [ V µ ( T − , X π ( T − , Y π ( T − = inf π ( · ) ∈A T − T − E (cid:20) µ r ( T − π ( T − σ ( T − σ ( T − (cid:62) ] π ( T − (cid:62) − r ( T − r ( T − x − r ( T − γ ( T − π ( T − (cid:62) + µ r ( T − (cid:2) r ( T − x − y ) + γ ( T − π ( T − (cid:62) − E [ π ( T − (cid:62) ]) (cid:3) (cid:21) − β ( T − µ . π ∗ ( T − , x , y ) = µ γ ( T − r ( T −
1) [ σ ( T − σ ( T − (cid:62) ] − , and the related value function is V µ ( T − , x , y ) = µ T − (cid:89) s = T − r ( s )( x − y ) − T − (cid:89) s = T − r ( s ) x − (cid:80) T − s = T − β ( s )4 µ . By Theorem 2.1, we have V µ ( t , x , y ) = inf π ( · ) ∈A T − t E [ µ (cid:0) X π ( T ) − Y π ( T ) (cid:1) − X π ( T )] = inf π ( · ) ∈A T − t inf π ( · ) ∈A T − T − E [ µ (cid:0) X π ( T ) − Y π ( T ) (cid:1) − X π ( T )] = inf π ( · ) ∈A T − t E [ V µ ( T − , X π ( T − , Y π ( T − = inf π ( · ) ∈A T − t inf π ( · ) ∈A T − T − E [ V µ ( T − , X π ( T − , Y π ( T − · · · · · · = inf π ( · ) ∈A tt E [ V µ ( t + , X π ( t + , Y π ( t + . Therefore, we can employ the above method to obtain the optimal strategy for 0 ≤ t < T , π ∗ ( t , x , y ) = γ ( t )2 µ (cid:81) T − s = t r ( s ) [ σ ( t ) σ ( t ) (cid:62) ] − , and the related value function V µ ( t , x , y ) = µ ( x − y ) (cid:18) T − (cid:89) s = t r ( s ) (cid:19) − x T − (cid:89) s = t r ( s ) − (cid:80) T − s = t β ( s )4 µ . This completes this proof. (cid:3)
Proof of Proposition 2.1.
For a given mean level L > (cid:81) T − h = t r ( h ) in constrained condition (2.3). Theoptimal strategy π ∗ ( · ) and π ∗ ( · ) satisfy E [ X π ∗ ( T )] = E [ X π ∗ ( T )] = L . By formulations (2.14) and (2.16), we haveVar[ X π ∗ ( T )] = (cid:18) E [ X π ∗ ( T )] − x (cid:81) T − h = t r ( h ) (cid:19) (cid:80) T − h = t β ( h )and Var[ X π ∗ ( T )] = (cid:18) E [ X π ∗ ( T )] − x (cid:81) T − h = t r ( h ) (cid:19) (cid:81) T − h = t [ β ( h ) + − .
30y Assumption H , we have β ( s ) > , t ≤ s < T , and T − (cid:88) h = t β ( h ) < T − (cid:89) h = t [ β ( h ) + − . Therefore, one obtains, Var[ X π ∗ ( T )] > Var[ X π ∗ ( T )] . For a given risk aversion parameter µ >
0, we have E [ X π ∗ ( T )] = x T − (cid:89) h = t r ( h ) + T − (cid:88) h = t β ( h )2 µ , and E [ X π ∗ ( T )] = x T − (cid:89) h = t r ( h ) + µ ( T − (cid:89) h = t [ β ( h ) + − . From β ( s ) > , t ≤ s < T , it follows12 µ T − (cid:88) h = t β ( h ) < µ (cid:18) T − (cid:89) h = t [ β ( h ) + − (cid:19) , which implies that x T − (cid:89) h = t r ( h ) < E [ X π ∗ ( T )] < E [ X π ∗ ( T )] . Again, by formulations (2.14) and (2.16), we haveVar[ X π ∗ ( T )] = (cid:80) T − h = t β ( h )4 µ < (cid:81) T − h = t [ β ( h ) + − µ = Var[ X π ∗ ( T )] . Therefore, Var[ X π ∗ ( T )] < Var[ X π ∗ ( T )] , E [ X π ∗ ( T )] < E [ X π ∗ ( T )] . (A.1)This completes the proof. (cid:3) References
D. Andersson and B. Djehiche. A maximum principle for SDEs of mean-field type.
Appl. Math.Optim. , 63:341–356, 2011.I. Bajeux-Besnainou and R. Portait. Dynamic asset allocation in a mean-variance framework.
Man-agement Science , 11:79–95, 1998.S. Basak and G. Chabakauri. Dynamic mean-variance asset allocation.
Review of Financial Studies ,23:2970–3016, 2010. 31. Bensoussan, K. Sung, and S. C. P. Yam. Linear-quadratic time-inconsistent mean field games.
Dyn. Games. Appl , 3:537–552, 2013.A. Bensoussan, K. Sung, S. C. P. Yam, and S. P. Yung. Linear-quadratic mean field games.
Journalof Optimization Theory and Applications , 169:496–529, 2016.C. Bernard and S. Vandu ff el. Mean-variance optimal portfolios in the presence of a benchmark withapplications to fraud detection. European Journal of Operational Research , 234:469–480, 2014.J. Bi, H. Jin, and Q. Meng. Behavioral mean–variance portfolio selection.
European Journal ofOperational Research , 271:644–663, 2018.T. R. Bielecki, H. Q. Jin, S. Pliska, and X. Y. Zhou. Continuous time mean-variance portfolio selectionwith bankruptcy prohibition.
Mathematical Finance , 15:213–244, 2005.T. Bj¨ork, A. Murgoci, and X. Y. Zhou. Mean-variance protfolio optimization with state-dependentrisk aversion.
Mathematical Finance , 24:1–24, 2014.T. Bj¨ork, M. Khapko, and A. Murgoci. On time-inconsistent stochastic control in continuous time.
Finance Stochastic , 21:331–360, 2017.R. Buckdahn, B. Djehiche, and J. Li. A general stochastic maximum principle for SDEs of mean-fieldtype.
Appl. Math. Optim. , 64:197–216, 2011.M. Dai, Z. Q. Xu, and X. Y. Zhou. Continuous-time markowitz model with transaction costs.
SIAMJournal on Financial Mathematics , 1:96–125, 2010.M. Dai, H. Jin, K. Steven, and Y. Xu. A dynamic mean-variance analysis for log returns.
Acceptedby Management Science , https: // ssrn.com / abstract = ffi cient is the1 / n portfolio strategy ? The Review of Financial Studies , 22:1916–1953, 2009.P. H. Dybvig. Ine ffi cient dynamic portfolio strategies or how to throw away a million dollars in thestock market. The Review of Financial Studies , 1:67–88, 1988.M. Fischer and G. Livieri. Continuous time mean-variance portfolio optimization through the meanfield approach.
ESAIM: Probability and Statistics , 20:30–44, 2016.Y. Hu, H. Jin, and X. Y. Zhou. Time-inconsistent stochastic linear-quadratic control.
SIAM Journalon Control and Optimization , 50:1548–1572, 2012.32. Huang, P. E. Caines, and R. P. Malhame. The Nash certainty equivalence principle and McKean-Vlasov systems: An invariance principle and entry adaptation.
Proceedings of the 46th IEEEConference on Decision and Control , pages 121–126, 2007.A. Ismail and H. Pham. Robust Markowitz mean-variance portfolio selection under ambiguous co-variance matrix.
Mathematical Finance , 29:174–207, 2019.D. Li and W. L. Ng. Optimal dynamic portfolio selection: Multi-period mean-variance formulation.
Mathematical Finance , 10:387–406, 2000.J. Li. Stochastic maximum principle in the mean-field controls.
Automatica , 48:366–373, 2012.A. E. B. Lim. Quadratic hedging and mean-variance portfolio selection with random parameters inan incomplete market.
Mathematics of Operations Research , 29:132–161, 2004.A. E. B. Lim and X. Y. Zhou. Quadratic hedging and mean-variance portfolio selection with randomparameters in a complete market.
Mathematics of Operations Research , 1:101–120, 2002.H. Markowitz. Portfolio selection.
Journal of Finance , 7:77–91, 1952.H. Markowitz.
Portfolio Selection: E ffi cient diversification of investment . John Wiley & Sons, NewYork, 1959.H. Markowitz. Mean-variance approximations to expected utility. European Journal of OperationalResearch , 234:346–355, 2014.R. C. Merton. An analytic derivation of the e ffi cient frontier. J. Finance Quant. Anal. , 7:1851–1872,1972.H. Pham and X. Wei. Dynamic programming for optimal control of stochastic McKean-Vlasov dy-namics.
SIAM Journal on Control and Optimization , 55:1069–1101, 2017.H. Pham and X. Wei. Bellman equation and viscosity solutions for mean-field stochastic controlproblem.
ESAIM: Control, Optimisation and Calculus of Variations , 24:437–461, 2018.H. R. Richardson. A minimum variance result in continuous trading portfolio optimization.
Manage-ment Science , 9:1045–1055, 1989.J. M. Xia. Mean-variance portfolio choice: Quadratic partial hedging.
Mathematical Finance , 15:533–538, 2005.S. Yang. A varying terminal time mean-variance model. arXiv:1909.13102 , pages 1–25, 2019.33. Yang. Bellman type strategy for the continuous time mean-variance model. arxiv:2005.01904 ,pages 1–30, 2020.J. Yong. Time-inconsistent optimal control problems and the equilibrium HJB equation.
Mathemati-cal control and related fields , 2:271–329, 2012.J. Yong and X. Y. Zhou.
Stochastic control: Hamiltonian systems and HJB equations . Springer, NewYork, 1999.X. Y. Zhou and D. Li. Continuous-time mean-variance portfolio selection: A stochastic LQ frame-work.