Sharp Target Range Strategy for Multiperiod Portfolio Choice ... - arXiv

Sharp Target Range Strategy for Multiperiod Portfolio Choice by Decensored Least Squares Monte Carlo

arXiv:1704.00416v2 [q-fin.PM] 20 Oct 2017

Rongju Zhang∗, Nicolas Langrené†, Yu Tian‡, Zili Zhu§, Fima Klebaner¶and Kais Hamzak First version: April 3, 2017 This revised version: October 17, 2017

Abstract A novel investment strategy is presented for portfolio choice problems. Our proposed strategy maximizes the expected portfolio value within a target range, composed of a conservative lower target representing capital guarantee and a desired upper target representing investment goal. This strategy favorably shapes the entire probability distribution of return, as it simultaneously seeks a desired expected return, cuts off downside risk, and implicitly caps volatility, skewness and other higher moments. To illustrate the effectiveness of our investment strategy, we study a multi-period portfolio selection problem with transaction cost, and develop a decensored regression approach that improves the classical least squares Monte Carlo algorithm when dealing with truncated and discontinuous payoff functions. Our numerical tests show that the resulting distribution of portfolio wealth mimics the shape of the objective function, and that the presented strategy dominates the classical utility approach in terms of the mean-variance efficient frontier and the trade-off between return and downside risk.

Keywords: target-based portfolio optimization; alternative risk measure; multiperiod portfolio choice; least squares Monte Carlo; decensored regression

JEL Classification: G11, D81, C63, C34, MSC Classification: 91G10, 91G80, 91G60

∗ Corresponding

author. Email: [email protected]. School of Mathematical Sciences, Monash University CSIRO ‡ School of Mathematical Sciences, Monash University § RiskLab, CSIRO ¶ School of Mathematical Sciences, Monash University k School of Mathematical Sciences, Monash University † RiskLab,

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1

Introduction

A crucial and long-standing problem in the theory and practice of portfolio allocation is the choice of an effective and transparent performance criteria that balances return and risk. This paper proposes a novel investment strategy that is versatile and intuitive to investors, coupled with a novel simulationand-regression algorithm for solving its corresponding multi-period problem. In the literature, several strands of research have addressed the problem of decision making under risk and uncertainty. A first strand of literature corresponds to the classical expected utility approach (Bernoulli (1954), von Neumann and Morgenstern (1944)) for which the investment preferences are characterized by an utility function. We refer to Hodges (1998), Stutzer (2000) and Bacon (2013) for exponential (constant absolute risk aversion, CARA) utility and Zakamouline and Koekebakker (2009) for general utility. Although theoretically elegant, these utility functions are abstract and impractical, due to the implicit risk-return trade-off being opaque to investors. In additions, the researchers who did try to calibrate utility functions for practical purposes found substantial inadequacies with how people make decisions in reality. For example, Tversky and Kahneman (1992)’s cumulative prospect theory has identified some practical features ignored by the classical utility theory, such as preferences relative to initial wealth, risk-seeking behavior when losing money, and overweighting of unlikely events, see also Barberis (2012) for example. As an offspring of classical utility theory, the mean-variance framework of Markowitz (1952) revisits the quadratic utility case, in which risk is measured by the variance of portfolio returns. Its objective is to minimize the variance of returns given a minimum expected return level, or equivalently, to maximize expected returns given a maximum variance level of returns. When asset returns are normally distributed, many other risk measures have been found equivalent to variance, for example, the equivalence to first and second order lower partial moments has been proved by Klebaner, Landsman, Makov, and Yao (2017). In the case of normal assumption, the mean-variance framework greatly benefits from its simple quadratic formulation, compared to other equivalent but more complicated representations. In situations when asset returns are not normally distributed, variance becomes an inadequate measure of portfolio risk. For such cases, the mean-variance approach has been extended to incorporate higher moments of return distribution, see for example Lai (1991) and Konno, Shirakawa, and Yamazaki (1993) incorporating a skewness component and Davis and Norman (1990) incorporating skewness and kurtosis. To solve a multi-objective optimization problem, the usual approach is to solve a linear objective function that accounts for higher moments through some reward/penalization weights, as weighted sum is the easiest way to combine multiple and possibly conflicting objectives, see Köksalan and Wallenius (2012). In general, such multi-objective optimizations might suffer from the difficulty of reaching a stable global solution. Moreover, these weights, designed as investment preferences for controlling risk-return tradeoffs, lack transparency for investors, making this approach shunned in practice. Another approach to address the non normality of returns is to use downside risk measures. The most common downside risk measures are Value-at-Risk (VaR, Longerstaey 1996) that estimates the investment losses associated with a specific likelihood of occurrence, and Conditional Value-at-Risk (CVaR, Rockafellar and Uryasev 2000) that evaluates the expected losses beyond the VaR threshold. These two measures have been widely incorporated into a return-downside risk approach where variance in Markowitz (1952)’s framework is replaced by a downside risk measure. We refer to Alexander and Bap-

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tista (2002) for the mean-VaR framework and Agarwal and Naik (2004) for the mean-CVaR framework. Another popular downside risk measure is the lower partial moment. First mentioned by Markowitz (1959), most early works focus on the semivariance, see, for example, Mao (1970b), Mao (1970a), Hogan and Warren (1972) and Porter (1974). Later, Fishburn (1977) considered a framework for higher-order lower partial moments. Interestingly, the formulation of lower partial moments naturally links to the utility theory and stochastic dominance, see, for example, Fishburn (1977) and Bawa (1978). The generalized mean lower partial moment model is presented in Harlow and Ramesh (1989). Compared to the classical mean-variance approach, it reduces downside risk while preserving the same or a greater level of expected return. A similar argument is supported by the empirical study in Harlow (1991), where the author emphasizes that traditional risk measures such as variance are special cases of the lower partial moment framework. Jarrow and Zhao (2006) introduce a downside loss-averse utility function and decompose it into three parts: expected return, standard deviation and a downside risk measure. Recently, with normality assumption, Klebaner et al. (2017) derive analytical solutions for the mean-lower partial moment portfolios and the lower partial moment portfolios when the mean constraint is not prespecified. The last main strand of literature corresponds to the target-based strategies that aim to track a prespecified profit target. Many of these target strategies are built upon the classical approaches discussed above. Regarding the classical utility approach, Teplá (2001) maximizes an expected hyperbolic absolute risk aversion (HARA) utility under constraint of exceeding a stochastic benchmark. Regarding the mean-variance approach, Li and Ng (2000) and Zhou and Li (2000) formulate an equivalent linearquadratic target-specified problem, Franks (1992) replaces the expected return in the mean-variance framework by expected return in excess of a predefined target, and Williams (1997) maximizes the probability of beating a target return for portfolios on the efficient mean-variance frontier. Regarding the (mean-) lower partial moment approach that penalizes the downside deviations from a prespecified target level, regarded as a special case of target-based strategies, we refer to Brogan and Stidham Jr. (2005) and Klebaner et al. (2017). In general though, the most popularly used target-based strategy is to maximize the probability of achieving the target. For example, Browne (1999a) and Browne (1999b) maximize the probability of exceeding a fixed target return and a stochastic benchmark respectively. Pham (2003) solves the problem of maximizing the probability of beating a stochastic benchmark by a given percentage. Gaivoronski, Krylov, and van der Wijst (2005) propose a dynamic benchmarktracking strategy with transaction cost, applicable to a variety of risk measures. Morton et al. (2006) maximize the probability of outperforming a benchmark with penalization of the expected downside deviation from the target. Symmetrically, one can instead minimize the probability of an undesirable outcome. Based on large deviation theory, Hata, Nagai, and Sheu (2010) and Nagai (2012) minimize the probability of underperforming a target return. As an application, Milevsky et al. (2006) derive an optimal annuitization strategy that minimizes the probability of lifetime ruin of a retiree. Unlike mean-variance, high moment penalization, expected utility and downside-based criteria, the explicitly specified investment target makes this strand of literature the most likely to be understood and applied in practice by practitioners. However, the difficulty lies in the choice of the target level. Overspecifying the return target may result in a risky allocation with an unspecified propensity to lose capital, and conversely, underspecifying the return target may forego potential gain opportunities. In this paper, we propose the Sharp Target Range Strategy (STRS) that maximizes the expected portfolio value under a prespecified target range. The idea is as follows: a target return range is specified by

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the investor: a conservative lower target representing the level of capital guarantee and an desired upper target corresponding to the return level the investor wishes to achieve. The STRS maximizes the expected portfolio value bounded within this target range, and the resulting optimal allocation therefore implicitly maximizes the probability that the investment return lies within the target range and as close to the upper target as possible. There are three main motivations behind the proposed STRS. The first motivation traces back to the primary purpose of an investment objective function, which is to carve a desirable shape for the probability distribution of returns. The STRS, seeking a desirable expected return while chopping off the tails of the distribution beyond the target range, monitors the entire shape of return distribution, and the statistical moments are implicitly contained. Our numerical results show that return realizations are well contained within the investor’s chosen range and at the same time tilted towards the upper target, and that our STRS dominates the CRRA utility in terms of the mean-variance efficient frontier and trade-off between return and downside risk. This practical success is attributed to the sacrifice of the upside potentials beyond the upper target: as the upside potential cannot be decoupled from downside risk, the proposed upper target disallows any excessive risk once the predefined investment goal is close to be achieved, which drastically improves downside risk management and makes the portfolio performance much more predictable. The second motivation corresponds to the difficulty of specifying a target return for the classical targetbased strategies, as one single target return parameter can hardly reconcile pursuit of return while simultaneously providing downside protection. In particular, the frail single-parameter compromise is vulnerable to misspecification of the target: for example, overspecifying the target (while expecting both high return and low downside risk) will result in excessively risky allocations with potentially extreme downside risks, which contradicts the initial investment requirements. By contrast, the STRS solves this dilemma by combining one upper target that accounts for return-seeking preference, with a lower target that accounts for loss-aversion preference. This separation improves the robustness to target misspecification. For example, overspecifying the upper target would not result in downside risk explosion, thanks to the capital protection provided by the lower target. Finally, the third motivation is to improve the communication and practical adoption of portfolio allocation techniques from academia to the industry. Performance criteria depending on abstract parameters with unforeseeable practical effects are unlikely to be adopted by investors. Our proposition of two explicit targets labeled in terms of returns, with intuitive purpose (capital protection for the lower target, desired investment return for the upper target), serves this quest for a simple and practical investment criteria that provides guidance towards better portfolio allocation decisions. To demonstrate the effectiveness of the presented STRS, we study a dynamic (multiperiod) portfolio choice problem with proportional transaction cost. In order to solve this multiperiod problem with the discontinuous STRS payoff function, we develop a decensored least squares Monte Carlo (LSMC) algorithm. The literature of LSMC methods as well as the details of our decensored technique will be discussed in details in Section 3. Our numerical tests show that the resulting simulated distribution of portfolio returns manages to mimic the precise shape of the target range function, and that the presented strategy dominates the classical utility approach in terms of the mean-variance efficient frontier and the trade-off between return and downside risk. We also find that the “best” level of lower target is to be the initial portfolio value, at which the standard deviation and downside risk of the portfolio are marginally

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minimized given a fixed level of upper target. Finally, we provide a five-year backtesting result and show that the real market performance does behave as what is expected from the design of the STRS. We provide two extensions to the STRS. The first extension deals with pure maximization of the probability that the final portfolio value lies within the target range, without further attempt to pursue a higher return. We call it Flat Target Range Strategy (FTRS). This FTRS generalizes the classical VaR minimization approach and is useful when maintaining solvency is more important than seeking high returns, for example for pension schemes, retirement funds and life-cycle management. The second extension concerns relative returns, namely Relative Target Range Strategy (RTRS), i.e., defining the bounds of the STRS or FTRS in terms of excess return over a stochastic benchmark, such as stock index, interest rate or inflation rate. The fixed-range STRS enables to protect the capital in bearish markets, but may underperform the market when it is bullish. Instead, the target range defined over a market benchmark aims to constantly outperform the market.

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Sharp Target Range Strategy

In this section, we formulate the sharp target range strategy (STRS) for portfolio optimization problems and discuss the benefits of this strategy. We consider a portfolio selection problem with d risky assets
available over a finite time horizon T . Let αt = αti 1≤i≤d be the portfolio weights in each risky asset at time t. Finally, let {Wt }0≤t≤T denote the portfolio value (or wealth) process. Assume that the investor tries to maximize the expectation of some function of final wealth E [f (WT )] over all the possible strategies {αt }0≤t≤T . Let {Ft }0≤t≤T be the filtration generated by all the state variables. At any time t ∈ [0, T ], the objective function simply reads sup E [f (WT ) |Ft ] ,

(2.1)

α

where the investment preference is characterized by the function f (·) . In this paper, we propose the following parametric shape: f (w) = (w − LW )1{LW ≤ w ≤ UW },

(2.2)

where LW ∈ R represents a conservative lower target, UW ∈ R represents a desired upper target, and

1{LW ≤ w ≤ UW } returns one if LW ≤ w ≤ UW otherwise returns zero. Throughout this paper, we normalize the portfolio value W and the  bounds [LW , U W ] by the initial wealth W0 . Indeed, formula w LW UW (2.2) shows that f (w; LW , UW ) = W0 × f W0 ; W0 , W0 , so we can assume without loss of generality that W0 = 1 and set the bounds LW and UW in the vicinity of 1. Figure 2.1 shows an example (2.2) with LW = 1.0 and UW = 1.2. Given the shape (2.2), the objective function (2.1) maximizes the expected final wealth within the interval [LW , UW ], while the values outside this interval are penalized down to zero. This strategy implicitly combines two objectives: maximizing the expected terminal portfolio value and maximizing the probability that the terminal portfolio value lies within the chosen target range [LW , UW ]. The distinctive feature of the STRS is the foregoing of the upside potential beyond an upper target UW , which seems to conflict with the non-satiation axiom that people prefer more to less. Everything else being equal (ceteris paribus assumption), it is true that people prefer more to less. In other words, if

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Figure 2.1: Sharp target range

getting more does not incur side effects, people will prefer more to less. This axiom in the context of stochastic portfolio optimization can be understood as follows: the downside risk being fixed (left tail of the portfolio returns), people will prefer a higher upside potential (longer right tail of returns). However, the classical non-decreasing utility cannot decouple downside risk and upside potential: both tails of the return distribution will be lengthened if higher upside potential is allowed. As the ceteris paribus assumption does not apply in the stochastic context, the choice between satiation or non-satiation solely depends on investor’s preference with respect to risk and return. As upside potential and downside risk are naturally intertwined, the proposed upper target eliminates the downside risk by cutting off its cause - the pursuit of excessive upside potential. As a result, the return realizations can be well contained within the target range and tilted towards the upper target, making the portfolio performance more predictable, which in many contexts is more important than allowing for the possibility of rare windfall returns at the cost of higher downside risk. In Section 5, we show that the return distribution produced by the STRS does mimic the shape presented in Figure 2.1. The STRS is preferable to other classical investment objective formulations (where γ denotes the corresponding risk preference parameter): 1. The mean-downside risk approach such as mean-semivariance, i.e., sup E [WT |Ft ] subject to E



α

2  Ft < γ2 , (WT − γ1 ) −

is similar to STRS with UW = ∞. The addition of a finite upper target generalizes this family and makes the investment return much more predictable (as discussed above). 2. The mean-variance framework, equivalently formulated as a target-specified strategy, i.e., inf E α



WT −

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 γ 2 Ft , 2

is vulnerable to target misspecification, for example an overly high target may cause extreme downside risks. Also the mean-variance optimization has been found highly sensitive with respect to the its input parameters of return, see for example Merton (1980). The STRS provides a lower target to protect the capital even if the investment goal (upper target) is unrealistically high, thus improves robustness of the optimization. 3. Compared to downside risk minimizations such as VaR minimization, i.e., inf VaRγ ( WT | Ft ) α

which ignores all the upside potential, the STRS cuts off only the extreme part of the upside potentials while maintaining active participation in market rallies to achieve the upper target. In Section 4.1, we extend STRS to a downside risk minimization strategy that maximizes the probability of the return lying within a target range, which generalizes the classical VaR minimization. 4. Compared to the multi-criteria objective, i.e., sup E [WT |Ft ] subject to Var [WT |Ft ] < γ1 , Skew [WT |Ft ] > γ2 , kurt [WT |Ft ] < γ3 α

with multiple conflicting and abstract moment constraints, STRS achieves multiple criteria by solving an unconstrained and intuitive optimization problem. By maximizing the expected portfolio value within a explicit return range, the STRS also carves the entire shape of return distribution, including the high order statistical moments. 5. Compared to non-linear utility functions such as constant relative risk aversion (CRRA), i.e.,  sup E α

 WT 1−γ Ft 1−γ

which is characterized by an abstract risk-aversion parameter γ, the STRS is expressed in term of two explicit investment targets, thus more transparent to investors. Most importantly, the STRS dominates the CRRA utility in terms of efficient frontier and of trade-off between return and downside risk, which will be shown in Figure 5.4 of our numerical experiment Section 5.

3

Dynamic Solution

In this section, we consider a dynamic portfolio selection problem and formulate it as a discrete-time dynamic programming problem for which we develop a decensored least-squares Monte Carlo (LSMC) method to solve it. The LSMC algorithm, originally developed by Carriere (1996), Longstaff and Schwartz (2001) and Tsitsiklis and Van Roy (2001) for the pricing of American options, has been extended to solve dynamic portfolio selection problems in Brandt, Goyal, Santa-Clara, and Stroud (2005), Garlappi and Skoulakis (2010) and Cong and Oosterlee (2016). Brandt et al. (2005) consider a CRRA utility function and determine a semi-closed form by solving the first order condition of the Taylor series expansion of the value function. Garlappi and Skoulakis (2010) claim that the convergence of Brandt et al. (2005)’s method is not stable and that it cannot handle problems where the control variable

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depends on the endogenous wealth variable. Instead, they introduce a state variable decomposition method to overcome this drawback. However, this decomposition relies on a linear separation between the observable component and stochastic deviation of returns, which cannot be applied to general return distributions. Cong and Oosterlee (2016) consider a mean-variance objective function and use a multistage strategy to perform forward simulation of control variables which are iteratively updated in the backward recursive program, where the admissible control sets are constructed as small neighborhoods of the solutions to the multi-stage strategy. Later, Cong and Oosterlee (2017) combine Jain and Oosterlee (2015)’s stochastic bundling technique with Brandt et al. (2005)’s method. Zhang, Langrené, Tian, Zhu, Klebaner, and Hamza (2016) consider a general class of utility functions and adopt Kharroubi et al. (2014)’s control randomization technique to the portfolio choice problem with switching costs including transaction cost, liquidity cost and market impact. To implement the LSMC method to solve the sharp target range strategy (STRS), the difficulty lies with the discontinuity of f (w) = (w − LW )1{LW ≤ w ≤ UW } at UW . In this section, we extend the classical LSMC method and propose a decensored regression method that approximates the value function in a manner as simple as approximating a linear function.

3.1

Dynamic programming

Denote Rf as the cumulative return of the investment on the risk-free asset. Let {Rt }0≤t≤T =  i 1≤i≤d Rt 0≤t≤T denote the excess returns of the risky assets over the risk-free rate and {Zt }0≤t≤T denote the vector of return predictors. Then, the optimization problem in (2.1) can be formulated as a stochastic control problem with exogenous state variables {Zt }0≤t≤T and endogenous state variable {Wt }0≤t≤T . Let A ⊆ Rd be the set of admissible portfolio strategies. The objective function in (2.1) now can be rewritten as vt (z, w)

:=

sup

E [f (WT ) |Zt = z, Wt = w ] .

(3.1)

{ατ ∈A}t≤τ ≤T

Consider an equally-spaced discretization of the investment horizon [0, T ], denoted as 0 = t0 < · · · < tN = T . Then, the wealth process evolves as Wtn+1

=


Wtn Rf + αtn · Rtn+1 ,

(3.2)

and the value function satisfies the following dynamic programming principle vtN (z, w)

= f (w),

vtn (z, w)

=


 sup E vtn+1 Ztn+1 , Wtn+1 |Ztn = z, Wtn = w

(3.3)

αtn ∈A

where f (w) = (w − LW )1{LW ≤ w ≤ UW }.

3.2

Standard least squares Monte Carlo

The first part of the LSMC algorithm consists in the forward simulation of all the stochastic state variables. Let M denote the number of Monte Carlo simulations. The return predictors {Zt }0≤t≤T and 8

the asset excess returns {Rt }0≤t≤T are generated through some calibrated return dynamics. By contrast, the portfolio value process {Wt }0≤t≤T is an endogenous state variable depending on the realizations  m 1≤m≤M ˜ tn 0≤n≤N , then compute the of the control {αt }0≤t≤T . We generate random portfolio weights α  m 1≤m≤M ˜t corresponding portfolio values W according to (3.2). The convergence proof of such control n 0≤n≤N randomization technique is provided in Kharroubi et al. (2014). The second part of the LSMC algorithm uses a dicretization procedure. We first discretize the control space as Ad = {a1 , ..., aJ } and define the conditional value function CVjtn as the expectation of the subsequent value function conditional on making the decision αtn = aj ∈ Ad , i.e., CVjtn (z, w)


 := E vtn+1 Ztn+1 , Wtn+1 Ztn = z, Wtn = w, αtn = aj ,

(3.4)

thus we can write  
vtn (z, w) = sup E vtn+1 Ztn+1 , Wtn+1 Ztn = z, Wtn = w ≈ max CVjtn (z, w) . aj ∈Ad

αtn ∈A

At time tN , the value function (3.3) is equal to vˆtN (z, w) = f (w) = (w − LW )1{LW ≤ w ≤ UW }. Assume n o1≤j≤J ˆ j (z, w) that the conditional value functions CV have been estimated. At time tn , for tn0 n+1≤n0 ≤N −1  each aj ∈ Ad , we reset the decisions αm tn 1≤m≤M to aj , and then recompute the endogenous wealth process from tn to tN : ˆ tm,(n,j) W n+1 ˆ tm,(n,j) W n+2

  ˜ tm Rf + aj · Rtm = W n n+1   n  o m,(n,j) m m ˆ l ˆ tm,(n,j) Rf + arg max CV ˆ = W Z , W · R tn+1 tn+1 tn+1 tn+2 n+1 al ∈Ad

.. . ˆ tm,(n,j) W N

  n  o m,(n,j) m,(n,j) f l m m ˆ ˆ ˆ = WtN −1 R + arg max CVtN −1 ZtN −1 , WtN −1 · RtN . al ∈Ad

(3.5)

ˆ tm,(n,j) := W ˆ tm where W k k

, k = n, . . . , N is the recomputed wealth from tn to tN , using ˆ m =W ˜ m ,αt =aj ) (W n tn tn the control aj at time tn and the estimated optimal controls at times tn+1 , . . . , tN −1 . Let {ψk (z, w)}1≤k≤K be the vector of basis functions of the state variables. To evaluate the conn  o ˆ tm,(n,j) ditional value functions CVjtn (z, w), the original LSMC algorithm regresses f W on N 1≤m≤M 
˜ m 1≤k≤K . However, one difficulty with such a least-squares regression lies with the disconψk Zm tn , Wtn 1≤m≤M tinuity of f at UW (equation (2.2)). To overcome thiso difficulty, we introduce a decensored least-squares n  m,(n,j) ˆ regression technique that approximates f WtN in a manner as simple as approximating 1≤m≤M   ˆ m W . tN αtn =aj

3.3

1≤m≤M

Decensored least squares Monte Carlo

n  o ˆ tm,(n,j) The classical LSMC estimates CVjtn (z, w) by regressing f W N

1≤m≤M


˜ m 1≤k≤K . on ψk Zm tn , Wtn 1≤m≤M

ˆ tm,(n,j) outside the target range [LW , UW ], it appears to be a censored reAs f censors the values of W N gression problem and an usual estimation approach is to use maximum likelihood estimation (MLE). 9

However, the main difference between our problem and n thecensored o regression problem is that we have m,(n,j) ˆ access to more information than the censored sample f WtN , i.e., we have access to 1≤m≤M n o ˆ tm,(n,j) both the censored and uncensored samples W . Thus, a direct use of MLE loses the N 1≤m≤M

ˆ tm,(n,j) which are in fact observable in our problem. The inadinformation of the censored values of W N ˆ m,(n,j) outside the target range [LW , UW ]. equacy of MLE motivates us to “decensor” o the values of WtN n 
ˆ tm,(n,j) ˜ m 1≤k≤K via MLE, we propose a on ψk Zm , W Then, instead of regressing f W N

1≤m≤M

n βˆj

k,tn

o

=

β∈RK

=

v u u t

tn

1≤m≤M


˜ m 1≤k≤K and obtain on ψk Zm tn , Wtn 1≤m≤M

arg min

1≤k≤K

σ ˆtjn

tn

1≤m≤M

decensored regression approach: n o ˆ tm,(n,j) • Firstly, regress W N

M X

K X

m=1

k=1

!2 ˜m βk ψk Zm tn , Wtn




−

ˆ tm,(n,j) W N

M K X X
1 j ˆ tm,(n,j) − ˜m W βˆk,t ψk Zm tn , W tn N n M − K m=1

!2 .

(3.6)

k=1

As a result, the terminal wealth can be modeled as ˆ t W N

Ztn =z,Wtn =w,αtn =aj

=µ ˆjtn (z, w) + σ ˆtjn ε,

µ ˆjtn (z, w) :=

K X

j βˆk,t ψk (z, w) , n

(3.7)

k=1

where ε is the regression residuals, which for demonstrative purpose we assume Gaussian. Let  2 φ(x) = √12π exp x2 represent the standard normal probability density function, and Φ(x) = Rx φ(x)dx represent the standard normal cumulative distribution function. Note that the as−∞ sumption of distribution is also required by MLE. • Secondly, plug equation (3.7) into the conditional value formula (3.4) to obtain a closed-form estimate. By combining (3.5), (3.4), (3.6) and (3.7), we obtain the following closed-form estimate of the conditional value function for each aj ∈ Ad at time tn : ˆ j (z, w) CV tn

 n o i ˆ t − LW 1 LW ≤ W ˆ t ≤ UW Zt = z, Wt = w, αt = aj W n n n N N h  n oi j j j j = Eε µ ˆtn (z, w) + σ ˆtn ε − LW × 1 LW ≤ µ ˆtn (z, w) + σ ˆtn ε ≤ UW )# " (   UW − µ ˆjtn (z, w) LW − µ ˆjtn (z, w) j ≤ε≤ = µ ˆtn (z, w) − LW Eε 1 σ ˆtjn σ ˆtjn " ( )# LW − µ ˆjtn (z, w) UW − µ ˆjtn (z, w) j +ˆ σtn Eε ε1 ≤ε≤ σ ˆtjn σ ˆtjn ! !!   LW − µ ˆjtn (z, w) UW − µ ˆjtn (z, w) j = µ ˆtn (z, w) − LW Φ −Φ σ ˆtjn σ ˆtjn ! !! LW − µ ˆjtn (z, w) UW − µ ˆjtn (z, w) j −ˆ σtn φ −φ , (3.8) σ ˆtjn σ ˆtjn = E

h

where the last inequality is obtained by direct integration.

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ˆ tn : (z, w) 7→ α ˆ tn (z, w) and vˆtn : (z, w) 7→ vˆtn (z, w) are estimated by Finally, the mappings α ˆ j (z, w) ˆ tn (z, w) = arg max CV α tn aj ∈Ad

and

ˆ j (z, w) . vˆtn (z, w) = max CV tn aj ∈Ad

(3.9)

In summary, thanks to the simple truncated linear shape of the sharp target range function (2.2), the conditional expectations in the dynamic programming equations (3.3) can be estimated by the ˆ tm,(n,j) in (3.6), the least-squares closed-form formula (3.8). Due to the linearity of the regressand W N

approximation is much more robust and stable than in the case of classical concave utility functions, for which the direct least-squares approximation E[U (WT )|Ztn = z, Wtn = w] ≈ β ·ψ(z, w) may lead to large cumulative statistical errors especially when risk-aversion is high (strong concavity), see Van Binsbergen and Brandt (2007), Garlappi and Skoulakis (2009) and Denault and Simonato (2017). Our sharp target range function (2.2) avoids this non-linearity problem, and the decensored LSMC algorithm converges much more quickly w.r.t. the number of Monte Carlo paths M . More generally, the approach proposed here (linear approximation (3.7) + truncation corrections (3.8)) can be adapted to the situations where residuals are significantly non-Gaussian: this would simply modify the correction terms (3.8). There is no restriction on the choice of return distribution, nor on the estimation methods to use for the residual distributions if needed (empirical distribution, kernel estimation, mixture normal, etc.). Nevertheless, without loss of generality, it is reasonable to assume normality for low-frequency trading such as monthly returns with monthly rebalancing considered in our numerical section 5. For this reason and for demonstration purpose, we assume normality of residuals, use the approach (3.7)-(3.8) for our implementation, and focus our analysis of the effects of the new investment strategy (2.2).

3.4

Upper target as stop profit

As discussed in Section 2, the main effect of the upper target UW in the performance measure is to reduce the downside risk. However, in multiperiod optimization, a paradox might occur when the realized wealth crosses the upper target: by default, the portfolio optimizer might tell the fund manager to pick the −(T −t)

assets most likely to fall. It is trivial to see that, when Wt ≥ UW Rf −(T −t)

maximized by henceforth investing UW Rf

−(T −t)

out the balance amount Wt − UW Rf

, the objective function can be

amount of wealth into the risk-free asset and by taking

from the problem. To implement such a correction more

rigourously, two approaches are possible: 1. One can replace T by T ∧τ in the objective function (2.1), where τ is the first (stopping) time when −(T −t)

Wt ≥ UW Rf −(T −t) UW Rf

. At time τ (if it occurs before T ), the dynamic allocation stops: the amount −(T −t)

is invested in the risk-free asset, and the balance amount Wt − UW Rf

is taken

out (consumption, investment in one of the assets or another portfolio, etc.) 2. A second approach is to add an extra dynamic control to the problem: dynamic withdrawal/consumption. For simplicity, we use the first approach in our numerical experiments, and invest the extra balance in the risk-free asset.

11

4

Extensions

The Sharp Target Range Strategy (STRS) (2.2) can be adjusted and extended to other types of portfolio allocation problems. This section provides two extensions: the Flat Target Range Strategy (FTRS) that purely maximizes the probability that the wealth lies within the prespecified target range without further attempts to rally for profits; and the target range strategies over a stochastic benchmark such that the absolute fixed target range is replaced by a relative target range.

4.1

Flat target range strategy

Our STRS (2.2) delivers a return distribution that was tilted towards the upper return target, so as to provide a balance between return seeking and downside risk protection. Yet, there exists some other types of portfolio allocation problems for which the ability to remain solvent prevails over the appetite for high returns. This includes insurance-related problems such as lifecycle portfolio choice. For such problems, one can adjust the sharp target range shape (2.2) to a flat target range shape given by f (w) = 1 {LW ≤ w ≤ UW } .

(4.1)

Figure 4.1 illustrates equation (4.1) with [LW , UW ] = [1.0, 1.2]. Figure 4.1: Flat target range

Then the portfolio optimization problem becomes vt (z, w)

=

sup {ατ ∈A}t≤τ ≤T

=

sup

E [1 {LW ≤ w ≤ UW } |Zt = z, Wt = w ] P [LW ≤ WT ≤ UW |Zt = z, Wt = w ] ,

(4.2)

{ατ ∈A}t≤τ ≤T

which is a pure probability maximizing strategy. The FTRS generalizes the classical Value-at-Risk (VaR) minimization approach: when UW = +∞, the 12

FTRS (4.2) and VaR minimizations are equivalent, the difference being a fixed, absolute cut-off level for the former and an implicit, relative cut-off level for the latter. In particular, the FTRS minimizes the probability of being below a particular level of “loss”, while the VaR procedure minimizes a particular loss quantile. Thus, one benefit of FTRS compared to VaR minimization is its intuitive transparency. When UW is finite, the FTRS provides greater flexibility for investors to devise their risk preferences, as the lower return target LW is an explicit input from the investor, and the option to fix an upper target UW broadens the range of possible risk profiles. The dynamic solution of the flat target range strategy can be found by the same procedure described in Section 3. oAs in Section 3, at time tn , for each aj ∈ Ad , we first regress the recomputed terminal wealth n 
m,(n,j) ˆt ˜ m 1≤k≤K and obtain W on the basis of ψk Zm t , Wt N

n

1≤m≤M

ˆ t W N

Ztn =z,Wtn =w,αtn =aj

n

1≤m≤M

=µ ˆjtn (z, w) + σ ˆtjn ε,

µ ˆjtn (z, w) :=

K X

j βˆk,t ψk (z, w) , n

k=1

where the residuals ε follow a standard normal distribution. From there the closed-form conditional value functions for FTRS are given by ˆ j (z, w) CV tn

= = =

4.2

h n o i ˆ t ≤ UW Zt = z, Wt = w, αt = aj 1 LW ≤ W n n n N h n oi Pε 1 LW ≤ µ ˆjtn (z, w) + σ ˆtjn ε ≤ UW ! ! UW − µ ˆjtn (z, w) LW − µ ˆjtn (z, w) −Φ . Φ σ ˆtjn σ ˆtjn P

(4.3)

Target range over a stochastic benchmark

So far, the target returns in the target range strategies (2.2) and (4.1) are defined in terms of fixed, absolute returns. It is also possible to define these threshold returns relatively to a stochastic benchmark, be it stock index, inflation rate, exchange rate or interest rate. We refer to Franks (1992), Browne (1999a), Brogan and Stidham Jr. (2005) and Gaivoronski et al. (2005) for classical investment strategies that aim to outperform a stochastic benchmark. Let B = (Bt )0≤t≤T denote the stochastic benchmark of interest, and define the relative excess wealth as W − B. We can then modify the target range (2.2) as: fB (w, b) := (w − b)1{LW ≤ w − b ≤ UW } ,

(4.4)

fB (w, b) := 1{LW ≤ w − b ≤ UW } ,

(4.5)

for STRS, and

for FTRS of (4.1). Then, the new objective function is vt (z, w, b)

=

sup

E [fB (WT , BT ) |Zt = z, Wt = w, Bt = b ] .

(4.6)

{ατ ∈A}t≤τ ≤T

The stochastic benchmark B can be simply modeled as one additional exogenous state variable, so that the new problem (4.6) can be solved using the same approach developed in Section 3. 13

5

Numerical experiments

In this section, we test the sharp target range strategy (STRS) (2.2) numerically, and illustrate how it is able to consistently meet the investor’s range objective. Table 5.1 summarizes the asset classes and exogenous state variables used for our numerical experiments. We consider a portfolio invested in five assets: risk-free cash, U.S. bonds (AGG), U.S. shares (SPY), international shares (IFA) and emerging market shares (EEM), the other assets listed in Table 5.1 being used as return predictors. The annual interest rate on the cash component is set to be 2%. We assume 0.1% proportional transaction cost and refer to Zhang et al. (2016) on how to deal with switching costs in the LSMC algorithm. A first-order vector autoregression (VAR) model is calibrated to the monthly log-returns of the assets in Table 5.1 from September 2003 to March 2016. By bootstrapping the residuals, M = 104 Monte Carlo simulation paths are generated for a twelve-month forward time period. Due to the strong stability of our decensored LSMC method, M = 104 paths is more than sufficient to reach numerical stability and convergence. For the same reason, we use a simple second-order multivariate polynomial as the basis functions for the linear least-squares regressions. Table 5.1: Risky assets and return predictors

5.1

Assets

Underlying

Data source

U.S. Bonds U.S. Shares International Shares Emerging Market Shares Japanese shares

AGG (ETF) SPY (ETF) IFA (ETF) EEM (ETF) Nikkei225

Yahoo Yahoo Yahoo Yahoo Yahoo

U.K. shares Australian shares Gold Crude Oil U.S. Dollar

FTSE100 ASX200 Spot Price Spot Price USD Index

Yahoo Finance Yahoo Finance World Gold Council U.S. Energy Info. Admin. Federal Reserve

Japanese Yen Euro Australian Dollar

JPYUSD USDEUR USDAUD

Federal Reserve Federal Reserve Federal Reserve

Finance Finance Finance Finance Finance

Wealth distribution

Figure 5.1 provides some examples of the estimated distribution of the terminal portfolio value when using the STRS. We recall that the portfolio value W and the bounds [LW , UW ] are scaled by the initial wealth, so that without loss of generality W0 = 1. The lower target LW is set to the initial wealth level 1.00, a natural choice representing investors’ preference for capital protection. Four different upper targets UW are tested: 1.05, 1.10, 1.20 and 1.30. UW can be viewed as representing the “appetite” for investment returns. Several comments can be made about the shape of the terminal wealth distribution produced by the STRS. The most striking observation, which is also the main result of the paper, is that the STRS (2.1) does confine most of the wealth distribution within the predefined target range, and for small upper targets UW = 1.05, 1.10, the wealth distributions do mimic the specific shape of the sharp target range function

14

(2.2), making downside risk negligible. As expected, setting the upper target UW to a higher level produces a higher expected final wealth with higher standard deviation and greater downside risk (as measured by the probability of losing capital). For the tails beyond the target range, the two low levels UW = 1.05 and UW = 1.10 produce a fatter right tail, while the two higher levels UW = 1.20 and UW = 1.30 produce a fatter left tail, which is consistent with the fact that the greater UW , the greater the willingness to take risk to achieve the upper target. This demonstrates the capability of the STRS to represent different risk appetites, and the capability of the decensored LSMC algorithm to handle the sharp, discontinuous payoff functions. An interesting quantity to monitor is the ratio R := (E [WT ] − LW )/(UW − LW ) which measures the location of the expected performance E [WT ] relative to the target range: R = 0% means E [WT ] = LW , while at the opposite R = 100% means E [WT ] = UW . In our experiments from Figure 5.1, R is a decreasing function of UW , from R = 72% for UW = 1.05 down to R = 38% for UW = 1.30. This illustrates the natural fact that the higher the desired upper target, the harder it is to achieve it. Figure 5.1: Terminal wealth distribution using sharp target range strategy

15

Figure 5.2 shows the time evolution of the wealth distribution (0.05 percentile to 99.95 percentile) over the whole investment horizon, for the STRS with [LW = 1.0, UW = 1.1] (top-left panel), [LW = 1.0, UW = 1.2] (top-right panel), [LW = 1.0, UW = ∞] (bottom-left panel) and [LW = 0, UW = ∞] (bottom-right panel), where the last approach is equivalent to maximizing the expected terminal wealth without taking risk into account. The results shows that the wealth distributions in the top panel are well tightened within the prespecified target ranges over the whole investment process, compared to the UW = ∞ case in the bottom panel. Once again, as upside potential and downside risk are naturally intertwined, one cannot contain the downside risk very well when the upper target is set to a very high level, as shown by the [LW = 1.0, UW = ∞] example (bottom-left panel). Figure 5.2: Time evolution of wealth distribution using STRS

16

5.2

Outperforming classical utility

Our next experiment shows that the STRS (2.2) outperforms classical utility functions such as CRRA utility (constant relative risk aversion U(w) = w1−γ /(1−γ)). The CRRA utility and other classical utility functions (CARA, HARA, etc.) have two main problems when it comes to solving portfolio selection problems: the aversion parameters are abstract and difficult to characterize for practical investment decisions; and for multiperiod problems, a simulation-and-regression approach generates large numerical errors when the utility function is highly nonlinear (high risk aversion), as noted in Van Binsbergen and Brandt (2007), Garlappi and Skoulakis (2009) and Denault and Simonato (2017). By design, the alternative target range “utility” (2.2) proposed in this paper does not suffer from these problems. Figure 5.3: Terminal wealth distribution comparison between STRS and CRRA

Figure 5.3 compares the distributions of terminal portfolio value generated by the STRS and the CRRA utility. Our main finding is that for each risk aversion level γ of the CRRA utility, there exists a target range [LW , UW ] such that the STRS produces a higher expectation, lower standard deviation and lower

17

downside risk. As an illustration, Figure 5.3 provides two examples on how the STRS [LW , UW ] = [0.9, ∞] and [LW , UW ] = [0.93, 1.53] outperform the CRRA utility with γ = 2 and γ = 10, respectively. As previously discussed in Section 3.3, the CRRA utility suffers from large numerical error in the LSMC algorithm when the risk-aversion parameter is high (high nonlinearity), while the STRS does not suffer from this problem at all (as the regression (3.6) only involves a linear regressand). In order to allow γ to be greater than 10 while ensuring numerical convergence within a manageable computational runtime, we now reduce the investment horizon to a three-month period. This allows us to obtain stable CRRA results for up to γ = 30. Figure 5.4 provides the efficient frontiers of the STRS (for different combinations of LW and UW ) and the CRRA utility (for different γ levels). The results show that the STRS (2.2) dominates CRRA utility in terms of the mean-variance efficient frontier and the trade-off between return and downside risk. Moreover, the STRS is much more flexible: in particular, it is naturally able to reach very low levels of standard deviation and downside risk (by setting a cautious, tight target range), which is hardly reachable by classical nonlinear utility functions. This is the reason for the shorter range of values on the efficient frontiers produced with the CRRA utility in Figure 5.4. Figure 5.4: Comparison with CRRA: efficient frontiers

A theoretical proof of the higher efficiency of STRS over classical utility strategies would be desirable to corroborate our numerical findings. However, given the difficulty to obtain an explicit optimal allocation for a single trading period with the simpler downside risk minimization objective (Klebaner et al. 2017), obtaining such a proof for STRS in a multiperiod, multi-asset setting might be out of reach. We thus leave it for further research.

5.3

Sensitivity analysis and choice of LW

The third experiment analyzes the sensitivities of the expected return, standard deviation and downside risk with respect to the bounds of the STRS. Figure 5.5 shows how the expected terminal wealth (E [WT ], first row), the standard deviation of terminal wealth (SD [WT ], second row) and the downside

18

risk (P [WT < 1], third row) are affected by changes in the upper bound UW (left column) and by changes in the lower bound LW (right column). Figure 5.5: Sensitivity analysis w.r.t. target bounds

19

The left column of Figure 5.5 shows how the expectation E [WT ], standard deviation SD [WT ] and downside risk P [WT < 1] increase in UW , though a plateau is reached around UW = 1.5 for P [WT < 1] and around UW = 1.8 for E [WT ]. On the right column, one can see that the standard deviation SD [WT ] and downside risk P [WT < 1] both increase whenever LW moves away from the initial wealth W0 = 1.0. When LW > 1.0, the risks increase with |W0 − LW | due to the additional risk required at the beginning of the trading period to force the portfolio value to grow from W0 = 1.0 to the lower target LW > W0 = 1.0. When LW < 1.0, the risks also increase |W0 − LW | due to the lack of penalization of losses. Nevertheless, the net effect of LW on E [WT ] is negligible. As a result, these observations suggest that LW = W0 = 1.0 is the “safest” choice for the lower bound of the target interval, from which the upper bound UW can be set according to the risk preference and return requirement of the investor. Figure 5.6: Terminal wealth distributions with flat target range strategy

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5.4

Extensions

This subsection discusses the wealth distributions produced by the modified target range strategies described in Section 4. Figure 5.6 provides examples for the flat target range strategy (FTRS) with LW = 1.0 and UW = 1.05, 1.10, 1.20 and +∞. The main observation is that, as expected, the probability of ending outside the predefined range [LW , UW ] is smaller than the STRS (2.2) (refer to Figure 5.1 for comparison). This is the main strength of the FTRS: downside risk is kept to a minimum, while the price to pay for this safety is the inability to generate high returns. Finally, the wealth distribution is less sensitive to the choice of UW : the distribution is tight even when UW = ∞. Figure 5.7: Excess terminal wealth distributions with relative target range strategies

Excess wealth distributions of sharp target range strategy (top row) and flat target range strategy (bottom row)

Figure 5.7 shows that our method can successfully deal with stochastic benchmarks as targets. The probability that the portfolio value underperforms the benchmark portfolio remains small (around 6% − 8% for the final excess return distributions), though higher than those provided by absolute targets.

21

This is because the passive equal weight benchmark already provides a high expected return, therefore outperforming it requires to take more risk than that was needed for the absolute targets.

5.5

Backtesting on real market

The last numerical experiments shows the out-of-sample backtesting results from March 2011 to March 2016. We calibrate the data from September 2003 to March 2011 and perform a portfolio allocation for the year from March 2011 to March 2012, then update the training data to include March 2011 to March 2012 to perform the allocation for the next year, then repeat this procedure until March 2016. We investigate five different portfolios: • P0 is the equal weight portfolio as our benchmark • P1 uses the conservative flat target range strategy (4.1) with LW = 1.0 and UW = 1.1 • P2 uses the sharp target range strategy (2.2) with LW = 1.0 and UW = 1.1 • P3 uses the sharp target range strategy (2.2) with LW = 1.0 and UW = 1.2 • P4 uses the sharp target range strategy (2.2) with LW = 1.0 and UW = 1.3 Figure 5.8 plots the cumulative wealths of these portfolios, with their terminal portfolio values marked in the legend box. The result shows that the real market performance does behave as what can be expected from the simulation studies. Both P1 and P2 are highly stable, suitable for conservative investments such as pensions, superannuations and life-cycle management. P3 delivers a higher return but a lower realized volatility, compared to the benchmark equal weight portfolio P0 . P4 is virtually as volatile as the benchmark portfolio, but delivers a much higher return. Thus, in the real market, the STRS combined with the decensored LSMC method does deliver what can be expected from the simulation studies. Figure 5.8: Backtesting cumulative wealth

22

6

Conclusion

This paper introduces the sharp target rage strategy (STRS) that maximizes expected portfolio return while simultaneously tightening the entire return distribution via an unconstrained optimization framework, replicating in a much simpler manner that what could be expected from constrained optimization methods, and its formulation involved with an indicator function has a potential to incorporate (multiple) constraints on the dynamic risk measures such as realized volatilities and maximum drawdowns. To demonstrate the effectiveness of the STRS, we study a multiperiod portfolio choice problem and propose the decensored least squares Monte Carlo (LSMC) method for handling the new objective function. Our numerical tests show that the resulting distribution of portfolio wealth mimics the shape of the objective function, and that the presented strategy dominates the classical utility approach in terms of the mean-variance efficient frontier and the trade-off between return and downside risk. This decensored regression technique can also be adopted for other discontinuous or truncated payoff functions in general stochastic control problems, such as pricing different types of exotic options.

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