# Active and Passive Portfolio Management with Latent Factors

We address a portfolio selection problem that combines active (outperformance) and passive (tracking) objectives using techniques from convex analysis. We assume a general semimartingale market model where the assets' growth rate processes are driven by a latent factor. Using techniques from convex analysis we obtain a closed-form solution for the optimal portfolio and provide a theorem establishing its uniqueness. The motivation for incorporating latent factors is to achieve improved growth rate estimation, an otherwise notoriously difficult task. To this end, we focus on a model where growth rates are driven by an unobservable Markov chain. The solution in this case requires a filtering step to obtain posterior probabilities for the state of the Markov chain from asset price information, which are subsequently used to find the optimal allocation. We show the optimal strategy is the posterior average of the optimal strategies the investor would have held in each state assuming the Markov chain remains in that state. Finally, we implement a number of historical backtests to demonstrate the performance of the optimal portfolio.

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## 1 Introduction

Problems in portfolio management can be divided into two types: active and passive. In the former, investors aim to achieve superior portfolio returns; in the latter, the investors’ goal is to track a preselected index; see, for example, Buckley and Korn (1998) or Pliska and Suzuki (2004). One can further separate active portfolio management objectives into two types: absolute and relative. There is a great deal of literature dedicated to solving various portfolio selection problems with absolute goals via stochastic control theory. The seminal work of Merton (1969) introduced the dynamic asset allocation and consumption problem, utilizing stochastic control techniques to derive optimal investment and consumption policies. Extensions can be found in Merton (1971), Magill and Constantinides (1976), Davis and Norman (1990), Browne (1997) and more recently Blanchet-Scalliet et al. (2008), Liu and Muhle-Karbe (2013) and Ang et al. (2014) to name a few. The focus in these papers is generally on maximizing the utility of discounted consumption and terminal wealth or minimizing shortfall probability, or other related absolute performance measures that are independent of any external benchmark or relative goal. Works on optimal active portfolio management with relative goals (i.e. attempting to outperform a given benchmark) can be found in Browne (1999a), Browne (1999b) and Browne (2000), Pham (2003) and, more recently, Oderda (2015).

There are also several works that address the question of achieving absolute portfolio selection goals when parameters are stochastic, including cases where the investor only has access to partial information and must rely on Bayesian learning or filtering techniques to solve for their optimal allocation. Merton (1971) solves for the portfolio that maximizes expected terminal wealth assuming that the instantaneous expected rate of return follows a mean-reverting diffusive process. Lakner (1998) extends this to the case where the drift processes are unobservable. In Rieder and Bäuerle (2005) the assets’ drift switches between various quantities according to an unobservable Markov chain; Frey et al. (2012) extends this to incorporate expert opinions, in the form of signals at random discrete times, into the filtering problem by using this observable information to obtain posterior probabilities for the state of the Markov chain. Bäuerle and Rieder (2007) introduces jumps to the asset price dynamics by including Poisson random measures with unobservable intensity processes. Latent models are also central to the work of Casgrain and Jaimungal (2018b) and Casgrain and Jaimungal (2018a) in the context of algorithmic trading and mean field games.

Many of the concepts discussed in this work, particularly the notion of functionally generated portfolios (FGPs) and rank-based models, are key concepts in Stochastic Portfolio Theory (SPT) (see Fernholz (2002) and Karatzas and Fernholz (2009) for a thorough overview). SPT is a flexible framework for analyzing portfolio behavior and market structure which takes a descriptive, rather than a normative, approach to addressing these issues, and emphasizes the use of observable quantities to make its predictions and conclusions. The appeal of SPT partially lies in the fact that it relies on a minimal set of assumptions that are readily satisfied in real equity markets and that the techniques it employs construct relative arbitrage portfolios that outperform the market almost surely without the need for parameter estimation. This is primarily done through the machinery of portfolio generating functions (PGFs), which are portfolio maps that give rise to investing strategies that depend only on prevailing market weights. A discussion of the relative arbitrage properties of FGPs and related approaches to achieving outperformance vis-à-vis the market portfolio can be found in Pal and Wong (2013), Wong (2015) and Pal and Wong (2016).

Although SPT focuses on almost sure outperformance, i.e. relative arbitrage with respect to the benchmark portfolio, we deviate from this criterion in favor of maximizing the expected growth rate differential. We present two justifications for this choice. First, certain rank-based models such as the first-order models admit equivalent martingale measures over all horizons implying the non-existence of relative arbitrage opportunities. This forces the investor to select an alternative performance criterion. Secondly, Fernholz (2002) argues for the use of functionally-generated portfolios such as diversity-weighted portfolios as benchmarks for active equity portfolio management given their passive, rule-based nature and ease of implementation. However, Wong (2015) notes that under certain reasonable conditions relative arbitrage opportunities do not exist with respect to these portfolios. Therefore, once again the investor must seek a substitute for almost sure outperformance if they decide to have a performance benchmark of this sort. One SPT-inspired work that uses an expectation-based objective function is Samo and Vervuurt (2016)

, in which machine learning techniques are utilized to achieve outperformance in expectation by maximizing the investor’s Sharpe ratio.

Active managers often dynamically invest in markets with the goal of achieving optimal relative returns against a performance benchmark while anchoring their portfolio to a tracking benchmark (in the sense of incurring minimal active risk/tracking error). They also often have in mind additional constraints on the investor’s portfolio, e.g. penalizing large positions in certain assets or excessive volatility in the investor’s wealth. In Al-Aradi and Jaimungal (2018), the authors formulate these goals and constraints by posing a portfolio optimization problem with log-utility of relative wealth, together with running penalty terms that incorporate the investor’s constraints on tracking a benchmark and total risk. They solve the problem in closed-form using dynamic programming under the assumptions that the benchmarks are differentiable maps that are Markovian in the asset values; this encompasses the market portfolio and, more broadly, the class of (time-dependent) functionally generated portfolios.

A shortcoming of Al-Aradi and Jaimungal (2018) is that when the investor values outperformance, the optimal solution relies crucially on the asset growth rate estimates, which are assumed to be bounded, differentiable, deterministic functions of time. However, returns are notoriously difficult to estimate robustly and the deterministic assumption does not provide adequate estimates. To address this shortcoming, here, we allow for growth rates to be stochastic and be driven by latent factors. This is essential to making the strategy robust to differing market environments. Our formulation also accommodates rank-based models; such models exploit the stability of capital distribution in the market to arrive at estimates of asset growth rates based on asset ranks.

Our modeling assumption is similar to that adopted in Casgrain and Jaimungal (2018a), who study the mean-field version of an algorithmic trading problem, where assets are driven by two components: a drift term and a martingale component both of which are adapted to an unobservable filtration. The investor’s strategy, on the other hand, is restricted to be adapted to a smaller filtration; namely, the natural filtration generated by the price process.

The approach we take to solve the stochastic control problem is based on techniques from convex analysis as in Bank et al. (2017) and Casgrain and Jaimungal (2018a), however these techniques date as far back as Cvitanić and Karatzas (1992). The reason we deviate from the dynamic programming approach taken in Al-Aradi and Jaimungal (2018) centers around the difficulty of extending that approach to more general market models. Although possible, it would be a difficult task to ensure that all the additional state variables (which would include various semimartingale local times in the case of rank-based models) satisfy the conditions for a Feynman-Kac representation to the HJB equation that arises from the control problem, which is a central aspect of the proof. A number of additional (possibly restrictive) assumptions would have to be made on the market model and, as such, the approach we adopt in the current work allows for a more succinct solution to a more general problem with fewer assumptions.

## 2 Model Setup

### 2.1 Market Model

We adopt a market model that generalizes the one in Al-Aradi and Jaimungal (2018) and is a multidimensional version of the one used in Casgrain and Jaimungal (2018a). Let be a filtered probability space, where is the natural filtration generated by all processes in the model. We assume that the market consists of assets defined as follows:

###### Definition 1

The stock price process for asset , for all , is a positive semimartingale satisfying:

 Xit=Xi0exp(∫t0γis ds+Mit) (2.1)

where is a -adapted process representing the asset’s (total) growth rate and is a -adapted martingale with representing the asset’s noise component.

It is convenient to work with the logarithmic representation of asset dynamics:

###### Proposition 1

The logarithm of prices, , satisfies the stochastic differential equation:

 (2.2)

This can also be expressed in vector notation as follows:

 dlogXt=γt dt+dMt, (2.3)

where

We make the following assumption on the growth rate and noise component of asset prices:

###### Assumption 1

The growth rate and martingale noise processes satisfy one of the two following conditions:

1. , ;

2. and ,

 where Lp ={f:Ω×[0,T]→Rn s.t.% E[∫T0(∥ft∥p)p dt]<∞},0

In the assumption above and for denote the -norm and -norm on , respectively. Furthermore, we will make use of the shorthand notation to denote the usual Euclidean norm.

We also assume the quadratic co-variation processes associated with the noise component satisfy

###### Assumption 2

Let be the matrix whose -th element is the quadratic covariation process between and , . We assume that, for each , there exists and such that

 ε∥x∥2 ≤ x⊺Σtx ≤ C∥x∥2,∀t≥0. (2.4)

This is an extension of the usual non-degeneracy and

bounded variance

conditions.

###### Remark 1

The constant in of Assumption 1 may depend on the constants and that appear in Assumption 2, but provided that is sufficiently large we can ensure that the candidate optimal solution that we obtain is in fact in the set of admissible controls.

### 2.2 Portfolios and Observable Information

The investor does not have access to the latent processes driving asset prices and observes asset prices alone (it is possible to allow other processes in addition to the price process, but here we restrict to this case). Let the filtration where denotes the investor’s filtration.

###### Definition 2

A portfolio is a measurable, -adapted, vector-valued process , where such that for all , satisfies:

 π1t+⋯+πnt=1a.s. (2.5)

Furthermore, we define the set of admissible portfolios as follows:

1. if Assumption 1(a) is enforced, we assume and define :

 A2={π:Ω×[0,T]→Rn s.t. π∈L2, F-adapted and π⊺t1=1, for t≥0  P-a.s.} (2.6)
2. if Assumption 1(b) is enforced, we assume and define:

 A∞={π:Ω×[0,T]→Rn s.t. π∈L∞,M, F-adapted and π⊺t1=1, for t≥0  P-a.s.} (2.7)

In the sequel, we write to denote either or depending on which part of Assumption 1 is being enforced.

###### Remark 2

The cost of allowing for noise processes is that both growth rate processes and admissible portfolios are rather than processes.

In the definition above, portfolios are adapted to the filtration , which is the information set generated by the asset price paths and not the full information set . The latter includes the noise component as well as its quadratic covariation process , both assumed unobservable. This ensures that strategies depend only on fully observable quantities which in our context are limited to the asset price processes.

Given the model dynamics, and portfolio assumptions, we next derive the dynamics of wealth associated with an arbitrary portfolio :

###### Proposition 2

The logarithm of the portfolio value process satisfies the SDE:

 dlogZπt=γπt dt+π⊺t dMt, (2.8)
 whereγπt =π⊺tγt+Γπt, Γπt=12[π⊺tdiag(Σt)−π⊺tΣtπt],

and and are the portfolio growth rate and excess growth rate processes, respectively.

Proof. The proof follows the same steps as the proof of Proposition 1.1.5. of Fernholz (2002). The proportional change in the value of portfolio is a weighted average of the simple return of each asset held in the portfolio:

 dZπtZπt=n∑i=1πit dXitXit.

From the asset dynamics in (2.2) and an application of Itô’s lemma we have:

 dXitXit=(γit+12⟨Mi⟩t)dt+dMit,

where is the quadratic variation process of . Another application of Itô’s lemma on the portfolio wealth process dynamics gives

 dZπtZπt=dlogZπt+12 d⟨logZπ⟩t.

The result follows by noting that the quadratic variation is given by:

 ⟨logZπ⟩t=n∑i,j=1πitπjt⟨logXi,logXj⟩t =π⊺tΣtπt

and then rearranging terms.

### 2.3 Market Model Examples

Al-Aradi and Jaimungal (2018) assume growth rates and volatilities are bounded, differentiable, deterministic functions and the only driver of asset prices was a multidimensional Wiener process. In this section we present two market models satisfying the assumptions in this paper that allow for more general asset growth rates. The two models are presented with the goal of improved growth rate estimation in mind.

#### 2.3.1 Diffusion-Switching Growth Rate Process

The diffusion-switching model assumes that asset growth rates switch between a number of possible diffusion processes according to an underlying Markov chain. That is:

 γt=γ(Θt)t, (2.9)

where is a continuous-time Markov chain with state space and is the growth rate diffusion process associated with state given as the solution to the SDE:

 dWγtγ(i)t\footnotesizen×1=Wγtϕ[rgb]1,0,0\footnotesizen×1(t,γt,i) dt+WγtΦ\footnotesizen×k(t,γt,i) dWγt\footnotesizek×1. (2.10)

In this formulation, is a -dimensional Wiener process driving the growth rate diffusions and and are the drift and volatility functions of the growth rate. We require and to be chosen so that for all . A sufficient set of conditions for this are the usual Lipschitz and polynomial growth conditions that guarantee the existence of a unique, square-integrable strong solution to the SDE (see Theorem 2.9 in Chapter 5 of Karatzas and Shreve (1998)). Figure 1 shows a simulation of this process when the possible diffusions are Ornstein-Uhlenbeck (OU) processes.

In Section 4, we take both and to be identically zero. This recovers the hidden Markov model (HMM) used in Rieder and Bäuerle (2005), where the growth rate switches between a number of possible constants rather than diffusion processes. This simplifies the calibration process and this is the model we employ in the implementation.

#### 2.3.2 Second-Order Rank-Based Model

An alternative model that may be considered is the second-order rank-based model of equity markets as described in Fernholz et al. (2013). In this model, an asset’s price dynamics depend on the rank of the asset’s market weights; typically, smaller assets have higher growth rates and volatilities than larger assets. The goal of this modeling approach is to better capture observed long-term characteristics of capital distribution in equity markets, such as average rank occupation times, by exploiting the inherent stability in the capital distribution curve.

Let be the rank of asset at time , the asset price is assumed to satisfy the SDE:

 dlogXit=(γi+n∑j=1gj1{rit=j}) dt+n∑j=1σj1{rit=j} dWjt (2.11)

That is, is the “name”-based growth rate of asset and is the additional growth an asset experiences when its capitalization occupies rank . Similarly, is the volatility of the asset in rank . We assume the model parameters satisfy the requirements for the market to form an asymptotically stable system; see Fernholz et al. (2013), which also provides an outline for parameter estimation for this class of models.

It is important to notice that when this model is assumed, the rank processes for each of the stocks must be incorporated in the optimization problem as state variables. This can vastly complicate the proof of optimality when using a dynamic programming approach. The approach we take in the present work does not suffer from these issues involving local times and non-differentiability. Finally, we note that it is possible to create a hybrid model that is rank-dependent and driven by an unobservable Markov chain, but this may lead to difficulties in the parameter estimation.

## 3 Stochastic Control Problem

### 3.1 Description

The stochastic control problem we consider is similar to the one posed in Al-Aradi and Jaimungal (2018). The investor fixes two portfolios against which they measure their outperformance and their active risk, respectively. That is, the investor chooses a performance benchmark , which they wish to outperform, and a tracking benchmark , which they penalize deviations from. The objective is to determine the portfolio process that maximizes the expected growth rate differential relative to over the investment horizon . Moreover, the investor is penalized for taking on excessive levels of active risk (measured against ). An additional penalty independent of the two benchmarks is also included to control absolute risk (as measured by quadratic variation of wealth) or penalize allocation to certain assets as discussed in Section 4 of Al-Aradi and Jaimungal (2018).

The main state variable in our optimization problem is the logarithm of the ratio of the wealth of an arbitrary portfolio relative to a preselected performance benchmark . Let denote the logarithm of relative portfolio wealth for the portfolios and . Then this process satisfies the SDE:

 dYπ,ρt=(γπt−γρt)dt+(πt−ρt)⊺dMt, (3.1)

which in turn implies

 Yπ,ρt=Yπ,ρ0+∫T0(γπt−γρt)dt+∫T0(πt−ρt)⊺dMt. (3.2)

Our main stochastic control problem is to find the optimal portfolio which, if the supremum is attained in the set of admissible strategies, achieves

 supπ∈Ac H(π) (3.3)

where is the performance criteria of a portfolio given by:

 H(π)=E[ζ0 Yπ,ρt−12∫T0ζ1s (πs−ηs)⊺Ωs(πs−ηs) ds−12∫T0ζ2s π⊺sQsπs ds]. (3.4)

Here, is a constant and with is an -adapted process defined on for some fixed . The vector represents the subjective preference parameters set by the investors to reflect their emphasis on three goals:

1. The first term is a terminal reward term which corresponds to the investor wishing to maximize the expected growth rate differential between their portfolio and the performance benchmark . It is also equivalent to maximizing the expected utility of relative wealth assuming a log-utility function.

2. The second term is a running penalty term which penalizes deviations from the tracking benchmark. When , the investor is penalizing risk-weighted deviations from the tracking benchmark, with deviations in riskier assets being penalized more heavily. Thus, this can be seen as the investor aiming to minimize tracking error/active risk.

3. The final term is a general quadratic running penalty term that does not involve either benchmark. One possible choice for is the covariance matrix , which can be adopted to minimize the absolute risk of the portfolio measured in terms of the quadratic variation of the portfolio wealth process, . Another option is to let be a constant diagonal matrix, which has the effect of penalizing allocation in each asset according to the magnitude of the corresponding diagonal entry. The investor can use this choice of as a way of imposing a set of “soft” constraints on allocation to each asset.

The reader is referred to Al-Aradi and Jaimungal (2018) for further interpretation of these terms.

###### Remark 3

The two preference parameters can be stochastic; e.g., they may depend on the investor’s wealth level or other factors. Furthermore, the preference parameters are restricted to for two reasons: firstly, it simplifies the proof of optimality; secondly, from Al-Aradi and Jaimungal (2018), the results are driven by the relative weights, rather than absolute weights, therefore restriction to the cube results in no loss of generality.

###### Remark 4

The benchmarks may be non-Markovian; if they are Markovian and can be represented as and , the functions and are not restricted to be differentiable. This allows for a much wider class of benchmarks including rank-based portfolios and portfolios constructed using additional information not related to asset prices, e.g., factor portfolios based on company fundamentals. Benchmarks from the class of functionally generated portfolios are allowed, including the market portfolio, as well as portfolios generated by rank-dependent portfolio generating functions, such as large-cap portfolios.

We also require the following assumption on the relative and absolute penalty matrices and :

###### Assumption 3

The penalty matrices and are -adapted matrix-valued stochastic processes such that, for each , there exists constants and satisfying

 ε∥x∥2 ≤ x⊺Ωtx ≤ C∥x∥, % andε∥x∥2 ≤ x⊺Qtx ≤ C∥x∥,∀t≥0. (3.5)

These bounds play an analogous role to the nondegeneracy and bounded variance assumptions made on the quadratic covariation , and ensure that the candidate optimal control we derive later is in fact admissible.

Allowing for stochastic penalty matrices is useful as it opens the door for stochastic volatility models in the case of (when choosing ) and stochastic transaction costs in the case of .

We next rewrite the control problem in terms of running reward/penalty terms. When either of the conditions in Assumption 1 is enforced, the expected value of the last integral in (3.2) is zero as the stochastic integral is in fact a martingale. Further, assuming that , the performance criteria becomes

 (3.6)

The generalizations achieved thus far compared to Al-Aradi and Jaimungal (2018) are summarized in Table 1 below.

### 3.2 Projection

To solve the the control problem (3.3) we follow the arguments in Casgrain and Jaimungal (2018a). The first step is to project the asset price dynamics onto the observable filtration , which enables us to rewrite the performance criteria (3.6) in terms of observable processes only. To this end, we first define the conditional expectation process so that

 ˆγt\coloneqqE[γt ∣∣ Ft].

This process represents the investor’s best estimate of the assets’ growth rates given all asset price information up to a given point in time. Similarly, we define a process corresponding to the projection of the unobservable quadratic covariation onto the same filtration, so that

 ˆΣt\coloneqqE[Σt ∣∣ Ft].
###### Proposition 3

Regardless of which part of Assumption 1 is enforced, the estimated growth rate process is -adapted with . Furthermore, the estimated quadratic covariation process is -adapted and satisfies:

 ε∥x∥2 ≤ x⊺ˆΣtx ≤ C∥x∥2

for some and for all and .

Proof. Assumption 2 implies that all entries of are bounded (see Appendix A of Al-Aradi and Jaimungal (2018)), and it follows that the conditional expectation is well-defined, bounded and -adapted. This also implies the required inequalities involving .

To prove the statements regarding , first, under either condition of Assumption 1, we have that for all and hence exists and is unique and integrable (see Durrett (2010), Lemma 5.1.1). Furthermore, by the definition of conditional expectations, is -measurable for all .

Next, suppose Assumption 1(a) is enforced, so that , i.e. that for each . By the same reasoning as above, this implies that exists, is unique, and integrable. By Jensen’s inequality for conditional expectations (Theorem 5.1.3 of Durrett (2010))

 (E[γit ∣∣ Ft])2 ≤ E[(γit)2 ∣∣ Ft] ⟹   E[(E[γit ∣∣ Ft])2] ≤ E[E[(γit)2 ∣∣ Ft]] ⟹   E[(ˆγit)2] ≤ E[(γit)2]<∞

As this is true for each , it follows that .

Finally, suppose Assumption 1(b) is enforced, so that , then we have that , and the conclusion follows.

The projection is measurable with respect to the investor’s filtration, and hence may be used to construct their portfolio.

###### Lemma 1

The innovations process, , defined by

 ˆMt\coloneqqlogXt−∫t0ˆγs ds (3.7)

is an -adapted martingale with . Furthermore, the asset dynamics satisfies an SDE in terms of -adapted processes as follows

 dlogXt=ˆγt dt+dˆMt. (3.8)

Proof. See Appendix A.1.

### 3.3 Optimization via Convex Analysis

The performance criterion (3.6) may be written in terms of the projected processes, so that

 H(π)=E[∫T0{ζ0(ˆγπt−ˆγρt)−12ζ1t (πt−ηt)⊺Ωt(πt−ηt)−12ζ2t π⊺tQtπt} dt]

where and are replaced with their conditional expectations and and where is the projected growth rate of portfolio defined analogously to the portfolio growth rate given in Proposition 2. This replacement is justified by Lemma 1 and the fact that we have and similarly . Following the steps in the proof of Proposition 2 of Al-Aradi and Jaimungal (2018), the performance criteria can be written in the following linear-quadratic form:

 H(π)=E[ ∫T0{−12π⊺tAtπt+π⊺tBt}dt]−E[∫T0{ζ0ˆγρt−12ζ1tη⊺tΩtηt}dt] where At =ζ0ˆΣt+ζ1tΩt+ζ2tQt Bt =ζ0(ˆγt+12diag(ˆΣt))+ζ1tΩtηt

As the second expectation does not depend on the control, it may be omitted in the optimization. It is convenient to define the assets’ instantaneous rate of return process given by

 αt\coloneqqγt+12% diag(Σt), (3.9)

along with its projected counterpart . With this, we can write the performance criteria that we aim to optimize as

 H(π)=E[∫T0{−12π⊺tAtπt+π⊺tBt}dt], (3.10a) where At =ζ0ˆΣt+ζ1tΩt+ζ2tQt (3.10b) Bt =ζ0ˆαt+ζ1tΩtηt (3.10c)

Next, we set our (unconstrained) search space to be

This forms a reflexive Banach space with norm - see Theorem 1.3 and Theorem 4.1 of Stein and Shakarchi (2011). The admissible set we wish to optimize over (the set of admissible portfolios ) is a subset of whether portfolios are defined to be bounded or just . Let be the dual space and let denote the canonical bilinear pairing over , i.e. . The performance criteria (3.10) can be viewed as a functional that maps elements from to a real number, i.e. . The following proposition ensures that the candidate optimal control we propose later is indeed optimal.

###### Proposition 4

The functional given by (3.10) is proper, upper semi-continuous and strictly concave.

Proof. See Appendix A.2.

The subsequent steps are to (i) compute the Gâteaux differential associated with the functional , (ii) find an element in the admissible set which makes it vanish, and (iii) use the strict concavity of the objective function to conclude that this element is a global maximizer. For more further details, the reader is referred to the first two chapters of Ekeland and Témam (1999).

###### Proposition 5

The functional is Gâteaux differentiable for all with Gâteaux differential given by

 ⟨˜π,H′(π)⟩=E[∫T0˜π⊺t[−Atπt+Bt] dt] (3.11)

Proof. See Appendix A.3.

Finally, we present our main result which gives the form of the optimal control for the stochastic control problem (3.3).

###### Theorem 1

The portfolio given by

 π∗t=A−1t [1−1⊺A−1tBt1⊺A−1t1 1+Bt], (3.12a) where At=ζ0ˆΣt+ζ1tΩt+ζ2tQt,andBt=ζ0ˆαt+ζ1tΩtηt. (3.12b)

is the unique solution to the stochastic control problem (3.3).

Proof. See Appendix A.4.

The optimal portfolio given by the above theorem resembles the solution in Al-Aradi and Jaimungal (2018), but the unobservable growth rate and quadratic covariation matrix are replaced by their projected counterparts and .

Furthermore, we can relate this optimal solution to the growth optimal portfolio (GOP) and the minimum quadratic variation portfolio (MQP). To recall: the GOP is the portfolio with maximal expected growth over any time horizon while the MQP is the portfolio with the smallest quadratic variation of all portfolios over the investment horizon. Formally, the GOP and MQP are the solutions to the optimization problems

 πGOP =argsupπ∈Ac E[log(ZπTZπ0)]and (3.13) πMQP =arginfπ∈Ac  E[∫T0π⊺tΣtπt dt], (3.14)

respectively.

###### Corollary 1

The growth optimal portfolio is given by:

 πGOPt=(1−1⊺ˆΣ−1tˆαt)πMQPt+ˆΣ−1tˆαt. (3.15)

where is the minimum quadratic variation portfolio given by:

 πMQPt=11⊺ˆΣ−1t1 ˆΣ−1t1. (3.16)

Proof. Use the optimal control given in Theorem 1 with and to obtain the MQP, and with to obtain the GOP.

Next, we show how the optimal portfolio may be split into sub-portfolios and how it may be written as a growth optimal portfolio for a modified asset price model.

###### Corollary 2

When (minimize relative and absolute risk), the optimal portfolio in Theorem 1 can be represented as

 π∗t=c0t πGOPt+c1t ηt+c2t πMQPt,

where for so that .

Proof. The results follow by adapting the proof in Appendix A.3 of Al-Aradi and Jaimungal (2018) to the current setting.

###### Corollary 3

The optimal portfolio is the growth optimal portfolio for a market with a modified (projected) rate of return process and a modified quadratic covariation matrix given by:

 ˆα∗t =ζ0tˆαt+ζ1tΣtηt (3.17a) Σ∗t =ζ0tΣt+ζ1tΩt+ζ2tQt (3.17b)

Proof. The result follows from direct comparison of the optimal portfolio in Theorem 1 with the form of the growth optimal portfolio in Corollary 1.

A few comments on the last two results:

1. When the investor wishes to minimize both relative and absolute risk, their optimal strategy is to invest in the GOP, MQP and tracking benchmark. The idea is to benefit from the expected growth rate of the GOP while modulating its high levels of absolute and active risk by investing in the MQP and tracking benchmark, respectively. The proportions are determined by the relative importance the investor places on outperformance, tracking and absolute risk as represented by and may vary stochastically through time. Notice also that the optimal solution does not depend on the performance benchmark and is myopic, i.e. independent of the investment horizon . This is expected as the GOP can be shown to maximize expected growth at any time horizon.

2. The absolute penalty term forces the optimal strategy towards the “shrinkage” portfolio given by . When is a diagonal matrix , then . From this, we see that the absolute penalty term forces us to shrink to a portfolio proportional to ; when is large is small and the shrinkage portfolio allocates less capital to asset . This can be used to tilt away from undesired assets; taking forces the allocation in asset to zero. Additionally, taking penalizes large positions in any asset by shrinking to the equal-weight portfolio, while setting forces shrinkage to the risk parity portfolio.

3. The second corollary implies that the investor is modifying the assets’ rates of return to reward those assets that are more closely correlated with the portfolio they are trying to track. This is because the term is a vector consisting of the quadratic covariation between between each asset and the wealth process associated with the tracking benchmark , i.e. . Moreover, if we consider the case where is a diagonal matrix, the modification to the covariance matrix amounts to increasing the variance of each asset according to the corresponding diagonal entry of . This in turn makes certain assets less desirable and is tied to the notion of tilting away from those assets as discussed in the previous point.

The points made above highlight the motivation for including the two running penalties. In principle, when an investor’s goal is simply to outperform a performance benchmark, they would hold the GOP to maximize their expected growth. However, it is well-documented that the GOP is associated with very large levels of risk (in terms of portfolio variance) as well as potentially large short positions in a number of assets. Adding the relative and absolute penalty terms mitigates some of this risk. It is also worth noting that the decompositions in Corollary 3 (ii) and (iii) can help guide the choice of the subjective parameters which can be used to express the proportion of wealth the investor wishes to place in the GOP, tracking benchmark and MQP.

## 4 Hidden Markov Model

The model presented above is quite general but not implementable without an explicit form for the conditional expectations and . In this section, we compute these conditional expectations for a hidden Markov model where the growth rate switches between a number of possible constant vectors according to an underlying Markov chain.

More specifically, let be a standard -dimensional Wiener process and let an unobservable, continuous-time Markov chain with state space and generator matrix . Next, suppose the asset prices satisfies the SDE

 dlogXt=γ(Θt) dt+ξ dWt, (4.1)

where for and is a constant matrix of sensitivities. Here, the growth rates are all bounded and the noise component is a square-integrable martingale.

###### Remark 5

To stabilize the estimation process, we choose to be time-independent. Moreover, cannot depend on the Markov chain in the same manner as the growth rates (i.e. switch between a number of fixed matrices according to the prevailing Markov chain state), otherwise the Markov chain becomes observable and there is no filtering problem; see Krishnamurthy et al. (2016).

If is a constant matrix that satisfies Assumption 2, then the market model above satisfies the requirements described earlier in the paper and the optimal portfolio is given by (3.12). Here, for all . It remains to compute an explicit form for . This filtering step involves deriving a posterior distribution for the current state of the Markov chain given all the observable information up to the present time, i.e., we are interested in computing .

First, let the investor’s prior distribution for the initial state of the Markov chain be denoted . Next, we define the posterior distribution of the state of the underlying Markov chain given the observable data as

 pjt=E[1{Θt=j} ∣∣ Ft] ,∀j∈M. (4.2)

The following lemma allows us to write the posterior probabilities in terms of un-normalized state variables.

###### Lemma 2

Let be an -adapted process satisfying Novikov’s condition

 E[exp(12∫T0∥λt∥2 dt)]<∞,

and define:

1. A probability measure via the Radon-Nikodym derivative

 d˜PdP=exp[−∫T0λ⊺t− dWt−12∫T0λ⊺t−λt−dt].
2. A stochastic process

 Υt=E[dPd˜P ∣∣∣ Ft]=exp[∫t0λ⊺t− dWt−12∫t0λ⊺t−λt−dt].

Then, the posterior probability processes can be written as:

 pjt=Pjt∑i∈MPit,∀j∈M, (4.3)

where .

Proof. See Appendix B.1.

The main theorem for this section gives the SDE system that governs the dynamics of the posterior probabilities.

###### Theorem 2

The state variables satisfy the following stochastic differential equations:

 dPjt = ∑i∈MPit−Gji dt+Pjt−(γ(j))⊺Σ−1 dlogXt, (4.4)

with initial conditions and .

Proof. See Appendix B.2.

In the sequel, we use the vector notation and . Note also that and are both -adapted stochastic processes.

Using the result above we arrive at the final form for the projected asset growth rates as

 ˆγt=m∑j=1pjtγ(j). (4.5)

In this model, the optimal portfolio (3.12) is a weighted average of optimal portfolios corresponding to each state of the Markov chain. In particular, the state-dependent portfolios are the optimal solutions assuming the Markov chain remains in the corresponding states at all times and the weights are given by the investor’s posterior probabilities that a given state prevails. In other words, the portfolio (3.12) is the posterior mean of optimal portfolios across states. This is summarized in the following corollary.

###### Corollary 4

For the HMM model (4.1) of asset price dynamics, the optimal portfolio (3.12) can be written as

 π∗t=∑i∈Mpitπ(i)t,
 where π(i)t withAt =ζ0Σ+ζ1tΩt+ζ2tQt, andB(i)t =ζ0γ(i)+ζ1tΣηt.

is the solution to the control problem (3.3) assuming the Markov chain state space is .

Proof. The result follows as a consequence of the linearity of the optimal control with respect to the inferred growth rate . Explicitly, we have

 m∑i=1pitπ(i)t

Now,

 m∑i=1pitB(i)t = m∑i=1pit[ζ0tγ</