Industry Portfolio Allocation with Asymmetric Correlations

We develop a new framework of optimal consumption and portfolio choice at industry portfolio level under dynamic and asymmetric correlations between industry and market portfolios. We derive in closed-form the optimal consumption and investment strategies under regime-dependent correlations environment. Overall, we ﬁnd that ignoring time-varying and asymmetric correlations between portfolios can be costly to investors when applied to a construction of the optimal portfolio. Finally, we empirically test the performance of the model-based investment strategy.


Introduction
What determines a satisfactory investment portfolio? Markowitz (1959) and Merton (1969Merton ( , 1971) provide a benchmark against which investment decisions can be rationalized: the risk (or volatility) and return must be balanced optimally. In this framework, an opti-mal investment decision is to achieve the best risk-return combination across all attainable combinations of risk and return (or in the entire investment opportunity set) offered by portfolios.
Standard investment literature (e.g., Markowitz, 1959;Merton, 1969Merton, , 1971) assumes that the investment opportunity is constant, and therefore faces a limitation by neglecting one major dimension of financial risk: time-varying and asymmetric correlations between portfolios. In our study, we overcome this limitation and solve the optimal consumption and investment model in a regime switching market with regime-dependent correlations between portfolios. We hope this paper will lend itself to the study of investment strategies for investors, industry, and academic professions as well.
Since the seminal work of Hamilton (1989), many studies have widely adopted his regime switching framework in economic modeling. As Cochrane (2017) arguably states that Asset prices and returns are correlated with business cycles. Stocks rise in good times, and fall in bad times. Real and nominal interest rates rise and fall with the business cycle. Stock returns and bond yield also help to forecast macroeconomic events such as GDP growth and inflation. In this paper, we develop a new framework of optimal consumption and portfolio choice at industry portfolio level under dynamic and asymmetric correlations between industry and market portfolios. We derive in closed-form the optimal consumption and investment strategies under regime-dependent correlations environment. We carry out an in-depth quantitative analysis to illustrate various properties of the optimal strategies. We find that time variations in correlation in the investment opportunity set, depending upon the current status of the regime, can play a role in explaining the asset pricing implications further. Intuitively, the risk associated with random fluctuations in correlation cannot be fully diversified and hence, investors should require a premium to compensate for their exposure to dynamic and asymmetric correlation. Such a risk compensation, thus, should be reflected in the optimal investment portfolio, adjusting its amount substantially in response to variations in correlation across the bull and bear regimes.
We empirically evaluate whether our model-based investment strategy can generate meaningful performance. We first provide empirical evidence supporting the use of regimedependent correlation for managing portfolios. Similar to Gomes et al. (2009), we consider two industry portfolios: durable sector A and non-durable sector B. We then examine the empirical performance of our proposed investment strategy, compared to other heuristic strategies such as 1/n, maximum diversification, inverse volatility, equal risk parity, and two tail-risk parity strategies.
Our analysis therefore suggests that consideration of dynamic and asymmetric correlations between industry and market portfolios is an important factor in the attainment of successful investment return in the crisis period. Loosely speaking, the greater the change in the investment opportunity set after regime switching from the bull (bear) regime to the bear (bull) regime just as economic recessions, the greater the benefit of considering timevarying correlation dynamics. This inversely implies that misestimating or overlooking such a regime-dependent correlation can be costly to investors. For instance, if an investor underestimates the correlation and adopts the corresponding heuristic investment strategies under the wrong estimation, the expected wealth loss from these trading strategies would be very high. thus, unclear what a model of optimal investment with such a joint consideration would deliver. We view our analytically tractable model as a complement for our better understanding of mainly numerically solved existing models with either sector-level investment or time-varying and asymmetric correlations between portfolios.
The paper is organized as follows. In Section 2, we develop a model of optimal consumption and portfolio choice under a regime switching environment with regime-dependent correlations between portfolios. In Section 3, we derive analytically tractable results for the optimal consumption and investment strategies. In Section 4, we carry out an in-depth quantitative analysis to investigate various properties of the optimal strategies further. In Section 5, we empirically test the performance of the model-based investment strategy. In Section 6, we conclude the paper.

The Model
Utility Function. As proposed by Duffie and Epstein (1992), an investor has the following continuous-time formulation of non-expected utility: where E t is the expectation taken at time t and f (c, V ) is the normalized aggregator for consumption c and utility V . The aggregator f (c, V ) follows Here, ψ > 0 is the elasticity of intertemporal substitution (EIS), γ > 0 is the coefficient of relative risk aversion (CRRA), and ρ * > 0 is the subjective discount rate. When γ = ψ −1 , implying θ = 1, the recursive utility f (c, V ) reduces to the widely used time-additive separable CRRA utility. In this case, f (c, and thus, it is additively separable in c and V . For θ = 1, the general specification of the recursive utility f (c, V ) is non-separable in c and V .
A Regime Switching Model. To examine the impact of macroeconomic conditions on optimal investment in the simplest possible environment, we assume that there are two regimes: "Bull" (regime 0) and "bear" (regime 1). The fundamental parameters in the financial market are regime dependent. We let i ∈ {0, 1} denote the current state of the regime, which is assumed to be governed by a two-state Markov chain with generators as 7 the following: Jang et al. (2007), we assume that an investor can observe the regime changing.
According to the assumed two-state Markov chain, regime i switches into regime j at the first jump time of a Poisson jump process with intensity λ i > 0, for i, j ∈ {0, 1}. Within the present model, the time T i to leave regime i follows an exponential distribution with intensity λ i : which implies that there is some probability of λ i dt that regime i switches into regime j over an infinitesimal time interval dt. Note that the expected duration of regime i is 1/λ i and the average fraction of time spent in regime i is λ j /(λ i + λ j ).

Financial
Market. An investor can trade the following assets in the financial market.
In regime i (i ∈ {0, 1}), the investor can invest in a bond (or a risk-free asset) growing at a continuously compounded, constant rate r i > 0. She can also trade three risky assets: Public market portfolio M , Industry A stock, and Industry B stock. In regime i, the value of public market portfolio, S M t , follows the widely adopted geometric Brownian motion (GBM): between market portfolio and industry stocks, the risks associated with industry stocks are not fully diversified by only dynamically trading market portfolio. In regime i (i ∈ {0, 1}), the beta of Industry k stock (k ∈ {A, B}) relative to market portfolio can be defined as Next, we define the systematic risk of industry stocks in terms of industry beta. The total volatility (or risk) of Industry k (k ∈ {A, B}) stock is σ k i . The part of this volatility (spanned by the market portfolio) isρ k i σ k i . The remaining volatility is denoted by i , which is given by: The undiversified volatility presents extra risk in an investor's overall portfolio. Thus, the investor requires different risk premia for bearing diversified and undiversified risks.
In the context of risk-return trade-off, an investor is able to earn excess risk-adjusted returns, known as called alphas, by investing in industry stocks. More specifically, the alphas are defined as follows: for Industry k ∈ {A, B}, Intuitively, the alphas are the capital asset pricing model (CAPM)-model-based risk-adjusted excess returns of the portfolio that consists of Industry A and Industry B stocks.
An Optimal Consumption and Portfolio Choice Problem. An investor's optimal consumption and portfolio choice problem is to maximize her recursive utility by controlling per-period consumption c and stock holdings π M , π A , and π B . This results in the following stochastic optimization problem with a nonnegative wealth constraint: subject to the following dynamic wealth constraints: where π M represents the dollar amount invested in the market portfolio M , and π A and π B are the dollar amount invested in the Industry A and Industry B stocks, respectively.
At time t, an investor consumes at the rate equal to c t and receives interest at the rate r i proportional to her wealth by investing in a risk-free bond, resulting in wealth accumulation at the rate of (r i X t − c t ). As the investor is exposed to systematic risk stemming from investments in market portfolio and industry stocks, i.e., when she faces random fluctuations of her wealth given by π k t σ k i dW k t for k ∈ {M, A, B}, the risk taking

Optimal Strategies
We derive in closed-form optimal consumption and investment strategies under a regime switching environment, where dynamic and asymmetric correlations between industry and market portfolios are incorporated.
Theorem 3.1. In regime i (i ∈ {0, 1}), optimal consumption c * t and optimal stock holdings (π M t ) * , (π A t ) * , and (π B t ) * are derived in closed-form: for any x > 0, where K i is a solution to the following system of equations for i, j ∈ {0, 1}: Here, the terms of cons M , con A , and con B are constants defined in Appendix A.
Proof. Refer to Appendix A. Q.E.D.
The optimal strategies given in (2) show that consumption and stock holdings have a linear relation with an investor's initial wealth. The investor formulates a non-myopic optimal consumption plan in the sense that future regime changes affect the consumption amount through the regime-dependent constant K i for i ∈ {0, 1}. Specifically, the consumption strategy is affected by not only the regime intensity λ i , but also key parameters involving the coefficient γ of CRRA and the EIS ψ.
Next, we determine some interesting implications of an investor's optimal investment strategies. To have a benchmark, we consider the simplest possible situation in which the market portfolio, Industry A and Industry B stocks are all independent i.e.,ρ k i = ρ i = 0.
Then, in regime i (i ∈ {0, 1}), the optimal stock holdings reduce to the following: These optimal investment strategies follow the traditional investment rule given by Merton 13 (1969Merton 13 ( , 1971: an investor tends to invest more in stocks as the Sharpe ratio κ k i (k ∈ {M, A, B}) increases or the coefficient γ of CRRA decreases.
When we consider only the systematic risk of Industry A and Industry B stocks i.e., Such undiversified risks have a significant influence on an investor's overall optimal portfolio strategy, inducing extra demand for hedging. More precisely, the optimal investment strategy reduces to the following: and The hedging demand against undiversified systematic risk is measured by the differences between optimal strategies given in (3) and (4), (5), (6). It is either increased or decreased depending upon the sign of correlationsρ A i andρ B i . Thus, an investor demands different risk premia for bearing diversified and undiversified risks.
When we consider the case in which systematic risk no longer exists, but dynamic and asymmetric correlations between industry portfolios exist, i.e., whenρ k i = 0 (k ∈ {A, B}, i ∈ {0, 1}), |ρ i | < 1, and ρ i = 0, 4 the optimal investment strategy reduces to the following: In this case, an optimal portfolio comprised of industry stocks is affected by both Sharpe ratios κ A i and κ B i . Compared to (3), the correlations ρ i between industry stocks would be crucial in deriving the optimal portfolio.

Quantitative Analysis
In this section, we perform an extensive quantitative analysis to discuss various properties of analytically tractable optimal investment strategies.
Baseline Parameter Values. Our modeling focus is on the effects of regime-dependent asymmetric correlations, abstracting away complex issues of other parameters. In light of such an objective, we assume symmetric values for other parameters. Reflecting today's low interest rate environment, we set risk-free interest rate to 1%, i.e., r 0 = r 1 = 0.01. We set equity premium to 7%, i.e., expected rates of returns on the public market portfolio are µ M 0 = µ M 1 = 0.08. We set market volatility to 23.5%, i.e., σ M 0 = σ M 1 = 0.235. Industry 4 By assumingρ k i = 0 (k ∈ {A, B}, i ∈ {0, 1}), industry portfolios do not have any exposure to systematic risk as industry beta becomes zero.
i.e., expected rates of returns on Industry A and B stocks are µ A 0 = µ A 1 = 0.05, and and ρ i are all zero. Then, the optimal portfolio proportions given in (3) are An investor is willing to invest more in Industry B stock than in the market portfolio and Industry A stock. In the spirit of Merton (1969Merton ( , 1971), the investor is highly dependent on Sharpe ratios when investing in the stock market.
Effects of Systematic Risk. Now, if we consider the systematic risk of Industry A and Industry B stocks, i.e., |ρ k i | < 1 (k ∈ {A, B}, i ∈ {0, 1}),ρ k i = 0, and ρ i = 0, the market portfolio and Industry A and Industry B stocks are not perfectly correlated. We set the across regime 0 ("Bull") and regime 1 ("bear") to 0.5 and 0.1, respectively. Then the optimal portfolio proportions given in (4), (5), and (6) are As a result of undiversified systematic risk given by (1), the optimal risky investment will reduce significantly compared to the benchmark case. This implies that the benchmark case does make sense only if an investor should be compensated by no more than the undiversifed systematic risk premium.

Effects of Dynamic and Asymmetric Correlations at Market Portfolio Level.
Next, we consider the case in which industry stocks are mutually independent i.e., ρ i = 0 respectively. The average correlationρ k (k ∈ {A, B}) is theñ The parameter values reflecting the case of dynamic and asymmetric correlations with the market portfolio can be chosen in the following two steps. First, correlationsρ A and ρ B across regime 0 ("Bull") and regime 1 ("bear") are set to 0.5 and 0.1, respectively.
Second, we obtain the values ofρ A 1 using the relationship (7) Table C.3. 5 [Insert Table C.3 here.] Interestingly, we find that when the current level of correlation pairs (ρ k 0 ,ρ k 1 ) is relatively high as in the Industry A case and when the sum of correlation pairs drift up from the baseline parameters, portfolio proportions (

Effects of Dynamic and Asymmetric Correlations at Industry Portfolio Level.
We allow for the case where there exist dynamic and asymmetric correlations between industry portfolios. We assume that the correlation between market portfolio and Industry stocks is zero, i.e.,ρ k i = 0 for Industry k ∈ {A, B} in regime i ∈ {0, 1}, but the correlations between industry stocks are nonzero, i.e., |ρ i | < 1 and ρ i = 0.
The parameter values reflecting this case can be chosen as follows. First, we set the correlation ρ across regime 0 ("Bull") and regime 1 ("bear") to 0.3. Second, we use the relationship between correlations in regime 0 and regime 1:  We emphasize that a similar pattern emerges to that in Table C.3. That is, when correlation pairs (ρ 0 , ρ 1 ) drift up from the baseline parameters, portfolio proportions, First, we consider more conservative equity premia of Industry A and B stocks than the baseline equity premia. We set expected rates of returns on Industry A and B stocks to 4% and 10% rather than 5% and 11%, i.e., µ A 0 = µ A 1 = 0.04 and µ B 0 = µ B 1 = 0.10. Second, 6 In Appendix, we also provide the portfolio proportions in regime 0 and regime 1, and the weighted average of portfolio proportions across both regimes (Table C.7). 20 we consider more volatile Industry A and B stocks than the baseline volatilities. We set volatilities of Industry A and B stocks to 21% and 26% rather than 20% and 25%, i.e., σ A 0 = σ A 1 = 0.21 and σ B 0 = σ B 1 = 0.26. Finally, we consider a more risk-averse individual by setting her risk aversion to 3 rather than 2, i.e., γ = 3.     adopts the corresponding heuristic investment strategies under the wrong estimation, the expected wealth loss from these trading strategies would be very high.

Conclusion
We develop a tractable investment model at industry portfolio level under dynamic and asymmetric correlations between portfolios. In our regime-dependent correlations environment, we derive in closed-form the optimal consumption and investment strategies. We find significant adjustments in the optimal investment portfolio, reflecting a compensation for the exposure to dynamic and asymmetric correlation. This implies that ignoring such time-varying and asymmetric correlations between portfolios can be costly to investors when applied to a construction of the optimal portfolio. Our empirical test of overall performance of the model-based investment strategy shows that ours can outperform other heuristic strategies such as such as 1/n, maximum diversification, inverse volatility, equal risk parity, and two tail-risk parity strategies. 26

Appendix A. Appendix
The dynamic programming approach leads to the following system of Hamilton-Jacobi-Bellman (HJB) equations for i, j ∈ {0, 1} (Merton, 1971): where V i (x) represents the value function with respect to an investor's initial wealth x in regime i. The optimality conditions for consumption c and stock holdings π M , π A , and π B follow We conjecture the form of V i as the following: for any x > 0, where K i is a constant to be determined for i ∈ {0, 1}. A straightforward calculation leads to optimal consumption and optimal stock holdings in the theorem in which K i is a solution to the following system of equations for i, j ∈ {0, 1}: for notational simplicity, we Appendix B. Parameter Estimation Algorithm Details

GJR-GARCH Model Parameters
Glosten, Jagannathan, and Runkle (1993, hereafter GJR) have proposed the GJR-GARCH (Generalized Autoregressive Conditional Heteroskedasticity) as the following: In the GJR-GARCH model, the market shock 2 t−1 at time t − 1 affects the volatility σ t at time t via a non-linear relation represented by the following key parameters: α, γ, ω, and β. More specifically, the market shock t at time t responds to the positive shocks (α) and the negative shocks (γ) in the stock market with the market shock t−1 at time t − 1. In particular, when γ > 0 such a response of the volatility to the market shocks becomes asymmetric. The other two parameters of ω and β summarize respectively the volatility mean and the linear coefficient of a relation between σ t and σ t−1 . The following table shows our estimation results of the parameters of the GJR-GARCH model.

DDC-and ADDC-GARCH Model Parameters
The dynamic conditional correlation (DCC)-GARCH model by Engle (2002) where P t is the conditional correlation matrix of the standardized errors t at time t,P =

Regime Switching Model Parameters
The two-state Markov chain regime switching model allows for the transition between bull and bear regimes, which captures an intrinsic property of the stock market with business cycles. For the dependent variable y t , we consider two different states of s: s = 0 (Bull) and s = 1 (bear). Then, where µ s is the state-dependent mean of the dependent variable y st . The transition of states is stochastic and governed by the following transition probability matrix P : where p i,j represents the probability of a switch from state i to state j for i, j ∈ {0, 1}.  Note. This figure illustrates two efficient frontiers using 12 industry portfolios obtained from Fama-French data library. Data span from July 1950 to July 2015. Panel A depicts efficient frontier using 3 month T-bill proxy for risk-free rate, S&P 500 for market return, 12 industry portfolios, and two aggregate industry portfolios constructed under the dynamic conditional correlation (DCC) and asymmetric DCC (ADCC) assumptions, respectively. Panel B exhibits an efficient frontier expansion after the asymmetric effect is employed to the forward-looking optimal risky asset weights. We plot the efficient frontier using the same risk-free rate, market return, and two aggregate industry portfolios as described in Panel A. Red dots represent the efficient frontier using a DCC aggregate industry portfolio and blue dots represent the efficient frontier using an ADCC aggregate industry portfolio. To capture the asymmetric effect, we apply 300 randomly generated weights.  Note. This figure displays the final wealth paths for various portfolios with the data period from January 1980 to July 2015.