Mean-variance principle of managing cointegrated risky assets and random liabilities
Introduction
Granger [15] discovered that a linear combination of two or more non-stationary time series could be stationary. Engle and Granger [14] further formalized the idea of integrated variables sharing an equilibrium relation that turned out to be either stationary or have a lower degree of integration than the original series. They denoted this property as cointegration, signifying co-movements among trending variables that could be exploited to test for the existence of equilibrium relationships within a fully dynamic specification framework. As the concept of cointegration has been shown to be very useful in a variety of economic models, Granger was thus awarded the Nobel Prize in Economics in 2003.
In finance, cointegration has been applied to asset pricing and portfolio management because it is an alternative concept for the co-movement of financial variables beyond the conventional correlation coefficient. In fact, the concept of cointegration is essential in hedging [2] and has potential application in index tracking [3]. Chiu and Wong [9] applied the continuous-time cointegration model originated in [22] to solve the mean-variance (MV) portfolio allocation of cointegrated risky assets in a complete market.
To the best of our knowledge, no previous theoretical work has been devoted to the asset-liability management (ALM) problem of cointegrated risky assets and random liabilities in a continuous-time economy. Although there are many criteria for ALM problems, the MV criterion [21] seems to be the best known in financial economics and resulted in a Nobel Prize in Economics for Markowitz in 1990. Thus, this paper concentrates on the MV criterion and hence extends the framework in [9] to ALM problems. Although Chiu and Wong in [10], [11] investigate the ALM problem of an insurer, the problem considered in this present paper is significantly different from that in that earlier paper which focuses on an insurance liability of a compound Poisson process. This paper postulates a log-normal process for the total value of liabilities which will be paid back to lenders at the terminal time of the investment horizon. Therefore, we concentrate on the operation of a mutual fund rather than an insurance company, and our approach is in line with [6], [13].
The last decade witnessed ground-breaking work on the MV portfolio theory beyond the one-period setting such as the problem in a multi-period setting [20], a generalization to the continuous-time economy [25] and the random parameters economy in a complete market setting [19]. Based on these results, Chiu and Li [6] explored the possibility of incorporating the notion of MV portfolio theory to the ALM problem in continuous time. They showed in [7] that the optimal surrogate safety-first ALM strategy can be subsumed as an efficient portfolio strategy of an equivalent MV ALM problem if the MV efficient frontier of a firm’s surplus constitutes of a quadratic form between mean and variance. Chiu et al. [13] further investigated the genuine safety-first principle for the ALM problem. However, the impact of cointegration to the ALM problem is yet to be considered.
Specifically, we use backward stochastic differential equation (BSDE) theory to study the continuous-time MV ALM problem of cointegrated risky assets and an uncontrollable random liability. We show that the optimal dynamic pre-commitment MV ALM strategy is related to a system of BSDEs with one of them being a nonlinear stochastic Riccati equation (SRE). However, solving a nonlinear SRE, regardless of whether it is analytical or numerical, is generally a very difficult task. Yet the particular structure of cointegration allows us to derive an explicit closed-form solution to the management policy. The key is the recognition of an exponential affine form in the solution process. That form is then solved from a system of ordinary differential equations (ODEs), which has a well-known solution.
The rest of the paper is organized as follows. Section 2 reviews the basic concept of cointegration and details the problem formulation. The corresponding mean-variance problem is completely solved in Section 3. The analytical result is applied to pair trading of cointegrated assets in Section 4, and Section 5 concludes the paper.
Section snippets
The financial market with cointegrated risky assets
Consider a financial market in which assets are traded continuously within the time horizon . These assets are labeled by for , with the 0-th asset being risk free. The risk-free asset satisfies the differential equation , where is the time-deterministic risk-free rate. The risky assets are defined through their log-price processes , where . The vector of log-prices satisfies the stochastic differential equation
Solution
The optimal solution of can be obtained by solving an alternative but equivalent max–min problem derived from the Lagrangian approach: where . This approach essentially moves the expectation constraint to the objective function of the optimization problem, but the transformation is not free of charge. The real price to pay is that we have to solve the additional outermost maximization problem. Thus, the solution process for (7)
Solutions to BSDEs
BSDE (9) is in fact a stochastic Riccati equation (SRE). A general SRE can be highly nonlinear, and its solvability is not absolutely clear so far. Lim [18] shows the existence and uniqueness of an SRE similar to (9) for the case of bounded random parameters in the asset dynamics. However, the drift term in (1) is linear in and hence unbounded. The results in [18] are then not suitable for us. This paper not only offers an explicit solution to each of the BSDEs in Theorem 3.1, but also
Efficient frontier
We develop an efficient frontier in an incomplete financial market with cointegrated assets and uncontrollable liability in this section. Let us begin with the optimal mean-variance trading strategy.
Numerical example
Example 6.1 Consider two risky assets whose log-asset values at time are with initial value and constant parameters: Clearly, is a cointegrating factor that exhibits mean reversion with a mean-reverting speed of 1. A risk-free asset is available in the market, which derives the risk-free interest rate to be 3%. It is assumed that the appreciation rate and volatility of liability are 0.8 and ,
Future research
Further research may consider an alternative objective function such as the maximization of expected utility of the terminal surplus. It would also be interesting to examine the effect of stochastic volatility (SV) on top of cointegration. A possible approach may combine the asymptotic SV framework of [8] and the approach presented herein.
Acknowledgments
We thank the Editor, Jussi Keppo, and an anonymous referee for their careful reading and valuable comments. H.Y. Wong acknowledges support by the Research Grants Council of HKSAR with GRF project number 403511. M.C. Chiu acknowledges a Small Research Grant by the Department of MIT at HKIEd.
References (25)
- et al.
Continuous-time mean-variance optimization of assets and liabilities
Insurance Math. Econom.
(2006) - et al.
Asymptotic expansion for pricing options on mean-reverting assets with multiscale stochastic volatility
Oper. Res. Lett.
(2011) - et al.
Mean-variance portfolio selection of cointegrated assets
J. Econom. Dynam. Control
(2011) - et al.
Mean-variance asset-liability management: cointegrated assets and insurance liabilities
European J. Oper. Res.
(2012) - et al.
Optimal investment for insurer with cointegrated assets: CRRA utility
Insurance Math. Econom.
(2013) Some properties of time series data and their use in econometric model specification
J. Econometrics
(1981)- et al.
Matrix Riccati Equations in Control and Systems Theory
(2003) Optimal hedging using cointegration
Philos. Trans. R. Soc. A
(1999)- et al.
Indexing and statistical arbitrage: Tracking error or cointegration?
J. Port. Management
(2005) - Y. Bao, A. Ullah, Expectation of quadratic forms in normal and nonnormal variables with econometric applications....
Dynamic mean-variance asset allocation
Rev. Fin. Stud.
Asset-liability management under the safety-first principle
J. Optim. Theory Appl.
Cited by (26)
Time-consistent mean-variance hedging of an illiquid asset with a cointegrated liquid asset
2019, Finance Research LettersCitation Excerpt :Using the linear-quadratic stochastic optimal control framework, Chiu and Wong (2011) derive the first explicit solution for dynamic MV portfolio selection with cointegration in continuous-time. Chiu and Wong (2013) generalize it to asset-liability management (ALM) problem with cointegrated assets and an uncontrollable stochastic liability. The MV-ALM problem is closely related to the MV hedging problem of Duffie and Richardson (1991) but the cointegration between the liability (or random endowment) and assets is yet to be investigated in the present paper.
Robust dynamic pairs trading with cointegration
2018, Operations Research LettersCitation Excerpt :The analytic tractability of the continuous-time cointegration model facilitates the theoretical development of optimal pairs trading strategies. The mean–variance pairs trading rules are derived for banks [4], insurance companies [5] and firms with liability [6]. The optimal pairs trading in [18] is based on maximizing the expected constant relative risk aversion (CRRA) utility.
Dynamic derivative-based investment strategy for mean–variance asset–liability management with stochastic volatility
2018, Insurance: Mathematics and EconomicsCitation Excerpt :Chen et al. (2008) and Chen and Yang (2011) extended the works of Chiu and Li (2006) and Leippold et al. (2004) to the case with Markovian regime switching market. Chiu and Wong (2012, 2013) applied the backward stochastic different equation (BSDE) method to study mean–variance ALM problems with cointegrated risky assets. Yao et al. (2013a, b) considered a continuous-time mean–variance ALM problem and a multi-period mean–variance ALM problem with uncertain time-horizon, respectively.
Dynamic mean-variance portfolio selection with liability and stochastic interest rate
2015, Economic ModellingDynamic cointegrated pairs trading: Mean-variance time-consistent strategies
2015, Journal of Computational and Applied MathematicsMean-variance portfolio selection under a constant elasticity of variance model
2014, Operations Research LettersCitation Excerpt :Recently, there has been an interest in studying the mean–variance portfolio selection problem in financial models with random parameters. See, for example, Lim and Zhou [22], Ferland and Watier [14] and Chiu and Wong [7], amongst others. The constant elasticity of variance (CEV) model was first introduced to the financial community by Cox [9].