Mean-variance principle of managing cointegrated risky assets and random liabilities

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Abstract

Using the diffusion limit of the discrete-time error correction model of cointegration for risky assets and geometric Brownian motion for the value of liabilities, we solve the asset-liability management (ALM) problem using the theory of backward stochastic differential equations. The solutions of the ALM policy and the efficient frontier in terms of surplus are obtained as closed-form formulas. We numerically examine the impact of cointegration to the trade-off between risk and return in 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 m+1 assets are traded continuously within the time horizon [0,T]. These assets are labeled by Si for i=0,1,2,,m, with the 0-th asset being risk free. The risk-free asset satisfies the differential equation dS0(t)=r(t)S0(t)dt,S0(0)=R0>0, where r(t) is the time-deterministic risk-free rate. The risky assets are defined through their log-price processes X1(t),,Xm(t), where Xj(t)=lnSj(t). The vector of log-prices satisfies the stochastic differential equation

Solution

The optimal solution of (P(R)) can be obtained by solving an alternative but equivalent max–min problem derived from the Lagrangian approach: maxλRminuUVar(Y(T))λ(E[Y(T)]R), where U={u():uΠ,(1), (2), (5)}. 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 X(t) 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 t are X(t)=[x1(t)x2(t)] with initial value [ln1ln2] and constant parameters: θ=(0.10.2),A=12(1111),σA=(0.20000.20)ΣA=(0.4000.4). Clearly, z(t)=x2(t)+x1(t) 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 αL and volatility σL of liability are 0.8 and (000.3),

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.

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