Dynamic Asset Allocation with Liabilities

• Published date: 01 March 2017
• Author: Athanasios Sakkas,Nikolaos Tessaromatis,Daniel Giamouridis
• DOI: http://doi.org/10.1111/eufm.12097
• Date: 01 March 2017

Dynamic Asset Allocation with

Liabilities

Daniel Giamouridis

Quantitative Solutions, Bank of America Merrill Lynch, United Kingdom

E-mail: daniel.giamouridis@baml.com

Athanasios Sakkas

Southampton Business School, University of Southampton, United Kingdom

E-mail: a.sakkas@soton.ac.uk

Nikolaos Tessaromatis

EDHEC Business School and EDHEC Risk Institute, France

E-mail: nikolaos.tessaromatis@edhec.edu

Abstract

We develop an analytical solution to the dynamic multi-period portfolio choice

problem of an investor with risky liabilities and time varying investment

opportunities. We use the model to compare the asset allocation of investors who

take liabilities into account, assuming time varying returns and a multi-period

setting with the asset allocation of myopic ALM investors. In the absence of

regulatory constraints on asset allocation weights, there are significant gains to

investors who have access to a dynamic asset allocation model with liabilities. The

gains are smaller under the typical funding ratio constraints faced by pension

funds.

The authors are grateful to the Editor, John Doukas and three anonymous referees for their

constructive comments. Furthermore, the paper has benefited from comments by Andrew

Ang, Michael Rockinger, Stephen Zeldes and participants at Fourth Joint BIS-World Bank

Public Investors Conference, Washington, DC 2012, at the Dauphine–Amundi Chair in

Asset Management 2012 Annual Workshop, Paris 2012 and Nineteenth International

Conference ‘Forecasting Financial Markets’, Marseille 2012. The authors are grateful for

financial support from the ‘Dauphine-Amundi Chair in Asset Management’. Daniel

Giamouridis also greatly acknowledges financial support from the Athens University of

Economics and Business Research Center (EP-1681-01, EP-1994-01) and notes that the

views expressed in this paper are his own and do not necessarily reflect those of Bank of

America Merrill Lynch. Daniel Giamouridis is also affiliated with LUMS, Cass Business

School, and the EDHEC-Risk Institute. This work was largely completed when Daniel

Giamouridis was an Associate Professor of Finance in the Department of Accounting and

Finance at Athens University of Economics and Business. Any remaining errors are the

responsibility of the authors.

European Financial Management, Vol. 23, No. 2, 2017, 254–291

doi: 10.1111/eufm.12097

© 2016 John Wiley & Sons, Ltd.

Keywords: strategic asset allocation, dynamic asset allocation, asset-liability

management, return predictability, myopic investors

JEL classification: G11, G12, G23

1. Introduction

It is known since the work of Samuelson (1969) and Merton (1969; 1971; 1973) that

long-term investors might judge risks very differently from short-term investors and

hence hold different portfolios. Although Merton (1971; 1973) developed an analytical

framework for understanding the portfolio demands of long-term investors, empirical

implementation of his work lagged far behind due to the difficulty of solving Merton’s

intertemporal model (Campbell and Viceira, 2002). The importance of liabilities in

strategic asset allocation has also been long recognised by both practitioners and

academics with analytical discrete-time solutions available only under static, one-period

horizons. In this paper we develop an analytical solution to the dynamic multi-period

portfolio choice problem of an investor under time-varying investment opportunities.

We then use the model to compare the asset allocation of investors who ignore the time-

varying properties of future asset risks and returns (myopic investors).

A brief taxonomy of multi-period portfolio choice solutions in an asset-only

framework is provided in Jurek and Viceira (2011, JV hereafter). JV identify three types

of approaches to optimal portfolio selection. First, optimal portfolios can be obtained as

exact analytical solutions for special cases of the multi-period portfolio choice problem

in a continuous time setting (Brennan and Xia, 2002; Chacko and Viceira, 2005; Kim and

Omberg, 1996; Liu, 2007; Sorensen, 1999; Wachter, 2002). Second, optimal portfolios

can be determined with numerical methods (Barberis, 2000; Brandt, 1999; Brandt et al.,

2005; Brennan et al., 1997; Dammon et al., 2001; Detemple et al., 2003; Koijen et al.,

2010; Lynch, 2001). Third, optimal portfolios can be computed in closed-form through

approximate analytical methods (Campbell and Viceira, 1999; 2001; 2002, Campbell

et al., 2003). JV also discuss the shortcomings of these approaches and propose an

analytical recursive approach which results in a closed-form solution for the dynamic

multi-period portfolio choice problem.

Multi-period portfolio choice is particularly relevant for investors who invest with the

objective to finance a long-term stream of liabilities. Best known examples of such

investors are pension funds, foundations and endowment funds. It is also relevant to

individual investors who manage their own retirement funds or build portfolios to

finance their children’s education. In this paper we study the portfolio choice problem of

this type of investor, a problem that is generally referred to as Asset Liability

Management (ALM hereafter).

Leibowitz (1987) and Sharpe and Tint (1990) are the first papers that incorporate

liabilities in a single period context. Since then the literature on portfolio choice in a

multi-period context in the presence of liabilities, parallels developments in the asset-

only portfolio choice literature. In a continuous setting Boulier et al. (1995) formulate a

dynamic programming model of pension fund behaviour. Sundaresan and Zapatero

(1997) formulate and solve a continuous time model for the strategic asset allocation of a

defined benefit pension fund taking into account the marginal productivity of the

employee under a constant investment opportunity set. Rudolf and Ziemba (2004)

© 2016 John Wiley & Sons, Ltd.

Dynamic Asset Allocation with Liabilities 255

assume time-varying investment opportunities and develop a four-fund theorem for

intertemporal surplus optimisation.

A second strand of the literature, closer to asset-liability modeling practice,

1

uses

stochastic programming techniques (Cari~

no et al., 1998; Cari~

no and Ziemba, 1998;

Geyer and Ziemba, 2008) to find optimal asset weights. A third approach is based on

simulation (Binsbergen and Brandt, 2016). These approaches are however subject to the

criticisms expressed in JV in the context of asset-only portfolio choice. Stochastic

programming techniques for example face the cost of tractability; the traditional

numerical methods cannot handle more than a few state variables, magnifying the effects

of estimation risk even further; and the ALM models carried out under a continuous time

context have been mostly studied under a constant investment opportunity set.

Another strand of the ALM literature, closer in spirit to our paper, uses approximate

analytical solutions and extends the asset-only models introduced by Campbell and

Viceira (1999; 2001; 2002) and Campbell et al. (2003) by incorporating liabilities.

Hoevenaars et al. (2008, HMSS hereafter) is a significant contribution in this literature.

HMSS assume that an investor with risky liabilities chooses the optimal portfolio

weights at the beginning of the investment horizon and rebalances to the initial allocation

at a rebalancing time shorter than the investment horizon. Barberis (2000) describes the

latter strategy as myopic rebalancing; it is myopic because the investor ignores the

information available at the rebalancing time. Myopic investors maintain the initial static

solution, through rebalancing, until the end of the horizon. In contrast, dynamic investors

follow a dynamic investment strategy where the portfolio weights take into account

changes in the investor’s investment opportunity set. Using the myopic asset-liability

model (M-ALM thereafter), HMSS find significant differences in the portfolio

allocations of investors who ignore liabilities.

Our principal contribution in this paper is that we develop a model which results in a

closed-form solution for the dynamic multi-period portfolio choice problem in the

presence of liabilities (D-ALM thereafter). Our model nests the asset-only model of JV.

It is analytically tractable and hence allows the study and interpretation of factors

affecting optimal portfolio choice in an ALM framework. The model we propose can

accommodate any number of assets and state variables. To develop the proposed model

we build on JV and HMSS. Our representative investor is endowed with power utility

over the ratio of assets and liabilities, as opposed to over assets only, at a fixed horizon;

she can invest in many risky assets and a riskless asset. The investment opportunity set is

time-varying and captures the variation in risk premia, interest rates and the state

variables. The optimal investment policy that we derive has a simple structure; it is

comprised of myopic and intertemporal hedging terms. The novel feature of our model is

that we show analytically how the intertemporal hedging demand which appears in the

asset-only multi-period portfolio choice literature is also driven by the correlation

between the innovations of liabilities and the innovations in the changes of the state

variables.

In our empirical investigation we take into account practical considerations. We

incorporate short-sales or borrowing constraints, as in Brandt et al. (2009). We also

1

ALM based on stochastic programming and used in practice include the Russel–Yasuda

Kasai model described in Cari~

no et al. (1998) and the Towers Perrin–Tillinghast ALM

approach presented in Mulvey et al. (2000).

© 2016 John Wiley & Sons, Ltd.

256 Daniel Giamouridis, Athanasios Sakkas and Nikolaos Tessaromatis