Selecting the best risk measure in multiobjective cash management

• DOI: http://doi.org/10.1111/itor.12580
• Date: 01 May 2019
• Published date: 01 May 2019
• Author: Francisco Salas‐Molina

Intl. Trans. in Op. Res. 26 (2019) 929–945

DOI: 10.1111/itor.12580

INTERNATIONAL

TRANSACTIONS

IN OPERATIONAL

RESEARCH

Selecting the best risk measure in multiobjective cash

management

Francisco Salas-Molina

Department of Management ‘Juan Jos´

e Renau Piqueras’, Universitat de Val`

encia, Av. Tarongers s/n, 46022 Val`

encia, Spain

E-mail: francisco.salas-molina@uv.es

Received 27 December 2017; receivedin revised form 13 July 2018; accepted 19 July 2018

Abstract

In this paper, we consider cash management from a multidimensional perspective in which cost and risk are

desired goals to minimize. Cash managers interested in minimizing risk need to select the most appropriate

risk measure according to their particular needs. In order to assess the quality of alternative risk measures,

we empirically compare eight different risk measures in terms of the combined cost–risk performance of a

cash management model. To this end, we rely on goal programming to derive optimal solutions for cash

management models. Our results show that risk measures based on cost deviations better capture risk in

comparison to those based on a reference cash balance. The methodology proposed in this paper allows cash

managers to propose and evaluate new risk measures.

Keywords:multidimensional finance; data-driven models; risk analysis; goal programming

1. Introduction

When facing cash management, we usually assume that risk control is implicit in decision making

by considering much higher penalty costs for negative cash balances than holding costs for positive

balances. Under the usual assumption of linear holding costs (see, e.g., Gormley and Meade, 2007;

da Costa Moraes and Nagano, 2014), the lower the balance, the lower the cost. However, low

balances may lead to high overdraft costs due to the uncertainty associated to future cash flows.

This situation can be partially solved by setting minimumcash balances for precautionary purposes

(Ross et al., 2002). However, cash managers can also derive better cash policies by including risk

analysis in their decision-making processes as recently proposed by Salas-Molina et al. (2018a,

2018b).

The cash management problem (CMP) is defined as an optimization problem that aims to find

the best sequence of control actions over a given planning horizon, what is called a policy, that

minimizes the cost of idle balances (holding/penalty costs) and the cost of controlling balances

C

2018 The Authors.

International Transactionsin Operational Research C

2018 International Federation ofOperational Research Societies

Published by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA02148,

USA.

930 F. Salas-Molina / Intl. Trans.in Op. Res. 26 (2019) 929–945

(transaction costs). Since Baumol (1952) and Miller and Orr (1966), the common two-asset setting

prevailed in many research works as surveyed in Gregory (1976), Srinivasan and Kim (1986), and

da Costa Moraes et al. (2015). This framework assumes the existence of a main bank account for

operational purposes, and a second account summarizing short-term assets, such as treasury bills

or marketable securities, ready to be converted in cash when needed.

Most cash management models in the literature focusedon a single objective, namely,minimizing

holding and transaction costs (Premachandra, 2004; Gormley and Meade, 2007; Baccarin, 2009;

Righetto et al., 2016). Recently, Salas-Molina et al. (2018a) introduced risk analysis in cash man-

agement by measuring both the cost and the risk of alternative policies. The authors measured cost

by the average daily cost and risk by the standard deviation of daily cost over a given planning

horizon. A different approach to cash management was proposed by Herrera-C´

aceres and Ibeas

(2016) by minimizing the sum of squared deviations from a cash balance reference signal, but with-

out considering costs. However, there is a lack of research about the goodness of such risk metrics

in the context of cash management.

Risk assessment is an ongoing issue in many scientific fields. On the one hand, risk has to be

defined from a qualitative point of view. Kaplan and Garrick (1981) define risk as the combination

of uncertainty and damage. The possibility of an unfortunate occurrence or the deviation from a

reference value and associateduncertainties are some of the additional definitions recentlyproposed

by Aven (2016). On the other hand, risk has to be quantitatively defined. In other words, some

particular metric has to be proposed in order to facilitate risk assessment. Furthermore, this metric

is usually domain specific. In this paper, we focus on this second aspect of risk analysis in an

attempt to evaluate alternative risk measures. First, we accept the fact that decision making in cash

management is enriched by considering multiplecriteria (Steuer, 1986; Ballestero and Romero, 1998;

Branke et al., 2008; Aouni et al., 2014) such as cost and risk. Then, we proposea method to select the

most appropriaterisk measure to be used as a key input to a multiobjective cash management model.

As a result, the main purpose of this paper is to provide a method to empirically compare different

risk measures within a multiobjective framework in which cost and risk are desired objectives to

minimize.

To this end, we first represent the common two-asset framework as a simple cash management

system with: (a) two accounts; (b) two possible transactions; and (c) a given cost structure with

holding and transaction costs. Once a particular cash management system is defined, we formulate

the CMP as a multiobjective goal program in order to ensure the optimality of solutions. By

establishing an achievement objective function with both a cost measure and a risk measure, we

aim to minimize a loss function expressed in terms of aggregated Manhattan distances to an ideal

point where cost and risk are minimum (Zeleny, 1982; Ballestero and Romero, 1998; Ballestero and

Pla-Santamar´

ıa, 2003; Jones et al., 2010; Yu, 2013). We consider linear and quadratic risk functions

for computational reasons to ensure the optimality of solutions as a desirable feature from a cash

manager point of view. It is important to say that the methodology presented in this paper can be

extended to consider additional measures of risk. This extension can be done either by linearizing

nonlinear risk functions or using heuristics such as evolutionary algorithms to obtain solutions.

Here, we focus on two kinds of risk measures:

rrisk measures based on a reference cash balance as in Herrera-C´

aceres and Ibeas (2016);

rrisk measures based on cost deviations as in Salas-Molina et al. (2018a).

C

2018 The Authors.

International Transactionsin Operational Research C

2018 International Federation ofOperational Research Societies