The Stochastic Cash Balance Problem with Fixed Costs: The Risk-averse Case

1 Introduction

The stochastic cash balance problem is an optimization problem faced by a firm, which has to decide how much cash to hold in order to meet its transaction requirement for a given planing horizon with multiple periods. Arrow et al[1] point out that the similarity between the motives of inventories of goods and those for keeping cash balances. In contrast to the usual inventory problem, the stochastic cash balance problem with a case where the cash level (i.e., the checking account level) during the period can either increase or decrease, depending on whether the income is larger or smaller than the expenses during that period. It also allows the decision maker to change the cash level in any direction at the beginning of each period. He can increase the checking account level by withdrawing money from his savings, or decrease it by transferring money to his savings. Therefore, the stochastic cash balance problem can be regarded as a special type of inventory control problems, where the customer demands may be positive or negative and the decision maker can increase or decrease it. Hence, we will use the term inventory level instead of cash level, and also use the terms “order” or “return” to indicate the increase or decrease of the cash levels. At the beginning of each period, the firm may decide to replenish the inventory or return excess stock. Both the ordering cost and the return cost may include a fixed component and a variable component which is proportional to the transaction amount. A holding or penalty cost is charged depending on whether the inventory level is positive or negative. The objective of the firm is to find an ordering or return policy so as to minimize the total expected cost, or equivalently, maximize the total expected profit over the entire planning horizon. Of course, this focus on optimizing expected profit or cost is appropriate for a risk-neutral decision maker, i.e., a firm that is insensitive to profit variations.

The stochastic cash balance problem received considerable amount of attention in the 1960s. Eppen and Fama[2], Whisler[3] consider a cash balance model with independent and identical distribution discrete demands with finite support and without fixed costs. They show the existence of order-up-to and return-down-to levels in the finite and infinite horizon models with discounted cost criterion. Feinberg and Lewis[4] justify the average cost case with the general demand distribution and study the problems with borrowing and lending options and no fixed costs, for which they establish the optimality of simple four-threshold policies. Girgis[5] investigates finite and infinite horizon discounted cost problems with continuous demand when there are fixed costs for increasing or decreasing demand (but not both). Neave[6] studies finite horizon problems with continuous demand when both transactions have fixed costs. However, Chen and Simchi-Levi[7] and Ye and Duenyas[8] notice that some of the claims in [6] are not proved. By using the notion of a (K, Q)-convex function introduced by [8], Chen and Simchi-Levi[7] describe the structural properties of optimal solutions of finite horizon cash balance problems when both transactions have fixed costs. Feinberg and Lewis[9] show that structural results stated by [7] indeed hold for finite horizon cash balance problems with discounted criteria and extend the results to the average cost per unit time criteria.

All the papers referenced above assume that the decision makers are risk-neutral. However, many are willing to tradeoff lower expected profit for downside protection against possible losses. Note that traditional stochastic cash balance models fall short of meeting the needs of risk-averse planners. For instance, traditional stochastic cash balance models do not suggest mechanism to reduce the chance of unfavorable profit levels. Thus, it is important to incorporate the notions of risk aversion in the stochastic cash balance problem.

A parallel stream of research studies risk-averse inventory models. Many of the risk-averse inventory models consider single period newsvendor type of models (see, for example, Chen, Xu and Zhang[10], Eeckhoudt, Gollier and Schlesinger[11], Lau[12], Wu, Zhu and Teunter[13]). Bouakiz and Sobel[14] characterize the inventory control strategy so as to minimize the expected utility of the net present value of costs over a finite planing or an infinite horizon. Assuming linear ordering cost, they prove that a base stock policy is optimal. Chen et al[15] propose a general framework to incorporate risk aversion into multi-period inventory models as well as multi-period models that coordinate inventory and pricing strategies. In both cases, they distinguish between models with fixed ordering costs and models with no fixed ordering costs. They show that the structure of the optimal policy for a decision maker with exponential utility function is almost identical to the structure of the optimal risk-neutral inventory (and pricing) policies. These structural results are extended to models in which the decision maker has access to a (partially) complete financial market and can hedge his operational risk through trading financial securities.

On the other hand, a decision maker may not know the exact demand distributions and have to estimate them from limited historical data. In this case, the decision maker is ambiguous about the probability distribution. Recently, Nilim and EI Ghaoui[16] study robust solutions to Markov decision problems with uncertain transition matrices. They propose the general idea on the ambiguity averse models, that is, the decision maker choose his policies assuming that nature is adversarial, choosing probability distributions from an ambiguity set to minimize the decision maker’s expected utility. Chen and Sun[17] adopt the robust dynamic programming modelling framework introduced by [16] to ambiguity and risk averse inventory and pricing models. They show that the optimal control policies share similar structure properties as Chen et al[15] for the finite horizon case and extend Chen et al[15] to including ambiguity aversion and considering infinite horizon models.

In this paper, we propose a framework for incorporating risk aversion in stochastic cash balance problem. We characterize the structure of the optimal policy on the risk-averse stochastic cash balance problem by using the consumption model introduced by Chen et al[15]. We show that the structure of the optimal policy for a decision maker with exponential utility function is almost identical to the structure of the optimal risk-neutral operations policy. Furthermore, we extend the results for the exponential utility function to the ambiguity aversion case.

The paper is organized as follows. In Section 2, we propose a model to incorporate risk aversion in the stochastic cash balance problem. In Sections 3 and 4, we focus on characterizing the structure of the optimal policies under the general increasing concave utility function and exponential utility function, respectively. In Section 5, we extend the results for the exponential utility function to the ambiguity aversion case. Finally, Section 6 is concluding section.

2 The basic model

Consider a risk-averse firm facing stochastic demand that has to make ordering or return decisions over a finite planning horizon with a total of T periods.

At the beginning of each period, an ordering or return decision is made. Let x_t be the inventory level at the beginning of period t before a decision is made and y_t be the inventory level at the beginning of period t after an ordering or return decision is made. Lead time for the ordering or return transaction is assumed to be zero. The transaction cost is denoted by c(x_t, y_t), which is calculated as follows:

where K ≥ 0, Q ≥ 0, k + q ≥ 0. Note that the assumption that k + q ≥ 0 implies that the unit refund is no more than the unit ordering cost.

For t = 1, 2, ⋯, T, let p_t be per unit “sale price” of product in period t and D_t(ϵ_t) (here ϵ_t is a random variable) be “stochastic demand” in period t, which consists of obligations paid less funds received (note that the demand in a period can be negative, which corresponds to receiving more funds than were paid out that period). Furthermore, demands in different periods are independent of each other. Unsatisfied demand is backlogged. Therefore, the inventory level carried over from period t to the next period, x_t+1, may be positive or negative. A cost h_t(x_t+1) is incurred at the end of period t which represents holding cost when x_t+1 > 0 and shortage cost if x_t+1 < 0. For technical reasons, we assume that function h_t(x) is convex and lim|x|→∞ht(x)=∞. Further, similar to Assumption 1 in Chen and Simchi-Levi[7], it is assumed that there are finite numbers x_t ≤ y_t ≤ v_t ≤ z_t such that (h_t(y_t) − h_t(x_t))/(y_t − x_t) < −k and (h_t(z_t) − h_t(v_t))/(z_t − v_t) > q.

To study the stochastic cash balance problem with fixed costs under risk aversion, we adopt the consumption model under uncertainty introduced by Chen et al[15]. The general idea is to directly model consumption, saving and borrowing decisions as well as inventory decisions for the stochastic cash balance problem. Specifically, assume that the decision maker has access to a financial market for borrowing and lending with a risk-free saving and borrowing interest rate r_f, or equivalently, the discount factor is γ=11+rf. At the beginning of period t, assume that the decision maker has initial wealth w_t and chooses an operations policy (order or return) that affects his income cash flow. At the end of period t, that is, after the uncertainty of this period has been resolved, the decision maker observes his current wealth level w_t + P_t and decides his consumption level f_t for the period, where P_t is the income generated at period t. Note that the income at period t is

where x⁺ = max{x, 0}, x⁻ = min{x, 0},

The remaining wealth, w_t + P_t − f_t, is then saved (or borrowed, if negative) for the next period, i.e., w_t+1 = (1 + r_f)(w_t + P_t − f_t), or, equivalently, f_t = w_t − γw_t+1 + P_t. The decision maker’s objective is to maximize his expected utility of the consumption flow E[Π(f₁, ⋯, f_T)] over the planing horizon 1,⋯, T. Moreover, at the last period T, we assume the decision maker consumes everything, which corresponds to w_T+1 = 0.

According to the consumption model, the decision maker’s problem is to find the inventory level y_t and decide the initial wealth level w_t (or equivalently, the consumption level f_t) for the following optimization problem.

When the utility function Π(f₁, f₂, ⋯, f_T) takes the linear form Π(f₁, f₂, ⋯, f_T) = ∑t=1Tγt−1ft, the consumption model reduces to the traditional risk-neutral stochastic cash balance problem. In this case, we denote V_t(x) to be the profit-to-go function at the beginning of period t with the initial inventory level x. A natural dynamic program for the risk-neutral stochastic cash balance problem is as follows:

with boundary condition V_T+1(x) = 0, where H_t(x) = E{pD_t(ϵ_t) − h_t(x − D_t(ϵ_t)) + γV_t+1(x − D_t(ϵ_t))}. Without loss of generality, we assume that k ≥ Q. Define Lt∈arg⁡maxx{Ht(x)−kx},lt=min{x|Ht(x)−kx=Ht(Lt)−kLt−K},lt′=min{x|Ht(x)−kx=Ht(Lt)−kLt−(K−Q)},Ut∈arg⁡maxx{Ht(x)+qx},ut=max{x|Ht(x)+qx=Ht(Ut)+qUt−Q},ut′=min{x|Ht(x)+qx=Ht(Ut)+qUt−(K−Q)}.

From Lemma 3 in Chen and Simchi-Levi[7], we have Lt≤Ut,lt′≤ut′. Moreover, ut′≤Ut due to K ≥ Q ≥ 0. Therefore, the above parameters satisfy the following relationship: l_t ≤ lt′≤Lt≤Ut≤ut,lt′≤ut′≤Ut≤ut.

Notice that these critical points have explicit implications in the stochastic cash balance problem. By definition, l_t is the largest value below which one always orders; lt′ is the smallest value above which one never orders; u_t is the smallest value above which one always returns; ut′ is the largest value below which one never returns. In particular, we call {lt,lt′}and{ut,ut′} the pairs of order- and return-associated critical points, respectively.

To provide a characterization of the optimal policy, Chen and Simchi-Levi use the following concept of (K, Q)-convexity, which is introduced by Ye and Duenyas[8].

See Lemmas 1 and 2 in [7] for the properties of the (K, Q)-convex function.

Note that the (K, 0)-convexity is exactly the K-convexity introduced by Scarf[18] for the classical stochastic inventory control problem with fixed ordering costs. Moreover, the (K, K)-convexity is the symmetric K-convexity, a concept introduced and applied in Chen and Simchi-Levi[19] to analyze a joint inventory control and pricing problem with fixed ordering costs and a general demand distributions.

Similar to the proof of Theorems 3.1 and 3.2 in [7], we have the following main results for the traditional risk-neutral stochastic cash balance problem.

3 Additive increasing concave utility model

In this section, we focus on the additive general increasing concave utility function. In this case, the objective function of (1) becomes Π(f1,⋯,fT)=∑t=1Tπt(ft), where the function π_t(⋅) is increasing and concave. That is, the utility of the consumption flow is the summation of the utility from the consumption in each period. According to the sequence of events as described before, the optimization model (1) can be solved by the following dynamic programming recursion.

where

with boundary conditions V_T(x, w) = π_T(w + P_T(x, y;ϵ_T)), V_T+1(x, 0) = 0. In contrast to risk-neutral stochastic cash problem, here the state variable is two-dimensional, i.e, the current inventory level x and the wealth level w.

Instead of working with the dynamic program (4) ∼ (5), we find that it is more convenient to work with an equivalent formulation. If y≥x,letΠt′(x,w)=Vt(x,w−kx), and the modified income in period t be Pt′(y;ϵt)=(γk−k)y+(p−γk)Dt(ϵt)−ht(yt−Dt(ϵt)). In this case, the dynamic program (4)∼(5) becomes

where

If y≤x,letΠt″(x,w)=Vt(x,w+qx), and the modified income in period t be Pt″(y;ϵt) = (q − γq)y + (γq − p)D_t(ϵ_t) − h_t(y_t − D_t(ϵ_t)). In this case, the dynamic program (4)∼(5) becomes

where

Therefore, The dynamic program (4)∼(5) becomes

4 Additive exponential utility function

In this section, we focus on a special case — the exponential utility function π_t(f) = −αte−fβt with parameters α_t, β_t > 0, where β_t is the risk tolerance factor, α_t reflects the decision maker’s attitude towards the utility obtained from different periods.

According to Chen et al[15], for a risk tolerance parameter R, denote the “certainty equivalent” operator with respect to a random variable ξ to be CEξR[ξ]=−Rln⁡E[e−ξR], which represents the amount of money a decision maker feels indifferent to a gamble with random payoff ξ. We also consider the “effective risk tolerance” per period defined as Rt=∑τ=tTγτ−tβτ. Further, we can obtain the expression R_t(1 + r_f) = (1 + r_f)β_t + R_t+1.

The next lemma states that we are able to separately make the operations decisions without considering the wealth/consumption decisions.

5 Additive exponential utility function with ambiguity aversion

In this section, we introduce the finite horizon ambiguity averse model under exponential utility function. Specially, assume that the decision maker does not know the exact probability distribution for the random variable ϵ_t. Rather, the decision maker is only aware of a set of probability distributions to which the probability distribution of ϵ_t belongs. According to Chen and Sun[17], in period t, the decision maker choose his policies assuming that nature is adversarial, choosing probability distributions g_ϵt from an ambiguity set Ω_t to minimize the decision maker’s expected utility. Thus, similar to (4)∼(5), a dynamic program for the risk and ambiguity averse stochastic cash balance problem is as follows:

where

with the boundary condition V_T(x, w) = π_T(w + P_T(x, y; ϵ_t)), V_T+1(x, 0)=0.

According to [17], we adopt the “general certainty equivalent” operator ϕ(⋅) defined on a function g(⋅) of an ambiguous uncertainty ξ, i.e, ΦΩR[ϕ(ξ)]=mingϵ∈Ω−Rln⁡Egϵ[e−ϕ(ξ)R]. Note that ΦΩR=CEξR when Ω is a singleton. Obviously, the operator ΦΩR generalizes the certainty equivalent operator CEξR in Section 4.

Assume that πt(f)=−αte−fβt, and the ambiguity sets satisfy certain technical conditions so that the minimization in the general certainty equivalent operator can always be attained. Similar to the proof of Lemma 3, the stochastic cash balance problem in the ambiguity and risk averse model (16)∼(17) can be calculated through the following dynamic programming

with boundary condition G_T+1(x) = 0.

To obtain the structure on the optimal policies, we need the following result, which implies the minimum envelope of (K, Q)-concave functions is still (K, Q)-concave.

6 Conclusions

In this paper, we propose a framework for incorporating risk aversion in stochastic cash balance problem. We characterize the structure of the optimal policy on the risk-averse stochastic cash balance problem according to the consumption model. We show that the structure of the optimal policy for a decision maker with exponential utility function is almost identical to the structure of the optimal risk-neutral operations policy. Furthermore, we extend the results for the exponential utility function to the ambiguity aversion case.