Maximizing the probability of realizing profit targets versus maximizing expected profits: A reconciliation to resolve an agency problem

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Abstract

It is generally accepted in the operations literature that a firm should strive to maximize its expected profit. However, in practice it is not uncommon for a firm to offer a bonus to managers for achieving some pre-established target profit, possibly yielding managerial actions that differ from the profit-maximizing approach (given a profit target, we assume managers will maximize the probability of reaching that target). We use the Newsvendor framework to illustrate how the firm's shareholders (e.g., through its board of directors) can align these two seemingly different decision approaches: maximizing expected profit versus maximizing the probability of reaching a target profit. Alignment is achieved by setting what we call an “aligned profit target” (APT) – a target profit that yields the same managerial action namely, the same stocking quantity, across both decision approaches. We find that the APT should typically be an aggressive profit target, one that is significantly higher than the maximum expected profit, with a corresponding low probability of achievement – this result is consistent across demand distributions with light tails (uniform), moderate tails (normal) and heavy tails (lognormal). Notably, the aggressive APT target should be distinguished from any target that the firm might set to signal future profit expectations to financial analysts.

Introduction

In the Operations literature it is commonly agreed that the principal objective for a firm is to maximize shareholder value, which in turn is achieved when the firm maximizes its expected profit. However, given that the shareholders of the firm typically do not directly observe and monitor the day-to-day actions of the firm's manager (we will use the singular “manager” throughout), an agency problem may arise – how can the shareholders (as represented by the Board of Directors) incentivize the manager to take the profit-maximizing action?

There is a rich literature addressing this type of principal-agent problem; we focus on one specific aspect of the problem as follows. In practice, one relatively straightforward incentive often offered by shareholders to motivate managerial alignment is for the shareholders (through the Board of Directors) to offer the manager a bonus for achieving some profit target (e.g., Guay et al., 2019). We explore a number of questions in such a setting. In particular, what are the implications of setting the profit target at the maximum expected profit (e.g., by how much does the expected profit drop)? Is it preferable to set the profit target above or below the maximum expected profit? And is there a profit target that will induce the profit-maximizing managerial action – and if so – how does this “Aligned Profit Target (APT)” compare with the expected maximum profit?

We use the Newsvendor setting to explore these questions. The standard Newsvendor solution is of course to set the stocking quantity at the level that maximizes the expected profit (we call this maximum the “Newsvendor profit,” or NP). In the situation where the shareholders set a profit target, we assume the manager will react by setting the stocking quantity at the level that maximizes her probability of hitting that profit target – we call this the Probability Maximizing Policy (PMP). In addition to the example of bonuses for managers within publicly held firms, Swinney, Cachon and Netessine (2011) argue that maximizing the probability of hitting a profit target is a sensible strategy with respect to the survival of start-ups.

It is important to contrast the internal profit target discussed herein with any external quarterly or yearly earnings forecast that the firm might try to influence or that might be determined by financial analysts. Firms may often try to avoid signaling an aggressive earnings number so that the stock price does not take an unexpected hit when earnings are released, for example. In contrast, the target that is set for internal consideration should be designed to ensure that the manager will seek to maximize expected profit, without regard to any externally focused earnings expectation. To reiterate, the analysis in this paper focuses on setting an appropriate target profit level for management to receive a bonus that will lead to maximum expected profits for the shareholder. Quoting Guay et al. (2019), “The vast majority of US executive compensation plans incorporate bonus payouts, and boards devote considerable time and effort to designing these often-complex plans.”

As previously mentioned, we analyze a Newsvendor setting; a typical example in the literature is where a firm has one chance (without reordering) to determine how many items to order from a manufacturer. We recognize that this specific example reflects a decision that may be made at some level below the C-suite, however our results can still be relevant to boards setting plans for executives, for the following two reasons. First, higher-level decisions, such as how big to build a plant (or how far or fast to expand), can be thought of in a Newsvendor framework – build your plant too big and you have idle capacity. Build it too small and you lose out on sales. And second, C-suite executives in turn set bonuses for managers reporting to them, so assuming C-suite executives are properly motivated by their board of directors to maximize expected profit, then they in turn will be motivated to set bonuses for the managers that face the more-classical Newsvendor decisions.

Our analysis focuses on situations where the cost-of-underage in the Newsvendor is significantly higher than the cost of overage – more specifically, where it is on the order of five to fifty times higher, such that the Newsvendor stocking quantity is around one to two standard deviations above the mean. This translates into service levels between roughly 84% and 98%. This range of service levels covers what seems typical for example in retail firms, where a typical service level might be 90% plus-or-minus 5% or so depending on the product (Lokad, 2021). In such a situation, we find a number of interesting results. First, we observe that the aligned profit target APT (see definition above) is generally about 1.5 times the Newsvendor profit (NP). In other words, the Board of Directors should set a profit target for the manager that is 50% higher than the profit the Board expects will actually be achieved (of course, the exact percentage depends on the specific parameters and can be up to triple the NP).

As this result might imply, we find that it is much preferred to set a profit target well above the NP as opposed to setting it “much too low.” This is similar to the admonition of Zoltners et al. (2017), who suggest that bonuses should not be wasted by setting easy targets. On the other hand, setting the profit target does not have to be pin-point accurate. Our results suggest that as long as the profit target is within 25% or so of the NP, the firm does not suffer unduly when the manager follows the probability-maximizing policy (PMP). Said another way, the expected profit is fairly robust to a range of profit targets, from around 75% of the NP all the way up to double the NP. In our results, the drop in expected profit over this range of profit targets is 15% or less.

The remainder of this paper is organized as follows. Following a literature review and the motivation behind the paper in section 2, in section 3 we present the base model given the objective of maximizing the probability of realizing or exceeding a target profit, given uncertainty in demand. In section IV, we present a numerical study. Our concluding remarks are presented in section V.

Section snippets

Literature review

There is a rich literature on the topic of agency theory, to determine how shareholders might influence the actions of executives, such that the interests of the firm's managers are more-closely aligned with those of the shareholders. For example, see Bennett et al. (2017), Feltham and Xie (1994), Healy (1985), Holmström (1979), and Lambert and Larcker (1987). Our contribution is to explore this issue in the context of one of the most prominent frameworks in operations management; that of the

Assumptions, definitions and the base model

Our model adds further context and specificity to the underlying concepts discussed above. Based on what is often observed in practice, we assume the manager's (annual) compensation is comprised of two parts, a fixed salary plus a bonus which is paid out if a target profit level is achieved. This compensation scheme is agreed upon by the shareholders and manager in stage 1 of our model. We assume the manager's participation constraint is such that the manager must receive total expected

Numerical results

We develop the baseline parameters shown in Table 1 to illustrate our results and to explore the sensitivity of the outcomes to changes in the relative costs of underage and overage. Results based on these parameter values are presented in the first row of Table 2, Table 3, Table 4 for three distributions: uniform, normal and lognormal, respectively. The upper tail of the distributions is lightest for the uniform and heaviest for the lognormal. In the subsequent rows of Table 2, Table 3, Table 4

Discussion and conclusion

Maximizing the expected profit is typically taken to be the optimal decision-making objective. However, many Boards of Directors incentivize the firm's managers by setting an explicit profit target (Guay et al., 2019), which may lead to a suboptimal managerial decision. If faced with a profit target, managers may seek to maximize the probability of meeting that target, instead of explicitly maximizing expected profit. Using the Newsvendor framework and a set of representative parameters

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