Can Trade-off Theory Explain Net Working Capital Management Decisions?

Haowen Luo*

Assistant Professor of Finance, Doermer School of Business

Purdue University Fort Wayne

luh@pfw.edu

Abstract

This study applies a partial-adjustment model to test how well trade-off theory explains net working capital management decisions and examines working capital management dynamics. The results indicate that firms have long run NWC targets and tend to gradually converge to the target from the firm’s initial net working capital level within each period. We estimate that typical firms close approximately 50% of the gap between their actual and target net working capital each year. Such high adjustment speed implies that a typical firm close half of a deviation from the target in about 13 months. Our results show that trade-off theory can well explain the management decisions on working capital holdings.

JEL classification: G 32

Keywords: Trade-off Theory; Net Working Capital;

* Corresponding author. 2101 E. Coliseum Blvd, Neff Hall 350C, Fort Wayne, Indiana 46805 (luh@pfw.edu).

Can Trade-off Theory Explain Net Working

Capital Management Decisions?

Abstract

This study applies a partial-adjustment model to test how well trade-off theory explains net working capital management decisions and examines working capital management dynamics. The results indicate that firms have long run NWC targets and tend to gradually converge to the target from the firm’s initial net working capital level within each period. We estimate that typical firms close approximately 50% of the gap between their actual and target net working capital each year. Such high adjustment speed implies that a typical firm close half of a deviation from the target in about 13 months. Our results show that trade-off theory can well explain the management decisions on working capital holdings.

JEL classification: G 32

Keywords: Trade-off Theory; Net Working Capital;

Introduction

As an aggregated measurement of a firm’s supply and demand of trade credits, net working capital

(NWC) plays an essential role in financial management. Previous studies have documented that net

working capital is tied to profitability and capital market access (Petersen and Rajan 1997), directly

related to financing deficits (Deloof and Jegers 1999) and affects firms’ risk and value (Smith 1980).

Survey evidence by Graham and Harvey (2001) shows that 49% of CFOs consider managerial issues

related to working capital management “important” or “very important.”

When determining the optimal level of NWC, management that maximizes shareholder wealth should

set the firm’s NWC at a level such that the marginal benefit of NWC equals the marginal cost. The costs

of NWC include the strong dependency on banks, suppliers, or internal funds to finance the working

capital gap. The fact that positive NWC must be funded either internally or externally also implies

opportunity costs. Moreover, high NWC may indicate potential high write-off costs of unsold or expired

inventory. There are three main benefits of holding NWC. First, firms are better prepared to fight against

bankruptcy due to liquidity shocks. Second, firms are better positioned to take advantage of new

business opportunities and increase sales because of higher investment in inventories and trade credit

granted. Lastly, because trade credit is generally more expensive than bank financing and the NWC gap

is financed by the relatively cheaper bank and internal funds, high NWC may indicate lower overall

financing costs.

The trade-off framework suggests that firms always maintain their target NWC in a frictionless world.

However, due to the adjustment costs, immediate adjustments to a firm’s target require managers to

compare adjustment costs against the costs of operating with the current NWC. Therefore, even if the

normative trade-off theory is at work, we would only observe firms partially adjust NWC holdings

toward their targets. The speed with which firms reverse deviations from target NWC depends on the

costs of adjustment.

Although we find trade-off theory appealing when explaining management decisions on NWC, there is a

lack of previous literature to support the argument. Despite extensive studies on how trade-off theory

explains capital structure decisions, we find very few studies in corporate finance literature testing how

well does trade-off theory explains working capital management decisions. Our research is motivated to

fill this gap and formally test the trade-off theory in the context of working capital management.

We utilize a dynamic partial adjustment model to account for NWC’s dynamic nature. We are interested

in testing two main research questions: (1) Is there indeed an NWC target and if so, (2) what is the

adjustment speed with which a firm moves toward its target. Our model specification allows each firm

to have a different target NWC based on its firm characteristics. Such target also varies over time to

reflect the target’s dynamic feature. More importantly, we recognize that the deviations from the target

NWC are not entirely adjusted toward the target due to frictions. Therefore, the estimated adjustment

speed can also reflect the effect of NWC adjustment costs. A high adjustment speed suggests low

adjustment cost, strengthening the trade-off argument that even if firms actively pursue target working

capital holdings over time, the movement toward the target is not complete due to the adjustment cost.

Our findings suggest that firms have long run NWC targets and tend to gradually converge to the target

from the firm’s initial NWC level within each period. For a typical firm, the adjustment speed toward its

long-run target is about 50% per year. Such high adjustment speed implies that a typical firm close half

of a deviation from the target in about 13 months. We use various model specifications and subsamples

to test the robustness of the estimated speed of adjustment, and we find that our results are generally

consistent and robust across specifications.

Our findings contribute to the current literature on NWC management in several ways. First of all, we

show that management decisions on working capital holdings are consistent with trade-off theory. The

empirically documented targeting behavior across firms also deepens our understanding of the

dynamics of working capital holdings over time. Second, our estimated speed of adjustment provides

direct information on the effect of working capital adjustment cost, which is a critical first step in testing

plausible alternative explanations on the movement of NWC. Lastly, our findings extend and

complement existing literature on determinants of NWC (Hill et al. 2010, Molina and Preve 2009). Unlike

previous studies that focus solely on how firms’ characteristics affect NWC in a static setting, we use

dynamic models to show that NWC level is also affected by unobservable target NWC. Depending on

how far away the current NWC level is from the target, firms exhibit heterogeneous adjustment speeds

moving toward the target.

The remainder of the paper is organized as follows. Section 2 provides some initial evidence on possible

target NWC holdings. Section 3 presents the partial adjustment framework used to test the trade-off

theory on NWC. Section 4 details the sample, data sources, and variable constructions. The empirical

results are presented and discussed in Section 5. Section 6 performs robustness tests to address

potential concerns with the empirical design. Section 7 concludes.

Do firms have target WCR? A first look

We start our investigation on whether firms have target NWC holding by visually inspect time-series

properties of changes in NWC. Specifically, if firms have the target, we should observe that NWC is

systematically adjusted to not rise too high or fall too low. In other words, negative autocorrelation (or

mean-reverting) in NWC is a necessary condition of the existence of target NWC. We can test the

hypothesis that NWC holdings are mean-reverting by estimating the following first-order autoregressive

model for each firm:

∆(𝑁𝑁𝑁𝑁𝑁𝑁)𝑡𝑡 = 𝛼𝛼 + 𝛽𝛽∆(𝑁𝑁𝑁𝑁𝑁𝑁)𝑡𝑡−1 + 𝜀𝜀𝑡𝑡

(1)

where NWC is the net working capital ratio and 𝜀𝜀𝑡𝑡 is an i.i.d distributed random variable with zero mean.

Following previous research (Hill et al. 2010), we define the NWC ratio as the sum of accounts receivable

and inventories net of accounts payable, scaled by sales. We estimate Equation (1) using all firms with at

least five years of Compustat data between 1962 and 2019. 10,496 firms satisfy the data requirement.

After running regressions for each of these firms, we obtained 10,496 estimated autoregressive

coefficients (𝛽𝛽), and we plot the distribution of autoregressive coefficients in Figure 1.

[Figure 1]

Our results show that the mean and median values of estimated coefficients on ∆(𝑁𝑁𝑁𝑁𝑁𝑁)𝑡𝑡−1 are -0.207

and -0.194, respectively. Besides, the coefficient of skewness is -0.482, suggesting the distribution is left-

skewed. The distribution also appeared to be leptokurtic with longer, fatter tails than the normal

distribution. In Figure 1, the red indicator line on the x-axis marks the point where β equals zero. As

shown, firms are more frequently distributed to the left of the zero-beta indicator line (negative β) in

our sample. The fraction of firms with a negative coefficient constitutes 78.81 percent of observations

(7,708 out of 10,496 firms), suggesting that there are systematic factors that cause firms to actively

manage NWC holdings so that NWC holdings do not drift too far away from the target.

As suggested by previous literature (Opler et al. 1999; Hamilton, 1994), the estimated coefficients

presented in Figure 1 are biased in small samples (e.g., firms with fewer time-series observations).

Although we require a minimum of 5-year observations for each firm when estimating Equation (1), it is

not clear from previous literature the minimum sample size to sufficiently reduce the bias. To mitigate

this issue, we re-estimate Equation (1) using survival firms with non-missing data in Compustat during

the sample period and plot the autoregressive coefficients’ distribution in Figure 2. The mean-reverting

pattern for surviving firms become even more substantial than those reported in Figure 1: the mean and

median values of estimated coefficients on ∆(𝑁𝑁𝑁𝑁𝑁𝑁)𝑡𝑡−1 are -0.202 and -0.204, respectively. The

majority of surviving firms (85.81 percent of surviving firms) exhibit mean-reverting behavior when

managing NWC holdings over time.

Overall, our initial results indicate that NWC holdings are systematically mean-reverting. Firms tend to

adjust their NWC as if NWC holdings are regulated by some unobservable upper and lower boundaries.

These results provide us with some good intuitions on the NWC target and serve as the starting point for

more rigorous statistical tests, which will be introduced next.

Model specification

In a perfect market, firms would adjust their NWC toward target immediately at no cost. However,

frictions may prevent firms from adjusting NWC ratios fully as firms must consider optimizing marginal

benefits conditional on changing marginal costs. Therefore, any testing model to study trade-off NWC

behavior must recognize that deviations from the target NWC holdings are not necessarily adjusted

quickly. Also, the model specification should be general enough to permit each firm has a different

target NWC, which may vary over time. To meet these requirements, we estimate the following partial

adjustment model to allow for incomplete adjustment of the firm’s initial NWC toward its target within

each period.

𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1 − 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 = 𝜃𝜃� 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1∗ − 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡� + 𝜀𝜀𝑖𝑖,𝑡𝑡+1

(2)

where 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1∗ is firm i’s unobservable target NWC holding at time t+1. The difference between

𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1∗ and 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 is the gap between firm i’s target and actual NWC holding. A positive gap suggests

that the firm’s actual NWC holding is below its target level, and trade-off theory suggests that the actual

NWC adjusts upward next year. In contrast, if the gap is negative, we expect managers to close the gap

by adjusting downward because the actual WCR is higher than the desired target. 𝜃𝜃 represents the

proportion of closed gap each year by a typical firm or the adjustment speed. Because 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡∗ might

differ across firms or over time, we estimate it using the following reduced-form model

𝑁𝑁𝑁𝑁𝑊𝑊𝑖𝑖,𝑡𝑡+1∗ = β𝑋𝑋𝑖𝑖,𝑡𝑡

(3)

where 𝑋𝑋𝑖𝑖,𝑡𝑡 is a vector of firm characteristics known to be determinants of optimal NWC holding from

previous literature. To model a target NWC, we use firm characteristics variables proposed by prior

literature (Hill et al. 2010; Molina and Preve 2009; Love et al. 2007). These variables include sales

growth, gross profit margin, sales volatility, operating cash flow, market-to-book ratio, firm size, market

power, financial distress, and firm age. Sales growth is measured as a percentage change in sales over

the period t-1 to t. gross profit margin is defined as sales net of costs of goods sold, scaled by sales. We

measure sales volatility as the five-year rolling standard deviation of a firm’s sales, standardized by net

assets (asset net of cash), for each firm-year observation. Operating cash flow is earnings before

depreciation (OIBDP) net of income taxes, scaled by asset net of cash. Market-to-book ratio (MB) is the

sum of the market value of equity and book value of liabilities net of payables, scaled by asset net of

cash. Firm size is calculated as the natural logarithm of the inflation-adjusted market value of equity.

Following previous literature (Hill et al. 2010; Molina and Preve 2009), we measure market power by

market shares, which is calculated as the ratio of sales to annual aggregate sales of the industry where

the firm belongs to. Industries are defined according to the Fama-French 48 industry portfolios.

Financial distress is constructed the same as in Molina and Preve (2009). Following Petersen and Rajan

(1997), we use the natural logarithm of one plus firm’s current age as our age measurement.

Substituting (3) into (2) gives the estimable model

𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1 = (1 − 𝜃𝜃)𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 + (𝜃𝜃𝛽𝛽)𝑋𝑋𝑖𝑖,𝑡𝑡 + 𝜀𝜀𝑖𝑖,𝑡𝑡+1

(4)

Equation (4) is the primary estimating model we will use throughout the paper. It assumes that firms will

adjust their NWC to target in the long run, but adjustment costs lead to small deviations from the target.

Our primary variable of interest, 𝜃𝜃, measures approximate adjustment speed, which assumes to be the

same for all firms. The estimated coefficients on 𝑋𝑋𝑖𝑖,𝑡𝑡 are the long-run impact of deterministic firm

characteristics on WCR, inflated by a common factor 𝜃𝜃.

we also control for fixed effects, and the final estimation model becomes:

𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1 = (1 − 𝜃𝜃)𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 + (𝜃𝜃𝛽𝛽)𝑋𝑋𝑖𝑖,𝑡𝑡 + 𝛼𝛼𝑖𝑖 + 𝛿𝛿𝑡𝑡 + 𝜀𝜀𝑖𝑖,𝑡𝑡+1

(5)

where 𝛼𝛼𝑖𝑖 is the firm-specific unobserved effects and 𝛿𝛿𝑡𝑡 is the year fixed effects. Following previous

literature (Flannery et al. 2006; Venkiteshwarran 2011), we use this specification to estimate the target

and speed of adjustment, 𝜃𝜃, simultaneously. Although the dynamic panel model described in Equation

(5) represents our main specification, we also report results estimated using other commonly used

methods such as Fama MacBeth (1973) (FM).

Data and descriptive statistics.

Our sample is constructed using all firms, excluding financial (SIC codes 6000-6999), utilities (SIC codes

4900-4999), and ADR firms, recorded in the Compustat database between 1962 and 2019. To

accommodate our models which include lagged variables, we exclude any firm with fewer than two

consecutive years of data. Observations with missing financial variables are also deleted. Given the

extended sample period, the influences of outliers are nontrivial. To mitigate the problem caused by

extreme values, we winsorize all firm-level ratios at the 1 percent level. Finally, for non-ratio variables,

we restate the values in inflation-adjusted 2019 dollars. The final sample has 160,949 firm-year

observations, consisting of 14,299 unique firms with a maximum, minimum, and median numbers of

appearance equal to 57, 2, and 9 years each. We report summary statistics for all variables in Panel A of

Table 1. The mean(median) NWC in our sample is 19.3% (19.8%), which is very similar to those reported

in Hill et al. (2010).

Given that our sample period is much more extended than that of Hill et al. (2010), similar NWC values

suggest that NWC levels are relatively stable over time. In comparison, although the median values are

similar to those reported in Hill et al. (2010), the mean values are quite different for all other firm

characteristic variables. Therefore, the stable NWC overtime may reflect firms’ efforts to maintain NWC

holdings within a specific range, consistent with targeting behavior implied by trade-off theory.

[Table 1]

Panel B shows the correlation coefficients for the key variables. As shown, these variables are not highly

correlated, and the signs of correlation coefficients are generally consistent with those reported by

previous literature (Hill et al. 2010; Molina and Preve 2009; Love et al. 2007). To formally test for

multicollinearity, we calculate the variance inflation factor (VIF) for our variables (unreported). The

results show that the average VIF is 1.43, and none of the variables have a VIF higher than 10. Therefore,

we conclude that multicollinearity is not a significant concern in our model.

Estimation of empirical results

Estimating NWC speed of adjustment Table 2 reports the regression results using various estimation methods. The critical variable of interest

is 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 and speed of adjustment is calculated as one minus estimated coefficient on 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡. Column

1 of Table 2 shows Fama and MacBeth (1973) estimates. The estimated coefficient on 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 is 0.749

and significant at 1 percent level, which corresponds to a speed of adjustment of 0.251. At this rate, it

takes 2.398 years for an average firm to close half of the deviation from the target. Although the

estimated speed of adjustment seems too slow to explain much variation in firms’ NWC based on trade-

off theory, it is expected. As demonstrated in Flannery et al. (2006), using Fama and MacBeth method

with simultaneous estimation of target produces the lowest estimate that significantly underestimates

the actual adjustment speed. One possible explanation is that Fama and MacBeth method failed to

recognize the data’s panel characteristics, e.g., stable, unobservable firm- or time- variables affecting

the NWC level. As a result, the ignored fixed effect variables are included in error terms. Because lagged

NWC is used as an independent variable and fixed effects are correlated with NWC in every period, the

error terms are correlated with the residual component of NWC. As a result, the coefficient on NWC is

upward biased, and the estimate of adjustment speed is downward biased (Anderson and Hsiao, 1981;

Baltagi, 2001; Bond, 2002).

To better utilize the information provided by our panel data sets and reduce the endogeneity problem

caused by ignoring fixed effects, in column 2, we performed a panel regression controlling for both firm-

and year- fixed effects. The estimated coefficient of 0.398 implies that average firms close 60.2% of the

gap between current and target NWC within one year, which is substantially faster than the estimate

produced in column (1). At this rate, firms can close half of the deviation from the target in about nine

months. According to Flannery and Hankins (2013), a significantly higher speed of adjustment estimated

using the fixed-effect model is also expected. Specifically, although the within-transformation

performed for calculating coefficients in column 2 eliminate fixed effects and provides us with consistent

estimates, it introduces a correlation between the transformed lagged dependent variable and error

term by construction. (Woodridge, 2002; Hsiao, 2003, etc.). Woodridge (2002) finds that due to such a

“mechanical endogeneity problem,” the estimated coefficient on lagged dependent variable is biased

downward by a factor of 1/T. In other words, the bias becomes insignificant for panel data set with a

large T. However, for our sample, the minimum number of years per firm is 2, the maximum is 57, and

the median is 9. Therefore, our sample constitutes a “large N, small T” data set, and the bias can be

substantial. As a result, our estimated coefficient on NWC is downward biased, suggesting the estimated

speed of adjustment is upward biased.

Therefore, the estimated speed of adjustments reported in columns 1 and 2 are downward and upward

biased, respectively. As suggested by Bond (2002), we can use these two estimates as the lower and

upper bound, and the actual coefficient must lie within the range created by these bounds. We adopt

two econometric strategies to circumvent the bias and obtain unbiased estimates: the two-stage least

square instrumental variables (IV-2SLS) approach and the Quasi-maximum likelihood (QML) method.

The estimated results from two-stage least-squares regression using instrumental variables (IV) are

reported in column 3. Following previous research (Anderson and Hsiao 1981, Green 2003, Flannery et

al. 2013), we use the second lag of dependent variable in the level regression as an instrument for the

endogenous variable of interest, 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡, to address the correlation between lagged dependent variable

and the error term. Column 3 reports the results of second-stage regression. As expected, after the

correction of endogeneity, the estimated coefficient on 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 increased from 0.398 to 0.513 while

estimated coefficients on other control variables barely changed. Accordingly, the estimated speed of

adjustment decreased to 0.487 from column 2 to column 3, suggesting firms can close half of the

target’s deviation in about 13 months. Such fast adjustment speed strongly supports the argument that

trade-off theory explains firms’ NWC decisions.

As pointed out by Arellano and Bond (1991), the IV-2SLS estimates performed in column 3 can be

further improved by first-differencing Equation (5) to eliminate fixed effects and use lagged dependent

variables as instruments for the first-differenced dependent variable in the context of generalized

method of moments (GMM). However, this estimation technique is not appropriate for our sample due

to weak instrument problems caused by persistence in the dependent variable (Flannery et al., 2013).

We find that the correlation in the lagged levels of NWC is 0.92 and statistically significant for our

sample. Given such high persistence in our dependent variable series, the first-differences of dependent

variables are close to zero, and instruments are weak. As an alternative, we follow Hsiao et al. (2002)

and use Quasi-maximum likelihood (QML) estimation modeling the unconditional likelihood function to

avoid biases caused by the lagged dependent variable’s correlation with the error term after the first-

difference transformation. As Hsiao et al. (2002) argued, the QML approach performs better than the

GMM approach to reduce estimators’ bias and improve the test statistics’ size and power. Column 4 of

Table 2 reports the results of QML estimation. The estimated speed of adjustment, 0.541, is slightly

higher than the one estimated using the IV-2SLS method but still within the range bounded by estimates

from pooled OLS (FM method) and fixed-effect model. At this rate, firms can close half of the deviation

from the target in about 11 months, which is very close to the one estimated in column 3.

[Table 2]

Deviations from target and speed of adjustment The results from Table 2 partially rely on the assumption that our estimated NWC targets are close to

the true unobservable targets. Although there is no formal statistical framework to test such a

hypothesis, it is reasonable to argue that if our estimated NWC targets are meaningful and appropriate,

we should observe that firms adjust toward these targets over time. Besides, trade-off theory suggests

that marginal cost increase with deviation from target, and firms with NWC holdings that are far away

from targets are more likely to actively adjust their NWC holdings toward target than firms with NWC

holdings around the target. Therefore, under trade-off theory, we should observe heterogeneous

adjustment speed across firms with various deviations from their NWC target. We investigate these

issues in this section.

[Figure 3]

To proceed, we first sort sample firms into quartiles based on their deviations from the estimated

targets. Quantile 4 (highest quantile) only includes firms with actual NWC holdings that are well under

the target. Accordingly, we expect the most substantial upward adjustment in the subsequent year for

these firms. In contrast, quantile 1 (lowest quantile) contains firms with the biggest negative gap

between estimated target and actual NWC holding. For such firms, trade-off theory suggests that

managers should actively adjust current NWC holdings downward in the subsequent year. We find

precisely that in Figure 3. The horizontal axis in Figure 3 measures the deviation quantiles (we report the

mean distance from the target for each quantile). The vertical axis is the subsequent year’s change in

NWC, reflecting managers’ decisions on moving toward the NWC target.

As shown, for firms within the lowest quantile (with mean distance from target equals -16.4%), the

mean(median) decrease of NWC in the subsequent year is 8.2% (3.1%). In contrast, for firms with

current NWC holding substantially below the target (with mean distance from target equals 14.3%), they

raise their NWC by a mean (median) of 6.1% (1.5%) during the subsequent year. For firms in the middle

two quartiles, we also find evidence that is consistent with target convergency. However, these firms

move toward their target NWC with much smaller adjustments.

[Figure 4]

Alternatively, the targeting behavior documented in Figure 3 can be evaluated by examining how firms

with high or low NWC holdings adjust their NWC subsequently. The tendency to move toward the

target should be stronger for those firms with the highest (lowest) levels of NWC holdings, even though

the target NWC may also higher (lower) for these firms. In Figure 4, we plot firms’ subsequent year’s

mean and median changes in NWC against their prior year’s absolute NWC. Similar to Figure 3, we sort

firms into quantiles based on their absolute NWC level, with quantile 1(4) represent the firms with the

least (highest) absolute NWC. As shown, firms with higher NWC holdings tend to reduce their NWC the

following year, while firms in the lower NWC quantile tend to increase their NWC during the subsequent

year, regardless of their position relative to the target. These results are consistent with Figure 3 as firms

with high(low) NWC holdings are more likely to be above(below) their target leverage. More

interestingly, Figures 3 and 4 suggest that firms exhibit asymmetric targeting behavior for an upward

and downward adjustment. In particular, we observe faster downward adjustment when absolute NWC

is high (or current NWC is far above target) than upward adjustment when absolute NWC is low (or

present NWC is far below target). One possible explanation is that downward adjustment costs less than

upward adjustments so that firms find it is easier to adjust NWC holdings downward. Alternatively, it

could be that firms consider above-the-target deviations more costly than below-the-target deviations,

so the managers are relatively more active in bringing NWC down to the target rather than boosting the

WCR when it is too low.

[Figure 5]

To further disentangle the effect of target convergency from the general tendency for firms with

extreme NWC holdings to actively reverse NWC holdings, we combine Figures 3 and 4 using a two-way

sorting method. Similar to Figure 3, we first sort firms into quartiles based on their deviations from the

estimated targets. Within each deviation quantile, we then form four quartiles based on the firm’s

absolute NWC holdings in the prior year and name each group “lowest NWC” (quantile 1), “low NWC”

(quantile 2), “high NWC” (quantile 3), and “highest NWC” (quantile 4), respectively. We plot the

subsequent change in NWC against the deviations from the estimated targets for each of these groups.

So, we have essentially replicated Figure 3 for each set of firms with similar absolute NWC levels. Figure

5.1 combines plotted lines for lowest versus highest NWC groups, while Figure 5.2 exhibits the results

for high and low NWC firms. Consistent with trade-off theory, we observe in both figures that firms with

positive deviation from the target (NWC below target) actively increase their NWC in the subsequent

year while firms with negative deviation from the target (NWC above target) actively adjust NWC

holdings downward, independent of their absolute NWC levels. As expected, the adjustment speed is

higher for firms with extreme NWC levels (“lowest NWC” group and “highest NWC” group).

When comparing firms with excessive NWC holdings in Figure 5.1, it appears that firms with the lowest

absolute NWC exhibit more volatile deviations from the target than firms with the highest absolute

NWC. Accordingly, these firms are also the most active ones to revert their NWC subsequently with

higher adjustment speed when deviations from the target are large. In comparison, at the other

extreme, firms with the most elevated NWC experience relatively less deviation from the target and

exhibit slower adjustment speed. The finding here suggests that, although deviations from target are

costly for firms on both ends of NWC distribution, the costs are higher for those with the lowest NWC.

The comparison between low NWC firms and high NWC firms, as shown in Figure 5.2, indicates that

these firms are very similar in deviations from the target and adjustment speed toward the target.

Although the mean-reverting pattern is still clearly shown, the magnitude is much smaller than those

with extreme NWC levels.

Alternative proxies for target NWC and speed of adjustment Although this paper’s primary purpose is not to compare and propose new regression specifications to

study the determinants of the firm’s optimal NWC holdings, our results in Table 2 rely on previous

literature to estimate target NWC holdings. Therefore, one may question how reliable our results are if

estimated target NWC changes, especially when earlier studies on determinants of optimal NWC impose

different restrictions on the data, and if our main findings in Table 2 are sample sensitive. In other

words, measurement error may bias our previous results. In this section, we perform more tests to see

how stable our empirical results are based on different estimates of target NWC.

[Table 3]

We use three alternative proxies for target NWC and test how the estimated speed of adjustment varies

under each alternative proxy using Equation (2). To make it easier to compare results across different

models, in column 1 of Table 3, we report the fixed effect regression results as in column 2 of Table 2.

The first alternative proxy of target NWC is generated using two-stage estimates similar to those

implemented by Fama and French (2002). Under this method, target leverages are estimated as fitted

values from Equation (3) in stage 1. The adjustment speed is calculated using Equation (2) with an

unobservable target proxied by the estimated target from stage 1. Column 2 reports the estimated

coefficients from the second-stage regression. As shown, the estimated speed of adjustment, 0.602, is

the same as the one estimated using the fixed-effect model. In column 3, we proxy target NWC by the

trailing average of a firm’s actual NWC, calculated as the average NWC for each firm over the past three

years. The estimated coefficient is slightly higher and significant at 1%, suggesting that the adjustment

speed is somewhat lower than those estimated previously. Lastly, we use industry median as a proxy for

target NWC and report the results in column 4, where industries are defined according to the Fama-

French 48 industry portfolios. The estimated speed of adjustment is now 0.575 and significant at 1%.

Overall, the results presented show that our findings in Table 2 are not sensitive to how target NWC is

estimated. The estimated speed of adjustment remains sufficiently high to justify the trade-off

explanation under various target NWC proxies.

To further investigate how robust our findings are in the presence of measurement error, we performed

a Monte Carlo simulation analysis by substituting a noisy proxy for target NWC into Equation (2):

𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1 − 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 = 𝜃𝜃�� 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1∗ + 𝜓𝜓𝑖𝑖,𝑡𝑡� − 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡�+ 𝜀𝜀𝑖𝑖,𝑡𝑡+1 (6)

We also control for firm and year fixed effects. Without further restrictions on coefficients, the model

can be rewritten as

𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1 = 𝜃𝜃1� 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1∗ + 𝜓𝜓𝑖𝑖,𝑡𝑡� + (1 − 𝜃𝜃2)𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 + 𝛼𝛼𝑖𝑖 + 𝛿𝛿𝑡𝑡 + 𝜀𝜀𝑖𝑖,𝑡𝑡+1 (7)

where 𝜓𝜓𝑖𝑖,𝑡𝑡 is the noise variable assumed to be normally distributed with zero mean. 𝛼𝛼𝑖𝑖 is the firm-

specific unobserved effects and 𝛿𝛿𝑡𝑡 is the year fixed effects. 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1∗ is estimated as the fitted value

from Equation (3). The variable of interest, 𝜃𝜃2, is the speed of adjustment. To quantify the effect of

measurement error, we increase the standard deviation of the noise, 𝜓𝜓𝑖𝑖,𝑡𝑡 from 0 percent to 50 percent

and report the results in each column in Table 4. As shown, the coefficients on lagged NWC are

significant across all models. As noise volatility increase from 0 percent to 50 percent, the estimated

speed of adjustment remains stable at around 60%, which is similar to what we find in Table 2.

[Table 4]

In conclusion, our analysis in this section strongly supports our central argument that firms adjust

rapidly toward time-varying target NWC as projected by trade-off theory.

Robustness tests

In this section, we test for the robustness of firm adjustment speed to the (1) boundary issues resulting

from firms with extreme NWC holdings, (2) estimation horizon, (3) firm size, and (4) alternative sample

period.

Speed of adjustment for less extreme firms

The rapid adjustment speed toward the target documented in this paper constitutes our empirical

results’ most notable feature. However, the estimation may be biased upward due to the possibility that

our results are dominated by firms that fall within either end of NWC distributions. These firms cannot

continue to increase or decrease their NWC holdings and therefore have to reverse the course. We

address such possibility by estimating the IV-2SLS and QMLE specifications as in column (3) and (4) of

Table 2 for firms fall within the middle part of observed NWC values each year. Table 5 presents the

results for both subsamples consisting of middle 50% and middle 60% of observed NWC values each

year. As shown, each subsample’s estimated adjustment speed is relatively stable at around 42%,

slightly lower than 48.7% estimated in Table 2 using the entire sample. The slower rate of adjustment

for middle firms is expected and consistent with findings in Figure 4, suggesting the tendency to move

toward the target is less intense for those firms with less extreme NWC holdings. However, because the

estimated speeds of adjustment for these subsamples are very similar to those using the entire sample,

we are confident that the boundary issue is not the cause of our high estimated adjustment speeds.

[Table 5]

Firm size impact

Although previous research (Whited 1992; Petersen and Rajan 1997; Deloof and Jegers 1999) suggest a

positive relationship between firm size and NWC holdings, it is not clear how size might affect

adjustment speed. We expect smaller firms to have higher adjustment speed because they grow fast

and prone to unstable sales, making these firms more like to deviate from their NWC targets. On the

other hand, even if deviations from the target are similar between smaller firms and larger firms, smaller

firms’ volatile cash flows make the deviations more costly. Therefore, small firms are more willing to

adjust NWC toward target than larger firms. Also, given their relatively small stock of NWC holdings,

smaller firms face smaller adjustment costs than larger firms and can quickly respond to deviations by

adjusting. To assess how firm size affects our results, we re-estimate our regressions for size-based

subsamples and report the results in Table 6. Specifically, each year we assign firms into quintiles based

on size. Quintile 1 contains the largest firms, and quintile 5 contains the smallest firms. The results

suggest that targeting behavior is common across firms of various sizes—the speed of adjustment range

between 26% and 60%. The largest firms adjust the least rapidly, suggesting larger firms may bear lower

deviation costs and less active to adjust NWC when they are away from their NWC target.

[Table 6]

Stability over estimation horizon

As Flannery and Rangan (2006) suggested, one way to check how well our partial adjustment model fits

the data is to compare theoretical and empirical estimates of adjustment. The partial adjustment model

assumes that the adjustment speed is constant each year, which implies that the gap closed over the

longer time interval can be calculated directly using one-year adjustment. Therefore, we can check how

close are the estimated adjustment over longer time intervals using partial adjustment models relative

to the ones directly calculated using one-year adjustment, which we call the theoretical adjustment.

Specifically, under the continuous rate of adjustment, we should observe the following relationship

between one-year adjustment and N-year adjustment:

�1 − Theoretical 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑁𝑁_𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦� = (1 − 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑜𝑜𝑜𝑜𝑦𝑦_𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦)𝑁𝑁

(8)

We can estimate the N-year adjustments using the partial adjustment model and compare it with

corresponding theoretical adjustments. A close correspondence between these two suggests that partial

adjustment specifications are appropriate to describe variation in the data. The results are reported in

Table 7. We estimated adjustments between one- and five- years. As shown in column 1, the one-year

adjustment speed is 48.7%. According to Equation (8), the theoretical adjustments for two-, three-,

four-, and five- years are 73.7%, 86.5%, 93.1%, and 96.4%, which are very close to those estimated using

partial adjustment model as shown between column 2 and 5. The result in column 5 also suggests that a

typical firm closes the entire NWC gap in less than five years.

[Table 7]

Alternative sample period

As the market conditions under which firms make their NWC decisions may change over time, the

estimated speed of adjustment may vary dramatically. Given that our extended sample period includes

multiple systematic events such as the tech bubble during the early 2000s and the financial crisis in

2008, it is necessary to check the robustness of our main results in alternative sub-sample periods. In

table 8, we report the estimated results under various periods. Column 1 presents the estimates during

the sample period before 2006, a period before the global financial crisis, to eliminate the effects of

market stress. Results in column 2 are estimated based on subsamples before 1999 to exclude the tech

bubble’s impact. We also divided our sample into two approximately equal periods, using 1995 as the

cutoff year. We estimate the speed of adjustment for each half of the original sample period and report

the results in columns 3 and 4. As shown, the estimated adjustment speeds are quite similar across

periods, range between 54.7% and 58.3%. These estimates are also very similar to 54.1%, the one

obtained from the QML method in Table 2.

[Table 8]

Conclusion

The purpose of this paper is to test how well the trade-off theory can explain the working capital

management decisions. Our analysis is built upon two basic questions inspired by implications of

normative trade-off theory: (1) Is there an NWC target, and if so, (2) what is the adjustment speed at

which a firm moves toward its target. We find strong evidence that firms do have a long-run NWC

target, and they tend to gradually converge to the target from the firm’s initial NWC level within each

period. We estimate that a typical firm’s adjustment speed is around 50% per year, suggesting that a

specific firm close half of a deviation from the target in about a year. Our main findings are not sensitive

to how target NWC is measured and are robust to various robustness checks. Our results show that

trade-off theory can well explain the management decisions on working capital holdings. Our estimated

speed of adjustment also provides direct information on the effect of working capital adjustment cost,

which is a critical first step in testing plausible alternative explanations on the movement of working

capital holdings.

Reference

ATANASOVA, C. 2007. Access to institutional finance and the use of trade credit. Financial Management, 36, 49-67.

BAKER, M. & WURGLER, J. 2002. Market timing and capital structure. The journal of finance, 57, 1-32. BALTAGI, B. H. & PINNOI, N. 1995. Public capital stock and state productivity growth: Further evidence

from an error components model. Empirical Economics, 20, 351-359. BATES, T. W., KAHLE, K. M. & STULZ, R. M. 2009. Why do US firms hold so much more cash than they

used to? The journal of finance, 64, 1985-2021. BERGER, A. N. & UDELL, G. F. 2006. A more complete conceptual framework for SME finance. Journal of

Banking & Finance, 30, 2945-2966. BOUGHEAS, S., MATEUT, S. & MIZEN, P. 2009. Corporate trade credit and inventories: New evidence of a

trade-off from accounts payable and receivable. Journal of Banking & Finance, 33, 300-307. CAMPBELL, J. Y., LETTAU, M., MALKIEL, B. G. & XU, Y. 2001. Have individual stocks become more volatile?

An empirical exploration of idiosyncratic risk. The Journal of Finance, 56, 1-43. CHEN, L. & ZHAO, X. 2006. On the relation between the market-to-book ratio, growth opportunity, and

leverage ratio. Finance Research Letters, 3, 253-266. DARIPA, A. & NILSEN, J. H. 2005. Subsidizing inventory: a theory of trade credit and pre-payment.

Available at SSRN 2342661. DECHOW, P. M. & DICHEV, I. D. 2002. The quality of accruals and earnings: The role of accrual

estimation errors. The accounting review, 77, 35-59. DELOOF, M. & JEGERS, M. 1996. Trade credit, product quality, and intragroup trade: some European

evidence. Financial management, 33-43. ELZINGA, K. G. & MILLS, D. E. 2011. The Lerner index of monopoly power: origins and uses. American

Economic Review, 101, 558-64. EMERY, G. W. 1987. An optimal financial response to variable demand. Journal of Financial and

Quantitative Analysis, 22, 209-225. FERRIS, J. S. 1981. A transactions theory of trade credit use. The Quarterly Journal of Economics, 96,

243-270. FRANK, M. Z. & GOYAL, V. K. 2009. Capital structure decisions: which factors are reliably important?

Financial management, 38, 1-37. GASPAR, J. M. & MASSA, M. 2006. Idiosyncratic volatility and product market competition. The Journal

of Business, 79, 3125-3152. HE, Z. & WINTOKI, M. B. 2016. The cost of innovation: R&D and high cash holdings in US firms. Journal of

Corporate Finance, 41, 280-303. HILL, M. D., KELLY, G. W. & HIGHFIELD, M. J. 2010. Net operating working capital behavior: a first look.

Financial management, 39, 783-805. LEE, G. & MASULIS, R. W. 2009. Seasoned equity offerings: Quality of accounting information and

expected flotation costs. Journal of Financial Economics, 92, 443-469. LERNER, A. P. 1934. Economic theory and socialist economy. The Review of Economic Studies, 2, 51-61. LONG, M. S., MALITZ, I. B. & RAVID, S. A. 1993. Trade credit, quality guarantees, and product

marketability. Financial management, 117-127. LOVE, I., PREVE, L. A. & SARRIA-ALLENDE, V. 2007. Trade credit and bank credit: Evidence from recent

financial crises. Journal of Financial Economics, 83, 453-469. MCNICHOLS, M. F. 2002. Discussion of the quality of accruals and earnings: The role of accrual

estimation errors. The accounting review, 77, 61-69.

MIAN, S. L. & SMITH JR, C. W. 1992. Accounts receivable management policy: theory and evidence. The Journal of Finance, 47, 169-200.

MOLINA, C. A. & PREVE, L. A. 2009. Trade receivables policy of distressed firms and its effect on the costs of financial distress. Financial management, 38, 663-686.

NA, H. J. 2019. Disappearing Working Capital: Implications for Accounting Research. NG, C. K., SMITH, J. K. & SMITH, R. L. 1999. Evidence on the determinants of credit terms used in

interfirm trade. The journal of finance, 54, 1109-1129. PETERSEN, M. A. & RAJAN, R. G. 1995. The effect of credit market competition on lending relationships.

The Quarterly Journal of Economics, 110, 407-443. PETERSEN, M. A. & RAJAN, R. G. 1997. Trade credit: theories and evidence. The review of financial

studies, 10, 661-691. STEWART, M. & MAJLUF, N. S. 1984. Corporate financing and investment decisions when firms have

information that investors do not have. Journal of Financial Economics, 13, 187-221. ZELLNER, A. 1962. An efficient method of estimating seemingly unrelated regressions and tests for

aggregation bias. Journal of the American statistical Association, 57, 348-368.

Figure 1: Mean reverting of NWC holdings

Figure 2 Mean reverting of NWC holdings for surviving firms

Figure 3 Subsequent year’s change in NWC ratio

Figure 4 Mean reversion in NWC ratio

Figure 5.1

Figure 5.2

Table 1 Summary statistics and sample correlations between main explanatory variables.

This table summarizes the variables used in the regression. Panel A provides summary statistics for key variables. Panel B exhibits Pearson correlation coefficients for all variables. Our sample is constructed using all firms, excluding financial (SIC codes 6000-6999), utilities (SIC codes 4900-4999), and ADR firms, recorded in the Compustat database between 1962 and 2019. The final sample has 160,949 firm-year observations, which consist of 14,299 unique firms. Sales growth is measured as a percentage change in sales over the period t-1 to t. gross profit margin is defined as sales net of costs of goods sold, scaled by sales. We measure sales volatility as the five-year rolling standard deviation of a firm’s sales, standardized by net assets, for each firm-year observation. Operating cash flow is earnings before depreciation (OIBDP) net of income taxes, scaled by asset net of cash. Market-to-book ratio (MB) is the sum of the market value of equity and book value of liabilities net of payables, scaled by asset net of cash. Firm size is calculated as the natural logarithm of the inflation-adjusted market value of equity. Following previous literature (Hill et al. 2010; Molina and Preve 2009), we measure market power by market shares, which is calculated as the ratio of sales to annual aggregate sales of the industry where the firm belongs to. Industries are defined according to the Fama-French 48 industry portfolios. Financial distress is constructed the same as in Molina and Preve (2009). Following Petersen and Rajan (1997), we use the natural logarithm of one plus firm’s current age as our age measurement.

Panel A: Descriptive statistics

variable N mean sd p50 min max

NWC 160949 0.193 0.308 0.198 -2.039 1.141

GrowthSale 160949 0.171 0.524 0.0879 -0.747 3.684

GPM 160949 0.198 1.136 0.317 -10.03 0.903

VolSalet 160949 0.373 0.565 0.223 0.0154 4.598

OCF 160949 -0.0518 0.745 0.106 -5.851 0.487

MB 160949 4.979 8.938 2.279 0.591 67

Size 160949 5.202 2.276 5.106 -0.192 10.76

MktPower 160949 0.0101 0.0275 0.0011 0 0.187

Distress 160949 0.0548 0.228 0 0 1

Age 160949 16.46 11.15 13 4 54

Panel B: Correlation coefficients of variables

NWC GrowthSale GPM VolSalet OCF MB Size MktPower Distress

GrowthSale -0.0265***

GPM 0.4195*** -0.0207***

VolSalet -0.2173*** -0.1148*** -0.1117***

OCF 0.4456*** -0.0537*** 0.5263*** -0.3786***

MB -0.3166*** 0.1509*** -0.3190*** 0.2868*** -0.6039***

Size 0.0658*** 0.0450*** 0.0872*** -0.2926*** 0.2131*** 0.0234***

MktPower 0.0306*** -0.0381*** 0.0422*** -0.1081*** 0.0923*** -0.0632*** 0.4555***

Distress -0.0491*** 0.0022 -0.0809*** 0.0818*** -0.1263*** 0.0151*** -0.2135*** -0.0735***

Age 0.0398*** -0.1228*** 0.0623*** -0.1028*** 0.1127*** -0.1619*** 0.3254*** 0.2720*** -0.0711***

Table 2 Estimating NWC speed of adjustment

This table reports the regression results to estimate NWC speed of adjustment using various estimation methods. The primary regression model is 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1 = (1 − 𝜃𝜃)𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 + (𝜃𝜃𝜃𝜃)𝑋𝑋𝑖𝑖,𝑡𝑡 + 𝛼𝛼𝑖𝑖 + 𝛿𝛿𝑡𝑡 + 𝜀𝜀𝑖𝑖,𝑡𝑡+1, where 𝑋𝑋𝑖𝑖,𝑡𝑡 is a vector of firm characteristics known to be determinants of optimal NWC holding from previous literature, 𝛼𝛼𝑖𝑖 is the firm-specific unobserved effects, and 𝛿𝛿𝑡𝑡 is the year fixed effects. Our primary variable of interest, 𝜃𝜃 , measures approximate adjustment speed. Our choice of 𝑋𝑋𝑖𝑖,𝑡𝑡 include sales growth, gross profit margin, sales volatility, operating cash flow, market-to-book ratio, firm size, market power, financial distress, and firm age. Sales growth is measured as a percentage change in sales over the period t-1 to t. gross profit margin is defined as sales net of costs of goods sold, scaled by sales. We measure sales volatility as the five-year rolling standard deviation of a firm’s sales, standardized by net assets, for each firm-year observation. Operating cash flow is earnings before depreciation (OIBDP) net of income taxes, scaled by asset net of cash. Market-to-book ratio (MB) is the sum of the market value of equity and book value of liabilities net of payables, scaled by asset net of cash. Firm size is calculated as the natural logarithm of the inflation-adjusted market value of equity. Following previous literature (Hill et al. 2010; Molina and Preve 2009), we measure market power by market shares, which is calculated as the ratio of sales to annual aggregate sales of the industry where the firm belongs to. Industries are defined according to the Fama-French 48 industry portfolios. Financial distress is constructed the same as in Molina and Preve (2009). Following Petersen and Rajan (1997), we use the natural logarithm of one plus firm’s current age as our age measurement. Column 1 shows results for Fama and MacBeth (1973) estimates. Column 2 reports the panel regression model’s result controlling for both firm- and year- fixed effects. Columns 4 and 5 report estimated results using the two-stage least square instrumental variables (IV-2SLS) and the Quasi-maximum likelihood (QML) method. Adjustment speed is 𝜃𝜃 and calculated one minus estimated coefficient on 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 . Half-life measures the number of years to close half of the deviation from the target. T-values are shown in parentheses. ***, ** and * represents significance at the 1%, 5%, and 10% levels, respectively.

Table 2

(1) (2) (3) (4)

FM FE IV QML NWC 0.749*** 0.398*** 0.513*** 0.459***

(39.66) (155.44) (67.45) (29.82) GrowthSale -0.005** -0.003*** -0.005*** -0.003

(-2.50) (-2.72) (-4.05) (-1.04) GPM 0.013*** 0.006*** 0.000 0.005

(2.69) (9.12) (0.32) (1.53) VolSale -0.018* -0.013*** -0.007*** -0.009***

(-1.80) (-10.30) (-5.48) (-2.90) OCF 0.025*** 0.027*** 0.020*** 0.020***

(2.94) (20.19) (13.73) (3.84) MB 0.000** -0.000 0.000*** 0.000

(2.13) (-1.12) (4.46) (1.49) size -0.001 0.009*** 0.007*** 0.007***

(-0.55) (13.88) (11.10) (7.23) MktPower -0.071*** -0.126** -0.080 -0.079*

(-3.87) (-2.54) (-1.64) (-1.83) Distress -0.021*** -0.015*** -0.018*** -0.014***

(-3.72) (-6.15) (-7.23) (-2.76) Age 0.001 -0.001 -0.001 -0.001***

(1.44) (-0.74) (-0.50) (-12.32) N 160949 160949 145398 129452 R2 0.638 0.193 0.177

Speed 0.251 0.602 0.487 0.541 Half-Life 2.398 0.752 1.038 0.891

Table 3 Alternative proxies for target NWC and speed of adjustment

This table reports regression results to estimate NWC speed of adjustment using alternative proxies for target NWC, 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1∗ . The estimation model is 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1 − 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 = 𝜃𝜃� 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1∗ − 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡� + 𝛼𝛼𝑖𝑖 +𝛿𝛿𝑡𝑡 + 𝜀𝜀𝑖𝑖,𝑡𝑡+1, where 𝛼𝛼𝑖𝑖 is the firm-specific unobserved effects, and 𝛿𝛿𝑡𝑡 is the year fixed effects. Column 1 reports the fixed effect regression results as in column 2 of Table 2. Column 2 reports the results using the proxy for target NWC estimated using the two-stage model as in Fama and French (2002). When evaluating target NWC in the first stage, we use the same determinants as before. These determinants include sales growth, gross profit margin, sales volatility, operating cash flow, market-to-book ratio, firm size, market power, financial distress, and firm age. Sales growth is measured as a percentage change in sales over the period t-1 to t. gross profit margin is defined as sales net of costs of goods sold, scaled by sales. We measure sales volatility as the five-year rolling standard deviation of a firm’s sales, standardized by net assets, for each firm-year observation. Operating cash flow is earnings before depreciation (OIBDP) net of income taxes, scaled by asset net of cash. Market-to-book ratio (MB) is the sum of the market value of equity and book value of liabilities net of payables, scaled by asset net of cash. Firm size is calculated as the natural logarithm of the inflation-adjusted market value of equity. Following previous literature (Hill et al. 2010; Molina and Preve 2009), we measure market power by market shares, which is calculated as the ratio of sales to annual aggregate sales of the industry where the firm belongs to. Industries are defined according to the Fama-French 48 industry portfolios. Financial distress is constructed the same as in Molina and Preve (2009). Following Petersen and Rajan (1997), we use the natural logarithm of one plus firm’s current age as our age measurement. Column 3 reports results using the trailing average of a firm’s actual NWC, calculated as average NWC for each firm over the past three years, as the proxy for target NWC. Column 4 reports results using industry median as a proxy for target NWC, where industries are defined according to the Fama-French 48 industry portfolios. Adjustment speed is 𝜃𝜃 and calculated one minus estimated coefficient on 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 . T-values are shown in parentheses. ***, ** and * represents significance at the 1%, 5%, and 10% levels, respectively.

Table 3

(1) (2) (3) (4)

NWC 0.398*** 0.398*** 0.420*** 0.425***

(155.44) (155.69) (163.42) (176.75) NWC_FF 0.220***

(31.79) L3NWC 0.019***

(6.90) Ind_Median 0.217***

(9.23) N 160949 160949 153514 160949 R2 0.114 0.111 0.100 0.105

speed 0.602 0.602 0.580 0.575

Table 4 Measurement error and speed of adjustment

This table presents the regression results to investigate how measurement error affects the estimated speed of adjustment. The estimation model is 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1 = 𝜃𝜃1� 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1∗ + 𝜓𝜓𝑖𝑖 ,𝑡𝑡� + (1 − 𝜃𝜃2)𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 + 𝛼𝛼𝑖𝑖 + 𝛿𝛿𝑡𝑡 + 𝜀𝜀𝑖𝑖,𝑡𝑡+1 , where 𝛼𝛼𝑖𝑖 is the firm-specific unobserved effects, 𝛿𝛿𝑡𝑡 is the year fixed effects, and 𝜓𝜓𝑖𝑖,𝑡𝑡 is the noise variable assumed to be normally distributed with zero mean. 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1∗ is estimated as the fitted value from equation (3). The variable of interest, 𝜃𝜃2, is the speed of adjustment. We increase the standard deviation of the noise, 𝜓𝜓𝑖𝑖 ,𝑡𝑡 from 0 percent to 50 percent and report the results in each column. Adjustment speed is 𝜃𝜃2 and calculated one minus estimated coefficient on 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 . T-values are shown in parentheses. ***, ** and * represents significance at the 1%, 5%, and 10% levels, respectively.

Table 4

(1) (2) (3) (4) (5) (6)

0% 5% 10% 20% 25% 50% NWC 0.394*** 0.413*** 0.421*** 0.425*** 0.425*** 0.426***

(154.65) (167.69) (173.67) (176.42) (176.73) (177.28) NWCNoise 0.397*** 0.160*** 0.060*** 0.018*** 0.012*** 0.004***

(35.49) (22.01) (13.66) (7.76) (6.29) (3.80) N 160949 160949 160949 160949 160949 160949 R2 0.192 0.187 0.186 0.185 0.185 0.185

Speed 0.606 0.587 0.579 0.575 0.575 0.574

Table 5 Speed of adjustment for less extreme firms

This table reports the results to estimate NWC speed of adjustment for subsamples consist of middle 50% and middle 60% of observed NWC values each year. The estimation model is 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1 = (1 − 𝜃𝜃)𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 + (𝜃𝜃𝜃𝜃)𝑋𝑋𝑖𝑖,𝑡𝑡 +𝛼𝛼𝑖𝑖 + 𝛿𝛿𝑡𝑡 + 𝜀𝜀𝑖𝑖,𝑡𝑡+1, where 𝑋𝑋𝑖𝑖,𝑡𝑡 is a vector of firm characteristics known to be determinants of optimal NWC holding from previous literature, 𝛼𝛼𝑖𝑖 is the firm-specific unobserved effects, and 𝛿𝛿𝑡𝑡 is the year fixed effects. Sales growth is measured as a percentage change in sales over the period t-1 to t. gross profit margin is defined as sales net of costs of goods sold, scaled by sales. We measure sales volatility as the five-year rolling standard deviation of a firm’s sales, standardized by net assets, for each firm-year observation. Operating cash flow is earnings before depreciation (OIBDP) net of income taxes, scaled by asset net of cash. Market-to-book ratio (MB) is the sum of the market value of equity and book value of liabilities net of payables, scaled by asset net of cash. Firm size is calculated as the natural logarithm of the inflation-adjusted market value of equity. Following previous literature (Hill et al. 2010; Molina and Preve 2009), we measure market power by market shares, which is calculated as the ratio of sales to annual aggregate sales of the industry where the firm belongs to. Industries are defined according to the Fama-French 48 industry portfolios. Financial distress is constructed the same as in Molina and Preve (2009). Following Petersen and Rajan (1997), we use the natural logarithm of one plus firm’s current age as our age measurement. Columns 1 and 2 report the estimation results using the IV-2SLS approach. Columns 3 and 4 report the estimation results using the QMLE method. Adjustment speed is 𝜃𝜃 and calculated one minus estimated coefficient on 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 . Half-life measures the number of years to close half of the deviation from the target. T-values are shown in parentheses. ***, ** and * represents significance at the 1%, 5%, and 10% levels, respectively.

Table 5

middle 50 percent middle 60 percent middle 50 percent middle 60 percent NWC 0.581*** 0.574*** 0.574*** 0.574***

(23.51) (28.48) (33.45) (39.99) GrowthSale -0.008*** -0.006*** -0.006** -0.003

(-8.99) (-6.80) (-2.36) (-1.10) GPM 0.002 0.007*** -0.001 -0.015

(1.64) (5.75) (-0.22) (-1.12) VolSale -0.001 -0.001 -0.004 -0.003

(-0.54) (-1.12) (-1.38) (-1.63) OCF 0.015*** 0.016*** 0.001 0.001

(9.93) (11.03) (0.12) (0.16) MB 0.001*** 0.000*** 0.001*** 0.000***

(7.77) (6.37) (3.12) (3) size 0.003*** 0.003*** 0.003*** 0.003***

(8.29) (8.24) (4.16) (5.45) MktPower -0.069** -0.079*** -0.054 -0.041

(-2.51) (-2.98) (-0.98) (-1.09) Distress -0.010*** -0.011*** -0.004 -0.009

(-5.50) (-6.66) (-0.70) (-1.61) Age 0.002 0.001 -0.001*** -0.001***

(1.6) (1.5) (-8.39) (-11.64) N 74406 89152 26955 39967 R2 0.147 0.154

speed 0.419 0.426 0.426 0.427 Half-Life 1.277 1.248 1.247 1.245

Table 6 Firms size and speed of adjustment

This table reports the results to investigate how firm size may affect estimated NWC speed of adjustment. To construct size-based subsamples, each year, we assign firms into quintiles based on size. Quintile 1 contains the largest firms, and quintile 5 contains the smallest firms. The estimation model is 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1 = (1 − 𝜃𝜃)𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 +(𝜃𝜃𝜃𝜃)𝑋𝑋𝑖𝑖,𝑡𝑡 + 𝛼𝛼𝑖𝑖 + 𝛿𝛿𝑡𝑡 + 𝜀𝜀𝑖𝑖,𝑡𝑡+1, where 𝑋𝑋𝑖𝑖,𝑡𝑡 is a vector of firm characteristics known to be determinants of optimal NWC holding from previous literature, 𝛼𝛼𝑖𝑖 is the firm-specific unobserved effects, and 𝛿𝛿𝑡𝑡 is the year fixed effects. Sales growth is measured as a percentage change in sales over the period t-1 to t. gross profit margin is defined as sales net of costs of goods sold, scaled by sales. We measure sales volatility as the five-year rolling standard deviation of a firm’s sales, standardized by net assets, for each firm-year observation. Operating cash flow is earnings before depreciation (OIBDP) net of income taxes, scaled by asset net of cash. Market-to-book ratio (MB) is the sum of the market value of equity and book value of liabilities net of payables, scaled by asset net of cash. Firm size is calculated as the natural logarithm of the inflation-adjusted market value of equity. Following previous literature (Hill et al. 2010; Molina and Preve 2009), we measure market power by market shares, which is calculated as the ratio of sales to annual aggregate sales of the industry where the firm belongs to. Industries are defined according to the Fama-French 48 industry portfolios. Financial distress is constructed the same as in Molina and Preve (2009). Following Petersen and Rajan (1997), we use the natural logarithm of one plus firm’s current age as our age measurement. Adjustment speed is 𝜃𝜃 and calculated one minus estimated coefficient on 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 . Half-life measures the number of years to close half of the deviation from the target. T-values are shown in parentheses. ***, ** and * represents significance at the 1%, 5%, and 10% levels, respectively.

Table 6

(1) (2) (3) (4) (5)

Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile 5 NWC 0.740*** 0.517*** 0.434*** 0.396*** 0.396***

(114.37) (20.33) (11.43) (13.82) (19.36) GrowthSale -0.012*** -0.000 -0.008*** -0.009*** -0.001

(-8.47) (-0.04) (-3.12) (-3.44) (-0.29) GPM -0.015*** -0.001 0.002 0.008*** 0.001

(-6.84) (-0.47) (0.58) (2.99) (0.49) VolSale -0.005* -0.004 -0.003 -0.008** -0.002

(-1.85) (-1.11) (-0.69) (-2.36) (-0.72) OCF 0.035*** -0.001 0.010** 0.011*** 0.028***

(7.40) (-0.33) (2.56) (2.79) (7.07) MB 0.000*** -0.000 0.001* 0.002*** -0.002***

(3.65) (-1.17) (1.81) (4.25) (-3.54) size 0.001 0.006*** 0.007** 0.007* 0.017***

(1.55) (2.65) (1.98) (1.84) (5.26) MktPower -0.014 0.024 0.004 -0.098 -0.032

(-0.62) (0.23) (0.01) (-0.16) (-0.01) Distress -0.007 -0.007 -0.002 -0.013** -0.027***

(-1.54) (-1.29) (-0.24) (-2.00) (-4.28) Age 0.001 0.001 -0.001 0.003 -0.022

(0.78) (0.56) (-0.12) (0.25) (-1.61) N 32122 30144 29242 28380 25510 R2 0.507 0.135 0.066 0.103 0.181

Speed 0.260 0.483 0.566 0.604 0.604 Half_Life 2.298 1.051 0.829 0.748 0.748

Table 7 Estimates over differing forecast horizons

This table reports the results to test how well our partial adjustment model fits the data. We estimate the one-, two-, three- four- and five-year adjustment speed respectively, using the estimation model: 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1 =(1 − 𝜃𝜃)𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 + (𝜃𝜃𝜃𝜃)𝑋𝑋𝑖𝑖,𝑡𝑡 + 𝛼𝛼𝑖𝑖 + 𝛿𝛿𝑡𝑡 + 𝜀𝜀𝑖𝑖,𝑡𝑡+1, where 𝑋𝑋𝑖𝑖,𝑡𝑡 is a vector of firm characteristics known to be determinants of optimal NWC holding from previous literature, 𝛼𝛼𝑖𝑖 is the firm-specific unobserved effects, and 𝛿𝛿𝑡𝑡 is the year fixed effects. Sales growth is measured as a percentage change in sales over the period t-1 to t. gross profit margin is defined as sales net of costs of goods sold, scaled by sales. We measure sales volatility as the five-year rolling standard deviation of a firm’s sales, standardized by net assets, for each firm-year observation. Operating cash flow is earnings before depreciation (OIBDP) net of income taxes, scaled by asset net of cash. Market-to-book ratio (MB) is the sum of the market value of equity and book value of liabilities net of payables, scaled by asset net of cash. Firm size is calculated as the natural logarithm of the inflation-adjusted market value of equity. Following previous literature (Hill et al. 2010; Molina and Preve 2009), we measure market power by market shares, which is calculated as the ratio of sales to annual aggregate sales of the industry where the firm belongs to. Industries are defined according to the Fama-French 48 industry portfolios. Financial distress is constructed the same as in Molina and Preve (2009). Following Petersen and Rajan (1997), we use the natural logarithm of one plus firm’s current age as our age measurement. We compare one-year adjustment speed with theoretical N-year adjustment speed using the equation: �1 − Theoretical 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑁𝑁_𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦� =(1 − 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑜𝑜𝑜𝑜𝑦𝑦_𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦)𝑁𝑁 . Adjustment speed is 𝜃𝜃 and calculated one minus estimated coefficient on 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 . Half-life measures the number of years to close half of the deviation from the target. T-values are shown in parentheses. ***, ** and * represents significance at the 1%, 5%, and 10% levels, respectively.

Table 7

(1) (2) (3) (4) (5)

k1 k2 k3 k4 k5 NWC 0.513*** 0.269*** 0.161*** 0.059*** -0.039***

(67.45) (32.07) (18.14) (6.43) (-4.06) GrowthSale -0.005*** -0.007*** -0.002 -0.002 0.004***

(-4.05) (-5.58) (-1.46) (-1.60) (2.88) GPM 0.000 -0.005*** -0.007*** -0.009*** -0.007***

(0.32) (-4.94) (-6.95) (-8.35) (-6.10) VolSale -0.007*** -0.011*** -0.011*** -0.009*** -0.006***

(-5.48) (-8.10) (-7.22) (-5.95) (-3.50) OCF 0.020*** 0.029*** 0.013*** 0.011*** 0.013***

(13.73) (17.59) (7.13) (5.97) (6.65) MB 0.000*** 0.000*** 0.001*** 0.001*** 0.000**

(4.46) (4.25) (5.47) (7.63) (2.49) size 0.007*** 0.009*** 0.010*** 0.009*** 0.006***

(11.10) (13.78) (13.84) (11.66) (8.23) MktPower -0.080 -0.086* -0.068 -0.038 -0.009

(-1.64) (-1.68) (-1.30) (-0.72) (-0.17) Distress -0.018*** -0.019*** -0.014*** -0.007** -0.010***

(-7.23) (-6.91) (-4.56) (-2.33) (-3.04) Age -0.001 -0.001 -0.001 0.001 0.001

(-0.50) (-0.43) (-0.16) (0.11) (0.12) N 145398 131937 120408 110178 101028

r2_w 0.177 0.058 0.024 0.017 0.011 Speed 0.487 0.731 0.839 0.941 1.039

Theoretical Speed 0.737 0.865 0.931 0.964 Half-Life 1.038 0.528 0.380 0.245 .

Table 8 Alternative sample period and speed of adjustment

This table report results to test how the speed of adjustment change under various periods. The estimation model is 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡+1 = (1 − 𝜃𝜃)𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 + (𝜃𝜃𝜃𝜃)𝑋𝑋𝑖𝑖,𝑡𝑡 + 𝛼𝛼𝑖𝑖 + 𝛿𝛿𝑡𝑡 + 𝜀𝜀𝑖𝑖,𝑡𝑡+1, where 𝑋𝑋𝑖𝑖,𝑡𝑡 is a vector of firm characteristics known to be determinants of optimal NWC holding from previous literature, 𝛼𝛼𝑖𝑖 is the firm-specific unobserved effects, and 𝛿𝛿𝑡𝑡 is the year fixed effects. Sales growth is measured as a percentage change in sales over the period t-1 to t. gross profit margin is defined as sales net of costs of goods sold, scaled by sales. We measure sales volatility as the five-year rolling standard deviation of a firm’s sales, standardized by net assets, for each firm-year observation. Operating cash flow is earnings before depreciation (OIBDP) net of income taxes, scaled by asset net of cash. Market-to-book ratio (MB) is the sum of the market value of equity and book value of liabilities net of payables, scaled by asset net of cash. Firm size is calculated as the natural logarithm of the inflation-adjusted market value of equity. Following previous literature (Hill et al. 2010; Molina and Preve 2009), we measure market power by market shares, which is calculated as the ratio of sales to annual aggregate sales of the industry where the firm belongs to. Industries are defined according to the Fama-French 48 industry portfolios. Financial distress is constructed the same as in Molina and Preve (2009). Following Petersen and Rajan (1997), we use the natural logarithm of one plus firm’s current age as our age measurement. Column 1 presents the estimates during the sample period before 2006. Results in column 2 are estimated based on subsamples before 1999. Using 1995 as the cutoff year, we divide the sample into two approximately equal periods. Columns 4 and 5 report estimation results for the first and second half of the original sample period. Adjustment speed is 𝜃𝜃 and calculated one minus estimated coefficient on 𝑁𝑁𝑁𝑁𝑁𝑁𝑖𝑖,𝑡𝑡 . Half-life measures the number of years to close half of the deviation from the target. T-values are shown in parentheses. ***, ** and * represents significance at the 1%, 5%, and 10% levels, respectively.

Table 8

(1) (2) (3) (4)

before06 before99 before95 after95 NWC 0.422*** 0.435*** 0.453*** 0.417***

(41.90) (39.00) (37.51) (33.09) GrowthSale -0.008*** -0.012*** -0.015*** 0.002

(-6.59) (-9.40) (-10.24) (1.38) GPM 0.004*** 0.011*** 0.012*** 0.003**

(3.25) (8.39) (7.36) (2.40) VolSale -0.011*** -0.007*** 0.003 -0.010***

(-7.41) (-4.41) (1.57) (-5.60) OCF 0.010*** 0.008*** 0.008** 0.023***

(5.22) (2.91) (2.36) (11.56) MB 0.000** 0.000** 0.000 0.000*

(2.16) (2.06) (1.10) (1.87) size 0.008*** 0.009*** 0.011*** 0.006***

(11.22) (12.08) (13.30) (6.13) MktPower -0.076 -0.089 -0.073 -0.118

(-1.32) (-1.44) (-1.14) (-1.14) Distress -0.023*** -0.026*** -0.025*** -0.007*

(-8.81) (-9.51) (-9.06) (-1.77) Age -0.001 -0.001 -0.001 0.001

(-0.63) (-0.66) (-0.57) (0.37) N 113435 87801 72413 76338 R2 0.139 0.127 0.129 0.150

Speed 0.578 0.565 0.547 0.583 Half-Life 0.803 0.833 0.875 0.794