Chapter 16
Evidence From the Corporate Statistics: $\kappa$ and $\mathrm{CCC}$ Across Firms

16.1 Position of this chapter

The $\kappa$ and $\mathrm{CCC}$ defined in Part I are applied to the industry and size tabulations of the Financial Statements Statistics of Corporations by Industry. Two cross-sections are taken.

The cross-section by size examines how the direction of credit corresponds to size class. Section 6.3 writes the credit layer as an image of bargaining power; if so, smaller firms should be the providers of credit.

The cross-section over time examines whether change in $\mathrm{CCC}$ indicates the state of an industry — whether demand is tight. But “the industry is hot” is not one-dimensional, and $\mathrm{CCC}$ touches only one of its axes. That is settled in the next section.

To state the conclusion first, both implications failed: the direction of credit has the opposite sign, and the leading property vanished entirely.

16.2 Hot and cold are not one-dimensional

Before measuring, what is to be judged must be settled. The $g^{\star }=(m+d-I∕r)∕\mathrm{CCC}$ of (5.6) consists of two separate axes, a numerator expressing profitability and a denominator expressing liquidity. Likewise, an industry being hot and entering it being profitable are different things.

Even where demand is tight, if an intermediary controls access to that demand, the surplus does not remain with the entrant. Chapter 17 finds the price of $T_{\mathit{𝑎𝑐𝑞}}$ reaching 20 points. Two axes are therefore needed.

Only the former can be handled here. Of the four quadrants, up and down can be distinguished but not left and right.

16.3 Data source

The time-series data of the Financial Statements Statistics of Corporations by Industry (e-Stat table 0003060791) were used. It is a designated statistical survey under the Statistics Act, and the annual survey covers all profit-making corporations. Capital classes extend down to below two million yen, so the bias towards listed companies raised in Section 12.9 does not arise.

Six items were obtained: notes receivable, accounts receivable, finished goods and merchandise, notes payable, accounts payable and sales. Section 16.10 adds inventory (period end) for a reconciliation. Raw figures rather than ready-made turnover indicators were obtained so that the definitions of Chapter 5 could be applied directly, without depending on whether the statistic’s own denominator is sales or cost of sales.

$$\begin{array}{lll}\hfill \mathrm{DSO}& =\frac{\text{notes receivable}+\text{accounts receivable}}{\text{sales}}\times 365& \hfill \text{(16.1)}\\ \hfill \mathrm{DPO}& =\frac{\text{notes payable}+\text{accounts payable}}{\text{sales}}\times 365& \hfill \text{(16.2)}\\ \hfill \mathrm{CCC}& =\mathrm{DSO}+\mathrm{DIO}-\mathrm{DPO}& \hfill \text{(16.3)}\end{array}$$

These express the credit positions of (3.2) in days. That is,

$${\kappa }_C\leftrightarrow \mathrm{DSO},{\kappa }_S\leftrightarrow -\mathrm{DPO},\sum _i{\kappa }_i\leftrightarrow \mathrm{DSO}-\mathrm{DPO}$$

(16.4)

A firm with ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_i{\kappa }_i>0$ is a provider of credit. The working capital of (5.1) is $W=(\text{DSO}+\text{DIO}-\text{DPO})\times r∕365$.

16.4 Size and the direction of credit

16.4.1 The implication tested

Registered implication: the smaller the size class, the larger $\mathrm{DSO}-\mathrm{DPO}$; that is, smaller firms are the providers of credit.

The ground was the claim of Section 6.3 that the credit layer is an image of bargaining power: small firms without bargaining power should be the ones kept waiting for payment.

16.4.2 Result

The implication is rejected; the sign is the opposite. Firms of 1bn yen and over sit at 27 days and those under 10m at 14, so large firms are the providers of credit. The ordering is consistent over the ten years 2015–2024, and the large-firm figure has in fact widened from 19 to 27 days.

16.4.3 The error in the reasoning

The error can be located. It was overlooked that DSO and DPO both move.

Small firms are kept waiting on receivables (DSO works against them). But at the same time they are not trusted by suppliers either and cannot stretch payables. As Figure 16.1 shows, the DPO of the under-10m class is 21 days, less than half the 44 of firms of 1bn and over.

A lack of bargaining power acts on both sides: collection is slow but payment is fast. On balance, weak bargaining power does not appear in the net credit position.

Under the $E\approx 0$ of Chapter 8, the requirement was the reverse, $\kappa <0$ (Corollary 5.4). Tolerating collection as long as a DSO of 71 days is possible because there is capital behind it.

16.4.4 The sign splits by industry

Only retail reverses sign. The $-16$ days of large retailers is exactly the negative credit position described in Part I: cash collected immediately and purchases paid in arrears. Here large firms are the recipients of credit and small retailers the providers.

The $+49$ days of information and communications and construction shows large firms bearing long advances in the position of contractor — the territory of families 1-3 (one-off arrears) and 5-3 (cost plus) of Part II.

The claim that “larger firms are the providers of credit” is therefore itself only an average across industries. What decides is not size but the $\mathrm{\Phi }$ dominant in that industry. The lesson recorded in Part IV — that levels of $\kappa$ differ structurally across industries so cross-sectional comparison is meaningless — holds on the size axis too.

16.4.5 Implication for Chapter 8

Chapter 8 deduced that a solo business without assets requires ${\kappa }_C<0$. The data show that requirement is not met: the under-10m class has DSO of 35 days and DPO of 21, a net 14 days as a provider.

The form differs from what was envisaged at registration, however. The small classes are not extremely $\kappa >0$; rather both DSO and DPO are short. The transactions themselves are small and short. Rather than lying outside the feasible region, they appear to sit at different coordinates.

16.5 Is $\mathrm{CCC}$ a leading indicator of the cycle?

16.5.1 The implication tested

Registered implication: the rate of change of $\mathrm{CCC}$ leads or is synchronous with the rate of change of sales. In particular, a worsening receivables turnover leads a slowdown in sales.

Correlations between the annual change $\mathrm{\Delta }\mathrm{CCC}(t)$ and the sales growth rate $g(t+k)$ were computed by industry and aggregated over 45 industries, excluding aggregate and short series.

16.5.2 Result

Synchrony is supported. A worsening $\mathrm{CCC}$ and a slowdown in sales occur in the same fiscal year: 31 of 45 industries show a negative correlation, with $p=0.008$ on a sign test.

Leading is rejected outright. 47% negative is no information at all ($p=0.72$). Change in $\mathrm{CCC}$ carries nothing about the following year’s sales.

16.5.3 The synchronous negative correlation may be a denominator effect

The mechanism is likely the reverse of what was assumed. $\mathrm{CCC}=W∕r$ has sales in the denominator. If sales fall, then unless inventory and receivables fall at the same rate, $\mathrm{CCC}$ worsens by definition.

When the panel by medium for family 2 was treated in Part IV, this was stated explicitly as the alternative: issuance shrinks and the ratio rises mechanically. Section 16.9 separates it here and shows that the denominator effect contributes about 17%.

16.5.4 That the sign splits by industry is a finding

Thirteen of the 45 industries have $|r|>0.4$ in the synchronous correlation.

The strongly negative industries are all make-to-order with large inventories and receivables, matching the conditions under which the denominator effect is strong.

What matters is that positive industries exist. A pure denominator effect would make every industry negative. The positive industries have a structure in which $\mathrm{CCC}$ grows as sales grow — they build working capital in order to grow — which is the region where the $g^{\star }$ of (5.6) binds.

Change in $\mathrm{CCC}$ therefore contains components other than the denominator effect. They cannot be separated in these data.

16.6 An observation not anticipated: the long-run rise in $\mathrm{CCC}$

Separately from the tests, a fact not envisaged in advance was observed.

As Figure 16.2 shows, the $\mathrm{CCC}$ of all industries rose from 24.9 days in FY2000 to 37.2 in FY2024, almost monotonically over 25 years. The net credit position also doubled, from 9.8 to 18.5 days. The Japanese corporate sector as a whole is moving towards holding more working capital.

16.6.1 The growth constraint has not worsened, however

It is tempting to conclude from this that the growth constraint has worsened, since the denominator of (5.6) has risen by half. That inference is wrong.

Over the same period the operating margin $m$ improved by 91%, from 2.62% to 5.01%, outrunning the worsening denominator. The ratio $m∕\mathrm{CCC}$ consequently rose from 38.5% to 49.2% (Proposition 14.2).

What is used here is $m∕\mathrm{CCC}$, not the $g^{\star }=(m+d-I∕r)∕\mathrm{CCC}$ of Corollary 5.7 (Remark 14.3), because $d$ and $I$ were not obtained far enough back.

The cause of the rise in $\mathrm{CCC}$ itself remains untested. Which of industrial composition, regulation of payment terms, the decline of trade bills, or the level of interest rates contributes is not separated here. The importance of the question is limited, however, since $g^{\star }$ has not worsened.

16.7 Levels by industry (for reference)

As the lesson of Part IV has it, cross-sectional comparison of levels cannot be interpreted; the figures are recorded as a reference on the distribution of $\mathrm{\Phi }$.

The ordering agrees with the types of Part II. Leasing (family 3-1) and real estate acquire an asset first and recover it over time; food services (family 1-1, immediate exchange) sits at $\kappa \approx 0$. The structure of the differences by industry described earlier appears directly.

16.8 Decomposing the rise in $\mathrm{CCC}$

The rise in $\mathrm{CCC}$ is decomposed into its causes. The following implication was fixed in advance.

Registered implication: the rise in $\mathrm{CCC}$ owes more to the composition effect than to within-industry change, because industries with long $\mathrm{CCC}$ have gained weight.

Alternative: if the within-industry effect dominates, working capital has increased in each industry irrespective of composition.

16.8.1 Shift-share decomposition

By Corollary 5.19, the overall $\mathrm{CCC}$ is a sales-weighted mean of industry figures, $\mathrm{CCC}={\sum <!--\; FUNCTION\; APPLICATION\; -->}_k{w}_k{\mathrm{CCC}}_k$. Its difference splits in two.

$$\mathrm{\Delta }\mathrm{CCC}=\underset{\text{within-industry}}{\underbrace{\sum _k{\overline{w}}_k\mathrm{\Delta }{\mathrm{CCC}}_k}}+\underset{\text{composition}}{\underbrace{\sum _k\mathrm{\Delta }{w}_k{\overline{\mathrm{CCC}}}_k}}$$

(16.5)

Taking $\overline{\cdot }$ to be the midpoint of the two periods, (16.5) holds exactly with no residual. Substituting $\overline{w}=(w_1+w_0)∕2$ and $\overline{\mathrm{CCC}}=({\mathrm{CCC}}_1+{\mathrm{CCC}}_0)∕2$, the right-hand side equals ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_k(w_{k1}{\mathrm{CCC}}_{k1}-w_{k0}{\mathrm{CCC}}_{k0})$. This is not an approximation introduced for measurement but a consequence of $\mathrm{CCC}$ being a ratio (Corollary 5.19).

The implication is rejected. The within-industry effect accounts for 83% to 98%. Since 2010, where the classification is stable, the composition effect is nearly zero. An explanation by the shift towards services does not hold.

The reason the composition effect is small is also clear. Information and communications rose in weight from 0.2% to 5.8%, pushing $\mathrm{CCC}$ up by $+3.05$ days, while electrical machinery shrank from 6.3% to 2.1% ($-1.62$ days) and advertising from 5.6% to 0.8% ($-1.59$ days): the pushes up and down nearly cancel.

16.8.2 Components of the within-industry effect

The within-industry effect splits into $\mathrm{DSO}$, $\mathrm{DIO}$ and $\mathrm{DPO}$.

Against Corollary 5.20, DSO and DPO correspond to ${\tau }_C$ and ${\tau }_S$ and DIO to ${\tau }_{\mathrm{inv}}$. Hence:

In both periods, change in contracts accounts for about 60%. The rise in $\mathrm{CCC}$ originates in changes to contractual terms rather than in demand or inventory management.

The main cause differs by period, and the path of trade bills explains it.

Bills fell to below a third during the 2000s. Notes payable lost 8.1 days, from 15.3 to 7.2, and the 6.9-day shortening of DPO is almost entirely explained by this. Bills are a device for creating ${\kappa }_S<0$, and their disappearance pushed $\mathrm{CCC}$ up.

In the 2010s the decline of bills had run its course and DSO lengthened by 5.26 days. The breakdown is notes receivable $-1.37$ days and accounts receivable $+6.62$ days. Although bills keep falling, accounts receivable grow by more than enough to offset them. By industry, $\mathrm{DSO}$ rose in 28 of 45 industries, with the largest contributions from wholesale (59$\to$70 days), retail (22$\to$28 days), information and communications (62$\to$78 days) and construction (65$\to$71 days).

16.9 Separating the contemporaneous negative correlation: the contribution of the denominator effect

For the contemporaneous negative correlation observed in Section 16.5, we separate information about demand from the denominator effect inherent in the definition $\mathrm{CCC}=W∕r$.

In log differences, $\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->\mathrm{CCC}=\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->W-\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->r$. Under a pure denominator effect $W$ would not move. Estimating the elasticity

$$\beta =\frac{\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->W}{\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->r}$$

(16.6)

therefore makes $1-\beta$ the contribution of the denominator effect.

The theoretical value of $\beta$ is $1$. By (5.1), $W=\mathrm{CCC}\cdot r$, so if $\mathrm{CCC}$ is constant then $\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->W=\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->r$. Hence

$$\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->\mathrm{CCC}=(\beta -1)\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->r$$

(16.7)

and the departure of $\beta$ from $1$ is exactly the sales elasticity of $\mathrm{CCC}$. Equation (5.10) of Corollary 5.20 decomposes this movement into change in contracts and change in operations.

Registered implication: $\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->W$ and $\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->r$ are positively correlated; that is, working capital falls when sales fall.

16.9.1 Result

The implication is supported. Working capital moves almost proportionally with sales, and only about 17% is attributable to the denominator effect. The contemporaneous negative correlation of Section 16.5 carries, for the most part, information from the demand side.

Substituting $\beta =0.830$ into (16.7) gives a sales elasticity of $\mathrm{CCC}$ of $-0.170$: $\mathrm{CCC}$ rises in phases where sales fall. The amplification factor of Remark 5.11 is then larger, so the degree to which a downturn in demand is transmitted to cash strengthens in recessions. Calling $1-\beta$ the “denominator effect” is a computational label, not a claim that $\mathrm{CCC}$ has not moved.

16.9.2 Lag structure

By Proposition 3.12, $W(t)$ corresponds to the integral of $r$ over the past $\tau$ periods. In industries with long $\mathrm{CCC}$ it should also depend on the previous year’s sales. The following was registered in advance.

Registered implication (1): the coefficient ${\beta }_1$ on $\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->r(t-1)$ is larger the longer that industry’s $\mathrm{CCC}$.

Registered implication (2): the total elasticity ${\beta }_0+{\beta }_1$ including the lag exceeds the contemporaneous-only estimate of $0.830$, so the contribution of the denominator effect is smaller than 17%.

The distributed-lag model $\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->W(t)={\beta }_0\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->r(t)+{\beta }_1\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->r(t-1)+𝜀$ was estimated on 886 observations across 43 industries.

Both implications are rejected. ${\beta }_1$ is nearly zero, and its correlation with $\mathrm{CCC}$ is $-0.101$ — the sign is also reversed. In industries with $\mathrm{CCC}<30$ days, ${\beta }_1=+0.217$; in those with $\mathrm{CCC}\ge 60$ days, $-0.267$. The total elasticity falls slightly from $0.830$ to $0.821$, leaving the contribution of the denominator effect essentially unchanged at 17% to 18%.

16.10 The effect of the scope of inventories on the conclusions

The $\mathrm{DIO}$ of this chapter uses only “finished goods or merchandise” and excludes work in process and raw materials. Since the Financial Statements Statistics carries “inventories (end of period)” as a separate item, the difference between the two measures the size of the understatement. This section measures that size.

This section is not a test of an implication. It substitutes a different measurement definition on the same data and checks how much the conclusions move.

16.10.1 Two cross-sections

“Inventories (end of period)” was obtained along two cuts. The two identify different variation, and must be read separately.

The estimate of this chapter (Section 16.9) uses 946 observations across 45 industries, the same design as the industry cut. The figures that correct this chapter are therefore those of the industry cut; the size cut answers a different question. Unless stated otherwise, the figures below are from the industry cut.

16.10.2 The level is understated roughly twofold

The ratio of $\mathrm{DIO}$ has a median of 1.99, and the median $\mathrm{CCC}$ rises from 28.0 to 43.9 days. For all industries (excluding finance and insurance), FY2000 goes from 24.9 to 38.2 days and FY2024 from 37.2 to 53.5 days.

The size cut gives a median ratio of 1.89 and a difference with median 4.1 days and mean 8.4 days — comparable in magnitude.

The difference is of a different order across industries. $\mathrm{CCC}$ rises by more than 30% in manufacturing and construction, while retail and employment placement barely move. Divergence arises only in industries that hold work in process — an obvious structure, but large enough to change the ordering of the industry levels in Section 16.7.

16.10.3 The effect on differences is small, but not negligible

Comparing the two definitions for $\mathrm{\Delta }\mathrm{CCC}$, the industry cut gives a correlation of 0.882 over 1,302 observations, with signs agreeing 86.8% of the time. The size cut (990 observations) gives a correlation of 0.976 and sign agreement of 91.0%: agreement is lower across industries. Because the weight of work in process differs by industry, including cross-industry variation produces a gap.

16.10.4 Correcting the definition worsens the elasticity slightly

Equation (16.6) is estimated on a common sample. Since the observations with $W>0$ differ by definition, the comparison is restricted to observations with $W>0$ under both definitions.

In both cuts, taking inventories at their total more than doubles the coefficient of determination. Working capital is more nearly proportional to sales when work in process and raw materials are included.

But the direction in which the denominator effect moves is opposite across the two cuts. In the size cut it falls from 0.100 to 0.040, while in the industry cut it rises slightly from 0.160 to 0.172.

16.10.5 Verdict

The conclusion of Section 16.9 is maintained, but it is merely maintained, not strengthened.

All three are supported, but none strongly. That $1-\beta$ is 0.172 means 17% of the increase in working capital is not proportional to sales: not dominant, but not negligible either.

16.11 An observation the simplified capacity constraint does not fit

Remark 2.23 introduced the simplification of treating $\mathrm{Cap}$ as a constant. This section records a case in which it does not fit reality. It is a record of a limitation, not a revision of the theory.

16.11.1 The observation

The price schedule of one major full-service fitness club chain was checked as of 2026. The company offers a weekday-daytime-only plan for those aged 60 and over alongside its ordinary plan. The two have identical structures — once a week (up to four visits a month), twice a week (up to eight), and unlimited — and differ only in price.

In all four categories the discount increases monotonically with usage. It stays around 4% for once a week but reaches about 22% on the unlimited plan. The price ratio from once a week to unlimited is 1.67–1.74 on the ordinary plan and 1.36–1.43 on the 60-and-over plan.

16.11.2 Inconsistency with the simplification

Under equation (2.17), a large discount on the unlimited plan is hard to explain, since it presses on the capacity constraint. Under the general form (2.16) , however, it is explicable: restricting the plan to weekday daytime places $\mathrm{supp}<!--\; FUNCTION\; APPLICATION\; -->{\delta }_i$ within the idle part of $\mathrm{Cap}(t)$, so total volume can be increased without worsening the constraint at the peak instant.

That is, the operator manipulates the distribution over time, not the total. The simplification of this work cannot express this device.

16.11.3 Why the theory is not revised from this observation

16.11.4 The simplified capacity constraint

In connection with Section 8.6, the following implication was fixed in advance.

Registered implication: older cohorts choose pay-per-use over flat-rate contracts. This follows from the preference predicted by socioemotional selectivity theory — that under a limited time horizon one chooses the certain. A flat-rate contract is a bundle of options; pay-per-use is certain.

Alternative: older cohorts have more disposable time and use the facility more often, so the flat rate is more advantageous. Under this account the preference for flat rates rises.

The signs are opposite, so the form is identifiable.

However, no data on consumer choice could be obtained, and the implication is untested. What is needed is the distribution of contract-form choice by age, which operators hold but do not publish.

Only the supply side’s price design was obtained. The discount structure above rests on the premise that older cohorts use the facility more, which is consistent with the alternative and points against the registered implication. But this is the operator’s belief, not the consumer’s choice. It requires the premise that the firm reads demand correctly, and so remains indirect evidence.

16.12 Limitations of this chapter

(1)

The separation of the denominator effect is of low confidence. The elasticity estimate puts the contribution at about 17%, but $R^2=0.076$, and most of the variation in $\mathrm{\Delta }\ln <!--\; FUNCTION\; APPLICATION\; -->W$ is unexplained.

(2)

The horizontal axis of the four quadrants is not measured. No cross-industry indicator exists for the ease of capturing surplus, so the classification remains one-dimensional, on the demand side alone.

(3)

The interpretation of the growth in accounts receivable is not settled. Section 16.8 excludes industrial composition and explains half by the disappearance of bills, but whether the growth of accounts receivable in the 2010s is a genuine lengthening or a shift between accounts cannot be separated (Remark 16.5).

(4)

The contract choice of older cohorts is untested. As in Section 16.11, no consumer-side choice data could be obtained. The supply side’s price design is consistent with the alternative but remains indirect evidence.

(5)

Inventories are “finished goods or merchandise” only. Work in process and raw materials are excluded, so $\mathrm{DIO}$ is understated and the level of $\mathrm{CCC}$ comes out shorter than it is. Section 16.10 measures the difference for ten industries and confirms that the effect on analyses of differences and correlations is small, but correcting the 45-industry estimate would require obtaining that item for all industries.

16.13 On both implications being rejected

The direction of credit had the opposite sign; the leading property vanished. Both registered implications failed.

Part IV recorded the effect of pre-registration, and it operated more clearly in this chapter. Had the implications not been fixed beforehand, the contemporaneous negative correlation would very likely have been read as “$\mathrm{CCC}$ works as a business-cycle indicator” and the credit position of large firms as “large firms support their counterparties”. Both are retrospective narratives.

The failure of the implications is not a rejection of the framework. The descriptions $\mathrm{CCC}$, $\kappa$ and $\mathrm{\Phi }$ function effectively; what failed were conjectures about what these quantities predict. Effectiveness as a tool of description and effectiveness as a tool of prediction are different things.