3 The Credit Position: κ and the Lag τ

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Chapter 3
The Credit Position: $κ$ and the Lag $τ$

3.1 The gap between delivery and settlement

Write the cumulants of the delivery $δ_{i}$ of Section 2.2 and of the settlement $π_{i}$ fixed by $Φ$ in (2.8) (with $π = Φ (δ)$ ) as

D_{i} (t) = \int_{0}^{t} v (δ_{i} (s)) 𝑑𝑠, P_{i} (t) = \int_{0}^{t} π_{i} (s) 𝑑𝑠

(3.1)

$D_{i}$ is value given and $P_{i}$ is consideration received.

Definition 3.1 (Credit position). The credit position towards party $i$ is defined as

κ_{i} (t) = D_{i} (t) - P_{i} (t)

(3.2)

By (3.1),

κ_{i} (t) = \int_{0}^{t} v (δ_{i} (s)) 𝑑𝑠 - \int_{0}^{t} π_{i} (s) 𝑑𝑠

so $κ_{i}$ is a function of the $Φ$ of (2.8): replacing $Φ$ for the same delivery $δ$ changes $κ_{i}$ . A positive $κ_{i}$ means the firm is extending credit to $i$ (delivered, not yet collected); a negative $κ_{i}$ means $i$ is extending credit to the firm (received, not yet delivered).

The sum of credit positions plus inventory is used repeatedly below.

Definition 3.2 (Working capital). With inventory $Inv (t) \geq 0$ , define

W (t) = \sum_{i \in N} κ_{i} (t) + Inv (t)

(3.3)

Since $κ_{i}$ is a function of $Φ$ , so is $W$ .

The $W$ of (3.3) is the amount of money tied up in the business; dividing it by the rate of sales gives $CCC$ in Chapter 5.

The merit of (3.2) is that it unifies.

Account	Party	Sign	Meaning
Receivables	$C$	$κ_{C} > 0$	the firm extends credit to the customer
Advances received, contract liabilities	$C$	$κ_{C} < 0$	the customer extends credit to the firm
Payables	$S$	$κ_{S} < 0$	the supplier extends credit to the firm
Advances paid	$S$	$κ_{S} > 0$	the firm extends credit to the supplier

Table 3.1: Four accounts unified by (3.2).

These are not four separate accounts but one quantity with different signs and indices.

3.1.1 Existing theory for each term

Each term of $κ_{i}$ has an established theory behind it. This text takes them as its foundation.

Party and sign	Phenomenon	Explanatory theory
$κ_{C} > 0$ (inter-firm)	selling on account	four motives for trade credit: financing advantage, price discrimination, quality assurance, liquidation advantage [14, 15]
$κ_{C} < 0$ (inter-firm)	advance payment	reverse trade credit: supplier finance, transaction assurance, bargaining power [12, 7]
$κ_{C} < 0$ (to consumers)	prepaid contracts	mental accounting, evidence on breakage [13]
$κ_{S} < 0$	payables	the mirror image of trade credit
$κ_{L} < 0$	wages in arrears	the bonding argument: discipline through deferred pay [1]
$κ_{K} < 0$	borrowing	standard corporate finance
$κ_{K}^{i}$	equity (residual claim)	party-dependent; Definition 3.21

Table 3.2: Existing theory explaining each term of

κ_{i}

Remark 3.3 (The four literatures do not cite one another). The theories in Table 3.2 developed in different fields to answer different questions.

Theory	Field	Central question
Trade credit	corporate finance	why a supplier can lend more cheaply than a bank
Bonding argument	labour economics	why involuntary unemployment exists
Signalling	economics of information	how ability is disclosed
Work on prepaid contracts	consumer behaviour	why consumers pay first

Table 3.3: The fields and central questions of the theories explaining each term of

κ_{i}

Because the questions differ, these literatures do not cite one another. Yet in (3.8) the customer’s prepayment, the supplier’s trade credit, the worker’s deferred wage, and the investor’s equity all appear as terms of the same sum.

Each theory explains one term of the sum; none has a framework that treats the sum. The contribution here is that integration, not the explanation of any individual term.

The relation between the cumulants is shown in Figure 3.1.

Figure 3.1: The credit position measured against cumulative delivery

D (t)

. Where the settlement curve

P (t)

lies above

D (t)

κ < 0

; where it lies below,

κ > 0

3.1.2 Both sides of a transaction: $κ$ is antisymmetric

Equation (3.2) is written on the firm’s own books. Writing the same transaction on the counterparty’s books exchanges value given for consideration received.

Proposition 3.4 (Antisymmetry). If two parties $i$ and $j$ share the valuation map $v$ , and $κ_{j}^{(i)}$ denotes the credit position towards $j$ on the books of $i$ , then

κ_{j}^{(i)} = - κ_{i}^{(j)}

(3.4)

Proof. Value given by $i$ to $j$ is value received by $j$ from $i$ , and consideration received by $i$ from $j$ is consideration given by $j$ to $i$ . Hence $D_{j}^{(i)} = P_{i}^{(j)}$ and $P_{j}^{(i)} = D_{i}^{(j)}$ ; substituting into (3.2) gives the result. □

Remark 3.5 (A shared $v$ is required). Equation (3.4) presumes both parties use the same $v$ . As Remark 3.23 notes, $v$ may be subjective; but the settlement amount $π$ is $𝒢$ -measurable and agreed, so any disagreement is confined to the $D$ side. Differences in acceptance criteria and disputed receivables are the cases in point.

Antisymmetry has a consequence for aggregation.

Corollary 3.6 (Inter-firm credit vanishes on aggregation). Let $𝒩$ be the set of parties in the corporate sector. The sum of credit positions closed within $𝒩$ is zero, so

\sum_{i \in 𝒩} W^{(i)} = κ_{vs. non-corporate} + \sum_{i \in 𝒩} {Inv}^{(i)}

(3.5)

where $κ_{vs. non-corporate}$ is the net position of the corporate sector against households, government and the rest of the world.

Proof. For each pair $(i, j)$ inside $𝒩$ , (3.4) gives $κ_{j}^{(i)} + κ_{i}^{(j)} = 0$ . Summing the $W$ of Definition 3.2 over all parties in $𝒩$ cancels the internal pairs and leaves the pairs with the outside together with inventory. □

Corollary 3.7 (Not every stratum can be a net provider of credit). Partition the corporate sector into disjoint strata. It is impossible for every stratum to be a net provider of inter-firm credit. If one stratum is a net provider, another is a net recipient.

Proof. By Corollary 3.6 the sum of inter-firm positions across strata is zero. If the net position of every stratum were strictly positive, the sum would be strictly positive, contradicting zero. □

Corollary 3.7 places a structural constraint on comparisons across size strata. When the direction of credit is measured by size in Part IV, part of the observed sign pattern follows automatically from this constraint.

Remark 3.8 (What the aggregate $CCC$ measures). Dividing (3.5) by total sales gives the aggregate $CCC$ . Since inter-firm positions have dropped out of the numerator, the aggregate $CCC$ is not an indicator of inter-firm credit. What remains is the position against the non-corporate sector, plus inventory. Chapter 16 reports the $CCC$ of all industries, and interpreting its level requires this distinction.

3.1.3 Additivity over parallel $Φ$

One party may run several $Φ$ in parallel.

Proposition 3.9 (Additivity of $κ$ ). If $Φ^{1}, \dots, Φ^{K}$ are run in parallel on disjoint deliveries $δ^{1}, \dots, δ^{K}$ , then

κ = \sum_{k = 1}^{K} κ^{k}, W = \sum_{k = 1}^{K} W^{k}

(3.6)

Proof. $D_{i}$ and $P_{i}$ in (3.1) are integrals and therefore additive in the integrand. Equation (3.2) is their difference and preserves additivity. □

What is additive is $κ$ and $W$ , not $CCC$ . Since $CCC = W ∕ r$ is a ratio, and ratios are not additive (Corollary 5.19).

Remark 3.10 (The decomposition is not unique). Equation (3.6) asserts that a decomposition exists, not that it is unique. Several families ${Φ^{k}}$ give the same $κ$ . The types of Chapter 7 are a generating set for $Φ$ , not a basis.

3.1.4 Expressing the position as a lag

$κ_{i}$ is a difference of cumulants, and what it contains is the time gap between delivery and settlement. Make this explicit.

Definition 3.11 (Settlement lag). For party $i$ , write $τ_{i}$ for the average lag from delivery to settlement. Its sign follows that of $κ_{i}$ : settlement later than delivery is positive.

Sign	State	Examples
$τ_{i} > 0$	settlement later than delivery (the firm extends credit)	receivables, selling on account
$τ_{i} < 0$	settlement earlier than delivery ( $i$ extends credit)	advances received, payables

Table 3.4: The sign of the settlement lag

τ_{i}

and the corresponding state.

Express the $κ_{i}$ of (3.2) through the lag $τ_{i}$ .

Proposition 3.12 (Decomposition of the credit position). In a steady state with a constant rate of delivery, as a time average

{\bar{κ}}_{i} = r τ_{i}

(3.7)

Proof. If settlement lags delivery uniformly by $τ_{i}$ , the unsettled cumulant at time $t$ is the delivery of the most recent $τ_{i}$ of time:

κ_{i} (t) = \int_{t - τ_{i}}^{t} r (s) 𝑑𝑠 .

If $r$ is constant the right-hand side equals $r τ_{i}$ . If the lag has a distribution, take $τ_{i}$ to be its mean. □

Remark 3.13 (It holds only on average). Actual settlement is discrete. Under monthly billing paid at the end of the following month, $π$ is a pulse once a month and $κ_{i} (t)$ moves in a sawtooth. Equation (3.7) holds only for its time average.

In statistics based on period-end balances, if fiscal year-ends cluster in particular months the phases of the sawtooth line up and the figures can deviate systematically from the mean. No correction for this bias is made here.

Example 3.14 ( $κ$ of an annually billed subscription). A service priced at 1,000 yen a month is paid annually, 12,000 yen received on 1 January. Delivery is spread evenly over twelve months, so after $t$ months

κ_{C} (t) = 1, 000 t - 12, 000 .

This is $- 12, 000$ at $t = 0$ , $- 6, 000$ at $t = 6$ , and $0$ at $t = 12$ . With 100 customers, the mid-period average is $κ_{C} = - 600, 000$ yen. That amount is interest-free funding.

3.2 Decomposing the cash balance

Use the $κ_{i}$ of (3.2) to decompose the cash balance.

Proposition 3.15 (Credit decomposition of cash). With equity $E (t)$ , inventory $Inv (t)$ and fixed assets $A (t)$ ,

M (t) = E (t) + \sum_{i \in N} \max {- κ_{i} (t), 0} - \sum_{i \in N} \max {κ_{i} (t), 0} - Inv (t) - A (t)

(3.8)

Proof. By the balance-sheet identity, total assets equal total liabilities plus net assets. Split assets into cash $M$ , positive credit positions (receivables, advances paid), inventory $Inv$ and fixed assets $A$ ; map liabilities to negative credit positions (payables, advances received) and net assets to $E$ . Then

M + \sum_{i} \max {κ_{i}, 0} + Inv + A = \sum_{i} \max {- κ_{i}, 0} + E .

Solving for $M$ gives (3.8). □

Equation (3.8) is only a rearrangement of an identity, but it fixes how to read it. The second term is the total credit drawn from others and the third the total credit extended to others.

Corollary 3.16 (What cash is). A firm’s cash is the total credit drawn from all counterparties, less the credit extended and the assets fixed in place.

3.2.1 The net form

The second and third terms of (3.8) are written separately in order to show the gross amounts. Looking only at the net, for any real $x$ we have $\max {- x, 0} - \max {x, 0} = - x$ , so the two terms cancel.

Proposition 3.17 (Cash is the reverse side of working capital). With the working capital $W$ of Definition 3.2, (3.8) can be written

M (t) = E (t) - W (t) - A (t)

(3.9)

Proof. Summing $\max {- κ_{i}, 0} - \max {κ_{i}, 0} = - κ_{i}$ over $i$ gives $\sum_{i} \max {- κ_{i}, 0} - \sum_{i} \max {κ_{i}, 0} = - \sum_{i} κ_{i}$ . Substitute into (3.8) and collect $Inv$ into $W$ . □

Checking against Example 3.19: $\sum_{i} κ_{i} = - 600 - 50 + 200 = - 450$ and $Inv = 0$ , so $W = - 450$ , and $100 + 450 - 30 = 520$ agrees.

Equation (3.9) says the same thing as (3.8), but it connects, through $W$ , to the $CCC$ of Chapter 5. How the constraint $M \geq 0$ of Section 4.2 bears on the choice of $Φ$ is settled by way of this form.

Remark 3.18 (The identity is a standard one). Equation (3.9) is the identity used in financial analysis, cash equals equity minus net operating assets. Its significance here is that $W$ is a function of $Φ$ through (3.2), so the cash constraint becomes a constraint on $Φ$ .

Example 3.19 (Funding from three sources). A business started with 1,000,000 yen of equity receives 6,000,000 yen of annual prepayments from customers ( $κ_{C} = - 6, 000, 000$ ), owes 500,000 yen to a cloud provider ( $κ_{S} = - 500, 000$ ), and has 2,000,000 yen of receivables from a large customer ( $κ_{C^{'}} = + 2, 000, 000$ ). There is no inventory and fixed assets are 300,000 yen. In units of 10,000 yen,

M = 100 + (600 + 50) - 200 - 0 - 30 = 520 .

Against 1,000,000 yen of equity, cash on hand is 5,200,000 yen. The difference of 4,200,000 yen is credit drawn from customers and suppliers, and it has a due date.

3.2.2 Equity and capital providers

The $E$ of Proposition 3.15 appears as a residual; its composition is made explicit here. Equation (3.8) uses this $E$ .

Definition 3.20 (Equity). With paid-in capital $P_{K} (t)$ and cumulative dividends $Div (t)$ , define

E (t) = P_{K} (t) + \int_{0}^{t} ϕ (s) 𝑑𝑠 - Div (t)

(3.10)

The second term is retained earnings.

Definition 3.20 links the $ϕ$ of Chapter 6 to the $E$ of this chapter. Substituting the three-way decomposition of $ϕ$ ,

retained earnings = \int (ϕ_{prod} + ϕ_{barg} + ϕ_{cog}) 𝑑𝑠 - Div

Chapter 6 treats the erosion of $ϕ_{cog}$ by learning and regulation, but what is eroded is the future flow; what has already accumulated remains in $E$ .

Definition 3.21 (Credit position with capital providers). For a capital provider $i \in K$ with probability measure $ℙ_{i}$ , let $V_{i} (t) = 𝔼_{i} [present value of future cash flows]$ be the value of the residual claim, and define

κ_{K}^{i} (t) = V_{i} (t) - P_{K} (t)

(3.11)

Proposition 3.22 ( $κ_{K}$ does not enter (3.8)). Because $κ_{K}^{i}$ depends on the capital provider $i$ , it cannot be included in the identity (3.8).

Proof. Equation (3.8) is a rearrangement of the balance-sheet identity and consists only of quantities on whose value all parties agree. Receivables and payables agree in amount between the parties, but the $V_{i}$ of (3.11) depends on $ℙ_{i}$ and does not. Including a party-dependent quantity would make the identity differ by party, and it would cease to be an identity. □

Remark 3.23 (Three degrees of subjectivity). Several unobservable quantities appear in this text, and they differ in character. Using the $𝒢$ of Chapter 4 they fall into three degrees.

Degree	Condition	Examples
Shared	$𝒢$ -measurable	the settlement amount $π$ , receivables
Private information	$F_{t}$ -measurable but not $𝒢$ -measurable	the valuation map $v$ , effort $a$
Measure-dependent	$F_{t}$ -measurable but dependent on $ℙ_{j}$	the value of the residual claim $V_{j}$

Table 3.5: Three degrees of subjectivity among unobservable quantities.

The subjectivity of $v$ is informational asymmetry and can be resolved by disclosure. The subjectivity of $V_{j}$ is disagreement about the probability measure; the same information yields different forecasts, so disclosure does not resolve it.

Two consequences follow. First, the value of $κ_{K}^{i}$ does not agree between the parties and so differs in character from the other $κ_{i}$ . Second, it is precisely that disagreement that makes equity investment possible: an investor invests because they value $V$ more highly than the manager does. This has the same structure as the heterogeneity of discount factors in Proposition 4.10.

The maturity of capital is discussed in Remark 3.24.

Remark 3.24 ( $τ_{K}$ and insolvency). For borrowing, $τ_{K}$ is fixed as the repayment date; for equity it is not fixed. Capital is the longest-dated credit and has no maturity.

Making dividends obligatory would fix $κ_{K}$ , but (4.3) would then break immediately upon non-payment. Leaving them non-obligatory as a residual claim, and forcing liquidation at the insolvency time of Section 4.2, avoids this. Bankruptcy is the mechanism that fixes $κ_{K}$ .

Remark 3.25 (Funds held on behalf of others). Among positions with $κ_{i} < 0$ , funds that legally belong to someone else — escrow, for instance — occupy the same place in (3.8) but differ in character. They drain rapidly when volume falls, and must not be treated as equivalent to other negative $κ$ .

Chapter 3 The Credit Position: κ and the Lag τ