Definition 5.1 (Cash conversion cycle). In a steady state, define

Its dimension is time.

Proposition 5.2 (The fundamental inequality). The cash constraint (4.3) is equivalent to

Corollary 5.3 (Upper bound on scale). When $\mathrm{CCC}>0$, the feasible rate of sales is bounded above by

When $\mathrm{CCC}\le 0$ and $E\ge A$, (5.3) holds for every $r$.

Corollary 5.4 (Without capital, only $\mathrm{\Phi }$ with $\mathrm{CCC}\le 0$ can be chosen). When $E=0$, (5.3) requires $\mathrm{CCC}\cdot r\le -A\le 0$, that is $\mathrm{CCC}\le 0$.

Proposition 5.5 (Growth and free cash flow). Writing $m$ for the margin and $I$ for capital expenditure, and taking $\mathrm{CCC}$ to be constant and independent of $m$,

Remark 5.6 ($\mathrm{FCF}$ is the time derivative of the cash balance). Differentiate (5.2). By (3.10), $\dot{E}=\varphi -\dot{\mathrm{Div}}$, and fixed assets satisfy $\dot{A}=I-dr$ (investment less depreciation). With no dividends and no external funding,

So $\mathrm{FCF}$ is not an independently introduced quantity but the time derivative of (3.9). The $g^{\star }$ of the next section is therefore the boundary of $\dot{M}\ge 0$.

Corollary 5.7 (Self-financeable growth rate). Writing $d$ for depreciation as a fraction of sales and $g=\dot{r}∕r$ for the growth rate, the condition $\mathrm{FCF}\ge 0$ is

When $\mathrm{CCC}<0$ there is no $g^{\star }$: $\mathrm{FCF}>0$ at any growth rate.

Remark 5.8 (The constraint on the level and the constraint on change are independent). Corollary 5.3 follows from $M\ge 0$ and Corollary 5.7 from $\dot{M}\ge 0$. Neither implies the other. Even with $M(0)>0$, if $g>g^{\star }$ then $M$ falls and (4.3) is violated in finite time.

Proposition 5.9 (Time to cash depletion). Let $I=0$, let $g$ be constant, and let $r(t)=r_0{e}^{\mathit{𝑔𝑡}}$. If $g>g^{\star }=(m+d)∕\mathrm{CCC}$, then starting from $M(0)=M_0>0$ the cash runs out at

and $T\mathrm{ins}\to \infty$ as $g\downarrow g^{\star }$.

Example 5.10 ($g^{\star }$ is a threshold, not a cliff). With $m=5\%$, $d=2\%$, $\mathrm{CCC}=0.25$ years, $M_0=100$ and $r_0=1,000$, we have $g^{\star }=28\%$.

A slight excess leaves more than a decade; a large one exhausts the cash within a year. What matters is not whether $g^{\star }$ has been exceeded but by how much.

Remark 5.11 ($\mathrm{CCC}$ is also an amplifier of demand variation). Equation (5.2) is linear in $r$ with coefficient $-\mathrm{CCC}$. If $E$ and $A$ are independent of $r$ in the short run,

So $\mathrm{CCC}$ not only fixes the bound on the level; it amplifies variation in demand into variation in cash. The larger a sector’s $\mathrm{CCC}$, the more readily (4.3) is violated for the same variation in demand.

Because no process is specified for $r$, no $\mathbb{P}(T\mathrm{ins}\le t)$ is given here. That the amplification coefficient is $\mathrm{CCC}$ can be stated without specifying the process.

Remark 5.12 (No assumption that $I\approx dr$). With a constant asset base, $I=dr$ and (5.6) reduces to $m∕\mathrm{CCC}$. That assumption is not made here. The measurements of Section 14.5 put $I∕r$ between $-1.8\%$ and $6.7\%$ across industries and size classes, which does not agree with $d$.

Remark 5.13 (Divergence when $\mathrm{CCC}$ is small). Equation (5.6) diverges as $\mathrm{CCC}\to 0$. Even for positive $\mathrm{CCC}$, if it is only a few days then a small movement in the numerator moves $g^{\star }$ a great deal.

At $\mathrm{CCC}=5$ days, for instance, a one-point change in the numerator moves $g^{\star }$ by about 73 points. In sectors where $\mathrm{CCC}$ is small, $g^{\star }$ does not work as an indicator. Section 14.5 shows this by reporting standard deviations alongside.

Example 5.14 (The magnitudes $g^{\star }$ actually takes). Consider a business with operating margin $m=5\%$, depreciation $d=2.5\%$ of sales, and capital expenditure $I∕r=3.5\%$ — close to the all-industry figures measured in Section 14.5. The numerator is

Computing $g^{\star }$ for several values of $\mathrm{CCC}$:

A business with $\mathrm{CCC}=120$ days necessarily needs outside funding to grow faster than 12% a year. A business with $\mathrm{CCC}=-30$ days accumulates cash the faster it grows. The sign fixes the qualitative constitution; the magnitude fixes the quantitative room.

The right-hand column gives the value when $d$ and $I$ are ignored. Here $d<I∕r$, so $m∕\mathrm{CCC}$ overstates $g^{\star }$. For businesses investing above depreciation, an assessment using $m$ alone is optimistic.

Remark 5.15 (Domain of definition). Equation (5.1) presumes a steady state and is undefined immediately after founding, where $r\approx 0$. This is taken up again in Section 12.4.

Proposition 5.16 (Structure of $\mathrm{CCC}$). In a steady state,

That is, $\mathrm{CCC}$ does not depend on the flow; it is a sum of lags.

Corollary 5.17 (Source of differences in level). ${\tau }_i$ is the quantity specified by degree of freedom (1) of Section 2.5, the order of $\mathrm{supp}<!--\; FUNCTION\; APPLICATION\; -->(\pi )$ and $\mathrm{supp}<!--\; FUNCTION\; APPLICATION\; -->(\delta )$. The level of $\mathrm{CCC}$ is therefore fixed by $\mathrm{\Phi }$.

Corollary 5.18 (Cross-sectional comparison is meaningless). Comparing the level of $\mathrm{CCC}$ across objects with different $\mathrm{\Phi }$ is comparing different contractual forms, and carries no information about demand or efficiency.

Corollary 5.19 (Aggregation is an $r$-weighted mean). In the parallel arrangement of Proposition 3.9, writing $r_k$ for the rate of sales of each part,

so $\mathrm{CCC}$ is not additive.

Corollary 5.20 (Decomposition of change). Differencing (5.8) splits it as

In (5.10), $\mathrm{\Delta }{\tau }_i\ne 0$ means a change in contractual terms, whereas $\mathrm{\Delta }{\tau }_{\mathrm{inv}}$ arises independently of contracts.

Proposition 5.21 (Working capital at the peak). With a constant rate of sales, a billing period $T_b$ and a payment lag ${\tau }_i$, the time average of ${\kappa }_i(t)$ is $r({\tau }_i+T_b∕2)$ and its maximum is $r({\tau }_i+T_b)$. Equation (5.3) therefore becomes

Proposition 5.22 (Lags are perfect substitutes at the margin). By (5.8), $\partial \ln <!--\; FUNCTION\; APPLICATION\; -->g^{\star }∕\partial {\tau }_i=-1∕\mathrm{CCC}$ for every $i$, independently of $i$.

Remark 5.23 (No comparable quantity can be constructed). Corollary 5.18 states why comparison fails; it does not supply a way to compare.

The natural candidate is the deviation from the ${\tau }_i^{\mathrm{contract}}$ that the contract specifies,

This quantity does not depend on $\mathrm{\Phi }$, so it can be compared across industries, and it measures whether performance follows the contract.

But ${\tau }_i^{\mathrm{contract}}$ is the payment term of an individual contract and does not exist in published statistics. The theoretical problem is solved and the measurement problem takes its place — category (iv), access constraint, in the taxonomy of Section 13.2.

Remark 5.24 (Independence of $m$ and $\mathrm{CCC}$). Corollary 5.7 gives $g^{\star }$ as a ratio but does not guarantee that numerator and denominator move independently.

The measurements of Section 14.5 show $\mathrm{CCC}$ rising 49% and $m$ rising 91% between fiscal 2000 and fiscal 2024 — both increasing. If holding more working capital and earning a higher margin are positively related, a change in $g^{\star }$ cannot be read as a loosening of the constraint.

This text does not formalize the relation between the two. $g^{\star }$ is reported as the value of a ratio and given no causal interpretation.