# Why does a business with an ROIC greater than its NOPAT growth rate means that the business is "generating excess cash"

In Michael Mauboussin's "Capital Allocation" paper (https://www.morganstanley.com/im/publication/insights/articles/article_capitalallocation.pdf) he frequently references the idea that a business with an ROIC greater than its NOPAT growth rate means that the business is "generating excess cash". Could anyone explain why this implication is true? That is, I don't see the connection on how A (ROIC > NOPAT growth) necessarily implies B (business is generating excess cash) here. Could anyone clarify this for me? Perhaps with some kind of mathematical proof to illustrate this plainly?

E.g.

Capital allocation is an important investment issue because the aggregate ROIC for public companies exceeds the aggregate growth rate This means that businesses generate excess cash

*It may be important to note that Mauboussin does not include cash in his definition of Invested Capital.

This is what I have so far:

``````Assume we know the IC[t=0], NOPAT growth rate = gN, and ROIC[t=0] > gN > 0.
After 1yr, NOPAT[1] = NOPAT[0] + (NOPAT[0] x gN).
To just maintain ROIC[1] = ROIC[0], incremental IC, IIC, would need to be the solution to...
(NOPAT[0] + (NOPAT[0] x gN)) / (IC[0] + IIC) = NOPAT[0] / IC[0]
==> IIC = IC[0] x gN
``````

We then need to prove that (NOPAT[0] + (NOPAT[0] x gN)) > IIC in order to imply excess capital left over (NOPAT[0] + (NOPAT[0] x gN)) - IIC > 0) in a constant ROIC situation. That is...

``````If n/c > g > 0, prove that (n + n*g) > c*g
``````

... At this point I'm stuck (and IDK that this can even be proven or is solvable given that we also would hold that n > 0 and c > 0 as well)

To put it another way...

``````Given:
1 >= ROIC[t=0] > gN > 0
IC[t=0]

After 1yr (t=1):
NOPAT[1] = NOPAT[0] x (1 + gN) = NOPAT[0] + NOPAT[0] x gN
To maintain same ROIC...
ROIC[1] = ROIC[0]
==> ROIC[1] = NOPAT[1] / (IC[0] + IIC) = (NOPAT[0] + NOPAT[0] x gN) / (IC[0] + IIC) = NOPAT[0] / IC[0] = ROIC[0]
==> IIC (the incremental invested capital to maintain ROIC given we generated NOPAT[t=1]) is whatever solves...
(n+ng) / (c+i) = n/c for i
==> IIC = IC[0] x gN
If IIC < NOPAT[1], then there is excess capital (ie. not all NOPAT[t=1] had to be reinvested to maintain constant ROIC ratio)

We are thus seeking to prove that the case where 1 >= ROIC[t=0] > gN > 0 must imply IIC < NOPAT[1]
... ie. 1 >= n/c > g > 0 ==> c*g < n + n*g
``````

*The reason I'm trying to proof this out is because Mauboussin seems to take it 1) for granted and 2) as a generality in that paper, so I'm trying to see it proved out as a generality myself.

For now I simply accept it as provable somehow because Wolfram Alpha tells me that final set of inequalities has the solutions space of `c>0, g>0, n>c*g`, ie. `IC[t=0] > 0, gN > 0, ROIC[t=0] > gN` (that last inequality being the thing I'd want to prove).

*I think I was stuck in the previous equations because I actually wrote the last inequality backwards (which I've edited now), so was trying to "prove" the wrong thing.