internal and sustainable growth rates

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Internal and sustainable growth rate estimation

David J. Moore, Ph.D.

October 16, 2012

Abstract

Given a firms ability to translate assets into sales and ability to con-vert sales into earnings I derive two growth rate estimates. The first es-timate, internal growth rate, measures how much sales can grow without

raising additional funds. The second estimate, sustainable growth rate,measures how much sales can grow while issuing debt and maintainingthe current debt-to-equity ratio (i.e., risk). These measures are useful indiscounted cash flow estimations and as a check of the true amount ofgrowth potential.

1 Additional funds needed

The additional funds needed for an increase in sales S is estimated by:

AFN1 =

A0

S0

S

L0

S0

S PMRR S1 (1)

with variables defined as follows:

Variable DefinitionS0 Sales in the current periodA0 Assets required to support sales S0

L0 Liabilities that increase spontaneously with sales S0

L0 = accounts payable + accruals

PM Profit marginPM= net income

sales= NI0

S0

RR Retention ratioRR = 1 dividends

net income= 1 Div0

NI0

S1 Sales forecasted in next periodS Change in sales

S= S1 S0

With an increase in sales S, the required asset increase (A

0/S0) S is paidby suppliers and employees (L

0/S0) S and retained profits PM RR S1.If assets are fully utilized then the assets required to support sales A

0equals

total assets A0. To generalize, given capacity utilization , A0 = A0. If we

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presume a growth rate g we can substitute S1 = (1 + g)S0 and S= S1S0 =(1 + g)S0 S0 = gS0 into Eq. (1):

AFN1 =

A0S0

gS0

L0

S0

gS0 PMRR S0 (1 + g)

= A0g L

0g PMRR S0 PMRR S0 g

Grouping the g terms together:

AFN1 = A0g L

0g PMRR S0 g PMRR S0 (2)

2 Internal growth rate

The internal growth rate IGR is the maximum growth rate attainable without

raising any additional funds. Assets must increase to support an increase insales. The question is, who is going to pay for those assets? With IGR wedetermine how much sales can grow using internal funds only. Those internalfunds are the retained profits from the new level of sales.

To find IGR set AFN1 = 0 in Eq. (2) and solve for g:

0 = A0g L

0g PMRR S0 g PMRR S0

= g (A0 L

0 PMRR S0) PMRR S0

IGR g =PMRR S0

A0 L0 PMRR S0(3)

Eq. (3) can be expressed in terms ofROA by noting the relationship betweenPM and ROA:

PM=NI0S0

=NI0A0

A0S0

= ROA A0S0

(4)

Substituting Eq. (4) into Eq. (3):

IGR =

ROA A0

S0

RR S0

A0 L0 ROA A0S0 RR S0

=ROA A0 RR S0

A0 L0 ROA A0 RR S0

IGR =ROA RR

L

0/A0 ROA RR(5)

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3 Sustainable growth rate

Bankruptcy risk increases with the amount of debt. However, the retainedprofits from additional sales can offset the bankruptcy risk. In particular, if weallow debt issuance to fund growth but fix the debt to equity ratio we can solveEq. (2) for g to obtain the sustainable growth rate SGR. Another way to lookat SGR is the maximum we can grow by using other peoples money withoutincreasing overall risk of the firm.

Let = AFN1 represent the additional debt issued to fund growth such thatD1 = D0 + . We are constrained such that the debt to equity ratio is fixed:

D1E1

=D0E0

Solving for :

D0 + E1

= D0E0

= E1D0E0

D0 (6)

The new level of equity E1 is the sum of the old level of equity E0 plus additionsto retained earnings ARE:

E1 = E0 + ARE

= E0 + PMRR S1

= E0 + PMRR S0 (1 + g)

= E0 + PMRR S0 + PMRR S0 g (7)

Substituting Eq. (7) into Eq. (6):

= E1D0S0D0

= (E0 + PMRR S0 + PMRR S0 g)D0E0

D0

= D0 + PMRR S0 D0E0

+ PMRR S0 D0E0

g D0

=NI0S0

RR S0 D0E0

+NI0S0

RR S0 D0E0

g

= ROERRD0 +ROERRD0 g (8)

Substituting AFN1 = and PM= ROE E0/S0 into Eq. (2):

AFN1 = A0g L

0g PMRR S0 g PMRR S0

= A0g L

0g (ROE E0/S0)RR S0 g

(ROE E0/S0)RR S0

= A0g L

0g ROERR E0 g ROERR E0 (9)

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Almost there! Now substitute Eq. (8) into Eq. (9):

ROERRD0 + ROERRD0 g = A0g L

0g ROERRE0 gROERRE0

ROERR (D0 + E0) = g (A0 L

0ROERR (D0 +E0))

g (A0 L

0ROERRA0) = ROERRA0

g =ROERRA0

A0 L0 ROERRA0

SGR g =ROERR

L

0/A0 ROERR

4 Conclusion

To increase sales a firm must increase assets. The increase in assets must bepaid for by internal or external funds. I derived the maximum growth rate basedon internal funding, internal growth rate or IGR as:

IGR =ROA RR

L

0/A0 ROA RR

I derived the maximum growth rate based on external debt financing whilemaintaining the current debt to equity ratio, the sustainable growth rate orSGR as:

SGR =ROERR

L

0/A0 ROERR

These growth rate estimates can be used in discounted cash flow (DCF) analysesand as a check for maximum possible growth rates given the firms current assetmanagement and operational efficiency. Use of growth rates in DCF modelshigher than IGR and SGR must be supported by evidence of future increasesin asset management or operational efficiency.

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internal and sustainable growth rates

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