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This topic falls under Financial Planning Models. Financial planning is a process consisting of the following:

Analyzing the investment and financing decisions available to a firm (debt or equity, dividend policy etc)

Projecting future consequences of current decisions (Capital Budgeting and Financing decisions – financial ratios come in hand here)

Deciding what alternatives to undertake (Financial managers have to choose among many financial plans)

Measuring subsequent performance against the goals set (Growth rate, return on capital etc)
When I was reading this particular topic, I was rusty on my linear algebra and some equations were not so obvious as to how they were deduced. Especially the Sustainable Growth Rate (rate at which firms can grow without changing the leverage i.e. maintaining the debtequity ratio).
The Sustainable Growth Rate is deduced from the Internal Growth Rate (Maximum rate of growth without external financing) which is not very difficult to derive:
Internal Growth Rate = Plowback ratio * ROE * (equity/assets)
Sustainable Growth Rate = Plowback ratio x ROE
It’s easy to use this formulas but the devil is in the detail. What is the real oracle that derives Sustainable Growth Rate from Internal Growth Rate? This is where my algebra was rusty. After dusting off my 1st year algebra textbook and perusing through a few topics, I had a Eureka moment that I would like to share with you! Here it is:
Required External Financing = New Investment – Retained Earnings
New Equity Issues + New Debt Issues = New Investment – Retained Earnings
New Equity Issues = New Investment – Retained Earnings – New Debt Issues
= (Growth Rate * assets) – Retained Earnings – New Debt Issues— (i)
Remember the Sustainable Growth Rate is attained by not offering any New Equity Issues. Therefore the New Equity Issues = Zero (0); NOTE: The sum of additional equity is New Equity Issues plus Retained Earnings. In our case, all additional equity will come from Retained Earnings since no new equity will be issued. To maintain the same debt equity ratio, the additional debt to additional equity will have to maintain the original debt:equity ratio. This is where it becomes confusing. So allow me to digress. By a property of ratios, if the original ratio was 400 : 600 , to maintain that ratio after adding the retained earnings to equity, you have to add to the debt the fraction of the retained earnings that maintains the 400 : 600 ratio. i.e. if you have a 1:2 ratio, if you add 1 to 2, the right hand side of the ratio becomes 3, to maintain the 1:2 original ratio, the left hand side need to be 1.5. How do you get there? take the ratio and multiply by left hand side (1/2 * 1 = .5).Therefore you have to add .5 to the left hand side, which is equal to 1.5. Now the new figures are 1.5 : 3 which has maintained the original 1:2 ratio!
Back to our problem, we add retained earnings to the original equity. therefore, if the original ratio is D:E, if you add retained earnings to the right hand side, you add ‘D/E * retained earnings’ to the left just like we did to the 1.5:3 ratio above!
Therefore, New Debt Issues = D/E * Retained Earnings. — (ii)
Please make sure you digest the above subtle concept before continuing. Once you understand this concept, everything else becomes very easy.
Okay, now that we know New Equity Issues = 0, from formula (i), simple arithmetic shows that,
Growth Rate = (Retained Earnings + New Debt Issues) / Assets
Using formula (ii) above,
Growth Rate = (Retained Earnings + D/E * Retained Earnings) / Assets
Growth Rate = (Retained Earnings (1+ D/E)) / Assets
From the balance sheet rules, the Assets = Debt + Equity(Liability) ————– (iii)
Using formula (iii),
Growth Rate = (Retained Earnings (1+ D/E)) / Debt + Equity
We know that Debt = D/E * Equity ————–(iv) , since the D:E is maintained
Using formula (iv)
Growth Rate = (Retained Earnings (1+ D/E)) / (Equity+( D/E * Equity))
Growth Rate = (Retained Earnings (1+ D/E)) /(Equity(1+D/E))
1+ D/E cancels out HENCE,
Growth Rate = Retained Earning / Equity,
Multiply the Denominator and the Numerator by Net Income
Growth Rate = Retained Earning / Net Income * Net Income/Equity
= Plowback Ratio * ROE
This is formula for Sustainable Growth Rate above!
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