In the following derivations the subscript E refers to end of period numbers

# In the following derivations the subscript e refers

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In the following derivations, the subscript “E” refers to end of period numbers, and the subscript “B” refers to beginning of period numbers. TE is total equity and TA is total assets. For the sustainable growth rate : Sustainable growth rate = (ROE E × b) / (1 ROE E × b) Sustainable growth rate = (NI/TE E × b) / (1 NI/TE E × b) We multiply this equation by: (TE E / TE E ) Sustainable growth rate = (NI / TE E × b) / (1 NI / TE E × b) × (TE E / TE E ) Sustainable growth rate = (NI × b) / (TE E NI × b) Recognize that the numerator is equal to beginning of period equity, that is: (TE E NI × b) = TE B Substituting this into the previous equation, we get: Sustainable rate = (NI × b) / TE B Which is equivalent to:
Sustainable rate = (NI / TE B ) × b Since ROE B = NI / TE B The sustainable growth rate equation is: Sustainable growth rate = ROE B × b For the internal growth rate: Internal growth rate = (ROA E × b) / (1 ROA E × b) Internal growth rate = (NI / TA E × b) / (1 NI / TA E × b) We multiply this equation by: (TA E / TA E ) Internal growth rate = (NI / TA E × b) / (1 NI / TA E × b) × (TA E / TA E ) Internal growth rate = (NI × b) / (TA E NI × b) Recognize that the numerator is equal to beginning of period assets, that is: (TA E NI × b) = TA B Substituting this into the previous equation, we get: Internal growth rate = (NI × b) / TA B Which is equivalent to: Internal growth rate = (NI / TA B ) × b Since ROA B = NI / TA B The internal growth rate equation is: Internal growth rate = ROA B × b

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