Guys, can someone please show me algebraically how the equations below are arrived at:

**RI = E - (r * B _{t-1}) = (ROE - r) * B_{t-1}**

Also, logically, what does this term imply: **ROE - r**?

Guys, can someone please show me algebraically how the equations below are arrived at:

**RI = E - (r * B _{t-1}) = (ROE - r) * B_{t-1}**

Also, logically, what does this term imply: **ROE - r**?

If you recall from L1 (dupont): ROE = NI / Equity… where NI is the same as E, and Equity is the same as B.

Now rearrange that equation, E = ROE*Book

Now from your equation RI = E-R*B… if we replace E with ROE*B, we get:

RI = ROE*B - R*B… Now do some alebra and you get

RI = (ROE-R)B

RI itself is very similar to EP and EVA but with just equity.

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I believe ROE - r, logically means the Return on Equity above the required return. Means the firm is growing and the value is greater than book (if ROE-r is positive)

This is how I understood it too.

RI = E − (*r* × B__{t}__{−}_{1})

= (ROE × B__{t}_{−}_{1}) − (*r* × B_{t}__{−}_{1})

= (ROE − *r*) × B__{t}__{−}_{1}

How much the company earns above the required return on common equity:

- If ROE –
*r*> 0, the company is adding value for the shareholders, and there is positive residual income - If ROE –
*r*= 0, the company is neither adding value nor destroying value for the shareholders, and residual income is zero - If ROE –
*r*< 0, the company is destroying value for the shareholders, and there is negative residual income

(Note: 125mph posted while I was finishing up making dinner (BBQ pork ribs) and feeding the dogs and cats, but I decided to leave this here anyway.)

Thanks. this makes sense now.

So if **ROE - r** is the return above the required rate.

Is **ROE - g** the return above the expected growth?

thanks Magician! You always add more explanation that what’s asked. very helpful!

Glad to be of service.

Yup.

Of course, *g* = ROE × *b*, so,

ROE – *g* = ROE – (ROE × *b*)

= ROE × (1 – *b*) = ROE × payout ratio

I would say **“the company is adding extraordinary value for the shareholders”** in the first case

Note that the required return ® is assumed to contain the opportunity cost of capital.

S2000magician:

- If ROE –
r< 0, the company isdestroying valuefor the shareholders, and there is negative residual income

This last case would be right, a ROE lower than the required return is destroying value in an economic sense. No necessarily financially, though.