Residual Income Formula

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

RI = E - (r * Bt-1) = (ROE - r) * Bt-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 itself is very similar to EP and EVA but with just equity.

1 Like

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_t1) − (r × Bt_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.


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 . This is because r itself is indeed the value created. If observed ROE materializes the required return, then value is truly added.

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

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