# Single Stage Residual Income Valuation

The formula of the single stage residual income valuation is the following:

V0=B0+ (ROE-R)B0/(r-g).

If this formula is similar as to when we do the constant dividend growth model where p=d0(1+g)/(r-g). Why isnt the numerator of the single stage residual income valuation also multiplied by (1+g). Arent we not assuming that the RI grows constantly forever?

For Gordon growth model. Note that the numerator is the next period dividend.

V_0 = \frac{D_1}{r-g} = \frac{D_0(1+g)}{r-g}

In the same way for the single stage RI model, the numerator must be the next period RI (in this case RI_1). And

RI_1 = E_1 - r \times B_0 = (ROE - r) \times B_0

There is no R_0 usually so you do not see R_0(1+g). But the formula assumes that in subsequent periods, the RI will grow at a constant rate of g, i.e. RI_2=RI_1(1+g) and so on.

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thanks

I just have another question.

The Residual Income Valuation is often compared to the justified price to book ratio and the curriculum explains how they are related to each other. I just dont see that relationship

can you pls help me

Let’s try. Assuming the stock is fairly priced. P_0 = V_0 . Then:

P_0 = B_0 + \frac{(ROE-r) \times B_0}{r-g}

Divide both sides by B_0

\frac{P_0}{B_0} = 1 + \frac{ROE-r}{r-g}

\frac{P_0}{B_0} = \frac{r-g}{r-g} + \frac{ROE-r}{r-g}

\frac{P_0}{B_0} = \frac{r-g + ROE - g}{r-g}

\frac{P_0}{B_0} = \frac{ROE - g}{r-g}

Up to this point, we have the Justified P/B ratio. The interpretations are aligned:

When ROE > r, then your P/B ratio SHOULD be greater than 1 (assuming the stock is correctly priced). In other words, P_0 > B_0

I have a video on it (towards the end): https://youtu.be/82GJu5umrB0