# Residual Income Valuation vs. Justified P/B Ratio

Residual income models can be used to estimate justified price multiples. From the justified P/B ratio based on fundamentals:

$$\frac{\text{P}_{0}}{\text{B}_{0}}=\frac{\text{ROE}-\text{g}}{\text{r}-\text{g}}$$

Which is mathematically equivalent to:

$$\frac{\text{P}_{0}}{\text{B}_{0}}=1+\frac{\text{ROE}-\text{r}}{\text{r}-\text{g}}$$

The approximated value of a share is the book value per share plus the present value of the expected stream of residual income.

A stock’s justified P/B ratio is related to expected future residual income. A closely related concept is Tobin’s q, the ratio of the market value of debt and equity to the replacement cost of total assets.

$$\text{Tobin’s q}= \frac{\text{Market value of debt and equity}}{\text{Replacement cost of total assets}}$$

The denominator uses total assets and is valued at replacement costs. Replacement costs consider the effects of inflation. Tobin’s q is expected to be higher the greater the productivity of a company’s assets. A difficulty in computing Tobin’s q is the lack of information on the replacement cost of assets.

If the return on equity (ROE) is higher than the required return on equity ($$r$$), the stock’s intrinsic value will be greater than the book value. The justified P/B ratio will be higher than one.

If the return on equity (ROE) is less than the required return on equity ($$r$$), the stock’s intrinsic value will be less than book value. The justified P/B ratio will be below one.

## Question

If the return on equity equals the required rate of return on equity, a share’s intrinsic value would most likely be equal to?

1. Book value.
2. Present value of the expected residual income.
3. Earnings per share.

### Solution

If the return on equity equals the required return on equity, the share’s intrinsic value will be equal to the book value. If the return on equity is greater than the required return on equity, the stock’s intrinsic value will be higher than the book value. If the return on equity is less than the required return on equity, the stock’s intrinsic value will be less than the book value.

B is incorrect. If the return on equity equals the required rate of return on equity, the residual income would be equal to zero.

C is incorrect. If the return on equity equals the required rate of return on equity, the share price would not be equal to its earnings per share.

LOS 26 (e) Explain the relation between residual income valuation and the justified price-to-book ratio based on forecasted fundamentals.

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