# Implied Growth Rate in Residual Income

The single-stage residual income equation can be rearranged to calculate the growth implied by the current market price. The current market price would be assumed to equal the stock’s intrinsic value. The intrinsic value is assumed to be the market price.

$$\text{g}=\text{r}-\bigg[\frac{(\text{ROE}-\text{r})\times\text{B}_{0}}{\text{V}_{0}-\text{B}_{0}}\bigg]$$

### Example

Assuming a company has the following information:

• Current book value per share = $8 • Expected long-term ROE = 15% • Required rate of return on equity = 8% If the current market price is$32, the implied growth rate can be calculated as:

\begin{align*}\text{g}&=\text{r}-\bigg[\frac{(\text{ROE}-\text{r})\times\text{B}_{0}}{\text{V}_{0}-\text{B}_{0}}\bigg]\\&=0.08-\bigg[\frac{(0.15-0.08)\times8}{32-8}\bigg]\\&=5.7\%\end{align*}

## Question

Assuming a company has the following information:

• Current book value per share = $12 • Expected long-term ROE = 14% • Required rate of return on equity = 9% • Current market price =$42

The implied growth rate is closest to?

1. 6%
2. 7%
3. 8%

#### Solution

\begin{align*}\text{g}&=\text{r}-\bigg[\frac{(\text{ROE}-\text{r})\times\text{B}_{0}}{\text{V}_{0}-\text{B}_{0}}\bigg]\\&=9\%-\bigg[\frac{(14\%-9\%)\times12}{42-12}\bigg]\\&=7\%\end{align*}

LOS 26 (g) Calculate the implied growth rate in residual income, given the market price-to-book ratio and an estimate of the required rate of return on equity.

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