# Single-Stage Residual Income Valuation

## Single-Stage Residual Income Valuation

The single-stage residual income (constant-growth) model assumes that a firm has a constant return on equity and constant earnings growth rate through time.

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

#### Example: Using a Single-stage Residual Income Model

A company’s current book value per share is $25.25, and the current price per share is$36.00. The company’s expected long-term ROE is 12%, and long-term growth is 6.5%. Assuming a cost of equity of 8%, the intrinsic value can be calculated as:

\begin{align*}\text{V}_{0}&=\text{B}_{0}+\frac{\text{ROE}-\text{r}}{\text{r}-\text{g}}\text{B}_{0}\\&=25.25+\bigg(\frac{0.12-0.08}{0.08-0.65}\bigg)\times25.25\\&=92.58\end{align*}

If the return on equity is lower than the required return on equity, the intrinsic value of a company’s share will be lower than its book value per share. When a company has no prospect of covering its cost of capital, a liquidation of the firm and redeployment of assets may be appropriate.

A drawback to the single-stage model is that it assumes the return on equity above the cost of equity will persist indefinitely. However, a company’s return on equity will likely revert to a mean value of return on equity over time. If an industry has a high return on equity, it will attract competition, eventually lowering the return on equity.

## Multistage Residual Income Valuation

The multistage residual income approach can be used to forecast residual income over a given time horizon and then compute a terminal value based on continuing residual income at the end of that period. Continuing residual income is residual income after the forecast period.

In the residual income model approach, the terminal value does not constitute a large portion of the intrinsic value. This is different from the DDM and DCF multistage approaches.

One of the following assumptions need to be made about the continuing residual income:

• The residual income continues indefinitely at a positive level.
• The residual income is zero from the terminal year forward.
• The residual income declines to zero as ROE reverts to the cost of equity over time.
• The residual income reflects the reversion of ROE to some mean level.

#### Example: Multistage Residual Income Model

A company has the following information:

• Current book value per share = 12 • ROE =16% • Expected EPS for Years 1-5 is calculated as ROE times the beginning book value per share. • Dividend payout ratio = 25% • Required rate of return on equity = 8% a) If the ROE falls to 8% after Year 5 and the continuing residual income falls to zero, the firm’s intrinsic value can be calculated as: $$\small{\begin{array}{l|c|c|c|c|c|c}\textbf{Year} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} &\textbf{5} & \textbf{6} \\ \hline \text{Beginning book value per share} & \12.00 & \13.44 & \15.05 & \16.85 & \18.87 & \21.14 \\ \hline\text{Add: EPS}\ (\text{ROE}\times\text{B}_{\text{t}-1}) & 1.92 & 2.15 & 2.40 & 2.70 & 3.02 & 3.38 \\ \hline \text{Less: Dividends}\ (25\%\times\text{EPS}) & 0.48 & 0.54 & 0.60 & 0.68 & 0.75 & 0.85 \\ \hline\textbf{Ending book value per share} & \textbf{13.44} & \textbf{15.05} & \textbf{16.85} & \textbf{18.87} & \textbf{21.14} & \textbf{23.67}\\ \end{array}}$$ $$\small{\begin{array}{l|c|c|c|c|c|c}\text{Earnings per share (EPS)} & 1.92 & 2.15 & 2.40 & 2.70 & 3.02 & 3.38 \\ \hline \text{Less: Equity charge per share}\ (\text{r}\times\text{B}_{\text{t}-1}) & 0.96 & 1.08 & 1.20 & 1.35 & 1.51 & 1.69 \\ \hline\textbf{Residual income per share} & \textbf{0.96} & \textbf{1.07} & \textbf{1.20} & \textbf{1.35} & \textbf{1.51} & \textbf{1.69}\\ \end{array}}$$ \begin{align*}\text{Intrinsic value}&=12.00+\frac{0.96}{(1.08)^{1}} +\frac{1.07}{(1.08)^2} +\frac{1.20}{(1.08)^3} +\frac{1.35}{(1.08)^4} +\frac{1.51}{(1.08)^5} \\&=16.78\end{align*} b) Assuming the continuing residual income remains constant at year five at0.90 into the future, the intrinsic value can be calculated as:

\begin{align*}\text{Terminal value}&= \frac{0.90}{0.08}\\&=11.25\\ \text{Intrinsic value}&=12.00+\frac{0.96}{(1.08)^1} +\frac{1.07}{(1.08)^2} +\frac{1.20}{(1.08)^3} +\frac{1.35}{(1.08)^4} \\& +\frac{1.51+11.25}{(1.08)^5}\\& =24.44\end{align*}

c) Assuming ROE declines to the required return on equity over time, the intrinsic value can be calculated using:

$$\text{V}_0= \text{B}_0+ ∑_{\text{t}=1}^{\text{T}-1} \frac{(\text{E}_{\text{t}}-\text{rB}_{\text{t}-1})}{(1+\text{r})^{\text{t}}}+\frac{\text{E}_{\text{T}}-\text{rB}_{\text{T}-1}}{(1+\text{r}-ω)(1+\text{r})^{\text{T}-1}}$$

Where:

$$ω=$$ Persistence factor.

A persistence factor of 1 indicates that residual income will not decline. It will remain at the same level indefinitely. A persistence factor of 0 indicates that there will be no residual income after the initial forecast horizon. The greater the persistence factor, the greater the stream of residual income in the final stage, and the greater the value of the stock.

Assuming a persistence factor of 0.4, the terminal value at the end of Year 5 is calculated as:

\begin{align*}\text{Terminal value}&=\frac{1.69}{(1+0.08-0.4)}\\&=2.49\\ \text{Intrinsic value}&=12.00+\frac{0.96}{(1.08)^1} +\frac{1.07}{(1.08)^2} +\frac{1.20}{(1.08)^3} +\frac{1.35}{(1.08)^4} \\ & +\frac{1.51+2.49}{(1.08)^5} \\&=18.47\end{align*}

## Question

Assuming a company has the following information:

• Current book value per share = $40 • Expected long-term ROE = 18% • Required rate of return on equity = 9% • Current market price =$95

The implied growth rate is closest to:

1. 2.5%.
2. 3.0%.
3. 2.0%.

### Solution

\begin{align*}\text{g}&=\text{r}-\bigg[\frac{(\text{ROE}-\text{r})\times\text{B}_{0}}{\text{V}_{0}-\text{B}_{0}}\bigg]\\&=0.09-\bigg[\frac{(0.18-0.09)\times40}{85-40}\bigg]\\&=2.5\%\end{align*}

LOS 26 (f) Calculate and interpret the intrinsic value of a common stock using single-stage (constant- growth) and multistage residual income models.

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