Continuing Residual Income

Previously, we mentioned the problem of forecasting residual income indefinitely into the future, which makes it difficult to calculate the present value of residual income and implement the residual income model. However, we can simplify the model by using the same multistage approach we used for DDM and free cash flow models. We’ll forecast residual income over a short term horizon (e.g., five years) and then make some simplifying assumptions about the pattern of residual income growth over the long term after five years.

Continuing residual income is the residual income that is expected over the long term.

Residual income will continue beyond a specified earnings horizon depending on the fortunes of the industry, as well as on the sustainability of a specific firm’s competitive prospects over the longer term. The projected rate at which residual income is expected to fade over the life cycle of the firm is captured by a persistence factor, ω, which is between zero and one.

To simplify the model, we typically make one of the following assumptions about continuing residual income at the end of the short term period:

• Residual income is expected to persist at its current level forever.
• Residual income is expected to drop immediately to zero.
• Residual income is expected to decline over time as ROE falls to the cost of equity (in which case residual income is eventually zero).
• Residual income is expected to decline to a long run average level consistent with a mature industry.

An analysis of the firm’s position in its industry and the structure of the industry will be necessary to justify one of these assumptions. The third scenario is the most realistic if we assume that over time, industry competition reduces economic profits to the point at which firms begin to leave the industry and ROE stabilizes at a long run normal level. The strength of the persistence factor will depend partly on the sustainability of the firm’s competitive advantage and the structure of the industry.

The more sustainable the competitive advantage and the better the industry prospects, the higher the persistence factor.

Higher persistence factors will be associated with the following:

• Low dividend payouts.
• Historically high residual income persistence in the industry.

Lower persistence factors will be associated with the following:

• High return on equity.
• Significant levels of nonrecurring items.
• High accounting accruals.

Think of the continuing residual income model as a multistage model similar to the multistage DDM and FCF models from earlier in the Equity materials. In the residual income model, intrinsic value is the sum of three components:

$V_0$ = $B_0$ + (PV of interim high growth RI) + (PV of continuing residual income)

Step 1: Calculate the current book value per share.

Step 2: Calculate residual income in each year 1 to T − 1 during the interim high- growth period and discount them back to today at the required return on equity.

Step 3: Calculate continuing residual income that begins at the end of the high growth period starting in year T, and then calculate the present value of continuing residual income as of the end of year T − 1 using the following formula:

Assumption #1: Residual Income Persists at the Current Level Forever

If ω = 1, residual income is expected to persist at the current level forever after year T − 1, so residual income in every year after T equals residual income in year T. The present value of continuing residual income at the end of year T − 1 is the present value of a perpetuity:

Assumption #2: Residual Income Drops Immediately to Zero

If ω = 0, residual income is expected to drop immediately to zero beginning in year T + 1, and the present value of continuing residual income in year T − 1 is:

Assumption #3: Residual Income Declines Over Time to Zero

If residual income is expected to decline over time after year T as ROE falls to the cost of equity capital, then the persistence factor, ω, is between zero and one, and the present value of continuing residual income in year T − 1 is equal to:

Assumption #4: Residual Income Declines to Long Run Level in Mature Industry There is another, simpler approach to calculating the PV of continuing residual income that does not rely on the formula or ω, the persistence factor, if residual income is expected to decline to a normal long run level consistent with a mature industry after year T.

First, recall from the single stage residual income model that market value equals book value plus the present value of residual income. Therefore, at any point in time (T), the present value of future residual income is the difference between market value ($P_T$) and book value ($B_T$):

How do we estimate $P_T$?

Given a forecasted price to book ratio and book value at the end of the year T, the value of the stock is:

To make this approach consistent with the first three that use the persistence factor equation, we can also calculate the present value of continuing residual income at time T − 1: