10 Derivation of the RIM
From the Dividend Discount Model to the Residual Income Model
10.1 Introduction
The Residual Income Model (RIM) — also known as the Edwards–Bell–Ohlson (EBO) model — reformulates equity valuation in terms of accounting quantities rather than dividends. The key insight is that, under the Clean Surplus Relation, dividends can be expressed as a function of book value and earnings, allowing the Dividend Discount Model (DDM) to be rewritten entirely in accounting terms.
This document derives the RIM step by step, starting from the DDM.
10.2 The Dividend Discount Model
Recall our starting point: the value of equity today (\(V_0\)) equals the present value of all future dividends:
\[ V_0 = \sum_{t=1}^{\infty} \frac{d_t}{(1+r)^t} \tag{DDM} \]
where:
- \(d_t\) is the dividend paid at time \(t\),
- \(r\) is the cost of equity (assumed constant).
While theoretically sound, the DDM is difficult to apply in practice because dividends are a policy choice and may not reflect fundamental value creation. We seek an equivalent formulation using book values and earnings. And we will get this using the so-called Clean Surplus Relation
\[ \underbrace{V_0 = \sum_{t=1}^{\infty}\frac{d_t}{(1+r)^t}}_{\text{DDM}} \;\xrightarrow{\text{CSR: }d_t = x_t - \Delta b_t}\; \underbrace{V_0 = b_0 + \sum_{t=1}^{\infty}\frac{x_t - r\,b_{t-1}}{(1+r)^t}}_{\text{RIM}} \]
10.3 The Clean Surplus Relation
The Clean Surplus Relation (CSR) is an accounting identity that links book value, earnings, and dividends. It states that the change in book value of equity equals earnings minus dividends (i.e., all changes in book value flow through the income statement):
\[ b_t = b_{t-1} + x_t - d_t \tag{CSR} \]
where:
- \(b_t\) is the book value of equity at the end of period \(t\),
- \(x_t\) is net income (earnings) for period \(t\),
- \(d_t\) is dividends paid during period \(t\).
Rearranging for dividends:
\[ d_t = x_t - (b_t - b_{t-1}) \tag{1} \]
Dividends equal earnings minus the increase in book value. This is simply the statement that net income not retained is paid out.
10.4 Substituting the CSR into the DDM
Substitute \((1)\) into the DDM:
\[ V_0 = \sum_{t=1}^{\infty} \frac{x_t - (b_t - b_{t-1})}{(1+r)^t} \]
Separate the sum:
\[ V_0 = \sum_{t=1}^{\infty} \frac{x_t}{(1+r)^t} - \sum_{t=1}^{\infty} \frac{b_t - b_{t-1}}{(1+r)^t} \tag{2} \]
We now simplify the second sum using a telescoping argument.
10.5 Telescoping the Book Value Sum
Consider the second sum in \((2)\). Write out the first few terms:
\[ \sum_{t=1}^{\infty} \frac{b_t - b_{t-1}}{(1+r)^t} = \frac{b_1 - b_0}{(1+r)^1} + \frac{b_2 - b_1}{(1+r)^2} + \frac{b_3 - b_2}{(1+r)^3} + \cdots \]
Regroup by collecting terms in \(b_t\):
\[ = -b_0 + b_1\!\left(\frac{1}{1+r} - \frac{1}{(1+r)^2}\right) + b_2\!\left(\frac{1}{(1+r)^2} - \frac{1}{(1+r)^3}\right) + \cdots \]
Factor each bracket:
\[ \frac{1}{(1+r)^t} - \frac{1}{(1+r)^{t+1}} = \frac{1}{(1+r)^t}\cdot\frac{r}{1+r} = \frac{r}{(1+r)^{t+1}} \]
Therefore:
\[ \sum_{t=1}^{\infty} \frac{b_t - b_{t-1}}{(1+r)^t} = -b_0 + \sum_{t=1}^{\infty} \frac{r \cdot b_{t-1}}{(1+r)^t} \]
provided the transversality condition holds:
\[ \lim_{T\to\infty} \frac{b_T}{(1+r)^T} = 0 \tag{TC} \]
This condition rules out weird dynamics — book value cannot grow forever at a rate exceeding the cost of equity.
10.6 Assembling the Result
Substituting back into \((2)\):
\[ V_0 = \sum_{t=1}^{\infty} \frac{x_t}{(1+r)^t} - \left(-b_0 + \sum_{t=1}^{\infty} \frac{r \cdot b_{t-1}}{(1+r)^t}\right) \]
\[ V_0 = b_0 + \sum_{t=1}^{\infty} \frac{x_t - r \cdot b_{t-1}}{(1+r)^t} \tag{3} \]
10.7 Defining Residual Income
Define residual income (also called abnormal earnings or economic profit) for period \(t\) as:
\[ x_t^a \equiv x_t - r \cdot b_{t-1} \tag{RI} \]
Residual income is the earnings that exceed the “normal” return on beginning-of-period book value. It measures value created beyond what equity investors require.
Substituting into \((3)\) yields the Residual Income Model:
\[ V_0 = b_0 + \sum_{t=1}^{\infty} \frac{x_t^a}{(1+r)^t} \tag{RIM} \]
10.8 Important Observations:
Equivalence to DDM. Under the CSR and the transversality condition (TC), the RIM is algebraically identical to the DDM. No new economic assumptions are required.
Accounting policy neutrality. Although book values and earnings individually depend on accounting choices, any change in accounting that affects \(b_t\) also affects future \(x_{t+1}^a\) in an offsetting way (provided the CSR holds), leaving \(V_0\) unchanged.
Special cases.
- If the firm earns exactly its cost of equity at all times, \(x_t^a = 0\) for all \(t\), and \(V_0 = b_0\) (price equals book value).
- If \(x_t^a > 0\) persistently, the firm trades at a premium to book (price-to-book \(> 1\)).
- If \(x_t^a < 0\) persistently, the firm trades at a discount to book (price-to-book \(< 1\)).
Practical tractability. Because residual incomes are anchored to near-term forecasts of earnings and book values — quantities that analysts readily forecast — the RIM is often more practical than the DDM, especially for non-dividend-paying firms.
10.9 Further Reading
The derivation draws on the foundational work of Edwards and Bell (1961), who introduced the concept of business income based on opportunity cost, and Ohlson (1995), who formalised the residual income valuation framework in an accounting context. The extension to operating and financial activities is developed in Feltham and Ohlson (1995). A comprehensive treatment of the model and its practical application can be found in Penman (2012).