The Ohlson and Feltham Ohlson Models

Abstract

This chapter analyses the phenomenon of “positive valuation of losses” in the new economy companies in the US. One of the potential explanations of this phenomenon is that these companies are start-up companies, mostly technology-based, that invest massively in intangible assets, in particular research and development (R&D) and advertising. Under Generally Accepted Accounting Principles (GAAP), these investments should be considered at full cost in the year they occur. Thus, in this chapter, we analyse the Ohlson (OM) (Contemp Acc Rev 11(2):661–687, 1995) and Feltham and Ohlson (FOM) (Contemp Acc Rev 11(2):689–731, 1995) valuation models. Feltham and Ohlson (Contemp Acc Rev 11(2):689–731, 1995) demonstrated analytically, using dynamic information, that losses, particularly at the stage of start-up in growth and technology-based companies, are considered to be costs that create an effect of conservatism accounting, consequently, there is an undervaluation of assets, hence the results and equity. However, this situation tends to be reversed over time, because given the principle of rationality, the investors continue to invest in the company if those investments are associated with abnormal profitability expectations.

Appendices

Appendix 2.1: Deduction of the OM Model (Eqs. 2.8 and 2.9)

• 1. Deduction of the Ohlson (OM) ( 1995 ) model—Eq. 2.8:

Ohlson (1995) defines the valuation function as:

$$P_{t} = {\text{bv}}_{t} + \alpha_{1} x_{t}^{a} + \alpha_{2} v_{t} .$$

with the parameters assuming the values:

$$\begin{aligned} \alpha_{1} & = \frac{w}{{R_{\text{f}} - w}} \ge 0\;{\text{and}} \\ \alpha_{2} & = \frac{{R_{\text{f}} }}{{\left( {R_{\text{f}} - w} \right)\left( {R_{\text{f}} - \gamma } \right)}} \succ 0. \\ \end{aligned}$$

To obtain this function we first assume:

• (i) The matrix $P = R_{\text{f}}^{ - 1} \left[ {\begin{array}{*{20}c} w & 1 \\ 0 & \gamma \\ \end{array} } \right]$;

• (ii) The dynamic of information is defined as:

  $\left[ {x_{t + 1}^{a} ,v_{t + 1}^{{}} } \right] = R_{\text{f}} P\left[ {x_{t}^{a,} ,v_{t} } \right] + \left[ {\varepsilon_{1,t + 1} ,\varepsilon_{2,t + 1} } \right]$ and,

• (iii) Assuming that the supranormal results are defined as:

  $$R_{\text{f}}^{ - \tau } E_{t} \left( {x_{t + \tau }^{a} } \right) = \left[ {1,0} \right]P^{\tau } \left[ {x_{t}^{a} ,v_{t} } \right].$$

Based on the Residual Income Valuation Model (RIV), the expression assumes:

$$P_{t} - {\text{bv}}_{t} = \sum\limits_{\tau = 1}^{\infty } {R_{\text{f}}^{ - \tau } E_{t} \left( {x_{t + \tau }^{a} } \right) = \left[ {1,0} \right]\;\left[ {P + P^{2} + \ldots } \right]\;\left[ {x_{t}^{a} ,v_{t} } \right]} .$$

A series of matrices P + P² + …are convergent, given the square root of the characteristic of the matrix is less than unit. Thus, we could conclude that:

$$\begin{aligned} & \left[ {1,0} \right]P\left[ {I - P} \right]^{ - 1} = \left[ {\alpha_{1} ,\alpha_{2} } \right],\;{\text{i}}.{\text{e}}.{:} \\ & \left[ {1,0} \right]\;\left[ {R_{\text{f}}^{ - 1} \left[ {\begin{array}{*{20}c} w & 1 \\ 0 & \gamma \\ \end{array} } \right]} \right]\;\left[ {\left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right] - R_{\text{f}}^{ - 1} \left[ {\begin{array}{*{20}c} w & 1 \\ 0 & \gamma \\ \end{array} } \right]} \right]^{ - 1} = \left[ {\alpha_{1} ,\alpha_{2} } \right] \\ & \left[ {\begin{array}{*{20}c} {R_{\text{f}}^{ - 1} w} & {R_{f}^{ - 1} } \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {1 - R_{\text{f}}^{ - 1} w} & { - R_{\text{f}}^{ - 1} } \\ 0 & {1 - R_{\text{f}}^{ - 1} \gamma } \\ \end{array} } \right]^{ - 1} = \left[ {\alpha_{1} ,\alpha_{2} } \right] \\ & \left[ {\begin{array}{*{20}c} {R_{\text{f}}^{ - 1} w} & {R_{\text{f}}^{ - 1} } \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {\frac{1}{{1 - R_{\text{f}}^{ - 1} w}}} & {\frac{{R_{\text{f}}^{ - 1} }}{{\left( {1 - R_{\text{f}}^{ - 1} w} \right)\left( {1 - R_{\text{f}}^{ - 1} \gamma } \right)}}} \\ 0 & {\frac{1}{{1 - R_{\text{f}}^{ - 1} \gamma }}} \\ \end{array} } \right] = \left[ {\alpha_{1} ,\alpha_{2} } \right] \\ & \left[ {\begin{array}{*{20}c} {\frac{{R_{\text{f}}^{ - 1} w}}{{1 - R_{\text{f}}^{ - 1} w}}} & {\frac{{R_{\text{f}}^{ - 1} wR_{\text{f}}^{ - 1} }}{{\left( {1 - R_{\text{f}}^{ - 1} w} \right)\left( {1 - R_{\text{f}}^{ - 1} \gamma } \right)}} + \frac{{R_{\text{f}}^{ - 1} }}{{1 - R_{\text{f}}^{ - 1} \gamma }}} \\ \end{array} } \right] = \left[ {\alpha_{1,} \alpha_{2} } \right] \\ & \left\{ {\begin{array}{*{20}l} {\alpha_{1} = \frac{{R_{\text{f}}^{ - 1} w}}{{1 - R_{\text{f}}^{ - 1} w}} = \frac{{R_{\text{f}}^{ - 1} w}}{{R_{\text{f}}^{ - 1} (R_{\text{f}} - w)}} = \frac{w}{{R_{\text{f}} - w}}} \hfill \\ {\alpha_{2} = \frac{{R_{\text{f}}^{ - 1} wR_{\text{f}}^{ - 1} }}{{\left( {1 - R_{\text{f}}^{ - 1} w} \right)\left( {1 - R_{\text{f}}^{ - 1} \gamma } \right)}} + \frac{{R_{\text{f}}^{ - 1} }}{{1 - R_{\text{f}}^{ - 1} \gamma }} = \frac{{R_{\text{f}}^{ - 1} wR_{\text{f}}^{ - 1} }}{{R_{\text{f}}^{ - 1} (R_{\text{f}} - w)R_{\text{f}}^{ - 1} (R_{\text{f}} - \gamma )}} + \frac{{R_{\text{f}}^{ - 1} }}{{R_{\text{f}}^{ - 1} (R_{\text{f}} - \gamma )}}} \hfill \\ \end{array} } \right. \\ & \left\{ {\begin{array}{*{20}l} {\alpha_{1} = \frac{w}{{R_{\text{f}} - w}}} \hfill \\ {\alpha_{2} = \frac{w}{{(R_{\text{f}} - w)(R_{\text{f}} - \gamma )}} + \frac{{R_{\text{f}} - w}}{{(R_{\text{f}} - w)(R_{\text{f}} - \gamma )}} = \frac{{R_{\text{f}} }}{{(R_{\text{f}} - w)(R_{\text{f}} - \gamma )}}} \hfill \\ \end{array} } \right.. \\ \end{aligned}$$

• 2. Deduction of the Eq. 2.9 of the OM Model

Assuming the Eq. 2.8:

$$P_{t} = {\text{bv}}_{t} + \alpha_{1} x_{t}^{a} + \alpha_{2} v_{t} ,$$

and substituting $x_{t}^{a}$, we obtain,

$$P_{t} = y_{t} + \alpha_{1} \left[ {x_{t} - \left( {R_{\text{f}} - 1} \right)y_{t - 1} } \right] + \alpha_{2} v{}_{t}.$$

Due the CSR principle:

$$\begin{aligned} P_{t} & = {\text{bv}}_{t} + \alpha_{1} x_{t} - \alpha_{1} (R_{\text{f}} - 1)({\text{bv}}_{t} - x_{t} + d_{t} ) + \alpha_{2} v_{t} \Leftrightarrow \\ P_{t} & = {\text{bv}}_{t} + \alpha_{1} x_{t} - \alpha_{1} \left( {R_{\text{f}} - 1} \right){\text{bv}}_{t} + \alpha_{1} \left( {R_{\text{f}} - 1} \right)x_{t} - \alpha_{1} \left( {R_{\text{f}} - 1} \right)d_{t} + \alpha_{2} v_{t} \Leftrightarrow , \\ \end{aligned}$$

$P_{t} = {\text{bv}}_{t} \left[ {1 - \alpha_{1} (R_{\text{f}} - 1)} \right] + \alpha_{2} v_{t} + \alpha_{1} x_{t} + \alpha_{1} (R_{\text{f}} - 1)x_{t} - \alpha_{1} (R_{\text{f}} - 1)d_{t}$ and $\kappa = \alpha_{1} \left( {R{}_{\text{f}} - 1} \right)$ thus,

$P_{t} = {\text{bv}}_{t} (1 - k) + \alpha_{2} v_{t} + \alpha_{1} \left[ {x_{t} + (R_{\text{f}} - 1)x_{t} - (R_{\text{f}} - 1)d_{t} } \right]$, $P_{t} = (1 - k){\text{bv}}_{t} + \alpha_{2} v_{t} + \alpha_{1} R_{\text{f}} x_{t} - \alpha_{1} (R_{\text{f}} - 1)d_{t}$, and $\varphi = \frac{{R_{\text{f}} }}{{R_{\text{f}} - 1}}$, we obtain:

$$P_{t} = \kappa \left( {\varphi x_{t} - d_{t} } \right) + \left( {1 - \kappa } \right){\text{bv}}_{t} + \alpha_{2} v_{t} .$$

Appendix 2.2: Deduction of the Equivalence of the Preposition Number 1 of Feltham and Ohlson (FOM) (1995) Model and Eqs. 2.15a, 2.15b and 2.15c

• (i) Equation 2.15a: $P_{t} = {\text{fa}}_{t} + \sum\nolimits_{\tau = 1}^{\infty } {R_{\text{f}}^{ - \tau } E_{t} \left( {c_{t + \tau } } \right)}$

  Assuming the FAR [fa _t  = fa _t−1 + i _{t   −}  (d _t  − c _t )] and the definition of i _t [i _t  = (R _f − 1)fa _t−1], the dividends could be expressed as:

  $$d_{t} = {\text{fa}}_{t - 1} + i_{t} + c_{t} - {\text{fa}}_{t} = {\text{fa}}_{t - 1} + (R_{\text{f}} - 1){\text{fa}}_{t - 1} + c_{t} - {\text{fa}}_{t} = R_{\text{f}} {\text{fa}}_{t - 1} + c_{t} - {\text{fa}}_{t.}$$

  Hence for any sequence of the variables c _t e fa _t ({c _t+τ , fa _t+τ } _τ≥1), valuation function is:

  $$P_{t} = \sum\limits_{\tau = 1}^{\infty } {R_{\text{f}}^{ - \tau } E_{t} \left( {d_{t + \tau } } \right)} = \sum\limits_{\tau = 1}^{\infty } {R_{\text{f}}^{ - \tau } E{}_{t}\left[ {R_{\text{f}} {\text{fa}}_{t + 1 - \tau } + c_{t + \tau } - {\text{fa}}_{t + \tau } } \right]} = {\text{fa}}_{t} + \sum\limits_{\tau = 1}^{\infty } {R_{\text{f}}^{ - \tau } E_{t} \left( {c_{t + \tau } } \right)} ,$$

  because $R_{\text{f}}^{ - \tau } E_{t} \left( {{\text{fa}}_{t + \tau } } \right) \to 0\;com\;\tau \to \infty$.

• (ii) Equation 2.15b: $P_{t} = {\text{bv}}_{t} + \sum\nolimits_{\tau = 1}^{\infty } {R_{\text{f}}^{ - \tau } E_{t} \left( {x_{t + \tau }^{a} } \right)}$

  This equations corresponds to the Residual Income valuation Model defines in Eq. 2.6.

• (iii) Equation 2.15c: $P_{t} = {\text{bv}}_{t} + \sum\nolimits_{\tau = 1}^{\infty } {R_{\text{f}}^{ - \tau } E_{t} \left( {ox_{t + \tau }^{a} } \right)}$

  Assuming operational and non-operational activities, the following expression corresponds to the operational abnormal results: ${\text{ox}}_{t}^{a} = {\text{ox}}_{t} - \left( {R_{\text{f}} - 1} \right){\text{oa}}_{t - 1}$, applying the OAR we obtain:

  $$\begin{aligned} {\text{oa}}_{t}& = {\text{oa}}_{t - 1} + {\text{ox}}_{t} - c_{t} , {\text{substituting the variable}} \; {\text{ox}}_{t}\\ c_{t} & = {\text{oa}}_{t - 1} + \left[ {{\text{ox}}_{t}^{a} + (R_{\text{f}} - 1){\text{oa}}{}_{t - 1}} \right] - {\text{oa}}_{t} \Leftrightarrow \\ c_{t} & = {\text{ox}}_{t}^{a} + R_{\text{f}} {\text{oa}}{}_{t - 1} - {\text{oa}}_{t} . \\ \end{aligned}$$

  For any sequences of the variables $\left( {\left\{ {{\text{ox}}_{t + \tau }^{a} ,\left. {{\text{oa}}_{t + \tau - 1} } \right\}_{\tau \ge 1} } \right.} \right)$, we obtain:

  $$\sum\limits_{\tau = 1}^{\infty } {R_{\text{f}}^{ - \tau } E_{t} \left( {c_{t + \tau } } \right) = \sum\limits_{\tau = 1}^{\infty } {R_{\text{f}}^{ - \tau } E_{t} \left( {{\text{ox}}_{t + \tau }^{a} + R_{\text{f}} {\text{oa}}_{t - 1 + \tau } - {\text{oa}}_{t + \tau } } \right)} = {\text{oa}}_{t} + \sum\limits_{\tau = 1}^{\infty } {R_{\text{f}}^{ - \tau } E_{t} \left( {{\text{ox}}_{t + \tau }^{a} } \right)} } ,$$

  thus, $R_{\text{f}}^{ - \tau } E_{t} \left( {{\text{oa}}_{t + \tau } } \right) \to 0\;com\;\tau \to \infty .$

  If we add fa _t to the above expression, and due the Eq. 2.15a, the valuation function is:

  $$P_{t} = \sum\limits_{\tau = 1}^{\infty } {R_{\text{f}}^{ - \tau } E{}_{t}\left( {d_{t + \tau } } \right) = ({\text{oa}}_{t} + {\text{fa}}_{t} ) + \sum\limits_{\tau = 1}^{\infty } {{\text{ox}}_{t + \tau }^{a} } } = {\text{bv}}_{t} + \sum\limits_{\tau = 1}^{\infty } {{\text{ox}}_{t + \tau }^{a} } .$$

Appendix 2.3: Deduction of the Feltham and Ohlson (1995)—Eq. 2.22

Assuming the definition of goodwill:

$$g_{t} = P_{t} - {\text{bv}}_{t}$$

Multiplying the above expression by R _f and according Ohlson (2000):

$$R_{\text{f}} g_{t} = E_{t} \left[ {g_{t + 1} + {\text{ox}}_{t + 1}^{a} } \right].$$

g _t could be defined as a linear function, such as:

$$P_{t} - {\text{bv}}_{t} = g_{t} = \alpha_{1} {\text{ox}}_{t}^{a} + \alpha_{2} {\text{oa}}_{t} + \beta \cdot v_{t}$$

with

$$\begin{aligned} & R_{\text{f}} g_{t} = R{}_{\text{f}}\left[ {\alpha_{1} {\text{ox}}_{t}^{a} + \alpha_{2} {\text{oa}}_{t} + \beta \cdot v_{t} } \right] = E_{t} \left[ {\alpha_{1} {\text{ox}}_{t + 1}^{a} + \alpha_{2} {\text{oa}}_{t} + \beta \cdot v_{t} + {\text{ox}}_{t + 1}^{a} } \right] \Leftrightarrow \\ & R_{\text{f}} \alpha_{1} {\text{ox}}_{t}^{a} + R_{\text{f}} \alpha_{2} {\text{oa}}_{t} + R_{\text{f}} \beta \cdot v_{t} = E_{t} \left[ {(\alpha_{1} + 1){\text{ox}}_{t + 1}^{a} + \alpha_{2} {\text{oa}}_{t + 1} + \beta v_{t + 1} } \right]. \\ \end{aligned}$$

Due the structure of Linear Information Model (LIM):

$$R_{\text{f}} \alpha_{1} {\text{ox}}_{t}^{a} + R_{\text{f}} \alpha_{2} {\text{oa}}_{t} + R_{\text{f}} \beta_{1} v_{1t} + R_{\text{f}} \beta_{2} v_{2t} = (\alpha_{1} + 1)E_{t} ({\text{ox}}_{t + 1}^{a} ) + \alpha_{2} E_{t} ({\text{oa}}_{t + 1} ) + \beta E_{t} (v_{t + 1} )$$

Substituting the expected values according the dynamic information:

$$\begin{aligned} R_{\text{f}} \alpha_{1} {\text{ox}}_{t}^{a} + R_{\text{f}} \alpha_{2} {\text{oa}}_{t} + R_{\text{f}} \beta_{1} v_{1t} + R_{\text{f}} \beta_{2} v_{2t} & = (\alpha_{1} + 1)(w_{11} {\text{ox}}_{t}^{a} + w_{12} {\text{oa}}_{t} + v_{1t} ) \\ & \quad + \alpha_{2} (w_{22} {\text{oa}}_{t} + v_{2t} ) + \beta_{1} \gamma_{1} v_{1t} + \beta_{2} \gamma_{2} v_{2t} \\ \end{aligned}$$

Solving the equation based on that the probability should be one, thus:

$$\begin{aligned} & \left\{ {\begin{array}{*{20}l} {R_{\text{f}} \alpha_{1} = (\alpha_{1} + 1)w_{11} } \hfill \\ {R_{\text{f}} \alpha_{2} = (\alpha_{1} + 1)w_{12} + \alpha_{2} w_{22} } \hfill \\ {R_{\text{f}} \beta_{1} = (\alpha_{1} + 1) + \beta_{1} \gamma_{1} } \hfill \\ {R_{\text{f}} \beta_{2} = \alpha_{2} + \beta_{2} \gamma_{2} } \hfill \\ \end{array} } \right. \Leftrightarrow \left\{ {\begin{array}{*{20}l} {R_{\text{f}} \alpha_{1} = \alpha_{1} w_{11} + w_{11} } \hfill \\ {R_{\text{f}} \alpha_{2} - \alpha_{2} w_{22} = (\alpha_{1} + 1)w_{12} } \hfill \\ {R_{\text{f}} \beta_{1} - \beta_{1} \gamma_{1} = (\alpha_{1} + 1)} \hfill \\ {R_{\text{f}} \beta_{2} - \beta_{2} \gamma_{2} = \alpha_{2} } \hfill \\ \end{array} } \right. \Leftrightarrow \left\{ {\begin{array}{*{20}l} {\alpha {}_{1}(R_{\text{f}} - w_{11} ) = w_{11} } \hfill \\ {\alpha_{2} (R_{\text{f}} - w_{22} ) = (\alpha_{1} + 1)w_{12} } \hfill \\ {\beta_{1} (R_{\text{f}} - \gamma_{1} ) = (\alpha_{1} + 1)} \hfill \\ {\beta_{2} (R_{\text{f}} - \gamma_{2} ) = \alpha_{2} } \hfill \\ \end{array} } \right. \\ & \Leftrightarrow \left\{ {\begin{array}{*{20}l} {\alpha_{1} = \frac{{w_{11} }}{{R_{\text{f}} - w_{11} }}} \hfill \\ {\alpha_{2} (R_{\text{f}} - w_{22} ) = (\frac{{w_{11} }}{{R_{\text{f}} - w_{11} }} + 1)w_{12} } \hfill \\ {\beta_{1} (R_{\text{f}} - \gamma_{1} ) = \frac{{w_{11} }}{{R_{\text{f}} - w_{11} }} + 1} \hfill \\ {\beta_{2} (R_{\text{f}} - \gamma_{2} ) = \alpha_{2} } \hfill \\ \end{array} } \right. \Leftrightarrow \left\{ {\begin{array}{*{20}l} {\alpha_{1} = \frac{{w_{11} }}{{R_{\text{f}} - w_{11} }}} \hfill \\ {\alpha_{2} (R_{\text{f}} - w_{22} ) = (\frac{{w_{11} + R_{\text{f}} - w_{11} }}{{R_{\text{f}} - w_{11} }})w_{12} } \hfill \\ {\beta_{1} (R_{\text{f}} - \gamma_{1} ) = \frac{{w_{11} + R_{\text{f}} - w_{11} }}{{R_{\text{f}} - w_{11} }}} \hfill \\ {\beta_{2} = \frac{{\alpha_{2} }}{{R{}_{\text{f}} - \gamma_{2} }}} \hfill \\ \end{array} } \right. \\ & \Leftrightarrow \left\{ {\begin{array}{*{20}l} {\alpha_{1} = \frac{{w_{11} }}{{R_{\text{f}} - w_{11} }}} \hfill \\ {\alpha_{2} = \frac{{R_{\text{f}} w_{12} }}{{\left( {R_{\text{f}} - w_{11} } \right)\left( {R_{\text{f}} - w_{22} } \right)}}} \hfill \\ {\beta_{1} = \frac{{R_{f} }}{{\left( {R_{\text{f}} - w_{11} } \right)\left( {R_{\text{f}} - \gamma_{1} } \right)}}} \hfill \\ {\beta_{2} = \frac{{\alpha_{2} }}{{R_{\text{f}} - \gamma_{2} }}} \hfill \\ \end{array} } \right. .\\ \end{aligned}$$