Residual income valuation and management remuneration under uncertainty: a note

Abstract

The so-called Residual Income Valuation theorem states that the value of a project or a firm can be determined either on the basis of cash flows between the firm and its owners or by using residual incomes, provided that cash flows and residual incomes are derived from a set of accounting data that fulfills certain regularity conditions. Residual income is defined as accounting earnings reduced by a capital charge on book equity capital. In this paper it is shown that this theorem also applies when residual incomes and in particular the discount factors are uncertain. Risk-aversion of principals and agents is taken into account on the basis of properly defined risk-adjusted discount rates. This approach is preferred as it facilitates practical application. Implications are drawn with regards to valuation but also to the design of management remuneration systems. It is shown that the capital charge rate used to determine the performance-related compensation component should be reduced below the risk-adjusted rate, if the fixed component falls below a certain threshold. Absent agency cost or other externalities, the reduction of the capital charge rate is required to avoid underinvestment.

Appendices

Appendix 1

1.1 1.1 Derivation of multi-period valuation models by forward iteration

The following analysis is used to provide background to three equations used in the text, namely Eqs. 8, 10, and 27. The starting point is the single-period SDF which is based on assumption 2.

$$ \tilde{V}_{t} = E_{t} \left[ {\left( {\tilde{C}_{t + 1} + \tilde{V}_{t + 1} } \right)\tilde{M}_{t + 1} } \right] $$

In a forward-iteration procedure the definition of $\tilde{V}_{t}$ is used and inserted similarly for $\tilde{V}_{t + 1}$ as well:⁴³

$$ \tilde{V}_{t} = E_{t} \left[ {\left( {\tilde{C}_{t + 1} + E_{t + 1} \left[ {\left( {\tilde{C}_{t + 2} + \tilde{V}_{t + 2} } \right)\tilde{M}_{t + 2} } \right]} \right)\tilde{M}_{t + 1} } \right] $$

As $\tilde{M}_{t + 1}$ will be known at t + 1, it can be integrated into the expectation operator as of that point in time. The two single-period discount factors $\tilde{M}_{t + 1}$ and $\tilde{M}_{t + 2}$ then can be aggregated to $\tilde{M}_{t,t + 2}$:

$$ \begin{aligned} \tilde{V}_{t} & = E_{t} \left[ {\tilde{C}_{t + 1} \tilde{M}_{t + 1} + E_{t + 1} \left[ {\left( {\tilde{C}_{t + 2} + \tilde{V}_{t + 2} } \right)\tilde{M}_{t + 2} \tilde{M}_{t + 1} } \right]} \right] \\ = E_{t} \left[ {\tilde{C}_{t + 1} \tilde{M}_{t + 1} + E_{t + 1} \left[ {\left( {\tilde{C}_{t + 2} + \tilde{V}_{t + 2} } \right)\tilde{M}_{t,t + 2} } \right]} \right] \\ \end{aligned} $$

Application of the law of iterated expectations leads to the following interim result:

$$ \tilde{V}_{t} = E_{t} \left[ {\tilde{C}_{t + 1} \tilde{M}_{t + 1} } \right] + E_{t} \left[ {\left( {\tilde{C}_{t + 2} + \tilde{V}_{t + 2} } \right)\tilde{M}_{t,t + 2} } \right] $$

Continuation of this procedure for T periods in total leads to the multi-period valuation formula as shown in Eq. 8.

$$ \tilde{V}_{t} = \sum\limits_{\tau = 1}^{T} {E_{t} \left[ {\tilde{C}_{t + \tau } \tilde{M}_{t,t + \tau } } \right] + E_{t} \left[ {\tilde{V}_{t + T} \tilde{M}_{t,t + T} } \right]} $$

The starting point of this analysis is set by assumption 2. This is not the case for the next valuation equation (i.e., Eq. 10), which is supposed to be equivalent to the SDF result and facilitated by using a modification of the starting point:

$$ \tilde{V}_{t} = E_{t} \left[ {\left( {\tilde{C}_{t + 1} + \tilde{V}_{t + 1} } \right)\tilde{M}_{t + 1} } \right] \Leftrightarrow 1 = E_{t} \left[ {{\frac{{\tilde{C}_{t + 1} + \tilde{V}_{t + 1} }}{{\tilde{V}_{t} }}}\tilde{M}_{t + 1} } \right] $$

The right-hand side states that the joint expectation of the gross rate of return and the SDF equals one. But $\tilde{M}_{t + 1}$ is not the only functional to achieve this. Trivially, $\tilde{D}_{t + 1} = \left( {1 + \tilde{R}_{t + 1} } \right)^{ - 1}$ constitutes also a valid discount factor, which leads to the value of one in any state of nature and thereby also on average.

$$ 1 = E_{t} \left[ {{\frac{{\tilde{C}_{t + 1} + \tilde{V}_{t + 1} }}{{\tilde{V}_{t} }}}\tilde{D}_{t + 1} } \right] \Leftrightarrow \tilde{V}_{t} = E_{t} \left[ {\left( {\tilde{C}_{t + 1} + \tilde{V}_{t + 1} } \right)\tilde{D}_{t + 1} } \right] $$

With the right-hand side of the above expression we have a new starting point for the forward iteration procedure. This leads to Eq. 10, which uses the firm-specific discount factors instead of the SDF:

$$ \tilde{V}_{t} = \sum\limits_{\tau = 1}^{T} {E_{t} \left[ {\tilde{C}_{t + \tau } \tilde{D}_{t,t + \tau } } \right] + E_{t} \left[ {\tilde{V}_{t + T} \tilde{D}_{t,t + T} } \right]} $$

The starting point for Eq. 27 is again a variation of this theme.

$$ 1 = E_{t} \left[ {{\frac{{\tilde{C}_{t + 1} + \tilde{B}_{t + 1} }}{{\tilde{B}_{t} }}}\tilde{H}_{t + 1} } \right] $$

Here, economic values are replaced by book values and a discount factor is used, which is based on accounting values. The product within the expectation operator is always one, as is in the previous case.

The following DCF model follows by the application of the forward iteration procedure, too:

$$ \tilde{B}_{t} = \sum\limits_{\tau = 1}^{T} {E_{t} \left[ {\tilde{C}_{t + \tau } \tilde{H}_{t,t + \tau } } \right] + E_{t} \left[ {\tilde{B}_{t + T} \tilde{H}_{t,t + T} } \right]} $$

In contrast to the two previous market-value versions, this model yields the book value of capital.

Appendix 2

2.1 2.1 Proof of the multi-period formulation of Eq. 33 by backward iteration

The multi-period value of the agent’s compensation can be formulated as follows:

$$ \tilde{S}_{t} = E_{t} \left[ {\left( {\alpha + \beta \tilde{\pi }_{t + 1} + \tilde{S}_{t + 1} } \right)\tilde{M}_{t + 1} } \right] $$

Without loss of generality we assume a constant shadow rate r. The compensation of the period t + 1 can be valued separately from the payments afterwards:

$$ \begin{aligned} \tilde{S}_{t} & = {\frac{\alpha }{1 + r}} + \beta E_{t} \left[ {\tilde{\pi }_{t + 1} \tilde{M}_{t + 1} } \right] + E_{t} \left[ {\left( {\alpha + \beta \tilde{\pi }_{t + 2} + \tilde{S}_{t + 2} } \right)\tilde{M}_{t,t + 2} } \right] \\ = \sum\limits_{\tau = 1}^{2} {{\frac{\alpha }{{\left( {1 + r} \right)^{\tau } }}}} + \beta \sum\limits_{\tau = 1}^{2} {E_{t} \left[ {\tilde{\pi }_{t + \tau } \tilde{M}_{t,t + \tau } } \right] + E_{t} \left[ {\tilde{S}_{t + 2} \tilde{M}_{t,t + 2} } \right]} \\ \end{aligned} $$

Continuation of this procedure for T periods in total leads with $\tilde{S}_{t + T} = 0$ to

$$ \tilde{S}_{t} = \sum\limits_{\tau = 1}^{T} {{\frac{\alpha }{{\left( {1 + r} \right)^{\tau } }}}} + \beta \sum\limits_{\tau = 1}^{T} {E_{t} \left[ {\tilde{\pi }_{t + \tau } \tilde{M}_{t,t + \tau } } \right]} , $$

which equals a constant (the value of the fixed salary payments) and a fraction of the value of future residual incomes. Next the value of residual incomes needs to be determined on the basis of the SDF:

$$ \begin{aligned} \sum\limits_{\tau = 1}^{T} {E_{t} \left[ {\tilde{\pi }_{t + \tau } \tilde{M}_{t,t + \tau } } \right]} & = \sum\limits_{\tau = 1}^{T} {E_{t} \left[ {\left( {\tilde{C}_{t + \tau } - \left( {1 + \tilde{R}_{t + \tau } } \right)\tilde{B}_{t + \tau - 1} + \tilde{B}_{t + \tau } } \right)\tilde{M}_{t,t + \tau } } \right]} \\ = \tilde{V}_{t} - \sum\limits_{\tau = 1}^{T} {E_{t} \left[ {\left( {1 + \tilde{R}_{t + \tau } } \right)\tilde{B}_{t + \tau - 1} \tilde{M}_{t,t + \tau } } \right]} + \sum\limits_{\tau = 1}^{T} {E_{t} \left[ {\tilde{B}_{t + \tau } \tilde{M}_{t,t + \tau } } \right]} \\ \end{aligned} $$

We define the partial value of the first sum above by $\tilde{W}_{t}$, which can be determined by recursion, starting in $t + T - 1$, i.e., one period before liquidation:

$$ \tilde{W}_{t + T - 1} = E_{t + T - 1} \left[ {\left( {1 + \tilde{R}_{t + T} } \right)\tilde{B}_{t + T - 1} \tilde{M}_{t + T} } \right] = \tilde{B}_{t + T - 1} $$

Moving one further period back in time to t + T − 2 leads to:

$$ \begin{aligned} \tilde{W}_{t + T - 2} & = E_{t + T - 2} \left[ {\left( {1 + \tilde{R}_{t + T - 1} } \right)\tilde{B}_{t + T - 2} \tilde{M}_{t + T - 1} } \right] + E_{T - 2} \left[ {\tilde{W}_{t + T - 1} \tilde{M}_{t + T - 1} } \right] \\ = \tilde{B}_{t + T - 2} + E_{T - 2} \left[ {\tilde{B}_{t + T - 1} \tilde{M}_{t + T - 1} } \right] \\ \end{aligned} $$

in t + T − 3:

$$ \begin{aligned} \tilde{W}_{t + T - 3} & = \tilde{B}_{t + T - 3} + E_{t + T - 3} \left[ {\left( {\tilde{B}_{t + T - 2} + \tilde{B}_{t + T - 1} \tilde{M}_{t + T - 1} } \right)\tilde{M}_{t + T - 2} } \right] \\ = \tilde{B}_{t + T - 3} + E_{t + T - 3} \left[ {\tilde{B}_{t + T - 2} \tilde{M}_{t + T - 2} } \right] + E_{t + T - 3} \left[ {\tilde{B}_{t + T - 1} \tilde{M}_{t + T - 3,t + T - 1} } \right] \\ \end{aligned} $$

and finally in t:

$$ \tilde{W}_{t} = \tilde{B}_{t} + \sum\limits_{\tau = 1}^{T - 1} {E_{t} \left[ {\tilde{B}_{t + \tau } \tilde{M}_{t,t + \tau } } \right]} $$

Collecting terms:

$$ \begin{aligned} \sum\limits_{\tau = 1}^{T} {E_{t} \left[ {\tilde{\pi }_{t + \tau } \tilde{M}_{t,t + \tau } } \right]} & = \tilde{V}_{t} - B_{t} - \sum\limits_{\tau = 1}^{T - 1} {E_{t} \left[ {\tilde{B}_{t + \tau } \tilde{M}_{t,t + \tau } } \right]} + \sum\limits_{\tau = 1}^{T} {E_{t} \left[ {\tilde{B}_{t + \tau } \tilde{M}_{t,t + \tau } } \right]} \\ = \tilde{V}_{t} - \tilde{B}_{t} + E_{t} \left[ {\tilde{B}_{t + T} \tilde{M}_{t,t + T} } \right] \\ \end{aligned} $$

with $\tilde{B}_{t + T} = 0$ we finally get:

$$ \tilde{S}_{t} = \sum\limits_{\tau = 1}^{T} {{\frac{\alpha }{{\left( {1 + r} \right)^{\tau } }}}} + \beta \left( {\tilde{V}_{t} - \tilde{B}_{t} } \right) $$

Thus, the value of the total compensation is proportional to the value of the firm, which is the parameter the principal seeks to maximize as well.

2.2 2.2 Proof of the multi-period formulation of Eq. 35 by forward iteration

Next we proof that the capital charge rate has to be the risk-free rate also for multi-period projects. For single-period projects the remuneration has the following value to the agent:

$$ \tilde{S}_{t} = \beta E_{t} \left[ {\left( {\tilde{C}_{t + 1} - \left( {1 + k} \right)I_{t} } \right)\tilde{M}_{t + 1} } \right] \Rightarrow \beta E_{t} \left[ {\tilde{C}_{t + 1} \tilde{M}_{t + 1} } \right] - {\frac{1 + k}{1 + r}}I_{t} > 0 $$

A two-period project has the cash flows $\tilde{C}_{t + 1}$ and $\tilde{C}_{t + 2}$. As the depreciation schedule is not relevant for the RIV theorem to hold we choose no depreciation in the first period and full depreciation in the last (second) period. Thus, the criterion for the agent to support a project becomes

$$ \begin{gathered} E_{t} \left[ {\left( {\tilde{C}_{t + 1} - kI_{t} } \right)\tilde{M}_{t + 1} } \right] + E_{t} \left[ {\left( {\tilde{C}_{t + 2} - (1 + k)I_{t} } \right)\tilde{M}_{t,t + 2} } \right] > 0 \quad \hfill \\ \Leftrightarrow \mathop \sum \limits_{\tau = 1}^{2} E_{t} \left[ {\tilde{C}_{t + \tau } \tilde{M}_{t,t + \tau } } \right] - {\frac{k}{1 + r}}I_{t} - {\frac{1 + k}{{\left( {1 + r} \right)^{2} }}}I_{t} > 0. \hfill \\ \end{gathered} $$

Continuation of this procedure for T periods in total yields:

$$ \begin{gathered} \sum\limits_{\tau = 1}^{T} {E_{t} \left[ {\tilde{C}_{t + \tau } \tilde{M}_{t,t + \tau } } \right]} - \sum\limits_{\tau = 1}^{T - 1} {{\frac{k}{{(1 + r)^{\tau } }}}I_{t} - {\frac{1 + k}{{(1 + r)^{T} }}}I_{t} > 0} \quad \hfill \\ \Leftrightarrow \tilde{V}_{t} - I_{t} \left( {\sum\limits_{\tau = 1}^{T - 1} {{\frac{k}{{(1 + r)^{\tau } }}} + {\frac{1 + k}{{(1 + r)^{T} }}}} } \right) > 0 \hfill \\ \end{gathered} $$

To establish the proof it remains to show that for k = r

$$ \sum\limits_{\tau = 1}^{T - 1} {{\frac{k}{{(1 + r)^{\tau } }}} + {\frac{1 + k}{{(1 + r)^{T} }}} = 1} $$

holds. This follows by algebraic manipulation:

$$ \begin{gathered} \sum\limits_{\tau = 1}^{T - 1} {{\frac{k}{{(1 + r)^{\tau } }}} + {\frac{1 + k}{{(1 + r)^{T} }}} = 1} \Leftrightarrow \sum\limits_{\tau = 1}^{T} {{\frac{k}{{(1 + r)^{\tau } }}} + {\frac{1}{{(1 + r)^{T} }}} = 1} \hfill \\ \Leftrightarrow \sum\limits_{\tau = 1}^{T} {{\frac{k}{{(1 + r)^{\tau } }}} = 1 - (1 + r)^{ - T} } \hfill \\ \Leftrightarrow \frac{k}{r}(1 - (1 + r)^{ - T} ) = 1 - (1 + r)^{ - T} \Rightarrow k = r \hfill \\ \end{gathered} $$

Thus, proposition 6 is true also for multi-period projects.