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Open Access 2020 | OriginalPaper | Chapter

6. Proofs

Authors: Lutz Kruschwitz, Andreas Löffler

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6.1 Williams/Gordon–Shapiro Formula (Theorem 3.​2) and Equivalence of Valuation Concepts (Theorem 3.​3)

We start with the proof of Theorem 3.​2. From the valuation equation (Theorem 3.​1) and the Assumption 3.​1
\displaystyle \begin{aligned} \widetilde{V}^u_t&=\sum_{s=t+1}^T\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{s}|\mathcal{F}_t\right]}{\left(1+k^{E,u}_{t}\right)\ldots\left(1+k^{E,u}_{s-1}\right)}\\ &=\sum_{s=t+1}^T \frac{(1+g_{t})\ldots(1+g_{s-1})\widetilde{\mathit{FCF}}^u_t}{\left(1+k^{E,u}_{t}\right)\ldots\left(1+k^{E,u}_{s-1}\right)}\\ &=\widetilde{\mathit{FCF}}^u_t\,\underbrace{\sum_{s=t+1}^T \frac{(1+g_{t})\ldots(1+g_{s-1})}{\left(1+k^{E,u}_{t}\right)\ldots\left(1+k^{E,u}_{s-1}\right)}}_{=\frac{1}{d^u_t}}. \end{aligned}
But the last equation says exactly that the firm value is a deterministic multiple of the cash flow. The relation between dividends and price $$d^u_t$$ is deterministic and positive if gt > −1. Using transversality this also applies for T →. Thus the Theorem 3.​2 is proven.
Now we prove Theorem 3.​3. The following results from the definition of the cost of equity of the unlevered firm if t + 1 < T
\displaystyle \begin{aligned} \widetilde{V}^u_t&=\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{t+1}+\widetilde{V}^u_{t+1}|\mathcal{F}_t\right]}{1+k^{E,u}_t} &&\text{by Definition 3.1} {}\\ &=\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{t+1}+(d^u_{t+1})^{-1}\widetilde{\mathit{FCF}}^u_{t+1}|\mathcal{F}_t\right]}{1+k^{E,u}_t} &&\text{by Theorem 3.2}\\ &=\frac{\left(1+(d^u_{t+1})^{-1}\right)\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{t+1}|\mathcal{F}_t\right]}{1+k^{E,u}_t} . \end{aligned}
(6.1)
The following is likewise valid if t + 1 < T
\displaystyle \begin{aligned} \widetilde{V}^u_t&=\frac{\operatorname*{\mathrm{E}}\nolimits_Q\left[\widetilde{\mathit{FCF}}^u_{t+1}+\widetilde{V}^u_{t+1}|\mathcal{F}_t\right]}{1+r_f} &&\text{by Theorem 2.2} {}\\ &=\frac{\left(1+(d^u_{t+1})^{-1}\right)\operatorname*{\mathrm{E}}\nolimits_Q\left[\widetilde{\mathit{FCF}}^u_{t+1}|\mathcal{F}_t\right]}{1+r_f}. &&\text{by Theorem 3.2} \end{aligned}
(6.2)
The comparison of both terms results in
\displaystyle \begin{aligned} \frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{t+1}|\mathcal{F}_t\right]}{1+k^{E,u}_t}=\frac{\operatorname*{\mathrm{E}}\nolimits_Q\left[\widetilde{\mathit{FCF}}^u_{t+1}| \mathcal{F}_t\right]}{1+r_f}\, \end{aligned}
and this also holds for t + 1 = T from transversality. And that is already the proposition of the theorem for s = t + 1.
We go back to Eqs. (6.1) and (6.2) and remove the terms already shown to be identical. There then remains
\displaystyle \begin{aligned} \frac{\operatorname*{\mathrm{E}}\left[\widetilde{V}^u_{t+1}|\mathcal{F}_t\right]}{1+k^{E,u}_t}=\frac{\operatorname*{\mathrm{E}}\nolimits_Q\left[\widetilde{V}^u_{t+1}|\mathcal{F}_t\right]}{1+r_f} \end{aligned}
or
\displaystyle \begin{aligned} \frac{\operatorname*{\mathrm{E}}\left[\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{t+2}+\widetilde{V}^u_{t+2}|\mathcal{F}_{t+1}\right]}{1+k^{E,u}_{t+1}}| \mathcal{F}_t\right]}{1+k^{E,u}_t}=\frac{\operatorname*{\mathrm{E}}\nolimits_Q\left[ \frac{\operatorname*{\mathrm{E}}\nolimits_Q\left[\widetilde{\mathit{FCF}}^u_{t+2}+\widetilde{V}^u_{t+2}|\mathcal{F}_{t+1}\right]}{1+r_f}|\mathcal{F}_t\right]}{1+r_f}. \end{aligned}
The law of the iterated expectation as well as the fact that the dividend-price relation is deterministic establishes
\displaystyle \begin{aligned} \frac{\operatorname*{\mathrm{E}}\left[(1+(d_{t+2}^u)^{-1})\widetilde{\mathit{FCF}}^u_{t+2}|\mathcal{F}_t\right]}{(1+k^{E,u}_{t})(1+k^{E,u}_{t+1})}=\frac{\operatorname*{\mathrm{E}}\nolimits_Q\left[ (1+(d^u_{t+2})^{-1})\widetilde{\mathit{FCF}}^u_{t+2}|\mathcal{F}_t\right]}{(1+r_f)^2}. \end{aligned}
After shortening of $$\left (1+(d_{t+2}^u)^{-1}\right )$$, that is the claim of the theorem for s = t + 2. The propositions for s = t + 3, … can now be proven analogously.

6.2 Valuation Formula with Investment Policy Based on Cash Flows (Theorem 3.​21)

With the following proof you have to make an effort to keep an overview. We begin with showing the difference between investments and accruals. Because of Definition 3.​10 and because there are only non-discretionary accruals, for the time being we could write
\displaystyle \begin{aligned}\widetilde{\mathit{Inv}}_t-\widetilde{\mathit{Accr}}_t = \widetilde{\mathit{Inv}}_t - \frac{1}{n}\left(\widetilde{\mathit{Inv}}_{t-1} + \ldots + \widetilde{\mathit{Inv}}_{t-n} \right)\,.\end{aligned}
Now it makes sense to use Definition 3.​13 and replace $$\widetilde {\mathit {Inv}}_{s}$$ with $$\alpha _{s} \widetilde {\mathit {FCF}}^u_{s}$$. That, however, fails in that for investment amounts that are not in the future, we have to take historical real numbers and can only use free cash flows in relation to future investments. With
\displaystyle \begin{aligned}\widetilde{H}_{s} = \left\{ \begin{array}{ll} \alpha_{s} \widetilde{\mathit{FCF}}^u_{s}, & \mbox{if }s>0 \\ \mathit{Inv}_{s}, & \mbox{else} \\ \end{array} \right. \end{aligned}
we get for all t ≥ 1 equation
\displaystyle \begin{aligned} \widetilde{\mathit{Inv}}_t-\widetilde{\mathit{Accr}}_t=\widetilde{H}_t-\frac{1}{n}\left(\widetilde{H}_{t-1}+\ldots+\widetilde{H}_{t-n} \right). \end{aligned}
(6.3)
Taking advantage of this relation, we get the following for the firm’s book value using Theorem 3.​17 (operating assets relation) and Assumption 3.​9 (subscribed capital) as well as Eq. (3.​28)
\displaystyle \begin{aligned} \widetilde{\underline{V}}^l_t &= \underline{V}^l_0+\underline{e}^l_{0,1}+\ldots+\underline{e}^l_{t-1,t} +\left(\widetilde{\mathit{Inv}}_1-\widetilde{\mathit{Accr}}_1\right)+\ldots+\left(\widetilde{\mathit{Inv}}_t-\widetilde{\mathit{Accr}}_t\right) \\ &= \underline{V}^l_0+\underline{e}^l_{0,t}+\left(\widetilde{\mathit{Inv}}_1-\widetilde{\mathit{Accr}}_1\right)+\ldots+\left(\widetilde{\mathit{Inv}}_t-\widetilde{\mathit{Accr}}_t\right) \\ &= \underline{V}^l_0+\underline{e}^l_{0,t}+\sum_{s=1}^t \left( \widetilde{H}_s-\frac{1}{n}\sum_{r=s-1}^{s-n}\widetilde{H}_r\right)\\ &= \underline{V}^l_0+\underline{e}^l_{0,t}+ \sum_{s=1}^t\widetilde{H}_s- \frac{1}{n}\sum_{s=1}^t\sum_{r=s-n}^{s-1}\widetilde{H}_{r}\\ &= \underline{V}_0+\underline{e}^l_{0,t}+ \sum_{s=1}^t\widetilde{H}_s- \frac{1}{n}\sum_{s=1}^t\sum_{r=0}^{n-1}\widetilde{H}_{r+s-n}\;.{} \end{aligned}
(6.4)
We will now rearrange the double sum. To this end we determine the number of possibilities to represent a given number a as a sum a = r + s such that the first summand r is between 0 and n − 1 and the second summand s is between 1 and t. We first show that A(a) is given by
\displaystyle \begin{aligned} A(a):=\begin{cases} a & \text{if}\quad 0< a < \min(t,n),\\ \min(t,n) & \text{if}\quad \min(t,n)\le a \le\max(t,n),\\ n+t-a & \text{if}\quad \max(t,n)< a < n+t,\\ 0 &\text{else}. \end{cases} \end{aligned}
To this end write a = r + s as a sum of ones with a separating vertical line between r and s
\displaystyle \begin{aligned} \underbrace{\overbrace{1\quad 1\quad 1\quad 1\;}^{r}|\overbrace{\;1\quad 1\quad 1\quad 1\quad 1\quad }^{s}}_{a}\qquad . \end{aligned}
The separating line cannot lie left from the first one (because r ≥ 0) and cannot lie right from the last one (because s > 0). Hence, there are exactly A(a) = a possibilities; this explains the first row of our definition. If a increases by one, the quantity A(a) increases by one as well.
If a gets above $$\min (n,t)$$, then A(a) remains at its current level. This is so because the vertical line cannot occupy all available positions. If, for example n ≤ t and therefore n < a, the vertical line cannot be right from the n-th one since we must have r < n. The argument is analog for n > t. This explains the second line of the definition.
The third line of the definition can be understood as follows. Look at a representation r + s where the vertical line is farthest to the left. Because a is greater than n and t by adding an additional one to the right this representation violates our rules since s gets too large. Hence, increasing a by one decreases the number A(a) by one. This finishes our proof of A(a).
Now (6.4) can be simplified to
\displaystyle \begin{aligned} \widetilde{\underline{V}}_t &=\underline{V}_0+\underline{e}^l_{0,t}+ \sum_{a=1}^t\widetilde{H}_a-\frac{1}{n}\sum_{a=1}^{n+t-1}A(a)\widetilde{H}_{a-n}\\ &=\underline{V}_0+\underline{e}^l_{0,t}-\sum_{a=1-n}^{0}\frac{A(a+n)}{n}\widetilde{H}_{a}+\sum_{a=1}^{t}\left(1-\frac{A(a+n)}{n}\right)\widetilde{H}_a\\ &=\underline{V}_0+\underline{e}^l_{0,t}-\sum_{s=1-n}^{0}\frac{A(s+n)}{n}\mathit{Inv}_{s}+\sum_{s=1}^{t}\left(1-\frac{A(s+n)}{n}\right)\alpha_s\widetilde{\mathit{FCF}}_s\,.{}\end{aligned}
(6.5)
This equation can be simplified even further. To this end we will look at the distinct cases.
First let n < t. Then from the definition of A(s + n)
\displaystyle \begin{aligned} A(s+n)&=\begin{cases} s+n & \text{if}\quad 0<s+n<n\\ n & \text{if}\quad n\le s+n \le t\\ n+t-s-n & \text{if}\quad t<s+n<n+t \end{cases}\\ &= \begin{cases} s+n & \text{if}\quad -n<s<0\\ n & \text{if}\quad 0\le s \le t-n\\ t-s & \text{if}\quad t-n<s<t \end{cases}\;\;.\end{aligned}
We will treat the second and the third summand separately. Because the index s in the first summand runs from 1 − n to 0 and the index in the second summand runs from 1 to t − 1 (6.5) simplifies to
\displaystyle \begin{aligned} \widetilde{\underline{V}}_t&=\underline{V}_0+\underline{e}^l_{0,t}-\sum_{s=1-n}^{0}\frac{A(s+n)}{n}\,\mathit{Inv}_{s}+\sum_{s=1}^{t-n}\left(1-\frac{A(s+n)}{n}\right)\alpha_s\widetilde{\mathit{FCF}}_s\\ & \qquad + \sum_{s=t-n+1}^{t}\left(1-\frac{A(s+n)}{n}\right)\alpha_s\widetilde{\mathit{FCF}}_s\\ &=\underline{V}_0+\underline{e}^l_{0,t}-\sum_{s=1-n}^{0}\frac{s+n}{n}\,\mathit{Inv}_{s}+\sum_{s=1}^{t-n}\left(1-\frac{n}{n}\right)\alpha_s\widetilde{\mathit{FCF}}_s\\ & \qquad + \sum_{s=t-n+1}^{t}\left(1-\frac{t-s}{n}\right)\alpha_s\widetilde{\mathit{FCF}}_s\\ &=\underline{V}_0+\underline{e}^l_{0,t}-\sum_{s=1-n}^{0}\frac{s+n}{n}\,\mathit{Inv}_{s}+\sum_{s=t-n+1}^{t}\frac{n-t+s}{n}\,\alpha_s\widetilde{\mathit{FCF}}_s\,. {} \end{aligned}
(6.6)
Let now n ≥ t. Then from the definition of A(s + n)
\displaystyle \begin{aligned} A(s+n)&=\begin{cases} s+n & \text{if}\quad 0<s+n<t\\ t & \text{if}\quad t\le s+n \le n\\ n+t-s-n & \text{if}\quad n<s+n<n+t \end{cases}\\ &= \begin{cases} s+n & \text{if}\quad -n< s<t-n\\ t & \text{if}\quad t-n\le s \le 0\\ t-s & \text{if}\quad 0<s<t \end{cases}\;\;.\end{aligned}
Again we will treat the second and the third summand separately. The index s in the first summand runs from 1 − n to 0 and in the second summand from 1 to t − 1. Hence, (6.5) simplifies to
\displaystyle \begin{aligned} \widetilde{\underline{V}}_t&=\underline{V}_0+\underline{e}^l_{0,t}-\sum_{s=1-n}^{t-n}\frac{A(s+n)}{n}\,\mathit{Inv}_{s}-\sum_{s=t-n+1}^{0}\frac{A(s+n)}{n}\,\mathit{Inv}_{s}\\ & + \qquad \sum_{s=1}^{t}\left(1-\frac{A(s+n)}{n}\right)\alpha_s\widetilde{\mathit{FCF}}_s\\ &=\underline{V}_0+\underline{e}^l_{0,t}-\sum_{s=1-n}^{t-n}\frac{s+n}{n}\,\mathit{Inv}_{s}-\sum_{s=t-n+1}^{0}\frac{t}{n}\,\mathit{Inv}_{s}+ \sum_{s=1}^{t}\left(1-\frac{t-s}{n}\right)\alpha_s\widetilde{\mathit{FCF}}_s\\ &=\underline{V}_0+\underline{e}^l_{0,t}-\sum_{s=1-n}^{0}\frac{\min(s+n, t)}{n}\,\mathit{Inv}_{s}+\sum_{s=1}^{t}\frac{n-t+s}{n}\,\alpha_s\widetilde{\mathit{FCF}}_s\,. {}\end{aligned}
(6.7)
Now we are able to rejoin both cases n < t and n ≥ t. The Eqs. (6.6) and (6.7) yield in compact notation
\displaystyle \begin{aligned} \widetilde{\underline{V}}_t=\underbrace{\underline{V}_0+\underline{e}^l_{0,t}-\sum_{s=1-n}^{0}\frac{\min(s+n,t)}{n}\mathit{Inv}_{s}}_{:=\underline{V}^{*l}_0}+ \sum_{s=1+\max(t-n, 0)}^{t}\frac{n-t+s}{n}\,\alpha_s\widetilde{\mathit{FCF}}_s\,. \end{aligned}
The first three summands will be designated as $$\underline {V}^{*l}_0$$. Economically, this term concerns the amount which the firm’s book value would be if up to time t, there are exclusively increases in subscribed capital and no single investment. That leads us to the representation
\displaystyle \begin{aligned} \widetilde{\underline{V}}^l_t &= \underline{V}^{*l}_0 + \sum_{s=1+\max(t-n,0)}^t \frac{n-(t-s)}{n}\,\alpha_s\,\widetilde{\mathit{FCF}}^u_s\,. \end{aligned}
If we use the agreement αs = 0 for s ≤ 0 (α was up to now only defined for future times), this equation becomes
\displaystyle \begin{aligned} \widetilde{\underline{V}}^l_t &= \underline{V}^{*l}_0 + \frac{n}{n} \widetilde{\mathit{FCF}}^u_t \alpha_t + \frac{n-1}{n} \widetilde{\mathit{FCF}}^u_{t-1} \alpha_{t-1} + \ldots+ \frac{1}{n} \widetilde{\mathit{FCF}}^u_{1+t-n} \alpha_{1+t-n} \end{aligned}
(6.8)
and we will from now on use this form.
The valuation with a policy based on book values is now finally successful by means of this representation. For that purpose we use everything that we have. Look at the valuation Eq. (3.​11) which is valid for every conceivable financing policy and onto which we want to fall back now. With means of financing based on book values, we have at all times
\displaystyle \begin{aligned}\widetilde{\underline{D}}_t = \underline{l}_t \widetilde{\underline{V}}^l_t,\end{aligned}
which with (6.8) and using Assumption 3.​7 leads us to
\displaystyle \begin{aligned} V^l_0 &= V^u_0+\tau r_f\frac{\underline{l}_0\underline{V}^l_0}{1+r_f} + \tau r_f\sum_{t=1}^{T-1} \frac{\underline{l}_t}{(1+r_f)^{t+1}} \,\, \text{E}_Q \left[ \underline{V}^{*l}_0 + \frac{n}{n} \widetilde{\mathit{FCF}}^u_t \alpha_t \right.\\ & \left. \quad + \frac{n-1}{n} \widetilde{\mathit{FCF}}^u_{t-1} \alpha_{t-1} + \ldots+ \frac{1}{n}\widetilde{\mathit{FCF}}^u_{t-n+1} \alpha_{t-n+1} \right] \\ &= \widetilde{V}^u_0+\tau r_f\frac{\underline{l}_0\underline{V}^l_0}{1+r_f} + \tau r_f\sum_{t=1}^{T-1} \underline{l}_t\,\left( \frac{\underline{V}^{*l}_0}{(1+r_f)^{t+1}} + \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_t\right]}{(1+r_f)^t}\frac{\frac{n}{n}\alpha_t}{1+r_f} \right. \\ & \left. \quad + \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_{t-1}\right]}{(1+r_f)^{t-1}}\frac{\frac{n-1}{n}\alpha_{t-1}}{(1+r_f)^2}+ \ldots+\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_{1+t-n}\right]}{(1+r_f)^{1+t-n}}\frac{\frac{1}{n}\alpha_{1+t-n}}{(1+r_f)^{n}} \right)\,. \end{aligned}
We are now using the Assumption 3.​1 and the Theorem 3.​3 based on it. That allows for the representation
\displaystyle \begin{aligned} V^l_0 &= V^u_0+\tau r_f\frac{\underline{l}_0\underline{V}^l_0}{1+r_f} + \tau r_f\sum_{t=1}^{T-1} \underline{l}_t\,\left( \frac{\underline{V}^{*l}_0}{(1+r_f)^{t+1}} + \frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_t\right]}{(1+k^{E,u})^t}\frac{\frac{n}{n}\alpha_t}{1+r_f} \right. \\ & \left. \quad + \frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{t-1}\right]}{(1+k^{E,u})^{t-1}}\frac{\frac{n-1}{n}\alpha_{t-1}}{(1+r_f)^2}+ \ldots+\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{1+t-n}\right]}{(1+k^{E,u})^{1+t-n}}\frac{\frac{1}{n}\alpha_{1+t-n}}{(1+r_f)^{n}} \right) \\ &= V^u_0+\tau r_f\frac{\underline{l}_0\underline{V}^l_0}{1+r_f} + \tau r_f\sum_{t=1}^{T-1} \underline{l}_t\,\frac{\underline{V}^{*l}_0}{(1+r_f)^{t+1}} \\ & \qquad + \tau r_f\sum_{t=1}^{T-1}\left( \frac{\alpha_t\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_t\right]}{(1+k^{E,u})^t}\frac{\frac{n}{n}\underline{l}_t}{1+r_f}+ \frac{\alpha_{t-1}\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{t-1}\right]}{(1+k^{E,u})^{t-1}}\frac{\frac{n-1}{n}\underline{l}_t}{(1+r_f)^2}\right.\\ & \qquad \qquad \left.+ \ldots+\frac{\alpha_{1+t-n}\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{1+t-n}\right]}{(1+k^{E,u})^{1+t-n}} \frac{\frac{1}{n}\underline{l}_t}{(1+r_f)^{n}}\right). \end{aligned}
Lastly, we only have the terms in the last two lines to concentrate on. We are obviously looking at a double sum. To simplify its representation, we have to consider, how often an expected cash flow comes up. It is recognized that the coefficient in front of $$\frac {\alpha _t \operatorname *{\mathrm {E}}\left [\widetilde {\mathit {FCF}}^u_t\right ]}{(1+k^{E,u})^t}$$ appears with the expressions
\displaystyle \begin{aligned} \frac{\frac{n}{n}\underline{l}_t}{1+r_f},\; \frac{\frac{n-1}{n}\underline{l}_{t+1}}{(1+r_f)^2},\;\ldots, \frac{\frac{1}{n}\underline{l}_{n+t-1}}{(1+r_f)^{n}}\,. \end{aligned}
Again, this requires that coefficients $$\underline {l}_s$$ with an index greater than T − 1 are set to zero. We get
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle V^l_0 = V^u_0+\tau r_f\frac{\underline{l}_0\underline{V}^l_0}{1+r_f} + \tau r_f\sum_{t=1}^{T-1} \underline{l}_t\,\frac{\underline{V}^{*l}_0}{(1+r_f)^{t+1}} +\\ &\displaystyle &\displaystyle \quad \qquad +\tau r_f\sum_{t=1}^{T-1} \frac{\alpha_t\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_t\right]}{(1+k^{E,u})^t}\left(\frac{\frac{n}{n}\underline{l}_t}{1+r_f}+ \frac{\frac{n-1}{n}\underline{l}_{t+1}}{(1+r_f)^2}+\ldots+\frac{\frac{1}{n}\underline{l}_{n+t-1}}{(1+r_f)^{n}}\right)\,, \end{array} \end{aligned}
where $$\underline {l}_s=0$$ for s ≥ T. With that Theorem 3.​21 is finally proven.

6.3 Adjusted Modigliani–Miller Formula (Theorem 3.​22)

Many factors remain constant in the theorem. More than anything, it affects the debt ratio $$\underline {l}$$, the investment parameter α, and the subscribed capital. Beyond that, it is assumed that the firm to be valued exists without end. By disregarding the time indices with the investment parameter and the debt ratio, we get the following using the sum of a geometric sequence:
\displaystyle \begin{aligned} V^l_0 &=V^u_0+\tau r_f\frac{\underline{l}\underline{V}^l_0}{1+r_f}+\tau r_f\sum_{t=1}^{\infty} \frac{\underline{l} \underline{V}^{*l}_0}{(1+r_f)^{t+1}}+\tau r_f\sum_{t=1}^{\infty} \frac{\alpha\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_t\right]}{(1+k^{E,u})^t}\left(\frac{\frac{n}{n}\underline{l}}{1+r_f}+\right.\\ & \qquad +\left. \frac{\frac{n-1}{n}\underline{l}}{(1+r_f)^2}+\ldots+\frac{\frac{1}{n}\underline{l}}{(1+r_f)^{n}}\right) \\ &= V^u_0+\frac{\tau\underline{l}\left(r_f\underline{V}^l_0+\underline{V}^{*l}_0\right)}{1+r_f}+\frac{\tau r_f\alpha\underline{l}}{n} \sum_{t=1}^{\infty} \frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_t\right]}{\left(1+k^{E,u}\right)^t} \left( \frac{n}{1+r_f}+\frac{n-1}{(1+r_f)^2}+ \right. \\ {} & \qquad \left. +\ldots+\frac{1}{(1+r_f)^n} \right). \end{aligned}
(6.9)
We make the effort now to get a compact representation of the expression
\displaystyle \begin{aligned}\frac{n}{1+r_f}+\frac{n-1}{(1+r_f)^2}+\ldots+\frac{1}{(1+r_f)^n}.\end{aligned}
To do so we look at the identity
\displaystyle \begin{aligned} (1+r_f)^{n}+(1+r_f)^{n-1}+\ldots+(1+r_f)=\frac{(1+r_f)((1+r_f)^n-1)}{r_f} \end{aligned}
and derive it according to rf,
\displaystyle \begin{aligned} n(1+r_f)^{n-1}+(n-1)(1+r_f)^{n-2}+\ldots+1=\frac{1+(nr_f-1)(1+r_f)^n}{r_f^2}. \end{aligned}
Multiplying by (1 + rf)n results in
\displaystyle \begin{aligned} n(1+r_f)^{-1}+(n-1)(1+r_f)^{-2}+\ldots+(1+r_f)^{-n}=\frac{nr_f-1+(1+r_f)^{-n}}{r_f^2}. \end{aligned}
Entering this into Eq. (6.9), results in
\displaystyle \begin{aligned} V^l_0 =V^u_0+\frac{\tau\underline{l}\left(r_f\underline{V}^l_0+\underline{V}^{*l}_0\right)}{1+r_f}+\frac{\tau r_f\alpha\underline{l}}{n}\,\frac{nr_f-1+(1+r_f)^{-n}}{r_f^2}\sum_{t=1}^\infty \frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_t\right]}{(1+k^{E,u})^t}. \end{aligned}
We recognize that the sum on the right-hand side exactly corresponds to the market value of the unlevered firm. This leads us to
\displaystyle \begin{aligned}V^l_0 =V^u_0+\frac{\tau\underline{l}\left(r_f\underline{V}^l_0+\underline{V}^{*l}_0\right)}{1+r_f}+\frac{nr_f-1+(1+r_f)^{-n}}{nr_f}\,\tau\alpha\underline{l} V^u_0\,.\end{aligned}
Lastly we turn our attention to the term
\displaystyle \begin{aligned}\frac{\tau\underline{l}\left(r_f\underline{V}^l_0+\underline{V}^{*l}_0\right)}{1+r_f}\end{aligned}
and consider that at time t = 0 the identity
\displaystyle \begin{aligned}\underline{V}_0^{*l}=\underline{V}_0^l-\sum_{s=1-n}^0 \frac{\min(n-s,t)}{n}\mathit{Inv}_s \end{aligned}
applies, so long as the influence of depreciation on those assets raised before time t = 0 is not excluded. But if we disregard this influence according to the gotten condition, then we get
\displaystyle \begin{aligned} \frac{\tau\underline{l}\left(r_f\underline{V}^l_0+\underline{V}^{*l}_0\right)}{1+r_f} &= \frac{\tau\underline{l}\left(r_f\underline{V}^l_0+ \underline{V}_0^l\right)}{1+r_f} \\ &= \tau\underline{l}\underline{V}^l_0 .\end{aligned}
We recognize that the product of debt ratio and book value of the firm corresponds to the book value of the debt, and can close up the proof considering Assumption 3.​7.

6.4 Valuation Formula with Financing Based on Cash Flows (Theorems 3.​23 and 3.​24)

From the definition of financing based on cash flows, the following first results for the amount of debt using the fact that debt is riskless:
\displaystyle \begin{aligned} \widetilde{D}_{t}=\left((1+\alpha r_f(1-\tau))D_{0}-\alpha\,\widetilde{\mathit{FCF}}^u_{1}\right)^+ \qquad \forall t \ge 1 \,. \end{aligned}
We enter this into Eq. (3.​11) and get
\displaystyle \begin{aligned} V^l_0&=V^u_0+\frac{\tau\operatorname*{\mathrm{E}}_Q[Z_1]}{1+r_f}+ \sum_{t=1}^{T-1} \frac{\tau r_f \operatorname*{\mathrm{E}}_Q\left[\left((1+\alpha r_f(1-\tau))D_{0}-\alpha\,\widetilde{\mathit{FCF}}^u_{1}\right)^+ \right]}{(1+r_f)^{t+1}}\\ &=V^u_0+\frac{\tau r_f D_0}{1+r_f}+ \frac{\tau r_f \operatorname*{\mathrm{E}}_Q\left[\left((1+\alpha r_f(1-\tau))D_{0}-\alpha\,\widetilde{\mathit{FCF}}^u_{1}\right)^+ \right]}{(1+r_f)r_f}\,\left(1-\frac{1}{(1+r_f)^{T-1}}\right)\;. \end{aligned}
With help of Theorem 3.​2, the third summand can be further simplified
\displaystyle \begin{aligned} V^l_0=V^u_0+\frac{\tau r_f D_0}{1+r_f}+ \tau \alpha d^u_1 \, \frac{\operatorname*{\mathrm{E}}_Q\left[\left(\frac{1+\alpha r_f(1-\tau)}{\alpha d^u_1}D_{0}-\widetilde{V}^u_1 \right)^+ \right]}{1+r_f}\,\left(1-\frac{1}{(1+r_f)^{T-1}}\right). \end{aligned}
Let us now look at a put on the value of the unlevered firm at time t = 1 with a strike of $$\frac {1+\alpha r_f(1-\tau )}{\alpha d^u_1}D_{0}$$. The bearer of this option receives the difference of the exercise price and the firm value, if this difference is positive. In the opposite case, the payment comes to zero. To determine the value of this put Π, we have to evaluate the expectation EQ[⋅] of the payments of the put and discount them with the riskless rate according to the duration of the option. This results exactly in
\displaystyle \begin{aligned} \varPi=\frac{\operatorname*{\mathrm{E}}_Q\left[\left(\frac{1+\alpha r_f(1-\tau)}{\alpha d^u_1}D_{0}-\widetilde{V}^u_1 \right)^+ \right]}{1+r_f}.\end{aligned}
But with that applies
\displaystyle \begin{aligned} V^l_0=V^u_0+\frac{\tau r_f D_0}{1+r_f}+ \tau \alpha d^u_1 \varPi\,\left(1-\frac{1}{(1+r_f)^{T-1}}\right),\end{aligned}
and that was what we wanted to show.
We come to the proof of Theorem 3.​24. Since the amount of debt of the first period is positive, expression $$(1+\alpha r_f(1-\tau ))D_{0}-\alpha \,\widetilde {\mathit {FCF}}^u_{1}$$ will not be negative in any case. Therefore we can write
\displaystyle \begin{aligned} \widetilde{D}_{t}=(1+\alpha r_f(1-\tau))D_{0}-\alpha\,\widetilde{\mathit{FCF}}^u_{1}\end{aligned}
for all t ≥ 1. From that we get the simpler valuation equation
\displaystyle \begin{aligned} V^l_0 &=V^u_0+\frac{\tau r_f D_0}{1+r_f}+ \frac{\tau r_f \operatorname*{\mathrm{E}}_Q\left[(1+\alpha r_f(1-\tau))D_{0}-\alpha\,\widetilde{\mathit{FCF}}^u_{1}\right]}{(1+r_f)r_f}\,\left(1-\frac{1}{(1+r_f)^{T-1}}\right)\\ &=V^u_0+\frac{\tau r_f D_0}{1+r_f}+\left(\frac{\tau (1+\alpha r_f(1-\tau))D_{0}}{1+r_f} -\tau \alpha\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_{1}\right]}{1+r_f}\right)\,\left(1-\frac{1}{(1+r_f)^{T-1}}\right)\,. \end{aligned}
Due to the equivalence of the valuation concepts (Theorem 3.​3), there results from that
\displaystyle \begin{aligned} V^l_0=V^u_0+\frac{\tau r_f D_0}{1+r_f}+\left(\frac{\tau (1+\alpha r_f(1-\tau))D_{0}}{1+r_f} -\tau \alpha\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{1}\right]}{1+k^{E,u}}\right)\,\left(1-\frac{1}{(1+r_f)^{T-1}}\right)\,. \end{aligned}
Lastly we make use of the fact that expected cash flows are constant. With that we end up with
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle V^l_0=V^u_0+\frac{\tau r_f D_0}{1+r_f}+\frac{\tau (1+\alpha r_f(1-\tau))D_{0}}{1+r_f}\,\left(1-\frac{1}{(1+r_f)^{T-1}}\right) -\\ &\displaystyle &\displaystyle \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad -\tau \alpha\frac{V^u_0k^{E,u}}{1+k^{E,u}} \frac{1-\frac{1}{(1+r_f)^{T-1}}}{1-\frac{1}{(1+k^{E,u})^{T-1}}}\,. \end{array} \end{aligned}
That agrees with the claim.

6.5 Valuation with Financing Based on Dividends (Theorem 3.​25)

Here we apply a different method to establish the value of the firm. For that we concentrate on the payments, which go to the debt and equity financiers. From the fundamental theorem the following first results for the levered firm:
\displaystyle \begin{aligned} V^l_0 &=\sum_{t=1}^T\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_t+\tau \widetilde{\,I\,}_{t}\right]}{(1+r_f)^t} \\ &=\sum_{t=1}^T \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_t-\widetilde{D}_{t-1}-(1-\tau) \widetilde{\,I\,}_{t}+\widetilde{D}_t\right]}{(1+r_f)^t} + \sum_{t=1}^T \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\,I\,}_t+\widetilde{D}_{t-1}-\widetilde{D}_t\right]}{(1+r_f)^t}. \end{aligned}
The second summand agrees with the sum of the discounted payments to the debt financiers. It should be exactly equal to D0, which can be easily proven,
\displaystyle \begin{aligned} \sum_{t=1}^T \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\,I\,}_t+\widetilde{D}_{t-1}-\widetilde{D}_t\right]}{(1+r_f)^t} & = \sum_{t=1}^T \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\,I\,}_t+\widetilde{D}_{t-1}\right]}{(1+r_f)^t}-\sum_{t=1}^T \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{D}_t\right]}{(1+r_f)^t}\\ &=\sum_{t=1}^T \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{D}_{t-1}\right]}{(1+r_f)^{t-1}}-\sum_{t=1}^T \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{D}_t\right]}{(1+r_f)^t}\\ &= D_0. \end{aligned}
With that we have the following equation for the value of the levered firm,
\displaystyle \begin{aligned} V_0^l=\sum_{t=1}^T \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_t-\widetilde{D}_{t-1}-(1-\tau) \widetilde{\,I\,}_{t}+\widetilde{D}_t\right]}{(1+r_f)^t}+D_0. \end{aligned}
During the first n periods, the shareholders get exactly the amount Div. Afterwards, the amount of debt remains constant. Using (3.​19) that leads to
\displaystyle \begin{aligned} V^l_0&=D_0+\sum_{t=0}^{n-1} \frac{Div}{(1+r_f)^{t+1}}+ \sum_{t=n}^{T-1} \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_{t+1}-(1-\tau)\widetilde{\,I\,}_{n+1}\right]}{(1+r_f)^{t+1}}- \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{D}_n\right]}{(1+r_f)^T}\\ &=D_0+\left(1-\frac{1}{(1+r_f)^{n}}\right)\frac{Div}{r_f} +\sum_{t=n}^{T-1} \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_{t+1}\right]}{(1+r_f)^{t+1}}-\\ &\qquad \qquad -\frac{(1-\tau)r_f\operatorname*{\mathrm{E}}_Q\left[\widetilde{D}_n\right]}{r_f(1+r_f)^{n}}\left(1-\frac{1}{(1+r_f)^{T-n}}\right) -\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{D}_n\right]}{(1+r_f)^T}. \end{aligned}
If we make use of the equivalence of the valuation concepts from (Theorem 3.​3) and further consider that the rate of growth of the expected cash flows is constant (g = const.), there then results
\displaystyle \begin{aligned} V^l_0&=D_0+\left(1-\frac{1}{(1+r_f)^{n}}\right)\frac{Div}{r_f}+ \sum_{t=n}^{T-1}\frac{(1+g)^{t}\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{1}\right]}{(1+k^{E,u})^{t+1}}-\\ &\qquad - \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{D}_n\right]}{(1+r_f)^{n}} \left(1-\tau\left(1-\frac{1}{(1+r_f)^{T-n}}\right)\right) \end{aligned}
(6.10)
\displaystyle \begin{aligned} &=D_0+\left(1-\frac{1}{(1+r_f)^{n}}\right)\frac{Div}{r_f}+ \frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{1}\right]}{k^{E,u}-g}\,\left[\left(\frac{1+g}{1+k^{E,u}}\right)^{n}- \left(\frac{1+g}{1+k^{E,u}}\right)^{T}\right]-\\ &\qquad \qquad - \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{D}_n\right]}{(1+r_f)^{n}} \left(1-\tau\left(1-\frac{1}{(1+r_f)^{T-n}}\right)\right)\;. {}\end{aligned}
(6.11)
We now turn to the amount of debt at time n. Since according to the condition that the credit always remains positive, we can simplify the formation law of the debt. With (3.​19) and rf = rf(1 − τ) applies
\displaystyle \begin{aligned} \text{E}_{Q} \left[\widetilde{D}_{t}|\mathcal{F}_{t-1}\right]=\text{E}_{Q} \left[Div-\widetilde{\mathit{FCF}}^u_{t}|\mathcal{F}_{t-1}\right]+(1+{r_f}^*)D_{t-1} \qquad \forall t\le n.\end{aligned}
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \text{E}_{Q} \left[\widetilde{D}_{n}\right]=\text{E}_{Q}\left[Div-\widetilde{\mathit{FCF}}^u_{n}\right]+(1+{r_f}^*)\,\text{E}_{Q}\left[Div-\widetilde{\mathit{FCF}}^u_{n-1}\right]+\ldots+\\ &\displaystyle &\displaystyle \qquad \qquad \qquad \qquad +(1+{r_f}^*)^{n-1}\,\text{E}_{Q}\left[Div-\widetilde{\mathit{FCF}}^u_{1}\right]+(1+{r_f}^*)^{n}D_0. \end{array} \end{aligned}
This equation can be brought into the form
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \ \text{E}_{Q}\left[\widetilde{D}_{n}\right]=\frac{(1+{r_f}^*)^n-1}{{r_f}^*}Div+(1+{r_f}^*)^{n}D_0-\\ &\displaystyle &\displaystyle -\text{E}_{Q}\left[\widetilde{\mathit{FCF}}^u_{n}\right]-(1+{r_f}^*)\;\text{E}_{Q}\left[\widetilde{\mathit{FCF}}^u_{n-1}\right]-\ldots-(1+{r_f}^*)^{n-1}\;\text{E}_{Q}\left[\widetilde{\mathit{FCF}}^u_{1}\right]\,. \end{array} \end{aligned}
From that the following is valid under the discounted expectation
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \frac{\operatorname*{\mathrm{E}}_{Q}\left[\widetilde{D}_{n}\right]}{(1+r_f)^n}=\frac{(1+{r_f}^*)^n-1}{{r_f}^*(1+r_f)^n}Div+ \left(\frac{1+{r_f}^*}{1+r_f}\right)^{n}D_0-\\ &\displaystyle &\displaystyle \ - \frac{\operatorname*{\mathrm{E}}_{Q}\left[\widetilde{\mathit{FCF}}^u_{n}\right]}{(1+r_f)^n}- \frac{1+{r_f}^*}{1+r_f}\frac{\operatorname*{\mathrm{E}}_{Q}\left[\widetilde{\mathit{FCF}}^u_{n-1}\right]}{(1+r_f)^{n-1}}-\ldots- \frac{(1+{r_f}^*)^{n-1}}{(1+r_f)^{n-1}}\frac{\operatorname*{\mathrm{E}}_{Q}\left[\widetilde{\mathit{FCF}}^u_{1}\right]}{1+r_f} \end{array} \end{aligned}
or due to Theorem 3.​3 as well as the constant rate of growth g and with $$\gamma =\frac {1+{r_f}^*}{1+r_f}$$ and $$\delta =\frac {1+g}{1+k^{E,u}}$$
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \frac{\operatorname*{\mathrm{E}}_{Q}\left[\widetilde{D}_{n}\right]}{(1+r_f)^n}=\frac{(1+{r_f}^*)^n-1}{{r_f}^*(1+r_f)^n} Div+ \gamma^{n}D_0-\\ &\displaystyle &\displaystyle \qquad \qquad \qquad - \delta^n\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{1}\right]}{1+g}- \gamma \delta^{n-1}\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{1}\right]}{1+g}-\ldots- \gamma^{n-1}\delta\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{1}\right]}{1+g}\;. \end{array} \end{aligned}
We combine the last summands in the equation and get
\displaystyle \begin{aligned} \frac{\operatorname*{\mathrm{E}}_{Q}\left[\widetilde{D}_{n}\right]}{(1+r_f)^n}=\frac{(1+{r_f}^*)^n-1}{{r_f}^*(1+r_f)^n}Div+ \gamma^{n}D_0- \left(1+ \frac{\gamma}{\delta} + \ldots + \left(\frac{\gamma}{\delta} \right)^{n-1} \right)\delta^{n}\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{1}\right]}{1+g} .\end{aligned}
or, after simplification,
\displaystyle \begin{aligned} \frac{\operatorname*{\mathrm{E}}_{Q}\left[\widetilde{D}_{n}\right]}{(1+r_f)^n}=\frac{(1+{r_f}^*)^n-1}{{r_f}^*(1+r_f)^n}Div+ \gamma^{n}D_0-\frac{\gamma^n -\delta^{n}}{\frac{\gamma }{\delta} - 1} \frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{1}\right]}{1+g} .\end{aligned}
We put this into (6.11) and get
\displaystyle \begin{aligned} &\ V^l_0=\left(1-\gamma^{n}\left(1-\tau\left(1-\frac{1}{(1+r_f)^{T-n}}\right)\right)\right)D_0+\\ &+ \left(1-\gamma^n\left(1-\tau\left(1-\frac{1}{(1+r_f)^{T-n}}\right)\right) -\tau\left(1-\frac{1}{(1+r_f)^T}\right) \right)\frac{Div}{r_f(1-\tau)}+\\ & +\left(\delta^n-\delta^{T}+ \frac{\gamma^{n}-\delta^{n}}{\frac{\gamma }{\delta} - 1} \,\frac{k^{E,u}-g}{1+g} \left(1-\tau\left(1-\frac{1}{(1+r_f)^{T-n}}\right)\right) \right)\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_1\right]}{k^{E,u}-g}.\end{aligned}
(6.12)
We now take advantage of the expected cash flow of the unlevered firm showing a constant rate of growth. From that results
\displaystyle \begin{aligned} V^u_0 &= \sum_{t=1}^T \frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_t\right]}{(1+k^{E,u})^t} \\ &=\sum_{t=1}^T \frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_1\right]}{1+g}\left(\frac{1+g}{1+k^{E,u}}\right)^t \\ &=\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_1\right]}{k^{E,u}-g}\left( 1-\left(\frac{1+g}{1+k^{E,u}}\right)^T\right). \end{aligned}
Plugging this into equation (6.12) gives
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \ V^l_0=\left(1-\gamma^{n}\left(1-\tau\left(1-\frac{1}{(1+r_f)^{T-n}}\right)\right)\right)D_0+\\ &\displaystyle &\displaystyle + \left(1-\gamma^n\left(1-\tau\left(1-\frac{1}{(1+r_f)^{T-n}}\right)\right) -\tau\left(1-\frac{1}{(1+r_f)^T}\right) \right)\frac{Div}{r_f(1-\tau)}+\\ &\displaystyle &\displaystyle \quad +\left(\delta^n-\delta^{T}+ \frac{\gamma^{n}-\delta^{n}}{\frac{\gamma }{\delta} - 1} \,\frac{k^{E,u}-g}{1+g} \left(1-\tau\left(1-\frac{1}{(1+r_f)^{T-n}}\right)\right) \right)\frac{V^u_0}{1-\delta^T}. \end{array} \end{aligned}
If we sort the terms around a bit, then we get our desired result.

6.6 Valuation with Debt-Cash Flow Ratio (Theorems 3.​26 and 3.​27)

From the definition of the dynamic leverage ratio we get
\displaystyle \begin{aligned} \widetilde{D}_s=\widetilde{L}^d_s\left(\widetilde{\mathit{FCF}}^u_s+\tau r_f\widetilde{D}_{s-1}\right), \end{aligned}
and hence
\displaystyle \begin{aligned} \widetilde{D}_s=\widetilde{L}^d_s\left(\widetilde{\mathit{FCF}}^u_s+\tau r_f\widetilde{L}^d_{s-1}\left(\widetilde{\mathit{FCF}}^u_{s-1} +\tau r_f\widetilde{D}_{s-2}\right)\right). \end{aligned}
Using induction this gives for s > t
\displaystyle \begin{aligned} \widetilde{D}_s=(\tau r_f)^{s-t}\widetilde{L}^d_s\ldots\widetilde{L}^d_{t+1}\widetilde{D}_t+\sum_{u=t+1}^s \widetilde{L}^d_s\ldots\widetilde{L}^d_u (\tau r_f)^{s-u}\,\widetilde{\mathit{FCF}}^u_u. \end{aligned}
Plugging this into the general valuation formula (3.​11) we get
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \widetilde{V}^l_t=\widetilde{V}^u_t+\frac{\tau r_f\widetilde{D}_t}{1+r_f}+\sum_{s=t+1}^{T-1}\frac{\tau r_f(\tau r_f)^{s-t}\widetilde{L}^d_s\ldots\widetilde{L}^d_{t+1}\widetilde{D}_t}{(1+r_f)^{s-t+1}}+\\ &\displaystyle &\displaystyle \qquad \qquad \qquad \quad \qquad + \sum_{s=t}^{T-1} \sum_{u=t+1}^s\frac{\operatorname*{\mathrm{E}}_Q\left[\tau r_f\left( \widetilde{L}^d_s\ldots\widetilde{L}^d_u (\tau r_f)^{s-u}\widetilde{\mathit{FCF}}^u_u\right)|\mathcal{F}_t\right]}{(1+r_f)^{s-t+1}}. \end{array} \end{aligned}
This simplifies to (let $$\widetilde {L}^d_s\ldots \widetilde {L}^d_{t+1}=1$$ if s = t)
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \widetilde{V}^l_t=\widetilde{V}^u_t+\widetilde{D}_t\sum_{s=t}^{T-1}\frac{(\tau r_f)^{s-t+1}}{(1+r_f)^{s-t+1}}\,\widetilde{L}^d_s\ldots\widetilde{L}^d_{t+1}\,+\\ &\displaystyle &\displaystyle \qquad \qquad \quad \qquad \qquad \qquad + \sum_{s=t}^{T-1} \sum_{u=t+1}^s\frac{\widetilde{L}^d_s\ldots\widetilde{L}^d_u (\tau r_f)^{s+1-u}} {(1+r_f)^{s+1-u}} \,\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_u|\mathcal{F}_t\right]}{(1+r_f)^{u-t}}. \end{array} \end{aligned}
Changing summation it yields
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \widetilde{V}^l_t=\widetilde{V}^u_t+\widetilde{D}_t\sum_{s=t}^{T-1}\frac{(\tau r_f)^{s-t+1}}{(1+r_f)^{s-t+1}}\,\widetilde{L}^d_s\ldots\widetilde{L}^d_{t+1}\,+\\ &\displaystyle &\displaystyle \qquad \qquad \quad \qquad \qquad \qquad + \sum_{u=t+1}^{T-1}\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_u|\mathcal{F}_t\right]}{(1+r_f)^{u-t}} \sum_{s=u}^{T-1} \frac{\widetilde{L}^d_s\ldots\widetilde{L}^d_u (\tau r_f)^{s+1-u}}{(1+r_f)^{s+1-u}} \end{array} \end{aligned}
or after using Theorem 3.​2
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \widetilde{V}^l_t=\widetilde{V}^u_t+\widetilde{D}_t\sum_{s=t}^{T-1}\frac{(\tau r_f)^{s-t+1}}{(1+r_f)^{s-t+1}}\,\widetilde{L}^d_s\ldots\widetilde{L}^d_{t+1}\,+\\ &\displaystyle &\displaystyle \qquad \qquad \qquad \qquad \qquad + \sum_{u=t+1}^{T-1}\left(\sum_{s=u}^{T-1} \frac{\widetilde{L}^d_s\ldots\widetilde{L}^d_u (\tau r_f)^{s+1-u}}{(1+r_f)^{s+1-u}}\right) \frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_u|\mathcal{F}_t\right]}{(1+k^{E,u})^{u-t}}. \end{array} \end{aligned}
Changing indices gives Theorem 3.​26.
Now let us turn to the case of infinite lifetime (Theorem 3.​27) and constant dynamic leverage ratio. In this case
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \widetilde{V}^l_t=\widetilde{V}^u_t+\widetilde{D}_t\sum_{s=t}^{\infty}\frac{(\tau r_f)^{s-t+1}}{(1+r_f)^{s-t+1}}\,(\widetilde{L}^d)^{s-t}+\\ &\displaystyle &\displaystyle \qquad \qquad \qquad \qquad \qquad + \sum_{s=t+1}^{\infty}\left(\sum_{u=s}^{\infty} \frac{(\widetilde{L}^d)^{u+1-s} (\tau r_f)^{u+1-s}}{(1+r_f)^{u+1-s}}\right) \frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_s|\mathcal{F}_t\right]}{(1+k^{E,u})^{s-t}}. \end{array} \end{aligned}
The sum of geometric series gives
\displaystyle \begin{aligned} \widetilde{V}^l_t=\widetilde{V}^u_t+\widetilde{D}_t\frac{\tau r_f}{1+r_f(1-\tau\widetilde{L}^d)}\,+\, \frac{\widetilde{L}^d\tau r_f}{1+r_f(1-\tau\widetilde{L}^d)}\sum_{s=t+1}^{\infty} \frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_s| \mathcal{F}_t\right]}{(1+k^{E,u})^{s-t}}. \end{aligned}
The sum on the right-hand side is just $$\widetilde {V}^u_t$$ and hence
\displaystyle \begin{aligned} \widetilde{V}^l_t=\widetilde{D}_t\frac{\tau r_f}{1+r_f(1-\tau\widetilde{L}^d)}\,+\, \widetilde{V}^u_t\left(1+ \frac{\widetilde{L}^d\tau r_f}{1+r_f(1-\tau\widetilde{L}^d)}\right). \end{aligned}

6.7 Fundamental Theorem with Income Tax (Theorem 4.​2)

We closely follow Löffler and Schneider (2003). Consider a model in discrete time t = 0, 1, …, T with uncertainty. The probability space is denoted by $$(\varOmega , \mathcal {F}, P)$$. The filtration $$\mathcal {F}$$ need not be finitely generated, it consists of the σ-algebras $$\mathcal {F}_0\subseteq \mathcal {F}_1 \subseteq \ldots \subseteq \mathcal {F}_T$$ that describe the information set of every investor.1 There are N tradeable risky financial assets that pay dividends (adapted random variables)
\displaystyle \begin{aligned}\widetilde{\mathit{FCF}}^1_{t}, \ldots, \widetilde{\mathit{FCF}}^N_{t}.\end{aligned}
The prices—also called values—of the risky assets at time t are adapted random variables
\displaystyle \begin{aligned}\widetilde{V}^1_{t}, \ldots, \widetilde{V}^N_{t}.\end{aligned}
There is one riskless asset, labeled n = 0. The prices of this asset are given by
\displaystyle \begin{aligned} V^0_{t} = 1 \end{aligned}
(6.13)
and the cash flows of the risk free asset are given by
\displaystyle \begin{aligned} FCF^0_{t} = r_f \end{aligned}
(6.14)
where rf is the riskless interest rate.2
At time t = 0 the investor selects a portfolio consisting of the available financial assets. This portfolio will be changed at every trading date t = 1, …, T. Immediately after time t the investors form a portfolio $$\widetilde {H}_{t}=\left (\widetilde {H}^0_{t},\ldots ,\widetilde {H}^N_{t}\right )$$ of all available assets, see Fig. 6.1. $$\widetilde {H}_t$$ is $$\mathcal {F}_t$$-adapted. Note that $$H_{-1} = \widetilde {H}_T = 0$$. At time t + 1 her portfolio has a value of
\displaystyle \begin{aligned} \widetilde{H}_{t} \cdot \widetilde{V}_{t+1} := \sum_{n=0}^N \widetilde{H}_{t}^n {\widetilde{V}}^n_{t+1}. \end{aligned}
We now introduce the tax system. We have to distinguish between the market value of a risky financial asset and the value that will be the underlying for the tax base. The underlying tax base will not be determined by the market alone but by the tax law. We denominate it the book value of a financial asset. The book value of a financial asset n at time t will be denoted by $$\widetilde { \underline {V}}^n_{t}$$ and is a random variable. We assume that the book value is an adapted random variable that will be zero at time t = T. It is not necessary for our model to incorporate other details from any actual tax law. At time t + 1 the portfolio $$\widetilde {H}_{t}$$ has the book value
\displaystyle \begin{aligned} \widetilde{H}_{t} \cdot \widetilde{\underline{V}}_{t+1} = \sum_{n=1}^N \widetilde{H}^n_{t} \widetilde{\underline{V}}^n_{t}. \end{aligned}
(6.15)
The tax base of the portfolio $$\widetilde {H}_{t}$$ at time t + 1 now consists of two parts. The first part (“riskless or interest income”) is given by the payments of the riskless asset $$r_f\widetilde {H}^0_{t}$$. The second part of the tax base (“risky or commercial income”) is given by the gains of the remaining risky financial assets
\displaystyle \begin{aligned}\sum_{n=1}^N\widetilde{H}^n_{t} \left( \widetilde{\mathit{FCF}}^n_{t+1} + \widetilde{\underline{V}}^n_{t+1} - \widetilde{\underline{V}}^n_{t} \right).\end{aligned}
Notice that the gain of a financial asset n consists of the cash flow $$\widetilde {\mathit {FCF}}^n_{t+1}$$ and the capital gain $$\widetilde { \underline {V}}^n_{t+1}- \widetilde { \underline {V}}^n_{t}$$. If the tax base is negative, there is an immediate and full loss offset. In t = 0 no tax is paid. We assume a proportional tax on both income which is time-independent and deterministic: the tax rate on riskless income (“tax on interest”) is τI and the tax rate on risky income (“tax on dividend”) is τD. Therefore, the tax payments in t + 1 are given by
\displaystyle \begin{aligned} \mathit{Tax}_{t+1}(\widetilde{H}_{t}) = \tau^I r_f\widetilde{H}^0_{t}+\tau^D \sum_{n=1}^N\widetilde{H}^n_{t} \left( \widetilde{\mathit{FCF}}^n_{t+1} + \widetilde{\underline{V}}^n_{t+1} - \widetilde{\underline{V}}^n_{t} \right) . \end{aligned}
(6.16)
We now turn to the characterization of the book value of financial assets. Our assumption concerning these book values is motivated by considering a riskless bank account with a closing balance equal to the book value. In every period the interest payment is added to and the cash flow (withdrawal) is subtracted from the opening balance. The evolution of the bank account from t to t + 1 is as follows:
 book value at the beginning of period t + 1 $$\underline {V}^0_{t}$$ + interest at t + 1 $$r_f \underline {V}^0_{t}$$ − withdrawal at t + 1 $$FCF^0_{t+1}$$ = book value at the end of period t + 1 $$\underline {V}^0_{t+1}$$.
We get
\displaystyle \begin{aligned} (1 + r_f) \underline{V}^0_{t} = FCF^0_{t+1} + \underline{V}^0_{t+1},\end{aligned}
(6.17)
which resembles to the fundamental Theorem 4.​2. Since at t = T book and market value will be equal to zero we conclude that this equation implies by induction that book value and market value are the same at every time t. Although other rules for the determination of book value could be incorporated, we make the assumption that the tax law requires investors to mark their financial assets to market in each period and the tax law is applied to that measure of value3:
Assumption 6.1
The book value $$\widetilde { \underline {V}}^n_{t}$$ of a financial asset is equal to its value $$\widetilde {V}^n_{t}$$
\displaystyle \begin{aligned} \widetilde{V}^n_{t} = \widetilde{\underline{V}}^n_{t}. \end{aligned}
(6.18)
We now show that if our tax system has no arbitrage opportunities an equivalent martingale measure exists. To this end let us define when a market is free of arbitrage.
Assumption 6.2 (No Arbitrage with Taxes)
There exists no trading strategy $$\widetilde {H}$$ that satisfies
\displaystyle \begin{aligned} \Delta_t\left(\widetilde{H}\right):=\widetilde{H}_t\cdot\left(\widetilde{\mathit{FCF}}_{t+1}+\widetilde{V}_{t+1}-\mathit{Tax}_{t+1}\right)-\widetilde{H}_{t+1}\cdot\widetilde{V}_{t+1} \ge 0\end{aligned}
(6.19)
for all t and
\displaystyle \begin{aligned} P\left(\Delta_t\left(\widetilde{H}\right) > 0\right) > 0 \end{aligned}
(6.20)
for at least one t.
Then the following holds.
Theorem 6.1
Under Assumptions 6.1 and 6.2 the following holds: There is an equivalent martingale measure Q such that
\displaystyle \begin{aligned} \widetilde{V}_{t} =\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}_{t+1}+\widetilde{V}_{t+1} - \mathit{Tax}_{t+1} | \mathcal{F}_t\right]}{1 +r_f(1-\tau^I)}. \end{aligned}
The proof of the fundamental theorem uses a result of Kabanov and Kramkov (1994). To this end we first notice that there is no one-period arbitrage in the market, i.e., there is no trading strategy $$\widetilde {X}$$ such that for one t
\displaystyle \begin{aligned} \widetilde{X}_{t}\cdot \widetilde{V}_{t}\le0\quad \text{and}\quad \widetilde{X}_{t}\cdot\left(\widetilde{\mathit{FCF}}_{t+1}+\widetilde{V}_{t+1}-\mathit{Tax}_{t+1}\right)\ge 0 \end{aligned}
with at least on inequality strict with probability greater than zero and $$\widetilde {X}_s=0$$ for all s ≠ t. But then there is also no random variable $$\widetilde {Y}_{t}$$ such that
\displaystyle \begin{aligned} \widetilde{Y}_t\cdot\left(\frac{\widetilde{V}_{t+1}+\widetilde{\mathit{FCF}}_{t+1}-\mathit{Tax}_{t+1}}{1+r_f(1-\tau^I)}-\widetilde{V}_t\right)\ge0 \end{aligned}
(6.21)
where the inequality is strict with probability greater than zero. This can be shown as follows: define
\displaystyle \begin{aligned} \widetilde{X}_t:=\widetilde{Y}_t-\left(\widetilde{Y}_t\cdot\widetilde{V}_{t}\right)\left( \begin{array}{c} 1\\ 0\\ \vdots\\ 0\\ \end{array} \right)\,. \end{aligned}
Using this strategy we get
\displaystyle \begin{aligned} \widetilde{X}_t\cdot \widetilde{V}_t=0 \end{aligned}
and
\displaystyle \begin{aligned} &\widetilde{X}_t\cdot\left(\widetilde{V}_{t+1}+\widetilde{\mathit{FCF}}_{t+1}-\mathit{Tax}_{t+1}\right)=\\ &\qquad \qquad =\widetilde{Y}_t\cdot\left(\widetilde{V}_{t+1}+\widetilde{\mathit{FCF}}_{t+1}-\mathit{Tax}_{t+1}\right) -\left(1+r_f\left(1-\tau^I\right)\right)\left(\widetilde{Y}_t\cdot\widetilde{V}_t\right)\\ &\qquad \qquad \ge0\, \end{aligned}
showing that $$\widetilde {X}$$ would be a one-period arbitrage.
Now apply the theorem 3 of Kabanov and Kramkov (1994), see also Irle (2012, section 5.4). From this there exists a bounded and Almost-everywhere positive Zt+1 such that
\displaystyle \begin{aligned} \operatorname*{\mathrm{E}}\left[Z_{t+1} \frac{\widetilde{V}_{t+1}+\widetilde{\mathit{FCF}}_{t+1}-\mathit{Tax}_{t+1}}{1+r_f(1-\tau^I)}|\mathcal{F}_t\right]=\widetilde{V}_t\,. \end{aligned}
The existence of Q follows now from standard arguments, see Kabanov and Kramkov (1994, p. 524) or Irle (2012, section 5.2).

6.8 Valuation Formula with Retention Based on Dividends (Theorem 4.​9)

The free cash flows are never lower than the dividend (see Assumption 4.​2). From (4.​2) and Definition 4.​4 it follows
\displaystyle \begin{aligned} \operatorname*{\mathrm{E}}\nolimits_Q\left[\widetilde{A}_s|\mathcal{F}_{s-1}\right]=\operatorname*{\mathrm{E}}\nolimits_Q\left[\frac{1}{1-\tau^D}\widetilde{\mathit{FCF}}^u_s-Div_s| \mathcal{F}_{s-1}\right]+(1+r_f)\widetilde{A}_{s-1}. \end{aligned}
Using induction, rule 4 and because dividends are certain, it follows for s > t
\displaystyle \begin{aligned} \operatorname*{\mathrm{E}}\nolimits_Q\left[\widetilde{A}_s|\mathcal{F}_t\right]=(1+r_f)^{s-t}\widetilde{A}_t+\sum_{v=t+1}^s (1+r_f)^{s-v} \left(\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_v|\mathcal{F}_t\right]}{1-\tau^D}-Div_v\right). \end{aligned}
We plug this term into (4.​4). It yields
\displaystyle \begin{aligned} \widetilde{V}^l_t&=\widetilde{V}^u_t+\left(1-\tau^D\right)\widetilde{A}_t+\sum_{s=t}^n\frac{\operatorname*{\mathrm{E}}_Q\left[\tau^Ir_f\left(1-\tau^D\right)\widetilde{A}_{s}| \mathcal{F}_t\right]}{\left(1+r_f\left(1-\tau^I\right)\right)^{s-t+1}}\\ &=\widetilde{V}^u_t+\left(1-\tau^D\right)\widetilde{A}_t+\sum_{s=t}^n\frac{\tau^Ir_f\left(1-\tau^D\right)(1+r_f)^{s-t}\widetilde{A}_t}{\left(1+r_f\left(1-\tau^I\right)\right)^{s-t+1}}+\\ &\qquad \qquad +\sum_{s=t+1}^n \sum_{v=t+1}^s \frac{\tau^Ir_f\,(1+r_f)^{s-v}\left(\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_v|\mathcal{F}_t\right]-\left(1-\tau^D\right) Div_v\right)}{\left(1+r_f\left(1-\tau^I\right)\right)^{s-t+1}}\\ &=\widetilde{V}^u_t+\tau^I\left(1-\tau^D\right)\left(\frac{1+r_f}{1+r_f\left(1-\tau^I\right)}\right)^{n-t+1} \widetilde{A}_t+\\ &\qquad \qquad +\sum_{v=t+1}^n\sum_{s=v}^n \frac{\tau^Ir_f\,(1+r_f)^{s-v}\left(\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_v| \mathcal{F}_t\right]-\left(1-\tau^D\right)Div_v\right)}{\left(1+r_f\left(1-\tau^I\right)\right)^{s-t+1}}, \end{aligned}
the last row by changing summands. Using geometric sums we get
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \ \widetilde{V}^l_u=\widetilde{V}^u_t+\tau^I\left(1-\tau^D\right)\left(\frac{1+r_f}{1+r_f\left(1-\tau^I\right)}\right)^{n-t+1} \widetilde{A}_t \\ &\displaystyle &\displaystyle + \tau^Ir_f\sum_{v=t+1}^n\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_v|\mathcal{F}_t\right]-\left(1-\tau^D\right)Div_v}{\left(1+r_f\left(1-\tau^I\right)\right)^{v-t}} \left(1 +\left(\frac{1+r_f}{1+r_f\left(1-\tau^I\right)}\right)^{n+1-v} \right) . \end{array} \end{aligned}
Lastly, we use Theorem 4.​4 and get
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \ \widetilde{V}^l_u=\widetilde{V}^u_t+\tau^I\left(1-\tau^D\right)\left(\frac{1+r_f}{1+r_f\left(1-\tau^I\right)}\right)^{n-t+1} \widetilde{A}_t \\ &\displaystyle &\displaystyle +\tau^Ir_f\sum_{v=t+1}^n\left(\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_v|\mathcal{F}_t\right]}{(1+k^{E,u})^{v-t}}- \frac{\left(1-\tau^D\right)Div_v}{\left(1+r_f\left(1-\tau^I\right)\right)^{v-t}} \right) \cdot\left(1 +\left(\frac{1+r_f}{1+r_f\left(1-\tau^I\right)}\right)^{n+1-v} \right) . \end{array} \end{aligned}
This was to be shown.
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Footnotes
1
These are standard assumptions in an uncertain economy, see Duffie (1988).

2
All our findings are also valid in an economy with a time-dependent interest rate. Therefore, interest is assumed to be constant in time without loss of generality.

3
The existing American tax law states under the Statements of Financial Accounting Standards (SFAS) 115 that “unrealized holding gains and losses for trading securities shall be included in earnings.” Hence, the American tax system contains elements similar to our assumption.

Literature
Duffie JD (1988) Security markets: stochastic models. Academic, Boston
Harrison JM, Kreps DM (1979) Martingales and arbitrage in multiperiod securities markets. J Econ Theory 20:381–408 CrossRef
Irle A (2012) Finanzmathematik: die bewertung von derivaten, 3rd edn. Springer Spektrum, Wiesbaden
Kabanov YM, Kramkov DO (1994) No-arbitrage and equivalent martingale measures: an elementary proof of the Harrison-Pliska theorem. Theory Probab Appl 39:523–527 CrossRef
Löffler A, Schneider D (2003) Martingales, taxes and neutrality. http://​ssrncom/​paper=​375060