Proportional loss distribution according to contractual debt payments Krause and Lahmann (
2017) derive an equation for the tax shield value for the case of a proportional loss distribution. Proportional loss distribution means that total losses
L are distributed proportionally or pro rata according to the contractually agreed debt payments. Losses on interest and principal payments are, respectively,
\(L^{{\text {Int}}}=L \times \frac{r^{\text {c}}}{R^{\text {c}}}\) and
\(C=L\times \frac{1}{R^{\text {c}}}\). The equation for the value of the tax shield is
$$\begin{aligned} {\text {VTS}}^{{\text {NC}}}&=\tau \times \frac{r^{\text {c}} \times D}{R^{\text {c}}}. \end{aligned}$$
(19)
Tax savings are given by
$$\begin{aligned} {\text {TS}}=\tau \times \left( r^{\text {c}}\times D - L\times \frac{r^{\text {c}}}{R^{\text {c}}}\right) . \end{aligned}$$
(20)
I substitute both equations into Eq. (
12) and obtain:
$$\begin{aligned} S\times E[R^{\text {E}}] + D\times E[R^{\text {D}}] = \left( S + D - \tau \times \frac{r^{\text {c}} \times D}{R^{\text {c}}}\right) \times E[R^{\text {U}}] + \tau \times r^{\text {c}} \times D - \tau \times E[L] \times \frac{r^{\text {c}}}{R^{\text {c}}}. \end{aligned}$$
(21)
I rewrite the term on the rhs:
\(\tau \times r^c \times D - \tau \times E[L] \times \frac{r^c}{R^c}=\tau \times r^c \times D \times \frac{E[R^D]}{R^c}\). I substitute this into the prior equation to obtain
$$\begin{aligned} S\times E[R^{\text {E}}] + D\times E[R^{\text {D}}]\times \left( 1-\frac{\tau \times r^{\text {c}}}{R^{\text {c}}}\right) = \left( S + D - \tau \times \frac{r^{\text {c}} \times D}{R^{\text {c}}}\right) \times E[R^{\text {U}}]. \end{aligned}$$
(22)
Now, I use the expected return equations from the mean-variance CAPM:
$$\begin{aligned} \nonumber&S\times (R^{\text {f}} + \beta _{{\text {E}},{\text {M}}}\times (E[R^{\text {M}}] - R^{\text {f}})) + D\times (R^{\text {f}} + \beta _{{\text {D}},{\text {M}}}\times (E[R^{\text {M}}] - R^{\text {f}}))\times \left( 1-\frac{\tau \times r^{\text {c}}}{R^{\text {c}}}\right) \nonumber \\&\quad = \left( S + D - \tau \times \frac{r^{\text {c}} \times D}{R^{\text {c}}}\right) \times (R^{\text {f}} + \beta _{{\text {U}},{\text {M}}}\times (E[R^{\text {M}}] - R^{\text {f}})). \end{aligned}$$
(23)
This simplifies to
$$\begin{aligned} S\times \beta _{{\text {E}},{\text {M}}} + D\times \beta _{{\text {D}},{\text {M}}}\times \left( 1-\frac{\tau \times r^{\text {c}}}{R^{\text {c}}}\right) = \left( S + D - \tau \times \frac{r^{\text {c}} \times D}{R^{\text {c}}}\right) \times \beta _{{\text {U}},{\text {M}}}. \end{aligned}$$
(24)
Rearranging for the levered beta I obtain
$$\begin{aligned} \beta _{{\text {E}},{\text {M}}} = \left( 1 + \frac{D}{S}\times \frac{1 + r^{\text {c}} \times (1-\tau )}{R^{\text {c}}}\right) \times \beta _{{\text {U}},{\text {M}}} - \beta _{{\text {D}},{\text {M}}}\times \frac{D}{S}\times \frac{1 + r^{\text {c}} \times (1-\tau )}{R^{\text {c}}}. \end{aligned}$$
(25)
or
$$\begin{aligned} \beta _{{\text {E}},{\text {M}}} = \beta _{{\text {U}},{\text {M}}} + (\beta _{{\text {U}},{\text {M}}} - \beta _{{\text {D}},{\text {M}}})\times \frac{D}{S}\times \frac{1 + r^{\text {c}}\times (1-\tau )}{R^{\text {c}}}. \end{aligned}$$
(26)
In most cases, the beta for the unlevered firm is bigger than the one for debt because of priorities of debt cash flows to be paid to debt holders. That means
\(\beta _{{\text {U}},{\text {M}}} - \beta _{{\text {D}},{\text {M}}}\) is usually positive and with that the levered beta is greater than the unlevered beta—something that one would intuitively expect. In the less likely case, if
\(\beta _{{\text {U}},{\text {M}}} < \beta _{{\text {D}},{\text {M}}}\), then
\(\beta _{{\text {U}},{\text {M}}} - \beta _{{\text {D}},{\text {M}}}<0\), and the levered beta is less than the unlevered beta.
Loss distribution not proportional to contractual debt payments As Krause and Lahmann (
2015) show, with a pro rata loss distribution according to contractual interest and principal payments, the expected rate of return on debt, i.e., the discount rate on debt
\(E[R^{\text {D}}]\), is equal to a weighted average of the expected rates of return on its components, i.e., the one on interest
\(E[R^{{\text {Int}}}]\) and the one on principal payments
\(E[R^{{\text {PP}}}]\). Notice that
\(p({\text {Int}})=\frac{E[{\text {Int}}]}{E[R^{{\text {Int}}}]}\) and
\(p({\text {PP}})=\frac{E[{\text {PP}}]}{E[R^{{\text {PP}}}]}\) define the discount rates for interest and principal payments. Krause and Lahmann (
2015) show that, with interest or principal prioritization, expected rates of return on debt, interest and principal payments regularly differ. Without a COD taxation, tax savings are just interest payments scaled by the tax rate. Thus, the rate of return and the expected rate of return on interest payments and on tax savings are equal:
\(E[R^{{\text {TS}}}]=E[R^{{\text {Int}}}]\).
5 The expected return on debt as a weighted average of the expected returns on interest and principal payments is
$$\begin{aligned} E[R^{\text {D}}]&= \frac{E[{\text {Int}}] + E[{\text {PP}}]}{D}\nonumber \\&= E[R^{{\text {Int}}}]\times \frac{p({\text {Int}})}{D} + E[R^{{\text {PP}}}]\times \frac{p({\text {PP}})}{D}. \end{aligned}$$
(27)
Since
\(D=p({\text {Int}}) + p({\text {PP}})\) and values are positive, the weights
\(\frac{p({\text {Int}})}{D}\) and
\(\frac{p({\text {PP}})}{D}\) are positive and add up to one. Due to this relation, possible relations of the three expected returns on debt cash flows are:
$$\begin{aligned} E[R^{{\text {TS}}}] = E[R^{{\text {Int}}}]< E[R^{\text {D}}] < E[R^{{\text {PP}}}] \end{aligned}$$
(28)
$$\begin{aligned} E[R^{{\text {TS}}}] = E[R^{{\text {Int}}}] = E[R^{\text {D}}] = E[R^{{\text {PP}}}] \end{aligned}$$
(29)
$$\begin{aligned} E[R^{{\text {TS}}}] = E[R^{{\text {Int}}}]> E[R^{\text {D}}] > E[R^{{\text {PP}}}]. \end{aligned}$$
(30)
In the mean-variance CAPM, the only parameter that leads to different expected returns between different assets is the beta of an asset. The risk-free rate and the equity premium are not dependent on what kind of asset is regarded. Therefore, the respective betas must follow the same ordering as the expected returns:
6$$\begin{aligned} \beta _{{\text {TS}},{\text {M}}} = \beta _{{\text {Int}},{\text {M}}}< \beta _{{\text {D}},{\text {M}}} <\beta _{{\text {PP}},{\text {M}}}\end{aligned}$$
(31)
$$\begin{aligned} \beta _{{\text {TS}},{\text {M}}} = \beta _{{\text {Int}},{\text {M}}}=\beta _{{\text {D}},{\text {M}}} = \beta _{{\text {PP}},{\text {M}}} \end{aligned}$$
(32)
$$\begin{aligned} \beta _{{\text {TS}},{\text {M}}} =\beta _{{\text {Int}},{\text {M}}}>\beta _{{\text {D}},{\text {M}}} > \beta _{{\text {PP}},{\text {M}}}. \end{aligned}$$
(33)
To derive equations for betas, I use Eq. (
12). I write it down in the form
$$\begin{aligned} S\times E[R^{\text {E}}] + D\times E[R^{\text {D}}] = (S + D - {\text {VTS}}) \times E[R^{\text {U}}] + {\text {VTS}} \times E[R^{{\text {TS}}}]. \end{aligned}$$
(34)
Rearranging, simplifying, and using the CAPM equations leads to
$$\begin{aligned} \beta _{{\text {E}},{\text {M}}}= \frac{S + D}{S}\times \beta _{{\text {U}},{\text {M}}} - \frac{D}{S}\times \beta _{{\text {D}},{\text {M}}} + \frac{{\text {VTS}}}{S}\times (\beta _{{\text {TS}},{\text {M}}} - \beta _{{\text {U}},{\text {M}}}). \end{aligned}$$
(35)
This equation allows the beta of the tax savings to be different than the one for total debt payments. In what follows, I establish equations that are comparable to the case with the pro rata loss distribution. I write the equation for the tax savings as
$$\begin{aligned} \nonumber {\text {VTS}}&=p({\text {TS}})=\tau \times p({\text {Int}})\nonumber \\&=\tau \times p(D \times r^{\text {c}} - L^{{\text {Int}}})\nonumber \\&=\tau \times \left( \frac{D\times r^{\text {c}}}{R^{\text {f}}} - p(L^{{\text {Int}}})\right) \nonumber \\&=\tau \times \frac{D\times r^{\text {c}}}{R^{\text {c}}}\times\left( \frac{R^{\text {c}}}{R^{\text {f}}} - \frac{p(L^{{\text {Int}}})}{D}\times \frac{R^{\text {c}}}{r^{\text {c}}}\right) \nonumber \\&=\tau \times \frac{D r^{\text {c}}}{R^{\text {c}}} + \tau \times \left( p(L)\times\frac{r^{\text {c}}}{R^{\text {c}}} - p(L^{{\text {Int}}}) \right) . \end{aligned}$$
(36)
The first term in the last equality is the equation for the value of the tax savings for a pro rata loss distribution according to contractual debt payments. In case of a pro rata loss distribution, the second term is always zero because then
\(p(L^{{\text {Int}}})=p(L)\times\frac{R^{\text {c}}}{r^{\text {c}}}\). With loss distributions not proportional to contractual debt payments, the second term is usually not zero. Using that in Eq. (
35), I obtain
$$\begin{aligned} \beta _{{\text {E}},{\text {M}}}=&\left( 1 + \frac{D}{S}\times \frac{1 + r^{\text {c}}\times (1-\tau )}{R^{\text {c}}} \right) \times \beta _{{\text {U}},{\text {M}}} - \frac{D}{S}\times \beta _{{\text {D}},{\text {M}}} + \frac{D}{S} \times \frac{\tau \times r^{\text {c}}}{R^{\text {c}}}\times \beta _{{\text {TS}},{\text {M}}} \nonumber \\&+ \frac{F}{S}\times (\beta _{{\text {TS}},{\text {M}}} - \beta _{{\text {U}},{\text {M}}}), \end{aligned}$$
(37)
with
\(F=\tau \times \left( p(L)\times \frac{r^{\text {c}}}{R^{\text {c}}} - p(L^{{\text {Int}}}) \right)\). It turns out that additional information is needed. The beta of the returns on tax savings, i.e., on interest payments is needed as well as the price of losses on interest payments.
7 I define
\(\beta _{\Delta {\text {TS}},{\text {M}}}=\beta _{{\text {TS}},{\text {M}}} - \beta _{{\text {D}},{\text {M}}}\). I use this relation and restate Eq. (
37) as
$$\begin{aligned} \beta _{{\text {E}},{\text {M}}}=&\left( 1 +\frac{D}{S}\times \frac{1 + r^{\text {c}}\times (1-\tau )}{R^{\text {c}}}\right) \times \beta _{{\text {U}},{\text {M}}} - \beta _{{\text {D}},{\text {M}}}\times \frac{D}{S}\times \frac{1 + r^{\text {c}}\times (1-\tau )}{R^{\text {c}}} \nonumber \\&+\left( \frac{D}{S} \times \frac{\tau \times r^{\text {c}}}{R^{\text {c}}} + \frac{F}{S}\right) \times \beta _{\Delta TS,M} + \frac{F}{S}\times (\beta _{{\text {D}},{\text {M}}} - \beta _{{\text {U}},{\text {M}}}). \end{aligned}$$
(38)
This allows for a better comparability with Eq. (
26), i.e., the equation for the pro rata loss distribution according to contractual principal and interest payments. I continue with two prominent cases of loss distributions: interest and principal prioritization.
Loss distribution not proportional to contractual debt payments—interest prioritization A reasonable non-proportional loss distribution is the case of interest prioritization. Interest prioritization means that principal payments will incur losses first. Only if losses are greater than principal payments, interest will incur losses as well. Relation (
28) is usually what we expect in this case.
8 I simplify further. I assume that interest payments will never incur losses. This is a reasonable assumption as long as interest payments are small relative to principal payments, which is what we mainly observe in practice. Under this assumption, interest payments are risk-free so that
\(L^{{\text {Int}}}=0\) in any state. The price of losses on interest payments must be zero as well. It follows that the beta of tax savings is zero. The equation for interest payments turns to
$$\begin{aligned} {\text {Int}}=r^{\text {c}} \times D. \end{aligned}$$
(39)
and tax savings are
$$\begin{aligned} {\text {TS}}^{{\text {NC}}}&=\tau \times r^{\text {c}}\times D. \end{aligned}$$
(40)
I discount this risk-free quantity at the risk-free rate, i.e.,
\(E[R^{{\text {TS}}}]=R^{\text {f}}\), to obtain the value of the tax savings
$$\begin{aligned} {\text {VTS}}^{{\text {NC}}}&=\tau \times \frac{r^{\text {c}}\times D}{R^{\text {f}}}. \end{aligned}$$
(41)
Equation (
37) condenses to
$$\begin{aligned} \beta _{{\text {E}},{\text {M}}} = \left( 1 + \frac{D}{S}\times \frac{R^{\text {f}} - \tau \times r^{\text {c}} }{R^{\text {f}}}\right) \times \beta _{{\text {U}},{\text {M}}} - \beta _{{\text {D}},{\text {M}}}\times \frac{D}{S}. \end{aligned}$$
(42)
As for the pro rata distribution the levered beta is also a combination of the unlevered beta and a scalar as well as the debt beta and a scalar. However, the scalars differ here.
Given total losses on debt
L, the losses on interest payments
\(L^{{\text {Int}}}\) must be greater than the pro rata share of total losses:
\(L\times\frac{r^{\text {c}}}{R^{\text {c}}} < L^{{\text {Int}}}\). Since this is true for any state in which losses occur, the factor
F is less than zero:
\(F=\tau \times \left( p(L)\times\frac{r^{\text {c}}}{R^{\text {c}}} - p(L^{{\text {Int}}}) \right) <0\). To parameterize, I assume that the price of losses on interest payments is equal to
\(p(L^{{\text {Int}}}) = p(L)\times\left( \frac{r^{\text {c}}}{R^{\text {C}}} + \alpha \right)\), with
\(\alpha \in \left( 0, 1-\frac{r^{\text {c}}}{R^{\text {c}}}\right)\) as the percentage that the price of interest losses is higher than the pro rata share of the price of total losses. Using this parameter in the equation for
F, I obtain
$$\begin{aligned} F&=-\tau \times \alpha \times p(L) \end{aligned}$$
(43)
$$\begin{aligned}&=-\tau \times \alpha \times D\times \frac{r^{\text {c}} - r^{\text {f}}}{R^{\text {f}}}. \end{aligned}$$
(44)
The second equality follows from the equation for the coupon rate,
10 which can be rearranged for the price of losses. Equation (
38) turns to
$$\begin{aligned} \beta _{{\text {E}},{\text {M}}}=&\left( 1 +\frac{D}{S}\times \frac{1 + r^{\text {c}}\times (1-\tau )}{R^{\text {c}}}\right) \times \beta _{{\text {U}},{\text {M}}} - \beta _{{\text {D}},{\text {M}}}\times \frac{D}{S}\times \frac{1 + r^{\text {c}}\times (1-\tau )}{R^{\text {c}}} \nonumber \\&+ \tau \times \frac{D}{S}\times \left( \frac{r^{\text {c}}}{R^{\text {c}}} - \alpha \times \frac{r^{\text {c}} - r^{\text {f}}}{R^{\text {f}}}\right) \times \beta _{\Delta {\text {TS}},{\text {M}}} + \tau \times \alpha \times \frac{r^{\text {c}} - r^{\text {f}}}{R^{\text {f}}}\times \frac{D}{S}\times (\beta _{{\text {U}},{\text {M}}} - \beta _{{\text {D}},{\text {M}}}). \end{aligned}$$
(45)
Notice that with
\(\alpha \in \left( 0, 1-\frac{r^{\text {c}}}{R^{\text {c}}}\right)\), it follows, for the term attached to
\(\beta _{\Delta {\text {TS}},{\text {M}}}\), that
\(\frac{r^{\text {f}}}{R^{\text {f}}}<\left( \frac{r^{\text {c}}}{R^{\text {c}}} - \alpha \times \frac{r^{\text {c}} - r^{\text {f}}}{R^{\text {f}}}\right) < \frac{r^{\text {c}}}{R^{\text {c}}}\).
11 Furthermore, it is reasonable to assume that
\(\beta _{\Delta {\text {TS}},{\text {M}}}>0\), i.e., the beta of the returns on tax savings is greater than the one for the returns on debt. Since the return on tax savings and on interest payments are the same, the betas of the two figures are the same as well. With losses first assigned to interest payments, their returns will regularly have a higher beta than the beta for returns on principal payments and the one for returns on debt payments as a whole.
12
In the next subsection, I will compare the differences of the equations more in detail.