A. Information Structure and Payoffs

For all three assets-in-place, there are two states: a good state and a bad state. The state of the economy is revealed at $ t=1 $ . We use notations $ G $ (good) and $ B $ (bad) to represent the two states of the assets-in-place for the stand-alone firm ( $ {F}_s $ ). For the multi-division firm, we denote the states for the assets-in-place of the first division ( $ {F}_m(1) $ ) as $ G1 $ and $ B1 $ , and for the assets-in-place of the second division ( $ {F}_m(2) $ ) as $ G2 $ and $ B2 $ . The probability of each state is $ \frac{1}{2} $ , and the outcomes are independent across all assets-in-place. For any given asset-in-place, the conditional probabilities of the cash flow at $ t=2 $ are as follows: If the state at $ t=1 $ is good (bad), the probabilities are $ \frac{1}{2} $ , $ \frac{1}{3} $ , $ \frac{1}{6} $ ( $ \frac{1}{6} $ , $ \frac{1}{3} $ , $ \frac{1}{2} $ ). The conditional distributions of $ {F}_s $ at $ t=2 $ , based on the information from $ t=0 $ and $ t=1 $ , areFootnote ⁹

(1) $$ \underset{t=0}{\underbrace{{\left.{F}_s\right|}_{t=0}=\left\{\begin{array}{cc}24,& \mathrm{with}\hskip0.33em \mathrm{prob}=1/3,\\ {}12,& \mathrm{with}\hskip0.33em \mathrm{prob}=1/3,\\ {}0,& \mathrm{with}\hskip0.33em \mathrm{prob}=1/3,\end{array}\right.}} $$

(2) $$ \underset{t=1}{\underbrace{{\left.{F}_s\right|}_{\operatorname{G},t=1}=\left\{\begin{array}{cc}24,& \mathrm{with}\ \mathrm{prob}=1/2,\\ {}12,& \mathrm{with}\ \mathrm{prob}=1/3,\\ {}0,& \mathrm{with}\ \mathrm{prob}=1/6,\end{array}\right.\hskip0.24em {\left.{F}_s\right|}_{\operatorname{B},t=1}=\left\{\begin{array}{cc}24,& \mathrm{with}\ \mathrm{prob}=1/6,\\ {}12,& \mathrm{with}\ \mathrm{prob}=1/3,\\ {}0,& \mathrm{with}\ \mathrm{prob}=1/2,\end{array}\right.}} $$

and the conditional expectation of $ {F}_s $ at $ t=2 $ , given the information from $ t=1 $ , is

(3) $$ \unicode{x1D53C}({\left.{F}_s\right|}_{\varPi_s,t=1})=\left\{\begin{array}{cc}16,& \mathrm{with}\hskip0.33em {\Pi}_s=G,\\ {}8,& \mathrm{with}\hskip0.33em {\Pi}_s=B,\end{array}\right. $$

where $ {\Pi}_s $ is the state variable for $ {F}_s $ , which can take two realizations, $ G $ and $ B $ , with equal probability.

Since $ {F}_m(1) $ and $ {F}_m(2) $ are independent and $ {F}_m={F}_m(1)+{F}_m(2) $ , we obtain $ {F}_m $ ’s conditional distribution by convolving the distributions of $ {F}_m(1) $ and $ {F}_m(2) $ :

(4) $$ \underset{t=0}{\underbrace{{\left.{F}_m\right|}_{t=0}=\left\{\begin{array}{cc}24,& \mathrm{with}\ \mathrm{prob}=1/9,\\ {}18,& \mathrm{with}\ \mathrm{prob}=2/9,\\ {}12,& \mathrm{with}\ \mathrm{prob}=1/3,\\ {}6,& \mathrm{with}\ \mathrm{prob}=2/9,\\ {}0,& \mathrm{with}\ \mathrm{prob}=1/9,\end{array}\right.}}\hskip1.2em \underset{t=1}{\underbrace{{\left.{F}_m\right|}_{G1G2,t=1}=\left\{\begin{array}{cc}24,& \mathrm{with}\ \mathrm{prob}=1/4,\\ {}18,& \mathrm{with}\ \mathrm{prob}=1/3,\\ {}12,& \mathrm{with}\ \mathrm{prob}=5/18,\\ {}6,& \mathrm{with}\ \mathrm{prob}=1/9,\\ {}0,& \mathrm{with}\ \mathrm{prob}=1/36,\end{array}\right.}} $$

(5) $$ \underset{t=1}{\underbrace{{\left.{F}_m\right|}_{B1B2,t=1}=\left\{\begin{array}{cc}24,& \mathrm{with}\ \mathrm{prob}=1/36,\\ {}18,& \mathrm{with}\ \mathrm{prob}=1/9,\\ {}12,& \mathrm{with}\ \mathrm{prob}=5/18,\\ {}6,& \mathrm{with}\ \mathrm{prob}=1/3,\\ {}0,& \mathrm{with}\ \mathrm{prob}=1/4,\end{array}\right.\hskip0.24em {\left.{F}_m\right|}_{G1B2\mathrm{or}B1G2,t=1}=\left\{\begin{array}{cc}24,& \mathrm{with}\ \mathrm{prob}=1/12,\\ {}18,& \mathrm{with}\ \mathrm{prob}=2/9,\\ {}12,& \mathrm{with}\ \mathrm{prob}=7/18,\\ {}6,& \mathrm{with}\ \mathrm{prob}=2/9,\\ {}0,& \mathrm{with}\ \mathrm{prob}=1/12,\end{array}\right.}} $$

the conditional expectation of $ {F}_m $ at $ t=2 $ , given the information at $ t=1 $ , is

(6) $$ \unicode{x1D53C}({\left.{F}_m\right|}_{\varPi_m,t=1})=\left\{\begin{array}{cc}16,& \mathrm{with}\hskip0.33em {\Pi}_m=G1G2,\\ {}12,& \mathrm{with}\hskip0.33em {\Pi}_m=G1B2\hskip0.33em \mathrm{or}\hskip0.33em B1G2,\\ {}8,& \mathrm{with}\hskip0.33em {\Pi}_m=B1B2,\end{array}\right. $$

and $ {F}_m $ ’s state variable $ {\Pi}_m $ has three realizations, and its distribution is given by

(7) $$ {\left.{\varPi}_m\right|}_{t=0}=\left\{\begin{array}{cc}G1G2,& \mathrm{with}\ \mathrm{prob}=1/4,\\ {}G1B2\hskip0.33em \mathrm{or}\hskip0.33em B1G2,& \mathrm{with}\ \mathrm{prob}=1/2,\\ {}B1B2,& \mathrm{with}\ \mathrm{prob}=1/4.\end{array}\right. $$

The binomial tree representations of the possible paths of $ {F}_s $ and $ {F}_m $ in the 2 periods are presented in Figures 1 and 2, respectively. It is worth noting that the firm value is invariant to corporate diversification when the firm is an all-equity firm, as $ \unicode{x1D53C}\left({\left.{F}_s\right|}_{t=0}\right)=\unicode{x1D53C}\left({\left.{F}_m\right|}_{t=0}\right)=12 $ . This is consistent with Mansi and Reeb (Reference Mansi and Reeb2002), in that for all-equity firms, corporate diversification is unrelated to excess total firm value. However, corporate diversification does reduce the conditional standard deviation of future firm value: $ \mathrm{Std}\left({\left.{F}_m\right|}_{t=0}\right)=6.93<9.80=\mathrm{Std}\left({\left.{F}_s\right|}_{t=0}\right) $ . This is consistent with the coinsurance effect of corporate diversification argued by Lewellen (Reference Lewellen1971), Galai and Masulis (Reference Galai and Masulis1976), and Hann et al. (Reference Hann, Ogneva and Ozbas2013).

Figure 1 Timeline of the Possible Paths of Stand-Alone Firm $ S $ ’s Assets-in-Place

Figure 1 plots all possible values of stand-alone firm $ S $ ’s assets-in-place on $ t=2 $ and two states $ \left\{G\hskip0.42em \mathrm{and}\hskip0.42em B\right\} $ on $ t=1 $ . The probability of each path is shown along the path. Long-term and short-term face values of debt are indicated in the figure as long and short lines with corresponding legends.

Figure 2 Timeline of the Possible Paths of Multi-Division Firm $ M $ ’s Assets-in-Place

In Figure 2, Graphs A and B plot all possible values of multi-division firm $ M $ ’s two assets-in-place $ {F}_m(1) $ and $ {F}_m(2) $ , respectively, on $ t=2 $ and two states $ \left\{G\hskip0.42em \mathrm{and}\hskip0.42em B\right\} $ on $ t=1 $ . The probability of each path is shown along the path. Graph C plots the same paths for the combined assets-in-place $ {F}_m $ for firm $ M $ . Long-term and short-term face values of debt are indicated in Graph C as long and short lines with corresponding legends.

B. Debt Overhang and Investment Incentives

We now introduce debt into the firm value to study the effect of corporate diversification on debt overhang and shareholders’ incentives. We follow Diamond and He (Reference Diamond and He2014) and assume that both $ S $ and $ M $ need to raise $ 8.25 $ at $ t=0 $ . The debt can be either long-term (maturing at $ t=2 $ ) or short-term (maturing at $ t=1 $ ). As shown in Diamond and He (Reference Diamond and He2014), given the payoffs above and the need to raise $ 8.25 $ , the short-term and long-term debt’s nominal values are $ {L}_s^{\mathrm{ST}}=8.5 $ and $ {L}_s^{\mathrm{LT}}=12.75 $ for $ S $ . Our model extension to the multi-division firm leads to $ {L}_m^{\mathrm{ST}}=8.33 $ and $ {L}_m^{\mathrm{LT}}=10.38 $ for $ M $ . Figures 1 and 2 illustrate the relationship between the payoffs and the nominal values of debt for $ S $ and $ M $ , respectively. The discrepancy between the nominal debt value of $ S $ and that of $ M $ arises from the reduction in default risk due to the coinsurance effect.Footnote ¹⁰ We assume that there is no cost to raise either short-term or long-term debt.Footnote ¹¹ For simplicity, we focus on an infinitesimal investment that only weakly increases or leaves unchanged the value of each of its debt and equity claims.Footnote ¹² Such investment occurs immediately after raising the debt at $ t=0 $ , and results in a marginal increment of the final cash flows at $ t=2 $ equal to $ \varepsilon >0 $ . The short-term ( $ {O}_i^{\mathrm{ST}} $ ) and long-term debt overhang ( $ {O}_i^{\mathrm{LT}} $ ) are

(8) $$ {O}_i^{\mathrm{ST}}={\unicode{x1D53C}}_0({\mathbf{1}}_{\{\unicode{x1D53C}({\operatorname{}{F}_i|}_{t=1})<{L}_i^{\mathrm{ST}}\}})\hskip0.33em \mathrm{a}\mathrm{n}\mathrm{d}\hskip0.42em {O}_i^{\mathrm{LT}}={\unicode{x1D53C}}_0({\mathbf{1}}_{\{{F}_i\ \mathrm{a}\mathrm{t}\ t=2<{L}_i^{\mathrm{LT}}\}}),\hskip0.6em \mathrm{f}\mathrm{o}\mathrm{r}\hskip0.42em i=s\hskip0.33em \mathrm{a}\mathrm{n}\mathrm{d}\hskip0.33em m, $$

where $ {\mathbf{1}}_{\left\{\cdot \right\}} $ is an indicator function that equals 1 when the condition in $ \left\{\cdot \right\} $ holds, and 0 otherwise. Combining (8) with the conditional distributions of $ {F}_s $ and $ {F}_m $ , we have

(9) $$ {O}_s^{\mathrm{ST}}=\frac{1}{2},\hskip1em {O}_s^{\mathrm{LT}}=\frac{2}{3},\hskip1em {O}_m^{\mathrm{ST}}=\frac{1}{4},\hskip1em \mathrm{and}\hskip1em {O}_m^{\mathrm{LT}}=\frac{1}{3}. $$

Comparing $ {O}_s^{\mathrm{ST}} $ with $ {O}_m^{\mathrm{ST}} $ and $ {O}_s^{\mathrm{LT}} $ with $ {O}_m^{\mathrm{LT}} $ shows that corporate diversification mitigates debt overhang in the short term and the long term. Moreover, $ {O}_m^{\mathrm{LT}}-{O}_m^{\mathrm{ST}}<{O}_s^{\mathrm{LT}}-{O}_s^{\mathrm{ST}} $ . Thus, corporate diversification reduces the difference between long-term and short-term debt overhang (the wedge).Footnote ¹³

Now, we describe how corporate diversification affects investment incentives. Denote the percentage investment cost by $ \lambda $ , and let

(10) $$ \left(1-\lambda \right)>{O}_i^j, $$

where $ i\in \left\{s,m\right\} $ and $ j\in \left\{ ST, LT\right\} $ . This condition implies that a firm invests only in projects with an NPV exceeding the debt overhang.

For $ S $ ’s shareholders, the condition above is satisfied if $ \lambda <1/3 $ , regardless of whether the firm raises short-term or long-term debt, implying an internal rate of return ( $ \mathrm{IRR}=\left(1-\lambda \right)/\lambda $ ), larger than $ 200\% $ . For $ M $ ’s shareholders, the condition becomes $ \lambda <2/3 $ , or equivalently $ \mathrm{IRR}>50\% $ . Focusing on the optimal choice for short-term debt only, the investment condition modifies to $ \lambda <1/2 $ ( $ \mathrm{IRR}>100\% $ ) for $ S $ ’s shareholders and $ \lambda <3/4 $ ( $ \mathrm{IRR}>33.\overline{3}\% $ ) for $ M $ ’s shareholders. Therefore, all else being equal, a multi-division firm is more likely to invest in new projects than a comparable stand-alone firm. This occurs because corporate diversification mitigates debt overhang through the coinsurance effect.

Although cash holding is not explicitly modeled here, our model implies that multi-division firms have more incentives to deploy excess cash (Opler, Pinkowitz, Stulz, and Williamson (Reference Opler, Pinkowitz, Stulz and Williamson1999)) for investment. This could result in a reduction in excess cash due to reduced debt overhang, providing an alternative explanation for Duchin (Reference Duchin2010), who finds that multi-division firms hold significantly less cash than stand-alone firms.Footnote ¹⁴

To recap, assuming that the market value of debt is the same for both $ S $ and $ M $ , the simple model above predicts that corporate diversification mitigates the debt overhang problem by decreasing both the extent of long-term and short-term debt overhang, as well as the wedge between them. However, the simplicity of this model comes at a cost: we assume that noninterest expenses are 0, and this leads us to conclude that short-term debt is preferred to long-term debt for both single-division and multi-division firms. This confirms Myers’s (Reference Myers1977) suggestion that short-term debt is a possible solution to the debt overhang problem in a frictionless scenario.Footnote ¹⁵

In Section II.C, we relax the assumption of zero noninterest expenses and we generalize our model to allow for a number of divisions, $ N $ , larger than 2. In line with the results of our 3-period model with only two divisions for $ M $ , we impose the following conditions:

(11) $$ \frac{\partial {O}^{\mathrm{LT}}(N)}{\partial N}<0,\hskip1em \frac{\partial {O}^{\mathrm{ST}}(N)}{\partial N}<0,\hskip1em \mathrm{and}\hskip1em \frac{\partial {\Delta}_O(N)}{\partial N}<0, $$

where $ 0<{O}^{\mathrm{ST}}(N)\le {O}^{\mathrm{LT}}(N)<1 $ and $ {\Delta}_O(N)={O}^{\mathrm{LT}}(N)-{O}^{\mathrm{ST}}(N) $ , i.e., the wedge. Therefore, by definition, $ {O}_s^{\mathrm{LT}}={O}^{\mathrm{LT}}(1) $ , $ {O}_s^{\mathrm{ST}}={O}^{\mathrm{ST}}(1) $ , $ {O}_m^{\mathrm{LT}}={O}^{\mathrm{LT}}(2) $ , and $ {O}_m^{\mathrm{ST}}={O}^{\mathrm{ST}}(2) $ .

C. Noninterest Debt Expenses and Debt Maturity

Short-term debt is known to have disadvantages over long-term debt. For example, short-term debt has higher issuance costs and higher rollover costs than long-term debt due to the higher frequency at which short-term debt needs to be issued or rolled over (Acharya et al. (Reference Acharya, Gale and Yorulmazer2011), He and Xiong (Reference He and Xiong2012), Cheng and Milbradt (Reference Cheng and Milbradt2012), Valenzuela (Reference Valenzuela2016)). To incorporate these additional noninterest costs in our model, we assume that the funding raised via short-term debt is proportional to investment size. Specifically, we denote such extra costs by $ \xi \ge 0 $ and define the overhang-adjusted NPV (Chen and Manso (Reference Chen and Manso2017)) as follows:

(12) $$ {R}_i^j=\left[1-\left(1+\xi {\mathbf{1}}_{\left\{j=\mathrm{ST}\right\}}\right)\lambda \right]\varepsilon -{O}_i^j\varepsilon, $$

which can be used to compare the investment incentives under different scenarios. When $ \xi =0 $ , as mentioned before, short-term debt is always preferred over long-term debt in terms of investment incentives in both $ S $ and $ M $ . When $ \xi >0 $ , however, debt maturity preferences depend on whether the firm is diversified or not. Due to the third condition in (11)—the wedge becomes smaller as $ N $ increases—when $ \xi >0 $ , the overhang-adjusted NPV for projects funded using long-term debt is more likely to be higher for $ M $ than for $ S $ . To formalize this intuition and generalize its validity to a broad range of realistic scenarios, we need to introduce a regularity assumption.

Assumption 1. The short-term debt expense $ \xi $ and the investment cost $ \lambda $ are independently and uniformly distributed on $ \left[0,\overline{\xi}\right] $ and $ \left[\underline{\lambda},1\right] $ , respectively. $ \overline{\xi} $ and $ \underline{\lambda} $ satisfy the following constraintsFootnote ¹⁶:

(13) $$ \frac{O_s^{LT}-{O}_s^{ST}}{1-{O}_s^{LT}}+\log \left(\frac{1-{O}_s^{ST}}{1-{O}_s^{LT}}\right)<\overline{\xi}\le \frac{2\left({O}_s^{LT}-{O}_s^{ST}\right)}{1-{O}_s^{LT}}, $$

(14) $$ 0<\underline{\lambda}<1-{O}_s^{LT}. $$

The uniform distribution assumption is common in the asset-pricing literature (Oehmke and Zawadowski (Reference Oehmke and Zawadowski2015), Glode and Opp (Reference Glode and Opp2016), Hollifield, Neklyudov, and Spatt (Reference Hollifield, Neklyudov and Spatt2017)). Both (13) and (14) are sufficient (albeit not necessary) conditions for the proposition we introduce below. Given reasonable values of $ {O}_s^{\mathrm{ST}} $ and $ {O}_s^{\mathrm{LT}} $ , equation (13) ensures $ \overline{\xi}>0 $ with a bounded upper limit, and equation (14) rise in plausible IRR scenarios.

Now, we use $ {R}^{ST} $ and $ {R}^{LT} $ to denote the NPV of projects funded with short-term debt and long-term debt, respectively. Moreover, $ {P}^{\mathrm{ST}} $ is the probability of raising short-term debt instead of long-term debt to invest in projects with a positive NPV: $ {P}^{\mathrm{ST}}=\unicode{x1D53C}\left({\mathbf{1}}_{\left\{{R}^{\mathrm{ST}}>\max \left(0,{R}^{\mathrm{LT}}\right)\right\}}\right) $ . Similarly, the probability of raising long-term debt instead of short-term debt is defined as $ {P}^{\mathrm{LT}}=\unicode{x1D53C}\left({\mathbf{1}}_{\left\{{R}^{\mathrm{LT}}>\max \left(0,{R}^{\mathrm{ST}}\right)\right\}}\right) $ . To understand the impact of corporate diversification, we use the subscript $ N $ , denoting the number of segments. Thus, $ {P}_N^{ST} $ and $ {P}_N^{LT} $ are the probabilities of investing using short-term and long-term debt, respectively, for a firm with $ N $ segments.Footnote ¹⁷

Given these definitions, we can now introduce Proposition 1, which is proved in Appendix A.

Proposition 1. Given a fixed market value of debt and conditions in (11) and Assumption 1, there exists a threshold $ {N}^{\ast } $ such that firms with more than $ {N}^{\ast } $ segments are more likely to invest using long-term debt, whereas those with fewer than $ {N}^{\ast } $ segments are more likely to invest using short-term debt. More formally:

$$ \mathrm{\exists}{N}^{\ast }:\{\begin{array}{cc}{P}^{LT}<{P}^{ST},& if\;N<{N}^{\ast },\\ {}{P}^{LT}\ge {P}^{ST},& if\;N\ge {N}^{\ast }.\end{array}\operatorname{} $$

Given the values of $ {O}_s^{\mathrm{LT}} $ and $ {O}_s^{\mathrm{ST}} $ in (9), the constraints in Assumption 1 for $ \overline{\xi} $ and $ \underline{\lambda} $ are

(15) $$ 0.91<\overline{\xi}\le 1\hskip0.62em \mathrm{and}\hskip0.62em 0<\underline{\lambda}<1/3. $$

For numerical illustration, we set $ \overline{\xi}=1 $ , which means the short-term debt expense is shared by each investment up to the total size of the initial investment outlay before the short-term debt expense; $ \underline{\lambda}=0.25 $ , which means $ \lambda \in \left[\mathrm{0.25,1}\right] $ , so that each investment’s IRR before the short-term debt expense is positive and no more than $ 300\% $ .Footnote ¹⁸ Given these settings and the values of $ {O}^{\mathrm{LT}} $ and $ {O}^{\mathrm{ST}} $ in (9), we have

(16) $$ {P}_s^{\mathrm{ST}}=0.112>0.047={P}_s^{\mathrm{LT}}, $$

(17) $$ {P}_m^{\mathrm{ST}}=0.116<0.447={P}_m^{\mathrm{LT}}, $$

where $ {P}_s^{ST} $ and $ {P}_m^{ST} $ are $ {P}^{ST} $ for the stand-alone and multi-division firms, respectively, and $ {P}_s^{LT} $ and $ {P}_m^{LT} $ are $ {P}^{LT} $ for the stand-alone and multi-division firms, respectively.

To see the above intuition more clearly, we plot in Figure 3 the probabilities of investing with short-term debt $ {P}_i^{\mathrm{ST}} $ and long-term debt $ {P}_i^{\mathrm{LT}} $ defined in Proposition 1, which are derived in Appendix A. The area of the different shapes represents the probability values. $ {P}^{\mathrm{ST}} $ ’s area clearly diminishes with $ {O}^{\mathrm{LT}}-{O}^{\mathrm{ST}} $ getting smaller, especially $ {P}_2^{\mathrm{ST}} $ , which has an upper bound of $ \frac{{\left({O}^{\mathrm{LT}}-{O}^{\mathrm{ST}}\right)}^2}{2\left(1-{O}^{\mathrm{LT}}\right)} $ .

Figure 3 Plots of the Numerical Examples of $ {P}_{\boldsymbol{s}}^{\mathbf{ST}} $ and $ {P}_{\boldsymbol{m}}^{\mathbf{ST}} $

In Figure 3, the areas of the different shapes represent probability values. Graphs A and B, respectively, plot stand-alone firm $ S $ and multi-division firm $ M $ ’s probabilities of investing with short-term debt $ {P}_i^{\mathrm{ST}} $ and long-term debt $ {P}_i^{\mathrm{LT}} $ defined in Proposition 1. The formulae of different areas are presented in Appendix A. The numerical values of parameters are set as follows: $ {O}_s^{\mathrm{ST}}=\frac{1}{2},{O}_s^{\mathrm{LT}}=\frac{2}{3},{O}_m^{\mathrm{ST}}=\frac{1}{4},\mathrm{and}\;{O}_m^{\mathrm{LT}}=\frac{1}{3} $ ; $ \overline{\xi}=1 $ and $ \underline{\lambda}=0.25 $ .

With this simple 3-period model, we gain valuable insights into how Hann et al.’s (Reference Hann, Ogneva and Ozbas2013) coinsurance effect of corporate diversification alleviates debt overhang and enhances investment incentives. This model elucidates a novel prediction accounting for higher noninterest expenses for short-term debt: a positive association between corporate diversification and debt maturity. However, this 3-period model is unable to incorporate more nuanced features for further analysis, such as the possibility of size heterogeneity for the segments of a multi-division firm, correlated payoffs for different segments, and continuous debt maturity. To offer further insights, in Section III, we develop a continuous-time structural model using option pricing.

A. A Basket Option Approach for Modeling Corporate Diversification

At time $ t $ , we have two potential outcomes for shareholders: if $ {V}_t<L $ , debt holders take over the defaulted firm and shareholders receive 0; if $ {V}_t\ge L $ , debt holders are repaid the full amount $ L $ and shareholders receive the residual value $ {V}_t-L $ . Thus, at time $ 0 $ , the market equity value of a levered firm with $ N $ divisions is

(19) $$ E\left(L,t\right)={\unicode{x1D53C}}_0^{\mathrm{\mathbb{Q}}}\left[{\left(\sum \limits_{i=1}^N{v}_{i,t}-L\right)}^{+}\right], $$

and the corresponding market value of debt is $ D\left(L,t\right)={V}_0-E\left(L,t\right) $ . Although the exact closed-form solution of the basket option is unavailable—to the best of our knowledge—highly accurate approximations exist. Here, we use Ju’s (Reference Ju2002) Taylor expansion approximation as the solution to equation (19) Footnote ²⁰:

(20) $$ E\left(L,t\right)=\left[{U}_1N\left({y}_1\right)- LN\left({y}_2\right)\right]+L\left({z}_1p(y)+{z}_2\frac{dp(y)}{dy}+{z}_3\frac{d^2p(y)}{dy^2}\right), $$

where $ N\left(\cdot \right) $ is the standard normal CDF and $ p\left(\cdot \right) $ is the normal PDF with mean $ \mu (1) $ and variance $ \nu (1) $ ,

$$ y=\mathit{\log}(L),{y}_1=\frac{\mu (1)-y}{\sqrt{\nu (1)}}+\sqrt{\nu (1)},\hskip1em {y}_2={y}_1-\sqrt{\nu (1)}. $$

Closed-form expressions for $ \mu (x) $ , $ \nu (x) $ , $ {z}_1 $ , $ {z}_2 $ , and $ {z}_3 $ are provided in Appendix B.

B. Revisiting the Corporate Diversification’s Effect on Debt Maturity and Overhang

Since we focus on debt overhang from infinitesimal investments, we define the debt overhang measure as follows:

(21) $$ O\left(N,t,W\right)=\sum \limits_{i=1}^N{W}_i\frac{\partial D\left(L,t\right)}{\partial {v}_{i,0}}=1-\sum \limits_{i=1}^N{W}_i\frac{\partial E\left(L,t\right)}{\partial {v}_{i,0}}, $$

where $ {W}_i $ is the $ i $ th element of the $ N\times 1 $ weighting vector $ W $ and $ {\sum}_i^N{W}_i=1 $ . The debt overhang measure defined by Diamond and He (Reference Diamond and He2014) under their Black–Scholes–Merton setting is a special case of equation (21) when $ N=1 $ . As argued by Galai and Masulis (Reference Galai and Masulis1976), corporate diversification increases firms’ debt capacity, which could result in higher leverage and/or longer debt maturity. Since the evidence on the relationship between corporate diversification and leverage is weak (see Berger and Ofek (Reference Berger and Ofek1995), Mansi and Reeb (Reference Mansi and Reeb2002), and Section IV), and our empirical results in Section IV provide strong evidence on the positive association between corporate diversification on debt maturity, we focus on a counterfactual analysis for debt maturity while constraining the face value of debt $ L $ to be the same for both $ S $ and $ M $ .Footnote ²¹

We use the Black–Scholes formula to price $ S $ ’s equity value, and we allow the debt maturity of $ M $ to vary until its equity value matches that of $ S $ . This enables us to calculate the implied debt maturity for $ M $ (i.e., implied by its equity value). Shareholders maximize the firm value by choosing the optimal debt maturity, conditional on their firm’s divisional structure. In other words, shareholders choose the optimal debt maturity with the minimal investment cost and debt overhang to achieve a given level of firm value and growth.

We impose conditions to ensure that $ M $ and $ S $ are strictly comparable. Specifically, we constrain $ M $ to have the same total assets-in-place, growth rate, market value of equity, and face value of debt as $ S $ . For simplicity, we also set each division’s assets-in-place within $ M $ to have equal weights and with the same volatility as $ S $ . To price $ M $ ’s equity value, we also need the pairwise correlation $ \rho $ between any two divisions, which negatively affects corporate diversification’s coinsurance effect. We set the pairwise correlation coefficient to three different levels: {0, 0.1, 0.3}. Thanks to the analytical formula in equation (20), we can easily solve for the value of debt maturity by equating $ M $ ’s equity value to that of $ S $ .

The numerical results are shown in Figure 4. From Graph A of Figure 4, we can clearly see that, given the same values of total assets-in-place, face value of debt, growth rate, and market value of equity, $ M $ ’s debt maturity is longer than that of $ S $ and increases with the number of segments, confirming the notion stated in Proposition 1 that $ M $ tend to issue long-term debt relative to $ S $ . This tendency becomes stronger as the number of segments grows. Graph B of Figure 4 shows the changing pattern of debt overhang with the increasing number of segments. The pattern matches nicely with the results in Section II that corporate diversification reduces debt overhang. We also find that as the average pairwise correlation between divisions decreases, the debt maturity increases and the debt overhang decreases even further. This observation reinforces the idea from Section II that corporate diversification increases debt maturity and mitigates debt overhang via the coinsurance effect.

Figure 4 Debt Maturity and Overhang Change with Number of Segments

Graph A (Graph B) of Figure 4 plots debt maturity (debt overhang) against the number of segments. The numerical values for the parameters are set as follows: $ {g}_i=r=5\% $ , $ {\sigma}_i=0.4 $ , $ V=100 $ , $ {v}_i=\frac{100}{N} $ , and $ L=60 $ . The three curves in Graph A (Graph B) represent debt maturity (debt overhang) with three pairwise correlation levels ( $ \rho =0,\ 0.1, $ and $ 0.3 $ ). The debt maturity and overhang of the comparable stand-alone firm are also plotted for reference purposes.

So far, the analysis using the basket option approach mirrors Section II.B with enhanced flexibility in modeling. However, we have not taken into account the maturity-sensitive noninterest debt expenses. Next, we consider the costs (analogous to Section II.C) and demonstrate that the intuition of Proposition 1 on the long-term debt and short-term debt separation in multi-division firms and stand-alone firms can also be shown under the basket option approach. Specifically, we use an exponential function to capture the maturity-sensitive noninterest debt expenses and extend the overhang-adjusted NPV in equation (12) to the following overhang and cost-adjusted NPVFootnote ²²:

(22) $$ R\left(N,t\right)=\left\{1-O\left(N,t,W\right)-\left[1+\exp \left(- bt\right)\right]\lambda \right\}\varepsilon, $$

where the cost function $ \exp \left(- bt\right) $ , with the maturity-sensitivity parameter $ b>0 $ , captures the negative correlation between noninterest debt expenses and maturity. Setting $ b=2.4 $ , $ \lambda =0.4 $ , $ \rho =0 $ , and $ N=6 $ alongside other numerical values already set in Figure 4, we present in Figure 5 an example of long-term debt and short-term debt separation consistent with Proposition 1. Maximizing the overhang-adjusted NPV in equation (22), there are cases where stand-alone firms’ optimal debt maturity is shorter than that of multi-division firms, when all else is equal. Figure 5 (x-axis and left y-axis) presents an example of such cases.

Figure 5 Optimal Debt Maturity in the Presence of Noninterest Debt Expenses

The left y-axis in Figure 5 visually compares the overhang and cost-adjusted NPVs, given various debt maturities of the multi-division firm with those of the comparable stand-alone firm. The right y-axis visually compares the corresponding market values of equity of the multi-division firm with those of the comparable stand-alone firm. The numerical values for the parameters are set as follows: $ {g}_i=r=5\% $ , $ {\sigma}_i=0.4 $ , $ N=6 $ , $ V=100 $ , $ {v}_i=\frac{100}{6} $ , $ L=60 $ , and $ {\rho}_{i,j}=0 $ .

C. Endogenous Debt Maturity in Corporate Diversification

We investigate how corporate diversification affects debt maturity and overhang by constraining the multi-division firm’s book value of debt to be the same as that of the stand-alone firm. This is intentional and consistent with Admati et al.’s (Reference Admati, DeMarzo, Hellwig and Pfleiderer2018) leverage ratchet effect—where shareholders resist book leverage reductions—and is confirmed empirically by Berger and Ofek (Reference Berger and Ofek1995), who document that there is no economically significant difference between the book leverage of multi-division and stand-alone firms. This controlled setting allows us to isolate the impact of corporate diversification on debt maturity and overhang while holding other determinants of firm value constant.

The coinsurance effect of corporate diversification on firm risk is well understood in the corporate diversification discount literature, but there is currently no formal investigation of its potential effects on debt maturity and overhang. There is, however, some evidence suggesting that corporate diversification might benefit debt holders relative to shareholders in levered firms. Specifically, Mansi and Reeb (Reference Mansi and Reeb2002) study the risk effects of corporate diversification and its impact on firm value in levered and all-equity firms. In all-equity firms, there is no corporate diversification discount, while in levered firms, shareholders’ losses due to corporate diversification increase with leverage. Moreover, the overall impact of corporate diversification on excess firm value tends to be negligible in levered firms. Thus, these results suggest that the coinsurance effect reduces the market value of equity and enhances the market value of debt in levered firms. Mansi and Reeb’s (Reference Mansi and Reeb2002) argument is essentially a restatement of a potential consequence of corporate diversification that has been put forward in earlier contributions, such as Higgins and Schall (Reference Higgins and Schall1975), Galai and Masulis (Reference Galai and Masulis1976), and Kim and McConnell (Reference Kim and McConnell1977). The coinsurance effect of corporate diversification may result in a wealth transfer from shareholders to debt holders. However, the implicit assumption in Mansi and Reeb (Reference Mansi and Reeb2002) is that when firms diversify, they maintain the same maturity and face value of debt. This is not necessarily the case in real capital markets.

Figure 5 illustrates the effect of corporate diversification on the market value of debt and equity for a given value of debt maturity. Specifically, the right y-axis in Figure 5 shows the market values of the stand-alone firm and the comparable multi-division firm. Assuming a debt maturity of 1 year for both firms, the market equity value of the multi-division firm is about 43, whereas that of the stand-alone firm is about 44. This result verifies Mansi and Reeb’s (Reference Mansi and Reeb2002) hypothesis: conditional on the assumption of the same debt maturity, the market value of equity for a stand-alone firm (red straight line in the graph) is higher than that of a multi-division firm (blue dashed straight line in the graph). However, this is a strong assumption in a world of imperfect ‘me-first’ rules where the shareholders control the investment decision (Galai and Masulis (Reference Galai and Masulis1976), Kim and McConnell (Reference Kim and McConnell1977)).

Now, let us consider what happens if we relax the assumption of constant debt maturity. The shareholders of the multi-division firm can increase the NPV of their investments (adjusted for debt overhang and investment cost) by choosing a longer debt maturity, as shown on the left y-axis of Figure 5. For example, choosing a debt maturity of 2.2 years—the optimal debt maturity maximizing the adjusted NPV for the multi-division firm in Figure 5—increases the market value of equity from about 43 (for 1-year maturity) to over 46. The optimal debt maturity (i.e., the debt maturity that maximizes the adjusted NPV) for the stand-alone firm is 1 year, which corresponds to a market value of equity of about 44. If both firms can choose their optimal debt maturity to maximize the adjusted NPV of their investments, diversifying can actually result in a premium. In other words, corporate diversification does not necessarily lead to a lower equity value if firms can increase their debt maturity when they decide to diversify, and increasing the debt maturity could lead to a diversification premium.

D. Implications for Corporate Diversification Discount and Premium

These results bear major implications on the interpretation of previous findings related to the existence of a diversification discount: if multi-division firms tend to have longer debt maturity than stand-alone firms—as we show below in our empirical exercise—due to the coinsurance effect, traditional measures of corporate diversification discount (Berger and Ofek (Reference Berger and Ofek1995), Mansi and Reeb (Reference Mansi and Reeb2002)) could be misleading. Our analysis here predicts that the debt maturity of multi-division firms can explain the traditional excess value measures (see Mansi and Reeb ((Reference Mansi and Reeb2002), equation (1))). Specifically, the benchmark used in conventional excess value measures is the sum of the market values of median stand-alone firms in each relevant industry segment. That is, for a multi-division firm $ i $ with $ m $ divisions, conventional excess value is measured as follows:

(23) $$ {V}_i^m-\sum \limits_{j=1}^m\mathrm{M}\mathrm{d}\mathrm{n}\{{V}_k^s,k\in j\mathrm{t}\mathrm{h}\ \mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\}, $$

where $ {V}_i^m $ is the value of firm $ i $ and $ \mathrm{M}\mathrm{d}\mathrm{n}\{{V}_k^s,k\in j\mathrm{t}\mathrm{h}\ \mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\} $ denotes the median value of stand-alone firms operating in sector $ j $ , controlled to be comparable (e.g., having the same level of sales) to division $ j $ of firm $ i $ .

However, we argue that an accurate measure of excess value should be based on a comparison of the firm’s value to the sum of the (hypothetical) values of its divisions—as if each division were operated independently—with financial structures optimized for stand-alone operation. Formally,

(24) $$ {V}_i^m-\sum \limits_{j=1}^m{V}_{i,j}^s, $$

where $ {V}_{i,j}^s $ represents the counterfactual value of division $ j $ if it were a stand-alone firm. Importantly, $ {V}_{i,j}^s $ is computed assuming the division adopts its own optimal debt maturity policy (the one that reflects its specific characteristics).

Replacing $ {V}_{i,j}^s $ with $ \mathrm{Mdn}\left\{{V}_k^s,k\in j\mathrm{th}\;\mathrm{sector}\right\} $ in the conventional excess value measure neglects the endogenous nature of debt maturity in corporate diversification. Therefore, the debt maturity of the benchmark in the conventional excess value measure is not adjusted to reflect the benefits of reduced debt overhang that comes with corporate diversification: the debt maturity of the benchmark has a much smaller cross-sectional variation and concentrates around the median debt maturity of stand-alone firms; the debt maturity of multi-division firms is much more heterogeneous. As a result, the conventional excess value measure may misrepresent the true economic implications of corporate diversification.

According to our analysis above, overlooking the debt maturity adjustment in the benchmark causes the excess value to increase with the debt maturity. A more appropriate excess value measure taking into account the endogenous debt maturity should not have a significant association with debt maturity. Unfortunately, there is no easy fix for the debt maturity misalignment issue in the traditional way (Berger and Ofek (Reference Berger and Ofek1995)) of measuring excess value. Without a structural model, it will be a daunting task to calculate the optimal debt maturity for the stand-alone counterpart of the multi-division firm. A radically different approach than the traditional one could be required to handle the endogenous debt maturity choice. The structural model we developed here shows potential. However, it is out of the scope of this article to explore this potential. We leave this for future research.

A. Data and Panel Regression Specifications

We obtain annual firm-level data and SIC industry classifications from Compustat – Fundamentals Annual, and annual division-level data from the Compustat – Historical Segments. The sample period commences in 1978, coinciding with the availability of Compustat segment data, and ends in 2022. Following Boguth et al. (Reference Boguth, Duchin and Simutin2022), we exclude firms with at least one division with SIC code within 6000–6999 (financial sector), below 1000 (agriculture), or equal to 8600, 8800, 8900, and 9000 (government, other noneconomic activities, or unclassified services). Our final sample includes 30,135 firms over 46 years of data. We also use Fama–French 48 industry classifications.Footnote ²³ We follow Chen et al. (Reference Chen, Xu and Yang2012) and Berger and Ofek (Reference Berger and Ofek1995) to construct the key variables for our empirical analysis. We report the details of the variable definitions in Table 2. Our proxy for debt maturity is the logarithm of debt maturity, denoted as DEBT_MATURITY, and our proxies for corporate diversification are a dummy variable identifying diversified firms (MULTI_DIVISION) and the logarithm of the number of segments (NUM_SEGMENTS).Footnote ²⁴ The summary statistics and pairwise correlations are presented in Table 3.

Table 3 Summary Statistics and Pairwise Correlations

B. Debt Maturity and Corporate Diversification

In this subsection, we present our key empirical results supporting our theoretical predictions regarding the positive association between corporate diversification and debt maturity.

1. Debt Maturity Before and After Switches in Divisional Structure

Previous literature on the determinants of the decision to diversify or refocus (Campa and Kedia (Reference Campa and Kedia2002)) neglects the role of debt maturity. In this section, we examine how DEBT_MATURITY changes around switches from being a stand-alone firm to being a multi-division firm (diversification), or vice versa (refocusing). Table 4 presents the results of 2-sample t-tests before and after the year of the switch. Regardless of the direction of the switch, the average difference in DEBT_MATURITY between the diversified and stand-alone state is statistically significant and positive in seven cases out of eight.Footnote ²⁵ This finding confirms that corporate diversification (refocusing) is always associated with subsequently longer (shorter) debt maturity. The results in the table are consistent with those reported in Figure 6, where we visualize the median DEBT_MATURITY over an 8-year window (4 years before and 4 years after) around the year of the switch.

Figure 6 Debt Maturity Before and After Diversification and Refocusing

Figure 6 plots the median DEBT_MATURITY of firms that change from stand-alone to multi-division or the other way around, or both, over an 8-year window around the year of the switch (4 years before and 4 years after). The dashed (solid) line shows the median DEBT_MATURITY dynamic of cases in which firms change from stand-alone to multi-division (multi-division to stand-alone).

Table 4 Debt Maturity Before and After Diversification and Refocusing

2. Primary Results

The central prediction of our theoretical models is that corporate diversification allows firms to have longer debt maturities. To test empirically whether this is true, we regress DEBT_MATURITY on MULTI_DIVISION and NUM_SEGMENTS and a set of control variables: MARKET_EQUITY, DEBT_TO_EQUITY, NET_INCOME, and CAPITAL_EXPENDITURE. We also include year-fixed effects and three types of cross-sectional fixed effects: firm-fixed effects, industry-fixed effects based on 4-digit SIC codes, and industry-fixed effects based on the 48 Fama–French industries.

In Table 5, we report the results of these regressions. The coefficients for MULTI_DIVISION (NUM_SEGMENTS) are around 0.052 (0.041) with individual firm-fixed effects, 0.098 (0.073) with 4-digit SIC industry-fixed effects, and 0.107 (0.078) with 48 Fama–French industry-fixed effects. All coefficient estimates are statistically significant at the 1% level. The coefficients on MULTI_DIVISION, which range between 0.052 and 0.107, suggest that diversification is associated with an increase in debt maturity of slightly more than 1 year.Footnote ²⁶ Since the median debt maturity for stand-alone firms is 4 years, corporate diversification allows a median stand-alone firm to increase debt maturity by around 25%.

Table 5 Debt Maturity Regression Results

The coefficients on NUM_SEGMENTS are positive and statistically significant, and range between 0.041 and 0.078. Since both variables are in logs, this means that a 10% increase in the number of segments increases debt maturity by around 0.39%–0.75%.Footnote ²⁷ Therefore, our results reported in Table 5 suggest that the effect of corporate diversification on debt maturity is mainly driven by the transition from being a stand-alone to a diversified firm, rather than investing in one additional segment. Once a firm has diversified, increasing the degree of diversification does not lead to substantial increases in debt maturity. In the Supplementary Material, we show that the results are robust to using Hoberg and Phillips’s (Reference Hoberg and Phillips2016) text-based network industry classifications Herfindahl-Hirschman Index (HHI) measures.

3. Incremental Explanatory Power

In Table 6, we examine the incremental explanatory power—in terms of adjusted $ {R}^2 $ —of NUM_SEGMENTS in regressions on DEBT_MATURITY. To facilitate the comparison with the results, including other explanatory variables, we report the results without firm fixed effects. Panel A of Table 6 reports the results for univariate regressions. Consistent with existing literature (Stohs and Mauer (Reference Stohs and Mauer1996)), the regression on MARKET_EQUITY yields the highest adjusted $ {R}^2 $ (14%). The regression on NUM_SEGMENTS has the second-highest adjusted $ {R}^2 $ , which is approximately 3.6%, corresponding to around 26% of the explanatory power of MARKET_EQUITY (as reported in the last row of Panel A of Table 6). The regressions on the other variables have an adjusted $ {R}^2 $ between 1.5% and 2.7%, corresponding to around 11%–19% of the explanatory power of MARKET_EQUITY.

Table 6 Incremental R ² Results

Panel B of Table 6 reports the results of multivariate regressions where we examine the incremental explanatory power of NUM_SEGMENTS relative to the others: DEBT_TO_EQUITY (first column), NET_INCOME (second column), CAPITAL_EXPENDITURES (third column), and MARKET_EQUITY(fourth column). In the first 3 columns of Panel B of Table 6, the incremental explanatory power of NUM_SEGMENTS ranges between 3% and 4%. The incremental explanatory power of the other variables ranges between 1.5% and 2.7%, consistent with the results of the univariate regression in Panel A of Table 6. However, the incremental explanatory power of NUM_SEGMENTS in the regression on MARKET_EQUITY is 0.3%. This result is not surprising: diversification and size are strongly connected, since diversified firms tend to be larger than stand-alone firms (Hund et al. (Reference Hund, Monk and Tice2024)), and the decision to diversify is influenced by recent asset growth (Campa and Kedia (Reference Campa and Kedia2002)). Consistent with this interpretation, the correlation between MARKET_EQUITY and NUM_SEGMENTS is 0.35, substantially higher than that between NUM_SEGMENTS and other control variables (ranging from −0.04 to 0.08; see Table 3). MARKET_EQUITY depends on both NUM_SEGMENTS and an average division size measure. Thus, the incremental value of NUM_SEGMENTS beyond MARKET_EQUITY lies in disentangling corporate diversification effects from overall firm size, offering additional insight into the determinants of DEBT_MATURITY.

4. Subsample Analysis

To further ease the concern that the results reported in Table 5 could be driven by firm size, we repeat the regression of Table 5 on three subsamples defined by at’s tertiles by years. The results are presented in Table 7. The positive effect of corporate diversification on debt maturity is observed in all three subsamples, confirming that the results reported in Table 5 are robust. However, we do find that the coefficients of MULTI_DIVISION and NUM_SEGMENTS become smaller and less significant as the firm size moves into larger tertiles, suggesting that corporate diversification has a more pronounced effect on small- and medium-sized firms’ debt maturity than on large size firms’. Although not modeled in our theory, it is intuitively sensible that firms are less keen to adjust debt maturity after diversification when they have lower growth options for investment. Therefore, a diminishing effect of corporate diversification on debt maturity is consistent with the fact that firm growth decreases with firm size (see, e.g., Evans (Reference Evans1987), Moeller, Schlingemann, and Stulz (Reference Moeller, Schlingemann and Stulz2004), Beck, Demirguc-Kunt, Laeven, and Levine (Reference Beck, Demirguc-Kunt, Laeven and Levine2008)).

Table 7 Debt Maturity Regression Results Conditional on Total Assets Tertiles

C. The Role of Debt Overhang

The central prediction of Proposition 1 is that corporate diversification leads to longer debt maturities because it mitigates the debt overhang problem. In this section, we test whether this is empirically verified using the proxy for overhang in Alanis et al. (Reference Alanis, Chava and Kumar2018) (OVERHANG).Footnote ²⁸ Unlike other measures of debt overhang that infer default probability using credit ratings, this proxy estimates default probabilities directly using the hazard model developed by Chava and Jarrow (Reference Chava and Jarrow2004), and can therefore be applied even to firms without credit ratings. Similar to our model, this proxy is based on a positive relationship between default probability and debt overhang and a positive relationship between the market value of debt and investment. For robustness, we also construct an alternative proxy, OVERHANG_ALT, for debt overhang from our data. For $ i $ th firm in year $ t $ , OVERHANG_ALT is defined as follows:

(25) $$ \mathrm{OVERHANG}\_\mathrm{AL}{\mathrm{T}}_{i,t}=\exp \left(-\frac{capx{v}_{i,t}}{d{t}_{i,t}}\right). $$

This definition ensures that OVERHANG_ALT is within [0, 1] and positively (negatively) related to leverage (long-term investment), consistent with Myers’s (Reference Myers1977) debt overhang theory and Cai and Zhang’s (Reference Cai and Zhang2011) empirical evidence.

To examine whether debt overhang is the channel through which corporate diversification affects debt maturity, we need to interact the proxy for debt overhang with our proxies for corporate diversification. We thus run regressions where we interact NUM_SEGMENTS with a dummy variable, OVERHANG_DUM, which is equal to 1 for firm $ i $ in year $ t $ if OVERHANG of firm $ i $ is higher than the median OVERHANG in year $ t $ , and 0 otherwise. Using OVERHANG_DUM, instead of OVERHANG, allows us to interpret the coefficients on the interaction term NUM_SEGMENTS  $ \times $  OVERHANG_DUM more easily.

The results in Panel A of Table 8 for OVERHANG_DUM suggest that debt overhang is positively related to debt maturity, consistent with the view that longer debt maturities are associated with higher debt overhang. The coefficient on NUM_SEGMENTS  $ \times $  OVERHANG_DUM is positive and statistically significant, confirming that corporate diversification’s positive relationship with debt maturity is stronger for firms with a higher degree of debt overhang. In other words, our results indicate that firms with higher debt overhang are more keen to take advantage of issuing longer-maturity debt when they diversify their businesses. In Panel B of Table 8, we replace OVERHANG_DUM with OVERHANG_DUM_ALT, and we find similar results: the coefficients on OVERHANG_DUM_ALT are positive and statistically significant, as are those on the interaction term NUM_SEGMENTS  $ \times $ OVERHANG_DUM_ALT.

Table 8 Overhang as a Channel for the Debt Maturity and Corporate Diversification Association

Taken together, the results reported in Table 8 provide corroborating evidence that debt overhang is a key factor channeling the interplay between corporate diversification and debt maturity.

D. Debt Maturity and Excess Value

To test our prediction of the positive relationship between debt maturity and the excess value (EV), we follow Berger and Ofek (Reference Berger and Ofek1995) and Mansi and Reeb (Reference Mansi and Reeb2002) and compute the EV for firms in our sample. More concretely, we measure the EV as the log difference between a firm’s capital value (the market value of equity  $ + $  the book value of debt) and the sum of imputed values for its segments as stand-alone entities. We calculate the imputed value of each segment by multiplying the median ratio, for stand-alone firms in the same industry, of CAPITAL_TO_SALES. The industry median ratios are based on the most refined SIC category that includes at least five single-line businesses with at least $20 million of sales and sufficient data for computing CAPITAL_TO_SALES. Specifically, for firm $ i $ with $ m $ division, its excess value is calculated as

(26) $$ {\mathrm{EV}}_i=\mathrm{TOTAL}\_{\mathrm{CAPITAL}}_i-\log \left[\sum \limits_{j=1}^m{sale}_{i,j}\times {\mathrm{Mdn}}_j\left(\mathrm{CAPITAL}\_\mathrm{TO}\_\mathrm{SALES}\right)\right], $$

where $ {sale}_{i,j} $ is the net sales of the $ j $ th division in firm $ i $ and $ {\mathrm{Mdn}}_j\left(\mathrm{CAPITAL}\_\mathrm{TO}\_\mathrm{SALES}\right) $ is the median CAPITAL_TO_SALES ratio of stand-alone firms in the $ j $ th division’s industry. As in Berger and Ofek (Reference Berger and Ofek1995), extreme EVs are excluded from the analysis. “Extreme” is defined as an absolute EV value above 1.386 (i.e., actual values either more than 4 times imputed or less than one-fourth imputed).

The cross-sectional distribution of EV over time is presented in Figure 7. It is clear that the EV value has turned more negative in recent years, consistent with recent studies using Berger and Ofek’s (Reference Berger and Ofek1995) EV measure. We regress the EV on DEBT_MATURITY and MULTI_DIVISION or NUM_SEGMENTS alongside the control variables and present the regression results in Table 9. The coefficients of DEBT_MATURITY are around 0.03 and highly significant at the 1% level in all versions of the regressions. In economic terms, the point estimates of the coefficients mean that a 1-standard-deviation increase in DEBT_MATURITY results in a 1.6% increase in the EV. These results provide strong evidence supportive of our prediction on the positive relationship between EV and debt maturity.

Figure 7 Excess Value Cross-Sectional Distribution over the Years

Figure 7 plots the cross-sectional distribution of Berger and Ofek’s (Reference Berger and Ofek1995) excess value measure from 1978 to 2022. The solid line represents the cross-sectional median of the excess value in each year. The shadowed area around the solid line represents the 25th to 75th percentiles of the cross-sectional distribution of the excess value in each year.

Table 9 Excess Value Regression Results

We also note that the coefficients of both MULTI_DIVISION and NUM_SEGMENTS are negative and significant. The negative sign of MULTI_DIVISION’s coefficient is consistent with Berger and Ofek ((Reference Berger and Ofek1995), Table 3) and Mansi and Reeb ((Reference Mansi and Reeb2002), Table II), indicating that the EV measure is more negative for multi-division firms than for stand-alone firms. The magnitude of MULTI_DIVISION’s coefficient captures the difference in average EV between multi-division and stand-alone firms. This difference is about −4.8% in our sample, which is very close to Mansi and Reeb’s (Reference Mansi and Reeb2002) −4.5% in their Table II. The coefficients of NUM_SEGMENTS are around −0.044 to −0.065 in all three versions of the regression with statistical significance at 1%, indicating, in economic terms, that for an average multi-division firm, increasing the degree of diversification by two segments will induce 7% decrease in its EV measure. These results are robust to sample selection as evidenced in Table 10, where we repeat the same regressions in subsamples before and after 2000 and find qualitatively the same conclusion in both subsamples. These are evidence replicating results found in typical studies of corporate diversification. The fact that the traditional EV measure is significantly correlated with debt maturity, which is consistent with our theoretical prediction, suggests that this measure could be misleading and likely to be overstated because it does not allow for the endogenous nature of debt maturity preferences in corporate diversification choices.

Table 10 Excess Value Subsample Regression Results