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Investment timing and capital structure with loan guarantees
Finance Research Letters xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Investment timing and capital structure with loan guarantees q Hua Xiang, Zhaojun Yang ⇑ School of Finance and Statistics, Hunan University, Changsha, China
a r t i c l e
i n f o
Article history: Received 18 November 2014 Accepted 24 January 2015 Available online xxxx JEL classification: C61 G11 G31 G24 Keywords: Loan guarantees Real options Capital structure
a b s t r a c t The equity-for-guarantee swap (EGS) is a new popular financial derivative. We derive closed-form solutions for the interaction of the optimal investment and financing with the swap in a realoptions framework. We find that there is an U-shaped relationship between investment timing and the coupon payment. In contrast to the classical model, EGS induces different investment and financing strategies. In particular, it delays investment. Whether the swap leads to debt overhang distortion depends on guarantee cost and the profitability of the project but it induces the borrower’s risk-shifting incentive. The larger the guarantee cost, the stronger the incentive. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction To solve financing constraints faced by small and medium-sized enterprises (SMEs), a new financial contract, called equity-for-guarantee swap (EGS), is introduced by entrepreneurs in China. This is an agreement among a bank/lender, an insurer and an SME/borrower, where a bank lends at a given interest rate to an SME and once the SME defaults on the loan, the insurer must pay all the outstanding interest and principal to the bank instead of the SME. In return for the guarantee, the SME must allocate a fraction of equity to the insurer, which is called the guarantee cost. Through the intervention of q The research for this paper was supported by National Natural Science Foundation of China (Project Nos. 71171078, 71371068 and 71221001). ⇑ Corresponding author. Tel.: +86 731 8864 9918; fax: +86 731 8868 4772. E-mail addresses: [email protected] (H. Xiang), [email protected] (Z. Yang).
http://dx.doi.org/10.1016/j.frl.2015.01.006 1544-6123/Ó 2015 Elsevier Inc. All rights reserved.
Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
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H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
insurers, the credit standing of borrowers is promoted, so that they can get financial services easier and cheaper from commercial banks. In particular, Chinese private online financial organizations have recently shaken the traditional financial markets monopolized by state-owned banks. For example, Alibaba’s Yu’e Bao has raised US$ 65.38 billion in eight months. To decrease financial risk originated from the Internet finance, it is believed that a credit guarantee system including the EGS would play an important role. Recently, much attention has actually been paid to EGS. For example, Yang and Zhang (2013) derive its equilibrium price while Yang and Zhang (forthcoming) discuss its utility-based price. Wang et al. (2015) consider a borrower who enters into EGS and invests in a take-it-or-leave-it project with a cash-out option. However, there is no literature considering the interaction of investment and financing with EGS and whether it leads to inefficiencies from debt overhang and asset substitution. Our paper is related to the literature on real options, which employs the real options approach to investigate the interplay between financing and investment, such as Sarkar (2011), Hackbarth and Mauer (2012) and Sundaresan et al. (forthcoming). However, they do not take financing constraints into account. We develop a real-options framework with the EGS to overcome financial constraints. The inefficiencies under the swap from debt overhang and asset substitution are discussed. The remainder of the paper is organized as follows. Section 2 describes the model setup and provides a benchmark model without the swap. Section 3 derives closed-form solutions for the investment, financing, default threshold, guarantee cost and the firm’s value. Section 4 provides numerical analysis. Section 5 concludes. 2. Model setup and an all-equity benchmark Model setup. We assume the only asset owned by an SME is an option to invest in a single project by paying a fixed investment I (sunk cost). The project generates cash flow QXðtÞ after the investment is exercised, where Q > 0 is constant, representing the product quantity of the project and XðtÞ is its product price. We suppose that the price follows a geometric Browmian motion given by
dXðtÞ ¼ ldt þ rdBðtÞ; t P 0; XðtÞ
Xð0Þ ¼ x0 > 0 given;
ð1Þ
where BðtÞ denotes the standard Brownian motion defined on the risk-neutral probability space ðX; F ; QÞ and l and r > 0 are constant. The cash flow QXðtÞ can be regarded as the firm’s earnings before interest and taxes at time t if the firm is in operation. Following Goldstein et al. (2001), we assume a simple tax structure that includes personal and corporate taxes, where interest payments are taxed at a personal rate si , effective dividends are taxed at sd , and corporate profits are taxed at sc , with full loss offset provisions. Let PðxÞ denote the after-tax value of the asset in place if the current price level is x, then we immediately get
PðxÞ ¼
1s Qx; rl
ð2Þ
where r is the constant risk-free interest rate, s is the effective tax rate defined by 1 s ¼ ð1 sc Þð1 sd Þ and l < r for a well-known reason. The benchmark model. We consider here the benchmark model where the SME finances the real investment with equity only. We denote the value of the unlevered firm by Eu ðxÞ if the current product price is x then the firm’s optimal investment problem can be formalized as follows:
"Z Eu ðxÞ ¼ supE T iU
þ1
T iU
rt
e
rT iU
ð1 sÞQXðtÞdt e
# IjXð0Þ ¼ x ;
ð3Þ
where T iU > 0 is the stopping time when the investment is exercised. Since this is an optimal stopping problem of Markovian dynamics, it suffices to consider the stopping times with the following form: Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
3
There is a trigger level xui , such that T iU :¼ infft P 0; XðtÞ P xui g. Therefore, by using a standard approach, the investment problem (3) can be equivalently written as
(
x sup ½Pðxui Þ I xui xui
b1 ) ;
1 l where b1 ¼ 2 þ 2 r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 l 2r 2 þ 2 > 1: 2 r r
After that, we directly derive the following proposition: Proposition 2.1. The optimal investment threshold and the value of the unlevered firm are given respectively by
xui ¼
b1 1 I; b1 1 Pð1Þ
Eu ðxÞ ¼ ½Pðxui Þ I
x xui
b1 ð4Þ
:
Throughout the text, we assume the current product price Xð0Þ ¼ x is so low that the investment should not be undertaken immediately. For example, we assume Xð0Þ ¼ x < xui in our benchmark model.
3. The pricing of corporate securities, capital structure and investment timing The pricing of debt, equity and guarantee cost. We assume the investment trigger level xi is given in advance. Just at the investment time, an entrepreneur makes financing decision by issuing debt and equity, where debt is perpetual with a guarantee from an insurer. Thanks to the guarantee, we assume the entrepreneur has the option to default on his debt obligations. For the same reason with the optimal investment, the optimal default policy is determined by a trigger level xd , i.e. the default time T d is given by T d ¼ infft P T i : XðtÞ ¼ xd g. Clearly, after investment has taken place, the levered firm should solve the following equity-valuation-maximization problem:
"Z El ðxi Þ ¼ supE Td
Td
# rðtT i Þ
e
ð1 sÞðQXðtÞ CÞdtjXðT i Þ ¼ xi ;
ð5Þ
Ti
where C is the coupon rate of the guaranteed debt paid by the borrower to the lender. Equivalently, it can be written as
( El ðxi Þ ¼ sup Pðxi Þ Pðxd Þ xd
b2 b2 !) xi C xi : ð1 sÞ 1 r xd xd
ð6Þ
Hence, the optimal default trigger level after investment is given by
xd
b2 r l C ¼ ; b2 1 Q r
1 l where b2 ¼ 2 2 r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 l 2r 2 þ 2 < 0: 2 r r
ð7Þ
Note that xd is given as a linear function of the coupon rate C. As aforementioned, we assume the entrepreneur has not enough money to invest in the project and he must borrow some money from a bank with the help of the EGS. We assume that once the borrower defaults on his debt, the lender takes the value of the asset minus bankruptcy costs. To determine the guarantee cost, let Dðxi Þ denote the value of debt at the investment time without insurance, we then obtain
"Z
Td
Dðxi Þ E
Ti
# erðtT i Þ ð1 si ÞCdt þ ð1 aÞerðT d T i Þ Pðxd ÞjXðT i Þ ¼ xi
b2 ! b2 C xi xi þ ð1 aÞPðxd Þ ¼ ð1 si Þ 1 ; r xd xd
Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
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H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
where a is the bankruptcy loss rate. Obviously, the last term represents the liquidation value of the firm at the investment time. We assume the insurer must pay the lender all the loss from the borrower’s default and so the lender’s asset is risk-free. Thus, its value is equal to ð1 si Þ Cr . For this reason, the value, denoted by Dguar ðxi Þ, of the insurer’s compensatory payment to the lender, must satisfy the following equation
Dðxi Þ þ ð1 si ÞDguar ðxi Þ ¼ D0 ðCÞ
C ð1 si Þ: r
ð8Þ
Therefore, we obtain
Dguar ðxi Þ ¼
b2 C 1a xi Pðxd Þ : r 1 si xd
ð9Þ
Clearly, to make the swap fair, an insurer’s compensatory payment Dguar ðxi Þ to a lender must be equal to the present value of an SME’s equity allocated to the insurer at the investment time. Consequently, it follows from (9) that
b2 C 1a xi Pðxd Þ ¼ /Eðxi Þ; r 1 si xd
ð10Þ
where / is called guarantee cost, which is the fraction of the SME’s equity allocated to the insurer in return for the guarantee. Hence, the guarantee cost / at the investment time is explicitly given by
b2 xi 1a 1 si Pðxd Þ xd /¼ b2 b2 ; Pðxi Þ Pðxd Þ xxi ð1 sÞ Cr 1 xxi C r
d
ð11Þ
d
which is a function of the product price xi when investment is exercised. Investment timing. Now, we turn to the optimal investment time. According to pecking order theory in corporate finance and due to asymmetric information, the borrower should only borrow the minimum money to start the project and so the coupon of debt issued is generally exogenously determined. As aforementioned, due to the Markovian dynamics we consider here, the optimal investment time is determined by a threshold value, xi , i.e. the firm optimally exercises the investment option once the price XðtÞ reaches xi from below. Therefore, for a given coupon rate C, the optimal threshold xi is the solution of the following optimization problem:
supfð1 /ÞEðxi Þ ðI D0 ðCÞÞg xi
b1 x : xi
ð12Þ
For this reason, we are led to the conclusion below. Proposition 3.1. For a given coupon rate C under EGS the borrowers optimal investment policy xi satisfies
ð1 b1 Þ
b2 x 1s C Qxi þ g i þ b1 I b1 ðs si Þ ¼ 0; r rl xd
ð13Þ
where
g ¼ ðb1 b2 Þ
a si sC : Pðxd Þ þ r 1 si
Furthermore, for a given investment threshold xi , the optimal debt level C is naturally given by
C ¼ argmaxfð1 /ÞEðxi Þ ðI D0 ðCÞÞg Therefore, letting u1 ¼
b2 1 b2
lr 1 r
Q
b1 x : xi
, we have
Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
a si 1 s s xi Q u1 þ ð1 b2 Þ 1 si r l r u1
b2
ðC Þ
b2
s si r
¼ 0:
5
ð14Þ
From (13) and (14), we derive the following conclusion. Proposition 3.2. With debt financing under EGS, the borrower’s optimal financing and investment policy are given by
C ¼
u2 b1 I b1 I ; xi ¼ ; u1 b1 1 k b1 1 k
ð15Þ
where
0 < u1 < 1;
0 < u2 ¼
a si 1 s sð1 b2 Þ ðb2 Þ þ 1 si s si s si
b1
2
< 1;
and
k¼
2 1s a si 1 s 1b2 1 si u2 s u1b 2 Q Q u2 þ : rl 1 si r l r u1 r u1
If si ¼ 0, this conclusion is actually implied by those derived by Sundaresan et al. (forthcoming) who however assume there are no financing constraints undergone by a borrower. In this way, we can say from another standpoint that thanks to EGS, the borrower’s financing constraints are effectively eliminated.
4. Numerical results To make an effective comparison, the baseline parameter values are selected as follows: the riskfree interest rate r ¼ 0:06, expected growth rate l ¼ 0:02, volatility r ¼ 0:3, personal tax rate si ¼ 0:25, corporate profit tax rate sc ¼ 0:35, dividend tax rate sd ¼ 0:20, bankruptcy loss rate a ¼ 0:5, initial product price x0 ¼ 1, sunk cost I ¼ 8 and the product quantity Q ¼ 1. With the baseline parameter values, the optimal coupon rate, investment threshold, default threshold, leverage and guarantee cost are given by C i ¼ 1:3325; xi ¼ 1:94; xd ¼ 0:4232; L ¼ 66:13% and / ¼ 0:2804 respectively. Investment timing and the firm’s values versus coupon rates. An entrepreneur might be reluctant to get loans at the cost of a fraction of equity since the value of equity would be significantly underestimated due to information asymmetry. His investment would therefore not take the optimal capital structure and the amount of money he borrows would just equal the financial gap for him to invest after he spends all his money. Consequently, we consider the case where the coupon rate is exogenously given. Fig. 1 reports the effect of the coupon rate on investment timing for three different volatilities (r ¼ 0:3; r ¼ 0:4; r ¼ 0:5). As we expected, no matter what capital structure is (i.e. how much the coupon rate C is), the bigger the risk, the larger the value of the option to invest and so the later the investment. Specifically, If SMEs undertake pure equity financing (i.e., C ¼ 0), then the optimal investment thresholds are xui ¼ 2:2144; 2:9099 and 3:7586 under the case of the volatilities r ¼ 0:3; 0:4 and 0:5 respectively. However, if debt financing is undertaken, there is an U-shaped relation between investment time and the coupon rate: Investment thresholds decrease first and then increase with the coupon rates. This is because a higher coupon rate has two opposite effects: One is to accelerate investment since it increases tax shields and the other is to delay investment because it increases bankruptcy costs and the guarantee cost as well. To investigate the effect of debt financing on the firm’s value, Fig. 2 presents the firm’s value as a function of the coupon payment for three different volatilities. The case of C ¼ 0 corresponds to an all-equity financing. It states that the value of the levered firm first rises and then declines with the coupon rate and the functions are concave, which indicates there is an optimal coupon rate to maximize the firm’s value under each given volatility. This happens just due to the same reason with Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
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H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
Investment trigger
4.5
σ=0.3 σ=0.4 σ=0.5
4
3.5
3
2.5
2
1.5
0
0.5
1
1.5
2
2.5
C Fig. 1. Investment thresholds as a function of the coupon rate with three different volatilities.
The value of option
7.2 7 6.8 6.6 6.4 6.2 6
σ=0.3 σ=0.4 σ=0.5
5.8 5.6 5.4 5.2
0
0.5
1
1.5
2
2.5
C Fig. 2. The firm’s value as a function of the coupon rate with three different volatilities.
why the curves in Fig. 1 are convex. In addition, Fig. 2 shows that under the baseline parameters, the optimal coupon rate lies between 1 and 1.5 and as we expected, the larger the volatility, the higher the firm’s value. Investment timing and the firm’s leverage versus volatilities under optimal capital structure. We set out to consider a situation where an entrepreneur has a deep pocket and thus the amount of money he borrows can be arbitrary instead of just the financial gap for him to invest. For this reason, the firm naturally takes the optimal capital structure, i.e. the coupon rate is endogenously decided to maximize the firm’s value rather than exogenously given. Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
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H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
Fig. 3 reports how investment threshold changes with the volatility r of the cash flow under the optimal capital structure for an unlevered firm, a levered firm with the swap and that without the swap, which is discussed by Sundaresan et al. (forthcoming). It says that the guaranteed firm always invests a little earlier than an all-equity financed firm but later than the firm discussed by Sundaresan et al. (forthcoming). As the volatility r increases, the optimal investment threshold for the guaranteed firm gets closer to that for an all-equity firm, and therefore, the overinvestment effect generated by EGS gradually disappears if the cash flow risk is high enough.
The optimal investment threshold 6 5.5 5
The levered firm with the swap The unlevered firm The levered firm without the swap
4.5 4 3.5 3 2.5 2 1.5 1 0.2
0.3
0.4
σ
0.5
0.6
0.7
Fig. 3. Optimal investment threshold versus the cash flow risk for an unlevered firm, a levered firm with and without the swap respectively.
The optimal leverage and guarantee cost 0.7 The guarantee cost Leverage with the swap Leverage without the swap
0.6
0.5
0.4
0.3
0.2
0.1
0 0.2
0.3
0.4
σ
0.5
0.6
0.7
Fig. 4. The leverage and guarantee cost under optimal capital structure versus the cash flow risk.
Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
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H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
Fig. 4 plots the firm’s optimal leverage and the guarantee cost as a function of the cash flow risk. The dotted line depicts the leverage of the firm with a guarantee, the dash line represents that derived by Sundaresan et al. (forthcoming), and the solid line gives the guarantee cost, respectively. It shows that as we expect, the optimal leverage ratio decreases monotonously in the volatility r. When the volatility is relatively low, the firm in our model should issue more debt than the firm considered by Sundaresan et al. (forthcoming). However, when the volatility gets large enough, the opposite holds true. Debt overhang and asset substitution. Under the EGS, the lender is well protected because the swap transfers his risk exposure entirely to the insurer. On account of that only the borrower (entrepreneur) has the right to manage the firm, naturally we wonder whether and how the swap induces inefficiencies from debt overhang and asset substitution. To achieve this aim, if the borrower invests an extra unit of account in the firm, we compute his net profit, which is reported in Fig. 5. It says that if the cash flow level is large and the guarantee cost is 1 0.8
Φ=0.1950
Φ=0.3324
0.6
Φ=0.5271
∂ (1−Φ)E/∂Π −1
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1
0
2
4
Cash flow X
6
8
10
Fig. 5. The sensitivity of the value of the shareholders’ claim to the firm’s value PðxÞ versus the cash flow level x. 5 4.5 Φ=0.1405
4
Φ=0.2520
Φ=0.4063
∂ (1−Φ)E/∂σ
3.5 3 2.5 2 1.5 1 0.5 0
0
5
10 Cash flow x
15
Fig. 6. The sensitivity of the value of the shareholders’ claim to the volatility
20
r as a function of the cash flow level x.
Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
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low enough, the net profit is positive, meaning that there is no debt overhang problem. But under the converse situation, the opposite holds true. Generally speaking, if debt is not protected, there must be a strong risk-shifting incentive for shareholders to invest in high risk project since they harvest all the profit if they succeed but transfer loss to the creditors if they fail. To measure the risk-shifting incentive, we compute the derivative of the value of the entrepreneur’s claim with regard to the volatility, which is reported in Fig. 6. As we expected, the derivative is always positive, meaning that the entrepreneur will invest in a higher project once he has such choice. The larger the guarantee cost, the stronger the incentive. For this reason, we propose that to enter into the swap, insurers should in advance specify the kind of projects the borrower must invest the loan in if the borrower has several investment choices in the future. 5. Conclusions The EGS is a new financial derivative, which is widely taken as small and medium-sized firm’s loan contracts in China and hopefully, it is believed that such swaps like this would play a vital role in Internet finance. However, there are few studies to consider the effect of the swap on investment and financing incentives of borrowers. We obtain closed-form solutions for the interaction of the optimal investment and financing and a linear relation between them is derived. We find that the levered firm would exercise its investment option earlier than the corresponding all-equity firm and there is an U-shaped relationship between investment timing and the coupon payment. In contrast to the classical real options theory, the EGS alters the investment and financing incentives of the borrower. In particular, the swap delays investment. Whether the swap induces debt overhang distortion depends on the guarantee cost and the profitability of the project but it surely gets rise to the borrower’s risk-shifting incentives. The larger the guarantee cost, the stronger the incentives. Accordingly, we suggest that while entering into the swap, the insurer had better in advance specify the kind of projects the borrower must invest the loan in if the borrower will have several investment choices after the swap contract is signed. References Goldstein, R., Ju, N., Leland, H., 2001. An EBIT-based model of dynamic capital structure. J. Bus. 74 (4), 483–512. Hackbarth, D., Mauer, D.C., 2012. Optimal priority structure, capital structure, and investment. Rev. Financ. Stud. 25 (3), 747– 796. Sarkar, S., 2011. Optimal size, optimal timing and optimal financing of an investment. J. Macroecon. 33 (4), 681–689. Sundaresan, S., Wang, N., Yang, J., 2015. Dynamic investment, capital structure, and debt overhang. Rev. Corp. Financ. Stud. (forthcoming). Wang, H., Yang, Z., Zhang, H., 2015. Entrepreneurial finance with equity-for-guarantee swap and idiosyncratic risk. Eur. J. Oper. Res. 241 (3), 863–871. Yang, Z., Zhang, C., 2015. Two new equity default swaps with idiosyncratic risk. Int. Rev. Econ. Financ. (forthcoming). Yang, Z., Zhang, H., 2013. Optimal capital structure with an equity-for-guarantee swap. Econ. Lett. 118 (2), 355–359.
Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Investment timing and capital structure with loan guarantees q Hua Xiang, Zhaojun Yang ⇑ School of Finance and Statistics, Hunan University, Changsha, China
a r t i c l e
i n f o
Article history: Received 18 November 2014 Accepted 24 January 2015 Available online xxxx JEL classification: C61 G11 G31 G24 Keywords: Loan guarantees Real options Capital structure
a b s t r a c t The equity-for-guarantee swap (EGS) is a new popular financial derivative. We derive closed-form solutions for the interaction of the optimal investment and financing with the swap in a realoptions framework. We find that there is an U-shaped relationship between investment timing and the coupon payment. In contrast to the classical model, EGS induces different investment and financing strategies. In particular, it delays investment. Whether the swap leads to debt overhang distortion depends on guarantee cost and the profitability of the project but it induces the borrower’s risk-shifting incentive. The larger the guarantee cost, the stronger the incentive. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction To solve financing constraints faced by small and medium-sized enterprises (SMEs), a new financial contract, called equity-for-guarantee swap (EGS), is introduced by entrepreneurs in China. This is an agreement among a bank/lender, an insurer and an SME/borrower, where a bank lends at a given interest rate to an SME and once the SME defaults on the loan, the insurer must pay all the outstanding interest and principal to the bank instead of the SME. In return for the guarantee, the SME must allocate a fraction of equity to the insurer, which is called the guarantee cost. Through the intervention of q The research for this paper was supported by National Natural Science Foundation of China (Project Nos. 71171078, 71371068 and 71221001). ⇑ Corresponding author. Tel.: +86 731 8864 9918; fax: +86 731 8868 4772. E-mail addresses: [email protected] (H. Xiang), [email protected] (Z. Yang).
http://dx.doi.org/10.1016/j.frl.2015.01.006 1544-6123/Ó 2015 Elsevier Inc. All rights reserved.
Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
2
H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
insurers, the credit standing of borrowers is promoted, so that they can get financial services easier and cheaper from commercial banks. In particular, Chinese private online financial organizations have recently shaken the traditional financial markets monopolized by state-owned banks. For example, Alibaba’s Yu’e Bao has raised US$ 65.38 billion in eight months. To decrease financial risk originated from the Internet finance, it is believed that a credit guarantee system including the EGS would play an important role. Recently, much attention has actually been paid to EGS. For example, Yang and Zhang (2013) derive its equilibrium price while Yang and Zhang (forthcoming) discuss its utility-based price. Wang et al. (2015) consider a borrower who enters into EGS and invests in a take-it-or-leave-it project with a cash-out option. However, there is no literature considering the interaction of investment and financing with EGS and whether it leads to inefficiencies from debt overhang and asset substitution. Our paper is related to the literature on real options, which employs the real options approach to investigate the interplay between financing and investment, such as Sarkar (2011), Hackbarth and Mauer (2012) and Sundaresan et al. (forthcoming). However, they do not take financing constraints into account. We develop a real-options framework with the EGS to overcome financial constraints. The inefficiencies under the swap from debt overhang and asset substitution are discussed. The remainder of the paper is organized as follows. Section 2 describes the model setup and provides a benchmark model without the swap. Section 3 derives closed-form solutions for the investment, financing, default threshold, guarantee cost and the firm’s value. Section 4 provides numerical analysis. Section 5 concludes. 2. Model setup and an all-equity benchmark Model setup. We assume the only asset owned by an SME is an option to invest in a single project by paying a fixed investment I (sunk cost). The project generates cash flow QXðtÞ after the investment is exercised, where Q > 0 is constant, representing the product quantity of the project and XðtÞ is its product price. We suppose that the price follows a geometric Browmian motion given by
dXðtÞ ¼ ldt þ rdBðtÞ; t P 0; XðtÞ
Xð0Þ ¼ x0 > 0 given;
ð1Þ
where BðtÞ denotes the standard Brownian motion defined on the risk-neutral probability space ðX; F ; QÞ and l and r > 0 are constant. The cash flow QXðtÞ can be regarded as the firm’s earnings before interest and taxes at time t if the firm is in operation. Following Goldstein et al. (2001), we assume a simple tax structure that includes personal and corporate taxes, where interest payments are taxed at a personal rate si , effective dividends are taxed at sd , and corporate profits are taxed at sc , with full loss offset provisions. Let PðxÞ denote the after-tax value of the asset in place if the current price level is x, then we immediately get
PðxÞ ¼
1s Qx; rl
ð2Þ
where r is the constant risk-free interest rate, s is the effective tax rate defined by 1 s ¼ ð1 sc Þð1 sd Þ and l < r for a well-known reason. The benchmark model. We consider here the benchmark model where the SME finances the real investment with equity only. We denote the value of the unlevered firm by Eu ðxÞ if the current product price is x then the firm’s optimal investment problem can be formalized as follows:
"Z Eu ðxÞ ¼ supE T iU
þ1
T iU
rt
e
rT iU
ð1 sÞQXðtÞdt e
# IjXð0Þ ¼ x ;
ð3Þ
where T iU > 0 is the stopping time when the investment is exercised. Since this is an optimal stopping problem of Markovian dynamics, it suffices to consider the stopping times with the following form: Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
3
There is a trigger level xui , such that T iU :¼ infft P 0; XðtÞ P xui g. Therefore, by using a standard approach, the investment problem (3) can be equivalently written as
(
x sup ½Pðxui Þ I xui xui
b1 ) ;
1 l where b1 ¼ 2 þ 2 r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 l 2r 2 þ 2 > 1: 2 r r
After that, we directly derive the following proposition: Proposition 2.1. The optimal investment threshold and the value of the unlevered firm are given respectively by
xui ¼
b1 1 I; b1 1 Pð1Þ
Eu ðxÞ ¼ ½Pðxui Þ I
x xui
b1 ð4Þ
:
Throughout the text, we assume the current product price Xð0Þ ¼ x is so low that the investment should not be undertaken immediately. For example, we assume Xð0Þ ¼ x < xui in our benchmark model.
3. The pricing of corporate securities, capital structure and investment timing The pricing of debt, equity and guarantee cost. We assume the investment trigger level xi is given in advance. Just at the investment time, an entrepreneur makes financing decision by issuing debt and equity, where debt is perpetual with a guarantee from an insurer. Thanks to the guarantee, we assume the entrepreneur has the option to default on his debt obligations. For the same reason with the optimal investment, the optimal default policy is determined by a trigger level xd , i.e. the default time T d is given by T d ¼ infft P T i : XðtÞ ¼ xd g. Clearly, after investment has taken place, the levered firm should solve the following equity-valuation-maximization problem:
"Z El ðxi Þ ¼ supE Td
Td
# rðtT i Þ
e
ð1 sÞðQXðtÞ CÞdtjXðT i Þ ¼ xi ;
ð5Þ
Ti
where C is the coupon rate of the guaranteed debt paid by the borrower to the lender. Equivalently, it can be written as
( El ðxi Þ ¼ sup Pðxi Þ Pðxd Þ xd
b2 b2 !) xi C xi : ð1 sÞ 1 r xd xd
ð6Þ
Hence, the optimal default trigger level after investment is given by
xd
b2 r l C ¼ ; b2 1 Q r
1 l where b2 ¼ 2 2 r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 l 2r 2 þ 2 < 0: 2 r r
ð7Þ
Note that xd is given as a linear function of the coupon rate C. As aforementioned, we assume the entrepreneur has not enough money to invest in the project and he must borrow some money from a bank with the help of the EGS. We assume that once the borrower defaults on his debt, the lender takes the value of the asset minus bankruptcy costs. To determine the guarantee cost, let Dðxi Þ denote the value of debt at the investment time without insurance, we then obtain
"Z
Td
Dðxi Þ E
Ti
# erðtT i Þ ð1 si ÞCdt þ ð1 aÞerðT d T i Þ Pðxd ÞjXðT i Þ ¼ xi
b2 ! b2 C xi xi þ ð1 aÞPðxd Þ ¼ ð1 si Þ 1 ; r xd xd
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H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
where a is the bankruptcy loss rate. Obviously, the last term represents the liquidation value of the firm at the investment time. We assume the insurer must pay the lender all the loss from the borrower’s default and so the lender’s asset is risk-free. Thus, its value is equal to ð1 si Þ Cr . For this reason, the value, denoted by Dguar ðxi Þ, of the insurer’s compensatory payment to the lender, must satisfy the following equation
Dðxi Þ þ ð1 si ÞDguar ðxi Þ ¼ D0 ðCÞ
C ð1 si Þ: r
ð8Þ
Therefore, we obtain
Dguar ðxi Þ ¼
b2 C 1a xi Pðxd Þ : r 1 si xd
ð9Þ
Clearly, to make the swap fair, an insurer’s compensatory payment Dguar ðxi Þ to a lender must be equal to the present value of an SME’s equity allocated to the insurer at the investment time. Consequently, it follows from (9) that
b2 C 1a xi Pðxd Þ ¼ /Eðxi Þ; r 1 si xd
ð10Þ
where / is called guarantee cost, which is the fraction of the SME’s equity allocated to the insurer in return for the guarantee. Hence, the guarantee cost / at the investment time is explicitly given by
b2 xi 1a 1 si Pðxd Þ xd /¼ b2 b2 ; Pðxi Þ Pðxd Þ xxi ð1 sÞ Cr 1 xxi C r
d
ð11Þ
d
which is a function of the product price xi when investment is exercised. Investment timing. Now, we turn to the optimal investment time. According to pecking order theory in corporate finance and due to asymmetric information, the borrower should only borrow the minimum money to start the project and so the coupon of debt issued is generally exogenously determined. As aforementioned, due to the Markovian dynamics we consider here, the optimal investment time is determined by a threshold value, xi , i.e. the firm optimally exercises the investment option once the price XðtÞ reaches xi from below. Therefore, for a given coupon rate C, the optimal threshold xi is the solution of the following optimization problem:
supfð1 /ÞEðxi Þ ðI D0 ðCÞÞg xi
b1 x : xi
ð12Þ
For this reason, we are led to the conclusion below. Proposition 3.1. For a given coupon rate C under EGS the borrowers optimal investment policy xi satisfies
ð1 b1 Þ
b2 x 1s C Qxi þ g i þ b1 I b1 ðs si Þ ¼ 0; r rl xd
ð13Þ
where
g ¼ ðb1 b2 Þ
a si sC : Pðxd Þ þ r 1 si
Furthermore, for a given investment threshold xi , the optimal debt level C is naturally given by
C ¼ argmaxfð1 /ÞEðxi Þ ðI D0 ðCÞÞg Therefore, letting u1 ¼
b2 1 b2
lr 1 r
Q
b1 x : xi
, we have
Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
a si 1 s s xi Q u1 þ ð1 b2 Þ 1 si r l r u1
b2
ðC Þ
b2
s si r
¼ 0:
5
ð14Þ
From (13) and (14), we derive the following conclusion. Proposition 3.2. With debt financing under EGS, the borrower’s optimal financing and investment policy are given by
C ¼
u2 b1 I b1 I ; xi ¼ ; u1 b1 1 k b1 1 k
ð15Þ
where
0 < u1 < 1;
0 < u2 ¼
a si 1 s sð1 b2 Þ ðb2 Þ þ 1 si s si s si
b1
2
< 1;
and
k¼
2 1s a si 1 s 1b2 1 si u2 s u1b 2 Q Q u2 þ : rl 1 si r l r u1 r u1
If si ¼ 0, this conclusion is actually implied by those derived by Sundaresan et al. (forthcoming) who however assume there are no financing constraints undergone by a borrower. In this way, we can say from another standpoint that thanks to EGS, the borrower’s financing constraints are effectively eliminated.
4. Numerical results To make an effective comparison, the baseline parameter values are selected as follows: the riskfree interest rate r ¼ 0:06, expected growth rate l ¼ 0:02, volatility r ¼ 0:3, personal tax rate si ¼ 0:25, corporate profit tax rate sc ¼ 0:35, dividend tax rate sd ¼ 0:20, bankruptcy loss rate a ¼ 0:5, initial product price x0 ¼ 1, sunk cost I ¼ 8 and the product quantity Q ¼ 1. With the baseline parameter values, the optimal coupon rate, investment threshold, default threshold, leverage and guarantee cost are given by C i ¼ 1:3325; xi ¼ 1:94; xd ¼ 0:4232; L ¼ 66:13% and / ¼ 0:2804 respectively. Investment timing and the firm’s values versus coupon rates. An entrepreneur might be reluctant to get loans at the cost of a fraction of equity since the value of equity would be significantly underestimated due to information asymmetry. His investment would therefore not take the optimal capital structure and the amount of money he borrows would just equal the financial gap for him to invest after he spends all his money. Consequently, we consider the case where the coupon rate is exogenously given. Fig. 1 reports the effect of the coupon rate on investment timing for three different volatilities (r ¼ 0:3; r ¼ 0:4; r ¼ 0:5). As we expected, no matter what capital structure is (i.e. how much the coupon rate C is), the bigger the risk, the larger the value of the option to invest and so the later the investment. Specifically, If SMEs undertake pure equity financing (i.e., C ¼ 0), then the optimal investment thresholds are xui ¼ 2:2144; 2:9099 and 3:7586 under the case of the volatilities r ¼ 0:3; 0:4 and 0:5 respectively. However, if debt financing is undertaken, there is an U-shaped relation between investment time and the coupon rate: Investment thresholds decrease first and then increase with the coupon rates. This is because a higher coupon rate has two opposite effects: One is to accelerate investment since it increases tax shields and the other is to delay investment because it increases bankruptcy costs and the guarantee cost as well. To investigate the effect of debt financing on the firm’s value, Fig. 2 presents the firm’s value as a function of the coupon payment for three different volatilities. The case of C ¼ 0 corresponds to an all-equity financing. It states that the value of the levered firm first rises and then declines with the coupon rate and the functions are concave, which indicates there is an optimal coupon rate to maximize the firm’s value under each given volatility. This happens just due to the same reason with Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
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H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
Investment trigger
4.5
σ=0.3 σ=0.4 σ=0.5
4
3.5
3
2.5
2
1.5
0
0.5
1
1.5
2
2.5
C Fig. 1. Investment thresholds as a function of the coupon rate with three different volatilities.
The value of option
7.2 7 6.8 6.6 6.4 6.2 6
σ=0.3 σ=0.4 σ=0.5
5.8 5.6 5.4 5.2
0
0.5
1
1.5
2
2.5
C Fig. 2. The firm’s value as a function of the coupon rate with three different volatilities.
why the curves in Fig. 1 are convex. In addition, Fig. 2 shows that under the baseline parameters, the optimal coupon rate lies between 1 and 1.5 and as we expected, the larger the volatility, the higher the firm’s value. Investment timing and the firm’s leverage versus volatilities under optimal capital structure. We set out to consider a situation where an entrepreneur has a deep pocket and thus the amount of money he borrows can be arbitrary instead of just the financial gap for him to invest. For this reason, the firm naturally takes the optimal capital structure, i.e. the coupon rate is endogenously decided to maximize the firm’s value rather than exogenously given. Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
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H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
Fig. 3 reports how investment threshold changes with the volatility r of the cash flow under the optimal capital structure for an unlevered firm, a levered firm with the swap and that without the swap, which is discussed by Sundaresan et al. (forthcoming). It says that the guaranteed firm always invests a little earlier than an all-equity financed firm but later than the firm discussed by Sundaresan et al. (forthcoming). As the volatility r increases, the optimal investment threshold for the guaranteed firm gets closer to that for an all-equity firm, and therefore, the overinvestment effect generated by EGS gradually disappears if the cash flow risk is high enough.
The optimal investment threshold 6 5.5 5
The levered firm with the swap The unlevered firm The levered firm without the swap
4.5 4 3.5 3 2.5 2 1.5 1 0.2
0.3
0.4
σ
0.5
0.6
0.7
Fig. 3. Optimal investment threshold versus the cash flow risk for an unlevered firm, a levered firm with and without the swap respectively.
The optimal leverage and guarantee cost 0.7 The guarantee cost Leverage with the swap Leverage without the swap
0.6
0.5
0.4
0.3
0.2
0.1
0 0.2
0.3
0.4
σ
0.5
0.6
0.7
Fig. 4. The leverage and guarantee cost under optimal capital structure versus the cash flow risk.
Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006
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H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
Fig. 4 plots the firm’s optimal leverage and the guarantee cost as a function of the cash flow risk. The dotted line depicts the leverage of the firm with a guarantee, the dash line represents that derived by Sundaresan et al. (forthcoming), and the solid line gives the guarantee cost, respectively. It shows that as we expect, the optimal leverage ratio decreases monotonously in the volatility r. When the volatility is relatively low, the firm in our model should issue more debt than the firm considered by Sundaresan et al. (forthcoming). However, when the volatility gets large enough, the opposite holds true. Debt overhang and asset substitution. Under the EGS, the lender is well protected because the swap transfers his risk exposure entirely to the insurer. On account of that only the borrower (entrepreneur) has the right to manage the firm, naturally we wonder whether and how the swap induces inefficiencies from debt overhang and asset substitution. To achieve this aim, if the borrower invests an extra unit of account in the firm, we compute his net profit, which is reported in Fig. 5. It says that if the cash flow level is large and the guarantee cost is 1 0.8
Φ=0.1950
Φ=0.3324
0.6
Φ=0.5271
∂ (1−Φ)E/∂Π −1
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1
0
2
4
Cash flow X
6
8
10
Fig. 5. The sensitivity of the value of the shareholders’ claim to the firm’s value PðxÞ versus the cash flow level x. 5 4.5 Φ=0.1405
4
Φ=0.2520
Φ=0.4063
∂ (1−Φ)E/∂σ
3.5 3 2.5 2 1.5 1 0.5 0
0
5
10 Cash flow x
15
Fig. 6. The sensitivity of the value of the shareholders’ claim to the volatility
20
r as a function of the cash flow level x.
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H. Xiang, Z. Yang / Finance Research Letters xxx (2015) xxx–xxx
9
low enough, the net profit is positive, meaning that there is no debt overhang problem. But under the converse situation, the opposite holds true. Generally speaking, if debt is not protected, there must be a strong risk-shifting incentive for shareholders to invest in high risk project since they harvest all the profit if they succeed but transfer loss to the creditors if they fail. To measure the risk-shifting incentive, we compute the derivative of the value of the entrepreneur’s claim with regard to the volatility, which is reported in Fig. 6. As we expected, the derivative is always positive, meaning that the entrepreneur will invest in a higher project once he has such choice. The larger the guarantee cost, the stronger the incentive. For this reason, we propose that to enter into the swap, insurers should in advance specify the kind of projects the borrower must invest the loan in if the borrower has several investment choices in the future. 5. Conclusions The EGS is a new financial derivative, which is widely taken as small and medium-sized firm’s loan contracts in China and hopefully, it is believed that such swaps like this would play a vital role in Internet finance. However, there are few studies to consider the effect of the swap on investment and financing incentives of borrowers. We obtain closed-form solutions for the interaction of the optimal investment and financing and a linear relation between them is derived. We find that the levered firm would exercise its investment option earlier than the corresponding all-equity firm and there is an U-shaped relationship between investment timing and the coupon payment. In contrast to the classical real options theory, the EGS alters the investment and financing incentives of the borrower. In particular, the swap delays investment. Whether the swap induces debt overhang distortion depends on the guarantee cost and the profitability of the project but it surely gets rise to the borrower’s risk-shifting incentives. The larger the guarantee cost, the stronger the incentives. Accordingly, we suggest that while entering into the swap, the insurer had better in advance specify the kind of projects the borrower must invest the loan in if the borrower will have several investment choices after the swap contract is signed. References Goldstein, R., Ju, N., Leland, H., 2001. An EBIT-based model of dynamic capital structure. J. Bus. 74 (4), 483–512. Hackbarth, D., Mauer, D.C., 2012. Optimal priority structure, capital structure, and investment. Rev. Financ. Stud. 25 (3), 747– 796. Sarkar, S., 2011. Optimal size, optimal timing and optimal financing of an investment. J. Macroecon. 33 (4), 681–689. Sundaresan, S., Wang, N., Yang, J., 2015. Dynamic investment, capital structure, and debt overhang. Rev. Corp. Financ. Stud. (forthcoming). Wang, H., Yang, Z., Zhang, H., 2015. Entrepreneurial finance with equity-for-guarantee swap and idiosyncratic risk. Eur. J. Oper. Res. 241 (3), 863–871. Yang, Z., Zhang, C., 2015. Two new equity default swaps with idiosyncratic risk. Int. Rev. Econ. Financ. (forthcoming). Yang, Z., Zhang, H., 2013. Optimal capital structure with an equity-for-guarantee swap. Econ. Lett. 118 (2), 355–359.
Please cite this article in press as: Xiang, H., Yang, Z. Investment timing and capital structure with loan guarantees. Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.01.006