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					              Convertibles in Sequential Financing1



                                               Susheng Wang2



                                                  June 2007



Abstract: Sequential financing is a well-used strategy in corporate finance. However,
depending on the type of financial instrument used to carry out the strategy, sequential
financing can have many potential problems. This paper shows that certain types of con-
vertibles can be deployed to resolve the problems completely. This may explain why con-
vertibles are widely adopted to implement sequential financing in reality, especially among
companies with many real investment options. In particular, we find that the call option and
some popular forms of call restrictions are necessary in the design of an efficient convertible.
Indeed, almost all real-world convertibles have a call feature and an overwhelming majority
of them has some form of call restrictions.



Keywords: Convertibles, Sequential Financing, Milestones, Call Options, Call Restrictions,
           Incentives



JEL Classification: G32, G31




     1   Acknowledgement: The refereeing process resulted in a major improvement to the paper and the input
from the referees and the Editor is gratefully acknowledged.
     2   Department of Economics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.
Email: s.wang@ust.hk.
1. Introduction
     Sequential financing has been a well-used strategy in corporate finance. In particular, it
is very popular among fast-growing firms, such as venture capital firms (Sahlman 1990) and
firms with many real investment options (Mayers 1998, 2000). This has been increasingly
true in recent years. Interestingly, sequential financing is mostly carried out by convertibles.
Why should sequential financing be carried out by convertibles? Further, real-world
convertibles almost always carry a call option together with some form of call restrictions.3
Why are these provisions necessary? What role do these special features play?

     A company often needs to raise funds for a planned expansion, expected growth
opportunities (real options), and emerging investment opportunities. The company may not
need the full amount right away; instead, it needs repeated investments over a period of time.
The company faces a tradeoff. If it raises the full amount upfront, the funding is secure, but it
risks alienating investors who fear that their money will be spent even if the project turns out
to be unworthy of investment. This is what Jensen (1986) calls the free-cash-flow problem.
Reluctant investors, fearing such over-investment, would demand terms that are more
advantageous to them, to the detriment of issuers. We argue that raising funds in stages
gives investors an opportunity to see how the project is doing, to derive incentives from the
manager, and to strike a good balance in risk sharing.

     However, sequential financing may cause some problems. First, since later installments
may not come, the manager may not have enough incentive to invest effort. Second, an in-
adequate initial investment may put the project at risk and the project may fail prematurely.
Third, the manager may try to boost early performance in order to impress investors to gain
further investment (Cornelli and Yosha 2003). Fourth, the company may pay issue costs re-
peatedly because the funds are raised in sequence (Mayers 1998, 2000). Finally, to induce
further installments, the manager may make sweet deals with investors, which may cause the
project to become unprofitable when the market fluctuates.

     Since convertibles tend to go hand in hand with sequential financing in reality (Mayers
1998, 2000), we naturally suspect that convertibles with their special features may be able to
resolve many of the associated problems in sequential financing. Based on the incomplete-
contract approach, this paper theoretically investigates whether or not convertibles can effec-




     3   In Lewis, Rogalski and Seward’s (1998) data set for the period 1978–1992, 95% of the convertibles are call-
able. As an example, Chakraborty and Yilmaz (2004) point out that, in the 2001 issues of convertibles, 95% of
them are callable and an overwhelming majority of these have restrictive call provisions (typically involving a
trigger price). See also Korkeamaki and Moore (2004).

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tively deal with the problems inherent to sequential financing. Indeed, we find that sequen-
tial financing using callable convertibles with call restrictions can be efficient. However, se-
quential financing using straight debt or straight convertibles is generally inefficient in a
risky environment.

    Our model emphasizes four popular features in real-world corporate finance: sequential
financing, convertibles, decision events, and milestones. Decision events include default,
conversion, call, and bankruptcy. Decision events can result from various ex-post options.
Investors would like to have ex-post options even though investors typically have no interest
in interfering with the daily management of the firm if the firm is doing well. One purpose is
to deal with decision events. If the firm is doing badly, to salvage their investments, investors
may use the conversion option to take control of the firm and replace the top manager or to
sell the firm; if the firm is just surviving, the investors may retain their loans to the firm to
earn interest; if the firm is doing well, the investors may use the conversion option to own a
share of equity at a low cost in order to benefit from the growth of the firm. This aspect of
our model follows the line of Aghion and Bolton (1992) and Hellmann (2001). The use of
milestones is another popular phenomenon in corporate finance. Convertibles, especially
those in venture capital funding, generally have statements on specific actions to be taken
conditional upon the firm achieving certain business or financial objectives, referred to as
milestones (Cuny and Talmor 2003). In our model, milestones are not imposed per se in the
model setup, but they appear in the equilibrium strategies. That is, investors will take
certain actions based on specific milestones in equilibrium. These milestones can either be
explicitly written into the contract or be implicit in the available choices provided by a
financial instrument.

    The conversion option plays a very different role in our model from its traditional role in
an agency model. In the literature, the two parties are supposed to engage in a joint venture
and they invest in turns. The conversion option allows a switch of ownership in an interim
stage so that each player becomes the full owner when it is time for him/her to invest. As
such, both parties invest efficiently. In contrast, we investigate the conversion option in a
model in which there is generally no switch in ownership. Instead, we use some special rights,
such as callability and call restrictions, to achieve efficiency. We allow the firm to force an
early conversion by calling its convertibles. Although the conversion option provides certain
protections and advantages to the investor, the call option limits the potential of the conver-
sion option to capture a high return by the investor. When the firm is doing very well, so that
the conversion value exceeds the call price, the manager can force conversion to protect the
existing shareholders. On the other hand, we also allow call protection for investors against
forced conversion when the firm is doing badly. We investigate whether or not these features
in convertibles are sufficient to induce both the investors and the firm to invest efficiently. It


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turns out that a callable option and some call restrictions can indeed play a key role in
achieving efficiency.

    The intuition is as follows. First, with straight debt, since the manager is the sole resid-
ual claimant, he/she will invest efficiently if the investor invests efficiently; but, the investor
will invest efficiently only if the risk of bankruptcy is minimal. Hence, debt financing is effi-
cient if and only if the manager’s incentive compatibility condition happens to be in align-
ment with the minimization of bankruptcy risk. Second, with a straight convertible, due to
the fact that the investor will share output when the firm does well, the manager will under-
invest even if the investor invests efficiently; hence, inefficiency is due to the manager’s in-
centive problem. Third, with a callable convertible, the manager’s incentive is improved suf-
ficiently; but, the investor will be concerned with the fact that he/she will be forced into con-
version when the firm does badly. On the other hand, an improvement in the manager’s in-
centive can improve the firm’s performance and reduce default and bankruptcy risks. Hence,
if the net effect of the negative and positive effects of the call option on the investor’s incen-
tive is neutral, efficiency is achievable. Finally, with proper call restrictions, a balance of the
negative and positive effects of the call option on the investor’s incentive can be achieved.
Interestingly, some popular forms of call restrictions are shown to be capable of achieving
such a balance.

    We briefly mention two key studies relating to our work here, leaving the detailed litera-
ture review until the next section. Mayers (1998, 2000) proposes the view that a convertible
is a cost-saving instrument in sequential financing. The costs include issue costs and agency
costs. This view is particularly relevant to companies with many real investment options.
Mayers cites survey evidence and provides empirical evidence in support of this view. There
are also a few other empirical studies in support of Mayers’ view. However, theoretical analy-
ses of convertibles in sequential financing are rare. The work by Cornelli and Yosha (2003) is
an exception. They emphasize the role of convertibles in dealing with window dressing in se-
quential financing. In contrast, we provide a theoretical analysis that emphasizes the role of
convertibles in dealing with many possible decision events in sequential financing and illus-
trates the efficiency properties of call options and call restrictions.

    This paper proceeds as follows. In Section 2, we present a literature review and set our
results in the context of the literature. In Section 3, we define a model of corporate finance in
an environment with uncertainty, moral hazards and sequential financing. To carry out se-
quential financing, we allow certain types of financial instruments as vehicles of investment.
In Section 4, we consider straight debt and three types of convertibles. An efficient callable
convertible with call restrictions is found. Finally, Section 5 concludes the paper with a few
remarks. All the proofs are in the Appendix.




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2. Literature Review
Popularity of Convertibles

    Convertibles are a major instrument in corporate finance (Trester 1998). According to
VentureXpert, a venture capital database, convertibles are the dominant instrument in ven-
ture capital financing and they have grown more popular in recent years. In 2005, 93% of
investments in all venture development stages were done by convertibles. The second most
popular financing instrument was debt, although debt financing accounts for only a small
percentage of investments and it has been decreasing in popularity in recent years. In 2005,
debt financing accounted for only 2.14% of financing in all venture development stages.

    There are many studies that seek to explain preferences of financial instruments in cor-
porate finance. They are based broadly on two approaches: the asymmetric-information ap-
proach and the incomplete-contract approach. There is also a recent literature on real op-
tions that explains the use of convertibles in sequential financing. We here give a short re-
view of the literatures on these three approaches in relation to our model.


The Asymmetric-Information Approach

    The main message from the asymmetric-information approach is that convertibles give
the firm a backdoor to equity and give investors an opportunity to wait and see if the project
is worth investing in.

    Myers and Majluf (1984) consider a firm that knows more than investors about its own
value. Due to this asymmetric information, investors tend to undervalue a good firm’s stock.
This may explain several aspects of corporate financing behaviors, including the tendency for
firms to rely on internal sources of funds and a preference for debt over equity. Under this
situation, one good way to sell equity is through convertibles.

    Stein (1992) considers three types of firms with the type being the firm’s private infor-
mation. A separating equilibrium is found in which a bad firm chooses equity financing, a
medium firm chooses convertible financing, and a good firm chooses debt financing. This
result follows from the large cost of financing distress from debt. The bad firm will not
choose convertible financing since, in a bad situation, it is not able to pay back the debt or to
force conversion in order to eliminate the debt. The potential cost of financial distress in-
duces the bad firm to avoid any form of debt financing. Similar arguments apply to the other
two types of firms.

    In contrast, our model is based on incentive inducement. In our model, under some mi-
nor conditions, any firm can efficiently employ convertibles in sequential financing. In par-
ticular, we do not require the firm to be of a medium type to issue convertibles. Our result is

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consistent with Stein (1992) in one aspect only: besides convertibles, a mature/good firm can
also efficiently issue debt. For this, Stein’s conclusion is based on a relatively large cost of
financial distress; only the good firm is able to avoid this cost. Our conclusion is based on
bankruptcy risk, which reduces the investor’s incentive to invest early; only a well-
established firm can convince the investor of minimum bankruptcy risk. Also, Stein (1992)
looks at corporate finance purely from the firm’s point of view. We look at it from both the
firm’s and the investors’ points of view, and the choice of the financial instrument is a bar-
gaining outcome or a balance of incentives and risks of the two parties. It is known that al-
most every convertible has a call feature in reality (Lewis et al. 1998). This call feature is es-
sential for Stein’s (1992) theory of backdoor equity financing. In our model, the call feature
offers a balance of bargaining power and hence a balance of incentives between the two par-
ties.

        Bagella and Becchetti (1998) provide a refinement of Stein’s (1992) asymmetric informa-
tion model. They show that a bond-plus-warrant issue is the optimal financing strategy in a
separating equilibrium.4 They also provide empirical evidence in support of their findings.

        However, the classification of firms into three types (bad, medium and good) is prob-
lematic. Two things immediately come to mind when we classify firms: risks and expected
returns. If we divide each of the two aspects (risks and returns) into three possibilities: low,
medium and high, then there are nine types of firms and most of the types cannot be ranked.
Also, the manager and investor have two things to determine in their investment decision: a
choice of financial instrument and a choice of investment strategy. In other words, when fac-
ing a specific combination of risks and expected returns, the manager and investor need to
determine the best combination of financial instrument with investment strategy. For exam-
ple, new startups in high-tech industries tend to be very risky and those that are able to find
funding are expected to have very high returns. It turns out that these firms have usually
been financed by convertibles using a sequential financing strategy. In other words, in corpo-
rate finance, the choice of financial instrument and the choice of investment strategy may be
tied together. This is actually the focal point of our model. Our point is that convertibles can
be efficient in a sequential investment strategy, given that the sequential investment strategy
is chosen for various reasons. This point applies to any type of firm.

        Chakraborty and Yilmaz (2004) and our paper have the common feature that empha-
sizes callability and call restrictions in convertibles. In both papers, call restrictions are es-
sential for efficiency. They use the asymmetric-information approach, while we use the in-
complete-contract approach. The key difference is that they show the efficiency of converti-
bles for a certain type of firms, while we show the efficiency of convertibles for a certain in-



        4   Our class of convertibles includes this bond-plus-warrant issue.

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vestment strategy. Specifically, they show that convertibles can be efficient for the good type
of firms; we show that convertibles can be efficient for the sequential investment strategy.

     In order to compare the agency model with the asymmetric-information model, in an
agency model, we call a firm a good firm if it has a high output ex post in equilibrium and we
call a firm a bad firm if it has a low output ex post. As such, in the agency model, the firm has
an endogenous type (determined in equilibrium by choices), while in an asymmetric-
information model, the firm has an exogenous type (assigned by nature).5 In the agency
model, the type is observable, while in an asymmetric information model, the type is not di-
rectly observable. However, the asymmetric-information model typically discusses a separat-
ing equilibrium (or a pooling equilibrium with a sufficiently reliable signal). If so, the two
classes of models can indeed be compared in equilibria.

     This difference in the endogeneity of type can result in very different behaviors. For ex-
ample, in Chakraborty and Yilmaz (2004), a good firm will call and a conversion happens,
while a bad firm cannot call and no conversion happens. In our paper, as shown for example
in the last figure (Figure 11), investors will voluntarily convert if it is a good firm; investors
will not convert and the firm cannot call if the firm is mediocre; and the firm will call and a
conversion happens if the firm is bad. That is, a bad firm keeps debt ex post in Chakraborty
and Yilmaz (2004), but it resolves debt ex post in our model.


The Incomplete-Contract Approach

     Our model is based on the incomplete-contract approach. This approach emphasizes in-
formation revelation during the production process and allows various ex-post options and
renegotiation possibilities. In addition to a revenue-sharing agreement, this approach allows
various mechanisms to deal with various problems such as information revelation, renego-
tiation, incentives, ex-post options, and holdups. In particular, this approach treats real-
world financial instruments, sequential financing and equity sharing as mechanisms de-
ployed by economic agents to deal with various corporate financing problems.

     This approach pays particular attention to two major issues: risk sharing and moral haz-
ard. First, there is a need for proper risk sharing between investors and managers. The risks
include various decision events, particularly default and bankruptcy, and various possible
shocks to the system. Second, there is also a need to deal with various incentive problems,
including the manager’s incentive to invest, the investor’s incentive to invest, the manager’s
incentive to keep the project going even if it is better to shut it down, and the investor’s in-




     5   A signal in an asymmetric information model can be endogenous. This is completely different from en-
dogenous types. In fact, the endogeneity of the signal does not play a role in a pooling equilibrium.

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centive to commit early. Third, the risk and incentive issues can be entangled. For example,
sequential financing can be used to handle risks, but sequential financing can itself create
certain agency problems. How should the various mechanisms be properly combined?

    A dominant view in the existing literature on convertibles and incentives is the owner-
ship-for-investment view. The two parties are supposed to invest in sequence. A convertible
allows a switch of ownership in the middle so that each investing party becomes the sole
owner when it is his/her turn to invest. Efficiency can be achieved under double moral haz-
ard since the sole owner has the incentive to invest efficiently. In such a model, a conversion
to equity will definitely happen in equilibrium. The pioneer of this view is Demski and Sap-
pington (1991), followed by Nöldeke and Schmidt (1995, 1998), Che and Hausch (1999),
Edlin and Hermalin (2000), Schmidt (2003), and many others.

    The ownership-for-investment view predicts an increase in capital expenditure following
a forced conversion. However, as pointed out by Alderson, Betker and Stock (2002), only
Mayers’ (1998) finding is consistent with this. Mayers (1998) presents an empirical study on
the real options theory. He argues that the firm faces investment opportunities at any mo-
ment of time. In comparison with debt, convertibles allow the firm to obtain cash flows for
profitable opportunities. Like Stein (1992), the call feature in convertibles is crucial. In May-
ers (1998), the call provision helps the firm obtain cash flows to finance profitable invest-
ment options, while in Stein (1992) the call provision helps the firm avoid financial distress.

    However, a more careful empirical study by Alderson, Betker and Stock (2002) does not
find that forced conversion leads to greater investments and financing activities. In fact, the
only thing that changes after forced conversion is the capital structure. This finding is consis-
tent with Baker and Wurgler’s (2002) finding that capital structure is less influenced by cor-
porate governance considerations than was previously thought. This makes sense, since, in
reality, investors have little interest in assuming control of the firm. For example, in the ven-
ture capital industry, investors are typically venture capital firms that invest in a portfolio of
small companies across industries. They have neither the technical expertise nor the mana-
gerial personnel to run a number of companies in diverse industries. They prefer to leave op-
erating control to the existing management. The investors do, however, want to participate in
any strategic decision that might change the basic product/market character of the company
and in any major investment decision that might divert or deplete the financial resources of
the firm. For this purpose, they will generally ensure some representation in the board of di-
rectors of the firm. Only if severe financial, operating, or marketing problems develop may
the investors want to be able to assume control and attempt to rescue their investments. For
this purpose, some protective provisions in their financing agreements will be sufficient.

    Our theoretical model is consistent with Alderson, Betker and Stock’s (2002) and Baker
and Wurgler’s (2002) empirical findings. In our model, the manager has full control of the

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project during its entire course. Only at an exit point may the conversion option be exercised,
depending on the firm’s performance and the rights defined in the financial instrument. In
particular, Alderson, Betker and Stock (2002) find that firms forcing conversion invest more,
borrow more, earn higher returns, and experience higher rates of total asset growth than the
median firm in their industry. Our theoretical prediction is consistent with this finding. By
comparing Propositions 4–7 with Proposition 3, we also predict that firms forcing conver-
sion will invest more (a larger x ), borrow more (a larger k ), and earn a higher return (more
efficient).

     Further, Baker and Wurgler’s (2002) empirical study shows that ‘market timing’ has
large and persistent effects on capital structure. Their main finding is that low leverage firms
are those that raised funds when their market valuations were high, and vice versa. That is,
the manager of the firm acts like a fund manager who manages the firm’s financial assets like
an investment portfolio. Baker and Wurgler’s contribution is to show that such an interpreta-
tion of a firm’s capital structure is consistent with data. In our model, conversion to equity is
not based on the need for the investor to become the owner of the firm (for him/her to invest
efficiently); instead, an ex-post decision on conversion is based on the comparison of the
market value of the firm with the conversion value of the convertible. As shown in Figures 6–
10, conversion to equity tends to happen when the firm’s market value is high and the firm
tends to keep debt when its market value is low. This is consistent with Baker and Wurgler’s
(2002) empirical findings, even though our theory is completely different from their inter-
pretation of their empirical findings.

     Finally, Lewis, Rogalski and Seward (1998) observe the importance of call restrictions in
their empirical study. They find that issuers even adjust call restriction features over busi-
ness cycles. Their finding is consistent with Mayers’ (1998, 2000) view and his empirical
study. Their empirical results confirm our theoretical findings. Indeed, we find that a call op-
tion is necessary and some forms of call restrictions are also necessary in designing an effi-
cient convertible. We also find that call restrictions should generally depend on market con-
ditions.


Sequential Financing

     Sequential financing is a well-used corporate strategy. For example, in venture capital,
almost all investment is through sequential financing instead of upfront financing (Sahlman
1990). According to the real options literature, when a company with many real investment
options raises funds for a project today, it must take into account a second fund raising in the
near future. It could raise funds today for both projects to save issue costs or raise them
when needed. Hence, the financing horizon of such a company is over multiple periods. In



                                                                                          Page 9
other words, as determined by the business environment, the financing strategy of a com-
pany with real options is inherently sequential.6

     As Mayers (2000, p.20) observes, 31% more convertible-calling companies raise new
capital around the time of the call than a typical company and most of them issue debt only.
Hence, a firm may have an optimal debt-equity ratio in the long run and fund raising for in-
vestment opportunities needs to take into account the capital structure in a multi-period set-
ting. For this purpose, the firm may need to choose a proper financial instrument and an ef-
fective financing strategy. As we show, a properly structured convertible coupled with a
properly designed sequential investment strategy is an efficient approach. Indeed, there are
many surveys and empirical studies (but only one theoretical study by Cornelli and Yosha
2003) that tie convertibles with sequential financing. This brings us to a recent explanation
for the popularity of convertibles by Mayers (1998, 2000).

     Mayers (1998, 2000) argues that convertibles can be a cost-saving instrument to carry
out sequential financing. Mayers finds a lot of survey and empirical evidence in support of
the use of convertibles in sequential financing, especially by corporations with large growth
opportunities. In particular, Mayers argues that convertibles are well suited to circumstances
where a firm needs immediate financing to undertake a project that is expected to be fol-
lowed by a second project at some time in the future. The firm can finance the first project by
issuing convertibles and then force a conversion to finance the second project. By this, the
firm does not have to make two back-to-back issues at double the cost. For small equity issu-
ers, as claimed by Smith (1977), the issue cost for raising a second fund can amount to a sub-
stantial 15% of funds raised.

     Asquith (1992) shows that about two-thirds of all convertibles are eventually converted.
Further, Asquith and Mullins (1991, p.1288) show that essentially all companies call their
convertibles if the conversion value exceeds the call price and if there are cash savings from
the conversion. As Mayers (2000) argues, these facts support the view that convertibles are
used in an anticipated/planned financing sequence.

     Jen, Choi and Lee (1997) show that the stock market responds more favorably to an-
nouncements of convertible issues by companies with high post-issue capital expenditures
and high market-to-book ratios (both are plausible proxies for growth potential), but with




     6   The literature on convertibles in sequential financing is different from the literature on the asymmetric-
information and agency models. The literature on the latter two addresses a financing problem at the time of is-
sue. The literature on convertibles addresses the firm’s financial needs today in conjunction with additional fi-
nancial needs in the future; the problem is how to minimize various costs (including agency costs) over the ex-
pected sequence of current and future financing.

                                                                                                         Page 10
low credit ratings and high (post-offering) debt-equity ratios. These findings support Mayers’
view on convertibles.

    Korkeamaki and Moore (2004, Table I) show that almost all convertibles are callable
and a large majority of callable convertibles has call restrictions. Their empirical study shows
that a firm’s subsequent capital expenditure occurs sooner after issuance of convertibles if
call restrictions are weaker. This finding supports the view that callability and call restric-
tions serve sequential financing, i.e., the firm plans to force conversion to finance new capital
expenditures.

    However, there are many unanswered questions in the literature. First, there are many
kinds of financial instruments in the market and many simple ways that can be applied to
avoid repeated issue costs. It is not clear why callable convertibles with call restrictions are
preferred. The design of such an instrument seems to be too complicated for the purpose.
Second, there are many types of convertibles (such as convertible bonds, convertible
preferreds, and debt with warrants) that can serve the same purpose. But, an industry tends
to choose its preferred type. For example, the dominant type in venture capital is convertible
preferreds (Trester 1998, Tables 1–5). Why this particular convertible? Third, many other
forms of financial instruments can also be consistent with the survey results and empirical
evidence. For example, as noted by Brennan and Her (1993) and Mayers (1998), there is am-
biguity with interpretations of existing evidence on the use of convertibles. Much of the exist-
ing evidence that supports the cost-saving argument in sequential financing (Mayers 1998)
also supports other known arguments, including after-issue risk shifting (Jensen and Meck-
ling 1976; Green 1984), risk estimation (Brennan and Kraus 1987; Brennan and Schwartz
1988), and asymmetric information (Constantinides and Grundy 1989; Stein 1992). These
problems call for theoretical analyses.

    In the existing literature, only Cornelli and Yosha (2003) offer a theoretical explanation
for the need to use convertibles in sequential financing. They argue that, with sequential fi-
nancing, the manager has the incentive to do window dressing in order to attract further in-
vestments from investors. However, with convertibles, if the manager overstates the com-
pany’s value, the investor will convert to equity and sell his/her shares in the market (or
equivalently sell his/her convertibles directly to the market). In other words, they emphasize
the role of convertibles in dealing with window dressing resulting from sequential financing.
We, on the other hand, emphasize the role of convertibles in dealing with various decision
events in sequential financing, including the possible problem of window dressing. In our
model, callability and call restrictions are crucial for efficiency.




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     Recent theoretical work by Wang and Zhou (2004) offers a similar model to ours. They
deal with straight equity in sequential financing, while we discuss straight debt and converti-
bles in sequential financing.7 They show that straight equity is inefficient and it is approxi-
mately efficient for “cost-efficient” firms. In contrast, we find a convertible to be efficient for
any firm under some minor technical conditions, and under an extreme case straight debt is
efficient. We argue that, by raising funds in stages, our solution gives investors an opportu-
nity to see how the project goes, to keep pressure on the manager to perform, and to strike a
balance in risk sharing. In particular, we argue that the various options in a convertible seem
fit to handle various possible decision events in sequential financing.

     Finally, one puzzling popular phenomenon in real-world venture capital is the milestone
financing strategy. That is, investors set performance targets or milestones to take certain
specified actions (Sahlman 1990). The existing literature lacks theoretical and empirical
studies on this phenomenon. Interestingly, our solutions are consistent with this phenome-
non. In fact, the variables y1 , y2 , and y3 in our model are precisely the milestones in equi-
librium, based on which certain actions are taken. The manager and investors can actually
specify these items in their agreement to implement explicitly a milestone strategy. Our
model does not impose a milestone strategy per se; instead, a milestone strategy appears
naturally in equilibrium, given the various options in a financial instrument under sequential
financing.



3. The Model
3.1. The Project
     Consider a firm that relies on an investor for investment. Here, the investor (she) is a
representative investor and the firm is managed by a manager (he) who represents the inter-
est of the existing shareholders by maximizing the expected profit of the firm. The project
lasts two periods. The manager provides an investment x, called effort, and the investor
provides the necessary funding of total amount K . After a contract between the manager
and the investor is signed, the manager provides his effort x at cost c( x ) (paid by the firm),
and this effort is applied throughout the two periods. The investor provides the funding K in



     7   Some technical details are useful to know here. Wang and Zhou (2004) actually allow any revenue-sharing
contract and show that straight equity is optimal among such contracts. However, their model does not include
our model as a special case. Although they allow all possible revenue-sharing contracts, their model is a standard
agency model. Our model is not a standard agency model; our model is an incomplete-contract model, in which
the admissible contracts include ex-post options such as ex-post call and conversion options. Since our solution (a
convertible) is efficient, our solution is also optimal among all possible incomplete contracts.

                                                                                                         Page 12
two stages with an initial installment k in the first period and a planned second installment
 K − k in the second period. Given effort x from the manager and the initial investment k
from the investor, an output y is produced at the end of the second period. This output is
random at the beginning of the first period (ex ante), with a distribution function Φ( y , x, k )
and a mean output μ( x, k ).

     The total investment K is the necessary amount for the project. Sequential financing is
used to allocate this amount between the two installments. Specifically, the investor provides
a total of k in funds at the beginning of the first period. After the uncertainty is realized at
the end of the first period, the two parties consider raising the second installment K − k .
The two parties have the option not to raise the second installment. Note that the decisions
on the investments are made by the two parties together through negotiation. Hence, the
statements “the investor provides the funds” and “the manager raises the funds” have the
same meaning and are used interchangeably in our paper.8

     The production process takes two periods to finish. Both the manager and the investor
are indispensable to the project. If the project is abandoned in the middle by either party
(only when it turns to be unprofitable), the firm/project is liquidated for a fraction θk of the
initial capital investment, where θ ∈ [0, 1). The manager and the investor share the revenue
at the end of the project based on the existing contract.


3.2. Timing of Events
     There is an information revelation process, as some information becomes known at the
end of the first period. The uncertainty of output is realized and publicly revealed at the end
of the first period (ex post). The manager’s effort is observable at time t = 1 but not verifi-
able. The amounts k and K − k are verifiable, but the decision on whether or not to con-
tinue the second investment at t = 1 is decided ex post and is not contractable ex ante. This
means that the decision on the option will be conditional on the observation of the manager’s
input x and on knowledge of the random shock. As some information becomes available ex
post, the two parties are allowed to renegotiate the contract.

     The timing of the events is illustrated in Figure 1.9 At t = 0, the two parties negotiate a
contract. If the contract is accepted by the two parties, the investor provides k and the man-
ager applies x and incurs c( x ). At t = 1, the uncertainty is resolved and the two parties con-



     8   In a typical agency model, an investment/effort variable is determined or controlled by one of the parties.
In contrast, the financial investments in our model are determined by both parties together. This is a unique fea-
ture. One advantage is that it allows our model to be applicable to various different applications.
     9   We can make the model more realistic by allowing a second random shock to the output in the second
period. However, there is no need for this complication for the current purpose.

                                                                                                          Page 13
sider the options to quit or to continue. If the project is bad, the two parties abandon the pro-
ject without investing K − k ; if the project is mediocre, they negotiate a new contract; if the
project is good, the second installment is raised and the project continues. Depending on the
nature of the contract, the two parties may renegotiate or decide on their options at any point
of time. At t = 2, the project is finished and the two parties divide the output based on the
existing contract.


            Contracting                    Information Revelation
            Effort:         x              Options
            Investment: k                   Investment: K − k                               Output: y

              0                                        1                                          2
            Ex ante                                Ex post                                       End

                                         Figure 1. The Timing of Events


3.3. Financial Instruments
     Admissible contracts in our model are special financial instruments. They are straight
bonds, straight convertibles, callable convertibles, and callable convertibles with call restric-
tions.

     A bond (a straight bond) allows an investor to invest in the form of debt that pays a
guaranteed rate of return r. A convertible allows an investor to invest in the form of debt
that pays a guaranteed rate of return r, and this security also provides an option for the in-
vestor to convert her investment into equity at any time (either at t = 1 or t = 2 ) at a guar-
anteed conversion ratio τ or at a given striking price. A convertible may also contain options
for the manager, such as a call option. We discuss three types of convertibles.10

     Each financial instrument also carries some other rights, especially when default or
bankruptcy happens. It also defines ex-ante and ex-post bargaining power. We specify these
rights in a few assumptions throughout the paper.




     10   Our convertible includes many types of convertibles in reality, including convertible preferred, convertible
debentures, convertible exchangeable preferred, and debt with warrants. All these convertibles have key features:
convertibility and a guaranteed fixed return before conversion. The minor differences among the convertibles
deal with more specific issues and contingencies in reality.

                                                                                                            Page 14
3.4. Assumptions
     Denote Φ( y , x, k ) as the distribution function of y conditional on investments ( x, k ).
Naturally, more initial investment k from the investor and more effort x from the manager
will increase the chance of success. Hence, this distribution function should be decreasing in
( x, k ), which is the so-called first-order stochastic dominance (FOSD).

Assumption 1 (FOSD). Φ x ( y , x, k ) < 0 and Φk ( y , x, k ) < 0 for all ( y , x, k ) .

     For convenience, we also impose the following assumption, which is imposed by almost
every agency model in the literature. This assumption is unnecessary but it substantially re-
duces the complexity of the derivation.

Assumption 2. The support of Φ( y , x, k ) is independent of investments ( x, k ).

     We take the interest rate r as given, which means that our bonds and convertibles can
be considered as having a so-called floating rate. Such an interest rate is determined or heav-
ily influenced by the prevailing safe rate of return in the market. In fact, the interest rate on
such a bond or convertible in reality is often pegged to Treasury security rates or the 3-
month LIBOR (slightly lower). Sometimes, the interest is accrued and paid with the principal
at maturity.11

     The total required investment K is not a choice variable; it is a given fixed number. We
may consider this amount as the necessary amount for a planned or expected business ex-
pansion in the coming years. We may also consider K as optimally determined by an earlier
problem and our current problem is to decide how to raise this total amount in multiple
stages. Specifically, given K , the manager can raise the funding in two installments. The two
parties have an option to abandon the project ex post by not raising the second installment.
The two parties may benefit from the resolution of uncertainty by investing ex post. But, the
allocation of investment across the two periods may affect the manager’s incentive and hence
affect the output indirectly. An ex-post investment decision imposes a risk on the manager
(due to the investor’s options to quit or to renegotiate), which may lead to lower effort.
Therefore, the allocation of investment needs to be arranged properly in order to balance the
manager’s incentive to expend effort and the investor’s benefit from late investment.




     11   See Gitman (2003, p.280). For example, in December 2003, Wyeth, a large pharmaceutical company, is-
sued $850 million in convertibles with an interest rate equal to the 6-month LIBOR minus 0.5%. The convertibles
matured on January 15, 2004, and they were convertible into shares of Wyeth common stock at a conversion
price $60.39 per share of common stock if certain conditions were met, including Wyeth common stock reaching
certain specified thresholds.

                                                                                                      Page 15
    The conversion ratio in reality is the number of common stock shares that a convertible
can be converted into. In our model, for convenience, the conversion ratio is the proportion
of the firm’s equity that a convertible can be converted into. We denote the conversion ratio
as τ , where τ ∈ [0, 1].

    Finally, assume that both parties are risk neutral in income. For simplicity, also assume
that there is no discount for time preferences and no interest payment at time t = 1. The in-
terest is paid at the end of the project with the proceeds.


3.5. Model Setup
    We do not restrict ourselves to the principal-agent setup, in which one of the parties is
given the full bargaining power ex ante. Instead, both parties in our model have certain bar-
gaining power both ex ante and ex post. The two parties negotiate and bargain over the terms
of a contract ex ante and possibly ex post. This means that, with the possibility of renegotia-
tion, the two parties will negotiate an agreement that ensures social welfare maximization ex
ante as well as ex post, subject to incentive conditions. An agreement is a financial instru-
ment that defines their trade, rights and bargaining positions.

    Specifically, let Π I and Π M be the ex-ante payoffs to the investor and the manager, re-
spectively. Assume that the investor’s initial investment k is contractable, but the decision
on whether or not to raise the second installment is not contractable ex ante. As usual, the
manager’s investment x is unverifiable. The contract is an outcome of negotiation. With the
assumption of an efficient bargaining outcome, the ex-ante problem is

                                    V = max Π M + Π I
                                          x, k ,

                                                   ∂Π M                                      (1)
                                            s.t.        = 0,
                                                    ∂x
where the constraint in (1) is the so-called incentive compatibility condition (IC condition).
Besides the choice variables x and k , there are other choice variables, depending on the
choice of financial instrument employed to carry out the sequential investments.

    Our tasks are (a) to identify those efficient mechanisms (financial instruments with cer-
tain features) that are popular in reality and (b) to identify those special features in the fi-
nancial instruments that are necessary for efficiency and also popular in reality. By this, we
show that the widely observed real-world phenomenon in corporate finance is indeed an effi-
cient behavior.

Remark 1. Individual rationality (IR) conditions are ignored. As long as the project is so-
cially viable (the joint payoff is positive in equilibrium), the two parties can make a monetary



                                                                                        Page 16
transfer to ensure their IR conditions. The size of this transfer depends on their relative bar-
gaining power and it will not affect their investment incentives.

Remark 2. There are two kinds of bargaining power for the two parties: the bargaining
power for ex-ante negotiation and the bargaining power for ex-post renegotiation. We do not
need to specify the ex-ante bargaining power explicitly, since the size of a monetary transfer
between the two parties will not affect their investment decisions. On the other hand, de-
pending on the type of financial instrument used to carry out investments, the ex-post bar-
gaining power is defined within the financial instrument. Each financial instrument carries
not only income rights but also some other rights (voting rights, default rights, bankruptcy
rights, call rights, etc.). We specify the relevant rights in the assumptions later, based on the
common practice in reality.

Remark 3. Although we limit ourselves to four types of financial instruments/contracts,
since we can find an efficient solution from these instruments, a larger set of admissible con-
tracts is unnecessary.



4. Sequential Financing
    In this section, we investigate how sequential financing can be efficiently carried out by
financial instruments. All the proofs are in the Appendix.


4.1. Benchmark: the First-Best Problem
    As a benchmark, we consider the first-best problem first. The first-best problem is a se-
quential financing problem in which there are no agency problems. Specifically, the man-
ager’s effort x is contractable and the second installment is provided if and only if it is ex-
post efficient to do so. Hence, at time t = 1, if and only if

                     y − ( K − k ) ≥ θk     or       y ≥ y1 ≡ θk + K − k ,                    (2)

it is ex-post efficient to continue the operation; otherwise, the firm defaults or the project is
abandoned at t = 1 (Figure 2). Note that we use the statements “the firm defaults” and “the
project is abandoned” interchangeably throughout the paper.


                default                          continuation
                           y1                                                    y

                      Figure 2. The Milestones for the First-Best Problem




                                                                                         Page 17
In other words, with probability Φ( y1 , x, k ), the firm defaults at t = 1. Hence, the first-best
problem is
                                                 ∞
                 max θk Φ( y1 , x, k ) + ∫
                 x , k ≥0
                                                     [ y − ( K − k )] d Φ( y, x, k ) − k − c( x ) .
                                                y1


The equations determining the solution are listed in the following proposition.


Proposition 1 (First Best). The first-best solution ( x ∗ , k ∗ ) is determined by two equations:
                                                             ∞
                            (1 − θ )Φ( y1 , x, k ) + ∫           Φk ( y , x, k ) dy = 0,                  (3)
                                                            y1
                                       ∞

                                   ∫   y1
                                            Φ x ( y , x, k ) dy + c ′( x ) = 0.                           (4)



       As shown in the proof of Proposition 1, (3) means that the marginal social welfare of ini-
tial investment k is zero, and (4) means that the marginal social welfare of the manager’s
effort x is zero.


4.2. Debt Financing
       In this section, we consider the situation when the manager raises funds in stages in the
form of straight debt. When sequential financing is carried out by debt, we call it debt financ-
ing.

       Under a debt contract, the investor loans the manager a specific amount and receives a
fixed rate of interest if the firm does not default or go bankrupt. As a common practice in re-
ality, debt holders have a special right called foreclosure. This means that, if the investor has
reasons to believe that the project is not worth continuing, she can foreclose the firm by forc-
ing the firm/project into bankruptcy and taking whatever assets are available less liabilities.

Assumption 3 (foreclosure). If the firm is expected to go bankrupt, a debt holder has the
right to foreclose the firm at t = 1 by selling off the firm for the liquidation value.

       Formally, suppose that the two parties negotiate a contract by which funding is raised in
stages in the form of debt. After accepting the contract, the investor provides k to the firm at
t = 0. At t = 1, with new information, the manager decides whether or not to raise the sec-
ond installment K − k in the form of debt. If the firm’s performance or the economic situa-
tion is not favorable, the investor has the right to foreclose the firm and take the liquidation
value θk . Denote this debt contract as ( r, K ) with principal K , interest rate r, and maturity
at t = 2.




                                                                                                      Page 18
Remark 4. We assume that there is no interest payment at t = 1 on either debt or converti-
bles. A model in which the interest payment rk is to be made at t = 1 can be transformed to
our current model by treating K − (1 + r ) k as the second installment (the investor’s net con-
tribution at t = 1 ) and treating (θ − r )k as the liquidation value of the firm at that time
(since the liabilities have to be paid before the liquidation value can be divided).

    Let us first derive the two parties’ ex-post decisions. Denote

                              y1 ≡ K − (1 − θ ) k ,        y2 ≡ (1 + r ) K ,

where y2 is the face value of the debt. At time t = 2 ,
             if (1 + r ) K ≤ y, the investor receives the principal plus interest,
             if (1 + r ) K > y , the firm is bankrupt (or the project fails).

At time t = 1, since the investor has the right of foreclosure, she will foreclose the firm if and
only if

                                         y − ( K − k ) < θk .                                    (5)

That is, if and only if y ≥ y1 , the manager is able to raise K − k to continue the project.
Condition (5) ensures ex-post efficiency. Hence, y1 is the threshold for default and y2 is the
threshold for bankruptcy. Note that, even if the firm is expected to go bankrupt, the investor
may still allow the project to continue (Figure 3). Note also that we use statements “the firm
is bankrupt,” “the project fails” and “the project turns out to be unprofitable to the firm” in-
terchangeably throughout the paper.


                  default          bankrupt                normal outcome

                             y1                       y2                             y

                            Figure 3. The Milestones for Debt Financing

Here is a summary of the choices made at t = 1 and t = 2 :
     The firm defaults at t = 1 if y < y1 .
     If y ≥ y1 , the amount K − k is raised at t = 1 and the firm continues its operation;
           the investor receives debt repayment at t = 2 when y ≥ y2 , and
           the firm is bankrupt (or the project fails) at t = 2 when y1 ≤ y < y2 .

This means that the firm receives income y − (1 + r ) K only when y ≥ y2 , while the inves-
tor receives θk when y < y1 , y when y1 ≤ y < y2 , and (1 + r ) K when y ≥ y2 . Denote
Π M and Π I as the manager’s and investor’s expected payoffs ex ante, respectively. Taking



                                                                                             Page 19
into account of the manager’s cost c( x ) and the investor’s investment k at t = 0 and K − k
if y ≥ y1 at t = 1, the above analysis implies:
                                 ∞
                      Π M ≡ ∫ [ y − (1 + r ) K ] d Φ( y , x , k ) − c( x ) ,
                                y2

                                                           y2
                       Π I ≡ θk Φ( y1 , x, k ) + ∫
                                                          y1
                                                                [ y − ( K − k )] d Φ( y, x, k )
                                      ∞
                                +∫        ⎡(1 + r ) K − ( K − k )⎤ d Φ( y , x, k ) − k .
                                     y2   ⎣                      ⎦

The ex-ante contractual problem is the optimization problem (1) with choice variables x and
k . The following proposition states the solution.


Proposition 2 (Straight Debt). Debt financing is efficient if the bankruptcy risk is minimal.
Specifically, let the solution of (1) be ( x, k ) and let y1 ≡ K − (1 − θ ) k . Then, debt financing is
                                           ˆ ˆ            ˆ                 ˆ
efficient if and only if
                                               y2

                                          ∫   ˆ
                                              y1
                                                              ˆ ˆ
                                                    Φ x ( y , x, k )dy = 0.                                              (6)

Further, if (6) fails, the manager will underinvest x and the investor will underinvest k .

                                                                                           y2
     In the definition of second-order stochastic dominance,                          ∫   y1
                                                                                                Φ( y , x, k )dy is used as a

risk measure, which measures the risk of bankruptcy in our model. Hence, condition (6)
means that more effort from the manager can no longer reduce the risk of bankruptcy fur-
ther. That is, condition (6) is the first-order condition (FOC) for minimizing the risk of bank-
ruptcy. A mature firm is relatively secure and its manager’s marginal contribution may not
be crucial to the firm’s survival. In this case, condition (6) may be satisfied or approximately
satisfied. If so, debt financing can be efficient or approximately efficient. This conclusion is
consistent with VentureXpert’s database, which indicates that a sizable portion of investment
is through debt financing when a firm is in its later/mature stages of development. This re-
sult is also consistent with Stein (1992), who shows that a good firm tends to use debt financ-
ing as opposed to equity or convertible financing.

     The intuition for Proposition 2 is as follows. The manager, since he is the sole residual
claimant, has full incentive to invest effort as long as the investor is willing to provide the ef-
ficient level of initial investment. The investor, however, with a fixed income, is not con-
cerned with the manager’s incentive; with the right of foreclosure, she is not concerned with
default either; hence, she cares only about bankruptcy. It turns out that, only when the risk
of bankruptcy is minimal, is the investor willing to invest efficiently (the initial investment).



                                                                                                                   Page 20
Therefore, if and only if the manager’s IC condition happens to be in alignment with the
minimization of bankruptcy risk, is debt financing efficient.

     However, condition (6) often fails. For example, consider a simple output process:

                                                      y = μ ( x , k ) + ε,                                          (7)

where μ( x, k ) is the mean output with μ x ( x, k ) > 0 and μk ( x, k ) > 0 (which is the FOSD),
and ε is a random shock with zero mean and density function φ(ε ). We have

                                                                                 φ ⎡⎣ y − μ ( x, k )⎤⎦dy.
                                 y2                                         y2

                            ∫   ˆ
                                y1
                                      Φ x ( y , x, k )dy = μx ( x, k ) ∫
                                                ˆ ˆ             ˆ ˆ
                                                                           ˆ
                                                                           y1
                                                                                              ˆ ˆ

Hence, condition (6) fails unless there is no chance of bankruptcy at all; that is, unless
φ( y ) = 0 for all y ∈ ⎡⎣ y1 + μ( x, k ), y2 + μ( x, k ) ⎤⎦ .
                          ˆ       ˆ ˆ             ˆ ˆ

Remark 5. We have adopted the first-order approach (FOA) in our model setting in (1). To
ensure the validity of FOA, additional conditions may be needed. Since we deal with simple
contracts only (either straight debt or convertibles), which pay either a fixed amount
s( y ) = (1 + r ) K or a fixed share s ( y ) = τ y with some decision events (such as default and
bankruptcy), with the usual assumption that the expected output is increasing and concave
in investments ( x, k ), we only need some simple conditions on the distribution function of
the milestones { y1 , y2 } to ensure the validity of FOA. For example, for the output process in
(7), the second-order condition (SOC) for the IC condition in (1) holds if the manager’s ex-
pected income is concave, which holds if

                                      ∂ ln {1 − Φ ⎣⎡ y2 − μ ( x, k )⎦⎤}
                                                              ˆ ˆ                         ˆ ˆ
                                                                                    μxx ( x, k )
                                                                           <−                    .                  (8)
                                                     ∂x                                   ˆ ˆ
                                                                                    μx ( x, k )

The left-hand side is the marginal percentage increase in Pr( y ≥ y2 ), and the right-hand
side is a measure of smoothness in expected output.12 This condition is a constraint on the
distribution function at the bankruptcy milestone y2 , which discourages overinvestment by
the manager. Hence, as long as there is no incentive for overinvestment by the manager, we
expect FOA to be valid. As the intuition for Proposition 2 indicates, the main problem is the
investor’s concern about the risk of bankruptcy; since the manager is the sole residual claim-
ant, he has no incentive problem. Hence, we do not expect an overinvestment problem by the
manager in this case. Since conditions for FOA are similar to (8) in other cases, we will not
discuss them one by one.




     12   This smoothness is the same as risk aversion for a utility function. An individual’s utility function tends to
be smooth if she is averse to risks and the smoothness can be measured by absolute risk aversion.

                                                                                                              Page 21
4.3. Straight Convertible Financing
    In this section, we suppose that the manager issues convertibles. When sequential fi-
nancing is carried out by convertibles, we call it convertible financing.

    In reality, although convertible holders receive a guaranteed rate of return, like a debt
holder, before conversion, some liquidation protection measures for debt holders do not ap-
ply to convertible holders. There are two key differences in a holder’s rights. First, a con-
vertible holder has no foreclosure rights. Second, in the case of default, a convertible holder
is not treated as a debt holder. These differences mean that whether or not the firm should
default is subject to negotiation and that, in the case of default, the investor is not entitled to
the firm’s worth up to her investment.

    Specifically, we denote a convertible as ( K , r , τ ) with principal K , interest rate r, con-
version ratio τ , convertible after t = 1, and maturity at t = 2. Again, as mentioned in
Remark 4, assume that there is no interest payment at t = 1. To distinguish it from other
more sophisticated forms of convertibles, we call it a straight convertible.

    A contract is renegotiable at any time. If renegotiation leads to default, a convertible
holder shares the proceeds with the manager according to their share holdings ( τ ,1 − τ ). In
equilibrium, renegotiation never needs to happen. Instead, the optimal contract will have a
clause indicating that the firm defaults at t = 1 if and only if the firm’s performance is lower
than a specified level.

    Let us derive the two parties’ decisions. At time t = 2,
          if (1 + r ) K < τ y ,     the investor converts;
          if τ y ≤ (1 + r ) K ≤ y , the investor holds debt;
          if (1 + r ) K > y,        the investor holds debt and the firm is bankrupt.

At time t = 1 , the two parties can renegotiate the existing agreement on the continuation of
the project, which implies ex-post efficiency. This means that, if and only if y ≥ y1 , the pro-
ject will continue and the second installment of funding will be raised. Let τ be the conver-
sion ratio. As mentioned above, if the firm defaults, the investor’s and the manager’s incomes
at t = 1 are τθk and (1 − τ ) θk , respectively. Let
                                                                      (1 + r ) K
                 y1 ≡ K − (1 − θ ) k ,    y2 ≡ (1 + r ) K ,    y3 ≡              .              (9)
                                                                          τ
Here is a summary of the choices made at t = 1 and t = 2 :




                                                                                          Page 22
   At t = 1, if y < y1 , the firm defaults;
                 if y ≥ y1 , the investor invests K − k and the firm continues its operation.
   At t = 2, if y > y3 , the investor converts to equity;
                 if y2 ≤ y ≤ y3 , the investor holds debt;
                 if y1 ≤ y < y2 , the investor holds debt and the firm is bankcrupt.

Hence, y1 is the threshold for default, y2 is the threshold for bankruptcy, and y3 is the
threshold for conversion. Here, y1 , y2 and y3 serve as milestones, by which certain actions
will be taken. In other words, sequential financing using convertibles manifests itself as a
milestone strategy in equilibrium (Figure 4).

                                                                              voluntary
               default                bankruptcy              debt            conversion
                              y1                         y2             y3                   y

                           Figure 4. The Milestones for Convertible Financing

    The above analysis indicates that the firm receives income (1 − τ )θk when y < y1 ,
 y − (1 + r ) K when y2 ≤ y < y3 , (1 − τ ) y when y ≥ y3 ; the investor receives τθk when
 y < y1 , y when y1 ≤ y < y2 , (1 + r ) K when y2 ≤ y < y3 , and τ y when y ≥ y3 . Taking
into account of the manager’s cost c( x ) and the investor’s investments k at t = 0 and
 K − k if y ≥ y1 at t = 1, the ex-ante payoffs are
                                                y3
     Π M ≡ (1 − τ )θk Φ( y1 , x, k ) + ∫ [ y − (1 + r ) K ] d Φ( y , x, k )
                                               y2
                 ∞
            +∫         (1 − τ ) yd Φ( y , x, k ) − c( x ),
                 y3

                                         y2
      Π I ≡ τθk Φ( y1 , x, k ) + ∫ [ y − ( K − k )] d Φ( y , x, k )
                                        y1
                  y3                                             ∞
            +∫ [ (1 + r ) K − ( K − k )] d Φ( y, x, k ) + ∫ [ τ y − ( K − k )] d Φ( y , x, k ) − k .
                 y2                                              y3


The ex-ante contractual problem is the optimization problem (1) with choice variables x, k
and τ . Besides the investment variables x and k , the conversion ratio τ is also determined
by the optimization problem. Although there is one more variable to choose in comparison
with debt financing, efficiency is not guaranteed; in fact, it becomes less likely that efficiency
will be achieved. As stated in the following proposition, the solution suggests that a straight
convertible can never achieve efficiency.


Proposition 3 (Straight Convertible). Straight convertibles are inefficient.




                                                                                                   Page 23
     This result is consistent with one stylized fact in reality: Although straight bonds are
popular in reality, straight convertibles are rare. Convertibles in reality typically include a
few other options.

     The intuition is clear. There are two milestones y1 and y2 in Figure 3 and three mile-
stones y1 , y2 and y3 in Figure 4. In comparison with straight debt, one key in a straight
convertible is the manager’s income. The manager gains a fraction of the liquidation value in
the case of y < y1 when the firm defaults, but he is no longer the sole residual claimant in
the case of y > y3 when the firm does very well. Clearly, the manager’s incentive is weaker
than with straight debt. Hence, with a straight convertible, the manager will underinvest
even if the investor invests efficiently. As we discussed before, inefficiency in straight debt is
mainly caused by the investor, who is satisfied (and invests efficiently) only if bankruptcy
risk is minimal; the manager is the sole residual claimant in that case who has no incentive
problems. In contrast, inefficiency in a straight convertible is mainly due to the manager’s
incentive problem.13

     With the above intuition, one obvious solution is to improve the manager’s incentive.
For this purpose, the next two sections introduce some popular real-world measures into our
convertibles.


4.4. Callable Convertible Financing
     In reality, almost every convertible has a call option (Lewis et al. 1998). That is, the is-
suer of a callable convertible has the option to force conversion after a certain period and the
conversion is to be based on a pre-determined conversion ratio and a call price in the con-
tract. In this section, we investigate whether or not this additional option can lead to effi-
ciency.

     Formally, as shown in Figure 5, suppose that the manager raises funds in stages by issu-
ing callable convertibles. The convertibles contain a call option for the manager to force con-
version after t = 1 and at call price q. That is, if the manager does not call, the investor has
the option to convert her investment into τ shares of equity or to demand payment
(1 + r ) K at maturity; but if the manager calls, the investor can either convert her invest-
ment into τ shares of equity right away or receive payment q. Denote this callable converti-
ble as ( K , r, τ , q) with principal K , interest rate r, conversion ratio τ , call price q, con-
vertible and callable after t = 1, and maturity at t = 2.




     13   The incentive problem on the investor’s side is complicated. With a straight convertible, the investor loses
the right of foreclosure and a fraction of the liquidation value if the firm defaults, but the investor can gain a bet-
ter return if the firm does well.

                                                                                                            Page 24
                                          .
             The two parties negotiate for an agreement
                                  t=0
        The investor provides k The manager provides x

                               t =1       .   uncertainty realized, renegotiation allowed

             Default                                     The investor provides K − k


        ⎛          ⎞
        ⎜(1− τ )θk ⎟
                                                                    .   t =1
                                                                                    The manager calls
        ⎜          ⎟                          No call
        ⎝ τθk ⎟
        ⎜          ⎠
                                          .    t=2                                               .     t=2

               Bankrupt          No convert Convert                      Bankrupt           No convert Convert

                 ⎛0 ⎞                      ⎞
                          ⎛ y − (1 + r ) K ⎟         ⎛(1− τ ) y ⎞            ⎛0 ⎞          ⎛ y − q⎞         ⎛(1− τ ) y ⎞
                 ⎜ ⎟
                 ⎜ ⎟      ⎜
                          ⎜                ⎟         ⎜          ⎟
                                                                ⎟            ⎜ ⎟           ⎜      ⎟         ⎜          ⎟
                 ⎜ y⎟     ⎜ (1 + r ) K ⎠   ⎟         ⎜
                                                     ⎜ τy ⎟                  ⎜ ⎟           ⎜      ⎟         ⎜          ⎟
                 ⎝ ⎠      ⎝                          ⎝          ⎠            ⎜ y⎟
                                                                             ⎝ ⎠           ⎜ q ⎟
                                                                                           ⎝      ⎠         ⎜ τy ⎟
                                                                                                            ⎝          ⎠

                  Figure 5. The Game Tree for Callable Convertible Financing

    We now derive the two parties’ ex-post decisions. Given a realized random shock and the
investor’s option to convert, without a call, the manager’s payoffs are indicated in Figure 6.

 Manager’s        (1− τ ) θk                         0                       y −(1 + r ) K                 (1− τ ) y
 payoff:
                   default                         bankruptcy                       debt               volunt. conversion

                                     y1                                 y2                            y3                   y
                         Figure 6. The Manager’s Payoffs without a Call

If the manager forces conversion (before the investor does it voluntarily), then the manager’s
payoffs are indicated in Figure 7. Here, we assume that, if a call is made with q > y , the in-
vestor gets y; that is, we treat this case as bankruptcy.

 Manager’s        (1− τ ) θk                   0                         y−q                               (1− τ ) y
 payoff:
                  default                 bankruptcy            no conversion                          forced conversion

                                y1                        q                                q/τ                                 y
                               Figure 7. The Manager’s Payoffs with a Call

By comparing the two figures, after careful analysis, we find:

   (a) If q / τ is in the equity interval ( y > y3 ) in Figure 6, the manager will never call.


                                                                                                                       Page 25
    (b) If q / τ is in the debt interval in Figure 6, the manager will always call.

    (c) If q / τ is in the bankruptcy interval in Figure 6, the manager will always call.

That is, if q ≤ y2 , where y2 ≡ (1 + r ) K , the manager will always call, and if q > y2 , the
manager will never call. This means that the case with q > y2 is the same as in the last sec-
tion and efficiency cannot be achieved. We will therefore consider only the case with q ≤ y2 .
With q ≤ y2 , the ex-ante payoffs are
                                                      q/τ                                    ∞
  Π M ≡ (1 − τ )θk Φ( y1 , x, k ) + ∫                       ( y − q ) d Φ( y , x , k ) + ∫         (1 − τ ) yd Φ( y , x, k ) − c( x ),
                                                  q                                          q/τ

                                         q                                               q/τ
   Π I ≡ τθk Φ( y1 , x, k ) + ∫
                                     y
                                       [ y − ( K − k )] d Φ( y, x, k ) + ∫ q [ q − ( K − k )] d Φ( y, x, k )
                                         1

                ∞
         +∫
              q/τ
                    [ τ y − ( K − k )] d Φ( y, x, k ) − k .

The ex-ante contractual problem is the optimization problem (1) with choice variables x, k ,
τ and q. It turns out that adding a call feature to convertibles does not necessarily guaran-
tee efficiency.


Proposition 4 (Callable Convertible). With a call option, for the first-best investments
( x ∗ , k ∗ ) with y1 ≡ K − (1 − θ ) k ∗ , we have:
                    ∗


                                                    ∗
(a) Any callable convertible with a call price q ≥ y1 is inefficient.

(b) A callable convertible is efficient if and only if
                                          ∗
                                         y1

                                 ∫            Φ x ( y , x ∗ , k ∗ )d y < θk ∗Φ x ( y1 , x ∗ , k ∗ ).
                                                                                    ∗
                                                                                                                                         (10)
                                     0


That is, under condition (10), we can find an efficient callable convertible ( K , r , τ ∗ , q∗ ) with
principal K , interest rate r, conversion ratio τ ∗ ∈ (0, 1), call price q∗ ∈ (0, y1 ) , converti-
                                                                                   ∗


ble and callable after t = 1, and maturity at t = 2, where K > 0 and r > 0 can take arbi-
trary values.


    The intuition for part (b) of Proposition 4 is as follows. Although a call option can im-
prove the manager’s incentive, since the manager is likely to call when the firm is not doing
well, it has a negative effect on the investor’s incentive. As explained following Proposition 2,
                                                                y1
the risk of default is measured by                      ∫   0
                                                                     Φ( y , x, k )d y. By FOSD, the right-hand side of (10) is

negative. Hence, condition (10) requires that increasing effort from the manager reduces the
risk of default sufficiently fast. By (10), the investor may not worry about the negative effect
of the call option on her, since the tendency to call when the output is low is compensated by
a smaller chance of default. That is, with a call option, the manager’s incentive is improved


                                                                                                                                  Page 26
sufficiently, and by (10) the net effect of the call option on the investor’s incentive is neutral.
Therefore, efficiency is achievable under (10).

     However, part (a) of Proposition 4 is disappointing. The call price q∗ for an efficient
callable convertible is too low. A real-world call price is typically higher than the capital K ,
which is intended to provide capital protection for an investor.14

     To improve the results, we go back to reality. We notice that real-world convertibles
typically contain provisions for call restrictions. This may well be the channel for us to find
an efficient callable convertible. This makes sense. A convertible without a call is not efficient
since it gives too much ex-post bargaining power to investors through the use of a conversion
option. A call feature offers the manager some ex-post bargaining power, which can improve
the situation. However, the call option may have given too much power to the manager. To
strike a balance, some call restrictions are needed. The following section investigates some
popular call restrictions in reality.


4.5. Callable Convertible Financing with Call Restrictions
     In reality, an overwhelming majority of callable convertibles have call restrictions (see
Footnote 3). The two most popular call restrictions are the so-called provisional calls and
floating conversion ratios. Interestingly, these call restrictions can indeed lead to efficiency.


A Provisional Call

     A provisional call means that the convertible cannot be called unless the stock trades
above a stated price level (for a certain period of time). This price level is called the trigger
price.

Assumption 4. Let y be the trigger price. In period 2, if and only if y ≥ y , the manager
has the option to call with call price q and conversion ratio τ .

     Since the firm either defaults or goes bankrupt when y < y2 , the trigger price should
not be below y2 . Also, since the investor will convert voluntarily when y > y3 , the trigger
price should not be above y3 . Hence, we should have y ∈ [ y2 , y3 ] . Further, since, in reality,

the manager always wants a successful conversion when he calls, we need q ≤ τ y . 15 Hence,
the ex-post decisions are shown Figure 8. When y ≥ y3 , the investor will convert voluntarily.



     14   See for example the discussion on the call premium by Gitman (2003, p.276).
     15   Asquith and Mullins (1991, p.1288) point out that, consistent with a high cost of a failed conversion, a firm
typically requires the conversion value to be 20% above the call price before it calls.

                                                                                                             Page 27
When y ∈ [ y , y3 ] , the manager will force conversion and the investor will choose to convert.

When y < y , the manager cannot force conversion and the investor will hold debt to matur-
ity. As a result, the firm goes bankrupt when y < y2 .


                 Default        Bankruptcy             Debt       Forced conversion           Conversion
                           y1                     y2          y                          y3                 y

          Figure 8. The Milestones for Convertible Financing with a Provisional Call


Then, the ex-ante payoffs are
                                           y                                        ∞
Π M ≡ (1 − τ )θk Φ( y1 , x, k ) + ∫ [ y − (1 + r ) K ] d Φ( y , x, k ) + ∫              (1 − τ ) yd Φ( y, x, k ) − c( x ),
                                          y2                                     y

                                  y2                                     y
 Π I ≡ τθk Φ( y1 , x, k ) + ∫          [ y − ( K − k )]Φ( y, x, k ) + ∫ ⎡⎣(1 + r ) K − ( K − k )⎤⎦ d Φ( y, x, k )
                                 y1                                     y2
            ∞
       +∫       [ τ y − ( K − k )] d Φ( y, x, k ) − k .
            y


The ex-ante contractual problem is the optimization problem (1) with choice variables x, k ,
τ , q and y . This problem implies an efficient callable convertible.


Proposition 5 (Provisional Call). With a provisional call option, for the first-best invest-
ments ( x ∗ , k ∗ ), with y1 ≡ K − (1 − θ ) k ∗ and any y ∈ [ y2 , y3 ] , if
                           ∗


                                                                      ∗
                                       φx ( y , x ∗ , k ∗ ) ≤ 0, for y1 < y < y ,                                      (11)

then sequential financing using convertibles can achieve efficiency. That is, we find an effi-
cient callable convertible ( K , r , τ ∗ , q∗ , y ) with principal K , interest rate r, conversion ratio
τ ∗ ∈ (0, 1), call price q∗ ≤ τ ∗ y , trigger price y ∈ [ y2 , y3 ] , convertible after t = 1, callable
after t = 1 if y ≥ y , and maturity at t = 2, where K > 0,                                    r > 0,     q∗ ≤ τ ∗ y and
y ∈ [ y2 , y3 ] can take arbitrary values. In particular, if y = y2 , condition (11) is unnecessary.


    Condition (11) means that an increase in the manager’s investment reduces the chance
of producing an output in ( y1 , y ) . With the possibility of ex-post renegotiation at t = 1, the
                             ∗

             ∗
risk of y < y1 is controllable. Hence, the two parties will be concerned with output in the
              ∗
range of y > y1 . The manager in particular prefers y ≥ y , in which case debt can be elimi-
                           ∗
nated. Conditional on y > y1 , condition (11) means that an increase in effort x enhances
the chance of y ≥ y . Hence, condition (11) encourages the manager to expend more effort.

    Condition (11) is easy to satisfy if the trigger price y is less than the mean output. This
situation is shown in Figure 9 where the red density curve is below the black density curve
for output below the mean output μ. For example, for the output process in (7), if φ(ε) is

                                                                                                                  Page 28
increasing when ε < 0 and decreasing when ε > 0, then output y satisfies condition (11) if
y is less than the mean output.




                                                                  as x increases

                                  φ( y , x , k )           φ( y , x ′, k )




                                 φx < 0            y   μ    φx > 0                 y


                    Figure 9. A Shift in the Density Function as x Increases


A Floating Conversion Ratio

    A floating conversion rate protects holders against dilution by giving them a greater
number of shares upon conversion if the common stock price is too low. Specifically, there
are two pre-determined conversion ratios, τ and τ , with τ > τ. If the stock price is lower
than a certain level y , the higher conversion ratio τ is applicable, otherwise the lower con-
version ratio is applicable. Denote this callable convertible as ( K , r, τ , q, τ , y ) with principal
K , interest rate r, call price q, conversion ratio τ if y ≥ y , conversion ratio τ if y < y ,
convertible and callable after t = 1, and maturity at t = 2.

    When y ≥ y3 , the investor will convert voluntarily and there is no need for the manager
to provide a higher conversion ratio for forced conversion. Hence, we can impose y ≤ y3 .
We will show that choosing y = y3 (actually, any y ≤ y3 ) is enough to ensure efficiency.

Assumption 5. In period 2, the manager has the option to call with call price q and with
conversion ratio τ if y ≥ y3 or conversion ratio τ , τ > τ , if y < y3 .


    In reality, when a firm makes a call, it always aims at a conversion (instead of paying
cash q) and a call almost always results in a conversion (Footnote 15). Hence, we let q sat-
isfy q ≤ τ y1 so that a forced conversion always leads to a conversion. Figure 10 indicates the

ex-post decisions. First, the possibility of renegotiation at t = 1 ensures ex-post efficiency,
implying that the firm defaults if and only if y < y1 . Second, in period 2, if and only if
y > y3 , the investor will voluntarily convert, by which the conversion ratio is τ . Third, de-
note y3 ≡ (1 + r ) K / τ . If y ∈ [ y3 , y3 ] , by Assumption 5, the manager will not call and the




                                                                                              Page 29
investor will hold debt to maturity. Finally, if y ∈ [ y1 , y3 ] , the manager will call; with
q ≤ τ y1 , the investor will choose to convert.


                 default          forced conversion                          debt                         vol. conversion

                             y1                                 y3                                   y3                     y

            Figure 10. The Milestones for Convertible Financing with a Floating Call

Hence, the ex-ante payoffs are
                                            y3                                             y3
Π M ≡ (1 − τ )θk Φ( y1 , x, k ) + ∫              (1 − τ ) yd Φ( y, x, k ) + ∫ [ y − (1 + r ) K ] d Φ( y, x, k )
                                           y1                                          y3
            ∞
       +∫        (1 − τ ) yd Φ( y , x, k ) − c( x ),
            y3

                                   y3                                                           y3
 Π I ≡ τθk Φ( y1 , x, k ) + ∫           [ τ y − ( K − k )] d Φ( y, x, k ) + ∫ ⎡⎣(1 + r ) K − ( K − k )⎤⎦ d Φ( y, x, k )
                                  y1                                                        y3
            ∞
       +∫        [ τ y − ( K − k )] d Φ( y, x, k ) − k .
            y3


The ex-ante contractual problem is the optimization problem (1) with choice variables x, k ,
τ , q, τ and y . This problem also implies an efficient callable convertible.


Proposition 6 (Floating Conversion Ratio). With a call option and a floating conversion
ratio, for the first-best investment ( x ∗ , k ∗ ) with y1 ≡ K − (1 − θ ) k ∗ and y3 ≡ (1 + r ) K / τ ,
                                                         ∗
                                                                                  ˆ                 ˆ
where τ is defined by (29), if
      ˆ
                                                      ˆ
                                                      y3

                                                 ∫    ∗
                                                     y1
                                                           yd Φ x ( y , x ∗ , k ∗ ) < 0,                                           (12)

then convertibles can achieve efficiency in sequential financing. That is, we find an efficient
callable convertible ( K , r , τ ∗ , q∗ , τ ∗ , y ∗ ) with principal K , interest rate r, threshold
                                   ∗
y ∗ = y3 , call price q∗ ≤ τ ∗ y1 , conversion ratio τ ∗ ∈ (0, 1) if y ≥ y ∗ , conversion ratio
        ˆ
τ ∗ ∈ ( τ ∗ , 1) if y < y ∗ , convertible and callable after t = 1, and maturity at t = 2, where
                                 ∗
K > 0, r > 0 and q∗ ≤ τ ∗ y1 can take arbitrary values.

    Condition (12) is implied by condition (11), which we have explained before.


A Floating Conversion Ratio and a Realistic Call Price

    A firm in reality typically offers a reasonably high call price in order to attract investors,
even though the firm rarely ends up paying the call price. We now consider the possibility of
achieving efficiency with a high call price.



                                                                                                                                Page 30
    Consider a callable convertible ( K , r , τ , q, τ , y ) with conversion ratio τ if y ≥ y and
conversion ratio τ if y < y . By ex-post rationality, we first know that the manager will not
call when y > y3 , whatever the value of q is. Second, if q ≥ τ y3 , the manager will not call
when y ≤ y3 . Hence, if q ≥ τ y3 , the manager will never call. As we know, an overwhelming
majority of convertibles in reality is called. Hence, we must have q < τ y3 . Also, if
q ∈ (τ y2 , τ y3 ) , then, when y ∈ ( y2 , q / τ ) , the manager will call with the knowledge that
the investor will take cash q instead of conversion. Such a situation never happens in reality;
when a firm calls, it always intends a conversion. Hence, the largest possible call price is
q = τ y2 , by which the investor will convert when called. Hence, the ex-post decisions are
shown in Figure 11.

                Default               Bankruptcy           Forced Conversion                  Debt    Volunt. Conversion
                                 y1                      y2                         y3               y3

           Figure 11. The Milestones for Convertible Financing with a Floating Call

Hence, the ex-ante payoffs are
                                                   y3                                          y3
   Π M ≡ (1 − τ )θk Φ( y1 , x, k ) + ∫                  (1 − τ ) yd Φ( y , x, k ) + ∫ [ y − (1 + r ) K ] d Φ( y, x, k )
                                                  y2                                          y3
               ∞
          +∫         (1 − τ ) yd Φ( y , x, k ) − c( x ),
               y3

                                          y2                                              y3
    Π I ≡ τθk Φ( y1 , x, k ) + ∫               [ y − ( K − k )]Φ( y, x, k ) + ∫ [ τ y − ( K − k )] d Φ( y, x, k )
                                         y1                                              y2
                y3                                                           ∞
          +∫
               y3
                     ⎡(1 + r ) K − ( K − k )⎤ d Φ( y , x, k ) +
                     ⎣                      ⎦                   ∫ y3 [ τ y − ( K − k )] d Φ( y, x, k ) − k.
The ex-ante contractual problem is the optimization problem (1) with choice variables x, k ,
τ , q, τ and y . This problem again implies an efficient callable convertible.


Proposition 7 (Realistic Call Price). With a call option and a floating conversion ratio, for
the first-best investment ( x ∗ , k ∗ ) with y1 ≡ K − (1 − θ ) k ∗ and y3 ≡ (1 + r ) K / τ , where τ is
                                              ∗
                                                                       ˆ                 ˆ         ˆ
defined by (31), if
                            y2                                               ˆ
                                                                             y3

                       ∫   y1
                                 yd Φ x ( y , x ∗ , k ∗ ) < 0 and       ∫   y2
                                                                                  yd Φ x ( y , x∗ , k ∗ ) < 0,             (13)

then convertibles can achieve efficiency in sequential financing. That is, we find an efficient
callable convertible ( K , r , τ ∗ , q∗ , τ ∗ , y ∗ ) with principal K , interest rate r, threshold
y ∗ = y3 , call price q∗ = τ ∗ y2 , conversion ratio τ ∗ ∈ (0, 1) if y ≥ y ∗ , conversion ratio
τ ∗ ∈ ( τ ∗ , 1) if y < y ∗ , convertible and callable after t = 1, and maturity at t = 2, where
K > 0 and r > 0 can take arbitrary values.



                                                                                                                       Page 31
        Again, condition (13) is implied by condition (11).

        The intuition for the efficiency results under the three types of call restrictions is clear.
When condition (10) fails, as discussed before, the call option has a negative net effect on the
investor’s incentive, even though it has a positive effect on the manager’s incentive. One way
to rectify this problem is to impose some restrictions on calls. A trigger price serves the pur-
pose. It prevents the manager from forcing conversion when the firm is doing badly. With
this trigger price, a proper balance of incentives between the manager and investor is
achieved for many output distribution functions satisfying some minor conditions such as
(11).



5. Concluding Remarks
        Sequential financing is a widely adopted strategy in corporate finance. To carry out this
strategy, various financial instruments can be employed. It turns out that convertibles are
particularly popular, especially among companies with many real investment options and
those with high potential and high risk.

        To investigate convertibles in sequential financing, we built a model that contains a few
basic features from real-world business financing. We intended to find out whether or not
some popular real-world measures imply efficiency. For example, for debt financing, we in-
vestigated foreclosure; and for convertible financing, we investigated forced conversion and
some popular call restrictions. We are particularly interested in incentive problems resulting
from sequential financing and in risks resulting from decision events, such as default, bank-
ruptcy and conversion.

        In a double moral hazard model with renegotiation, we show first that straight debt can
achieve efficiency when bankruptcy risk is minimal [condition (6)]. Second, straight con-
vertibles cannot achieve efficiency since the manager is not the sole residual claimant when
the firm does well. Third, a call option provides the manager with sufficient incentive, but
the investor is concerned with being forced into conversion when the firm does badly. Hence,
callable convertibles may achieve efficiency only if the manager’s increased effort can suffi-
ciently reduce the chance of default [condition (10)]. Finally, using a trigger price, callable
convertibles with call restrictions can achieve efficiency by restricting the manager’s power of
forcing conversion when the firm does badly.

        There are many studies on sequential financing and there are also many studies on con-
vertibles. However, there are few studies that explain why sequential financing needs to be
implemented by convertibles and why other financial instruments, such as straight bonds,
are not suitable for sequential financing. Our study provides a theoretical explanation for a


                                                                                            Page 32
popular real-world financing strategy. Further, our study is the first to show efficiency in se-
quential financing using convertibles with call restrictions.

    Finally, we do not consider straight equity, since Wang and Zhou (2004) have studied it
in sequential financing and they show that straight equity financing is inefficient.



Appendix

Proof of Proposition 1

    With y1 = K − (1 − θ ) k being dependent on k , the first-order condition (FOC) for k is
      ∂(Π M + Π I )                                                                         ∞
 0=                 = θΦ( y1 , x, k ) + θk Φk ( y1 , x, k ) − θkφ( y1 , x, k ) (1 − θ ) + ∫ d Φ( y, x, k )
          ∂k                                                                               y1
                               ∞
                         +∫        [ y − ( K − k )] d Φk ( y, x, k ) + ⎡⎣ y1 − ( K − k )⎤⎦ φ( y1 , x, k ) (1 − θ ) − 1
                              y1
                                                                                                       ∞
                       = (θ − 1)Φ( y1 , x, k ) + (θk + K − k ) Φ k ( y1 , x, k ) + ∫                       y d Φk ( y , x, k )
                                                                                                      y1
                                                                ∞
                       = −(1 − θ )Φ( y1 , x, k ) − ∫                Φk ( y , x, k ) dy.
                                                               y1


The FOC for x is
              ∂(Π M + Π I )                            ∞
         0=                 = θk Φ x ( y1 , x, k ) + ∫ [ y − ( K − k )] d Φ x ( y, x, k ) − c ′( x )
                   ∂x                                 y1
                                           ∞
                                   = −∫        Φ x ( y , x, k )d y − c ′( x ).
                                          y1


Here, for simplicity of presentation, we have implicitly assumed that, for z = x, k ,
                                                                ∞
                lim yΦ z ( y , x, k ) = 0 and
                y →∞                                       ∫   y1
                                                                    Φ z ( y , x, k )d y is well defined.

                                                                         ∞
Without these assumptions, we should replace                        ∫   y1
                                                                             Φ z ( y , x, k ) dy by

                                                    ∞
                        y1Φ z ( y1 , x, k ) + ∫         y d Φ z ( y , x, k )         for z = x, k ,
                                                   y1


and all the results in this paper still hold.


Proof of Proposition 2

    We have
       ∂Π M     ∞                                                     ∞
            = ∫ [ y − (1 + r ) K ] d Φ x ( y , x, k ) − c ′( x ) = −∫ Φ x ( y , x, k )dy − c ′( x ).                             (14)
        ∂x     y2                                                    y2




                                                                                                                            Page 33
By (4) and (14), the IC condition holds for the first-best solution if and only if
                                                          y2

                                                     ∫   y1
                                                               Φ x ( y , x, k ) dy = 0.                                                            (15)

Hence, debt financing yields the first-best solution if and only if (15) holds.

    We now show underinvestment by the two parties. We have
∂Π I                                                                         y2
     = θΦ( y1 , x, k ) + θk φ( y1 , x, k )(θ − 1) + θk Φ k ( y1 , x, k ) + ∫ d Φ( y, x, k )
 ∂k                                                                         y1
                                                                               y2
         − ⎡⎣ y1 − ( K − k )⎤⎦ φ( y1 , x, k )(θ − 1) + ∫                            [ y − ( K − k )] d Φk ( y, x, k )
                                                                             y1
              ∞                          ∞
         +∫        d Φ( y , x, k ) + ∫         ⎡ (1 + r ) K − ( K − k )⎤ d Φk ( y , x, k ) − 1
              y2                         y2    ⎣                       ⎦
                                                                                    y2
      = (θ − 1)Φ( y1 , x, k ) + θk Φk ( y1 , x, k ) + ∫                                  [ y − ( K − k )] d Φk ( y, x, k )
                                                                                  y1
              ∞
         +∫        ⎡(1 + r ) K − ( K − k )⎤ d Φk ( y , x, k )
              y2   ⎣                      ⎦
                                                                                                    y2
      = (θ − 1)Φ( y1 , x, k ) + (θk + K − k )Φk ( y1 , x, k ) + ∫                                        yd Φk ( y, x, k ) − y2Φk ( y2 , x, k )
                                                                                                   y1
                                               y2
      = −(1 − θ )Φ( y1 , x, k ) − ∫                 Φk ( y , x, k )dy.
                                              y1


Since Φk ( y , x, k ) < 0, we have
                              ∂Π I                               ∞
                                   < −(1 − θ )Φ( y1 , x, k ) − ∫ Φk ( y , x, k )dy ,
                               ∂k                               y1


implying
                                                               ∂Π I
                                                                                         < 0.                                                      (16)
                                                                ∂k        ( x∗ ,k ∗ )


We further have
         ∂2ΠI                                  y2                      ∂                                              y2
              = −(1 − θ )Φ x ( y1 , x, k ) − ∫ Φ xk ( y , x, k )dy = −                                           ∫         Φ x ( y , x, k )dy
         ∂x∂k                                 y1                       ∂k                                            y1

                   ∂ 2      y2
              =−
                  ∂x∂k   ∫ y1 Φ( y, x, k )dy.
An increase in k and x should reduce the chance of bankruptcy, i.e.,

                                                    ∂2              y2

                                                   ∂x∂k        ∫   y1
                                                                         Φ( y , x, k )dy < 0.

Hence,

                                                                   ∂2ΠI
                                                                        > 0.                                                                       (17)
                                                                   ∂x∂k


                                                                                                                                                Page 34
                                                   ˆ ˆ
Condition (16) implies that the optimal solution ( x, k ) must be inefficient. Hence, the man-
ager will put in less effort, i.e., x < x ∗ . By (17), this implies
                                    ˆ
                                                              ∂Π I
                                                                                     < 0.
                                                               ∂k       ( x ,k ∗ )
                                                                          ˆ


Therefore, the optimal initial investment k must also be less than the first-best level k ∗ .
                                          ˆ


Proof of Proposition 3

     We have
     ∂Π M                                    y3                           ∞
          = (1 − τ )θk Φ x ( y1 , x, k ) + ∫ [ y − (1 + r ) K ] d Φ x + ∫ (1 − τ ) yd Φ x − c ′( x )
      ∂x                                    y2                           y3
                                                                                                                        y3
            = (1 − τ )θk Φ x ( y1 , x, k ) + ⎣⎡ y3 − (1 + r ) K ⎤⎦ Φ x ( y3 , x, k ) − ∫                                     Φ x ( y , x, k )dy
                                                                                                                       y2
                                                                                ∞
              −(1 − τ ) y3Φ x ( y3 , x, k ) − (1 − τ ) ∫                             Φ x ( y , x, k )dy − c ′( x )
                                                                              y3
                                                              ∞                                           ∞
            = (1 − τ )θk Φ x ( y1 , x, k ) − ∫                    Φ x ( y , x, k )dy + τ ∫                    Φ x ( y, x, k )dy − c ′( x ).
                                                          y2                                          y3


Hence, the IC condition is
                                                    ∞                                            ∞
           (1 − τ )θk Φ x ( y1 , x, k ) − ∫              Φ x ( y , x, k )dy + τ ∫                    Φ x ( y , x, k )dy = c ′( x ).                  (18)
                                                    y2                                          y3


If (4) holds, then (18) becomes
                                                          y2                                         ∞
             (1 − τ )θk Φ x ( y1 , x, k ) + ∫                  Φ x ( y , x, k )dy + τ ∫                    Φ x ( y , x, k )dy = 0.                   (19)
                                                         y1                                          y3


By Assumption 1, this equation can never hold. Hence, straight convertibles cannot achieve
the first best.


Proof of Proposition 4

     We have
                                                                    ∞
             Π M + Π I = θk Φ( y1 , x, k ) + ∫                          [ y − ( K − k )] d Φ( y, x, k ) − k − c( x ) .
                                                                   y1


Since this objective function has nothing to do with τ and q, we choose ( x ∗ , k ∗` ) to maxi-
mize it without the IC condition. We can then find τ and q to satisfy the IC condition.
The IC condition is
                                          q/τ                                                                   ∞
   (1 − τ )θk Φ x ( y1 , x, k ) + ∫             ( y − q)d Φ x ( y , x, k ) + (1 − τ ) ∫                              yd Φ x ( y , x, k ) = c ′( x ). (20)
                                      q                                                                        q/τ




                                                                                                                                                  Page 35
Hence, the problem now is to find a pair (τ , q) with τ ∈ [ 0, 1] and q ∈ [ 0, y2 ] that satisfies
the above equation for ( x ∗ , k ∗ ) . Condition (20) can be changed to
                                                              q/τ                                            ∞
        c ′( x ) = (1 − τ )θk Φ x ( y1 , x, k ) + ∫                 yd Φ x ( y, x, k ) + (1 − τ ) ∫               yd Φ x ( y , x, k )
                                                          q                                                 q/τ

                 −q [Φ x ( q / τ , x, k ) − Φ x ( q, x, k ) ]
                                                              ∞                               ∞
               = (1 − τ )θk Φ x ( y1 , x, k ) + ∫                 yd Φ x ( y , x, k ) − τ ∫         yd Φ x ( y , x, k )
                                                          q                                   q/τ

                 −q [Φ x ( q / τ , x, k ) − Φ x ( q, x, k ) ]
                                                           ∞                                  ∞
               = (1 − τ )θk Φ x ( y1 , x, k ) − ∫                 Φ x ( y , x, k )dy + τ ∫          Φ x ( y, x, k )dy.
                                                          q                                  q/τ


By (4), this becomes
                                                     q                                 ∞
              (1 − τ ) θk Φ x ( y1 , x, k ) +
                                                ∫   y1
                                                         Φ x ( y , x, k )dy + τ ∫
                                                                                      q/τ
                                                                                            Φ x ( y , x, k )dy = 0.                        (21)

This is impossible to satisfy unless q < y1 . If the project is not liquidated at t = 1, it means
that output is larger than y1 . Hence, we always have y > q by the time when the manager
decides whether or not to call. This means that the manager will always be able to pay q
when it has to. Hence, bankruptcy can never happen when the convertible can be called.

     We now need to show that, for the first-best solution ( x ∗ , k ∗ ) , we can find a pair (τ , q)
with τ ∈ (0, 1) and q < y1 that satisfies condition (21). Let
                                                                      q                               ∞
           ψ( q, τ ) ≡ (1 − τ )θk Φ x ( y1 , x, k ) + ∫ Φ x ( y, x, k )dy + τ ∫                             Φ x ( y , x, k )dy.
                                                                     y1                               q/τ


By (4), we have
                                    ∞
                    ψ( q,1) = ∫         Φ x ( y , x, k )dy = −c ′( x ) < 0, for any q ≤ y1 .                                               (22)
                                   y1


By (10), we also have
                                                                             0
                           ψ(0, 0) = θk Φ x ( y1 , x, k ) + ∫ Φ x ( y , x, k )dy > 0.                                                      (23)
                                                                            y1


Hence, by (23) and the continuity of ψ( q, 0) in q, there is q∗ ∈ (0, y1 ) such that

                                                         ψ( q∗ , 0) > 0.                                                                   (24)

Given this q∗ , again by the continuity of ψ( q∗ , τ ) in τ , since ψ ( q∗ ,1) < 0 and ψ ( q∗ , 0) > 0,
there is τ ∗ ∈ (0, 1) such that

                                                         ψ( q∗ , τ ∗ ) = 0.

That is, we find a pair ( q∗ , τ ∗ ) ∈ (0, y1 )×(0, 1) such that the IC condition is satisfied. There-
fore, ( x ∗ , k ∗ , τ ∗ , q∗ ) is a solution of problem (1). This solution is efficient.



                                                                                                                                        Page 36
Proof of Proposition 5

    We have
                                                               ∞
               Π M + Π I = θk Φ( y1 , x, k ) + ∫                   [ y − ( K − k )] d Φ( y, x, k ) − k − c( x ) .
                                                              y1


Since this objective function has nothing to do with τ , q and y , we choose ( x ∗ , k ∗` ) to
maximize it without the IC condition. We can then find τ , q and y to satisfy the IC con-
dition. The IC condition is
                         ∂Π M                                    y
                0=            = (1 − τ )θk Φ x ( y1 , x, k ) + ∫ [ y − (1 + r ) K ] d Φ x ( y , x, k )
                          ∂x                                    y2
                                                                                                                                                    (25)
                                           ∞
                                    +∫         (1 − τ ) yd Φ x ( y , x , k ) − c ′( x ).
                                           y


Then,
                                                                                                              y
         0 = (1 − τ )θk Φ x ( y1 , x, k ) − y2 [Φ x ( y , x, k ) − Φ x ( y2 , x, k ) ] + ∫ yd Φ x ( y , x, k )
                                                                                                             y2
                               ∞
              +(1 − τ ) ∫          yd Φ x ( y , x, k ) − c ′( x )
                              y
                                                                                                             ∞
            = (1 − τ )θk Φ x ( y1 , x, k ) − y2 [Φ x ( y , x, k ) − Φ x ( y2 , x, k ) ] + ∫                            yd Φ x ( y, x, k )
                                                                                                             y2
                      ∞
              −τ ∫         yd Φ x ( y , x, k ) − c ′( x ).
                     y


We can then solve for τ :
                                                                                                  ∞
              θk Φ x ( y1 , x, k ) + y2 [Φ x ( y2 , x, k ) − Φ x ( y , x, k ) ] + ∫                    yd Φ x ( y , x, k ) − c ′( x )
        ∗
     τ =
                                                                                                 y2
                                                                           ∞
                                                                                                                                             .
                                               θk Φ x ( y1 , x, k ) + ∫        yd Φ x ( y , x, k )
                                                                          y


By (5), we have
                                     ∞                                                             ∞
                 c ′( x ) = −∫           Φ x ( y , x, k ) dy = y1Φ x ( y1 , x, k ) + ∫                 y d Φ x ( y , x, k ).
                                    y1                                                            y1


Hence,
                                                                                                                  y2
              −( K − k ) Φ x ( y1 , x, k ) + y2 [Φ x ( y2 , x, k ) − Φ x ( y , x, k ) ] − ∫                            yd Φ x ( y , x, k )
        ∗
     τ =
                                                                                                             y1
                                                                           ∞
                                                                                                                                             .      (26)
                                               θk Φ x ( y1 , x, k ) + ∫        yd Φ x ( y , x, k )
                                                                          y


We can also write τ ∗ as
                                                                                          y2
                                  θk Φ x ( y1 , x, k ) − y2 Φ x ( y , x, k ) + ∫               Φ x ( y , x, k )dy
                         τ∗ =
                                                                                      y1
                                                                                      ∞
                                                                                                                          .                         (27)
                                   θk Φ x ( y1 , x, k ) − y Φ x ( y , x, k ) − ∫               Φ x ( y, x, k )dy
                                                                                      y




                                                                                                                                                 Page 37
We need to show that τ ∗ ∈ (0, 1) .

    For y ∈ [ y2 , y3 ] , first, by (11), we have
                                  y2                                     y2

                             ∫   y1
                                           yd Φ x ( y , x, k ) = ∫
                                                                        y1
                                                                              yφx ( y , x, k ) dy ≤ 0.

Also by (11), we have

                                              Φ x ( y2 , x, k ) − Φ x ( y , x, k ) ≥ 0.

Hence, the numerator of the expression in (26) is positive. Second, by Assumption 1, by
comparing the three terms in the numerator of (27) with the corresponding three terms in
the denominator, it is obvious that the numerator of the expression in (27) is less than the
denominator of the expression. Hence, τ ∗ ∈ (0, 1) .


Proof of Proposition 6

    We have
                                                                 ∞
             Π M + Π I = θk Φ( y1 , x, k ) + ∫                       [ y − ( K − k )] d Φ( y, x, k ) − k − c( x ) .
                                                             y1


Since this objective function has nothing to do with τ , q, τ and y , we choose ( x ∗ , k ∗` ) to
maximize it without the IC condition. We can then find τ , q, τ and y to satisfy the IC
condition. The IC condition is
               ∂Π M                                    y3
         0=         = (1 − τ )θk Φ x ( y1 , x, k ) + ∫ (1 − τ ) yd Φ x ( y , x, k ) − c ′( x )
                ∂x                                    y1
                                                                                                                                      (28)
                                      y3                                                   ∞
                          + ∫ [ y − (1 + r ) K ] d Φ x ( y, x, k ) + ∫                          (1 − τ ) yd Φ x ( y , x, k ).
                                  y3                                                       y3


In an extreme case with τ = τ , using (4), condition (28) becomes
                                                             ∞
    0 = (1 − τ )θk Φ x ( y1 , x, k ) + (1 − τ ) ∫                 yd Φ x ( y , x, k ) − c ′( x )
                                                            y1
                                                                                                  ∞
      = (1 − τ )θk Φ x ( y1 , x, k ) − (1 − τ ) y1Φ x ( y1 , x, k ) − (1 − τ ) ∫                      Φ x ( y, x, k )d y − c ′( x )
                                                                                                 y1

      = (1 − τ )θk Φ x ( y1 , x, k ) − (1 − τ ) y1Φ x ( y1 , x, k ) − τ c ′( x )
      = −(1 − τ ) ( K − k ) Φ x ( y1 , x, k ) − τ c ′( x ).

The above equation implies a solution for τ :
                                                    ( K − k ∗ ) Φ x ( y1∗ , x ∗ , k ∗ )
                                      τ=
                                      ˆ                                                         .                                     (29)
                                               ( K − k ∗ ) Φ x ( y1∗ , x ∗ , k ∗ ) − c ′( x ∗ )

We have τ ∈ (0, 1) . By (28), taking a total differentiation for (τ , τ ) yields
        ˆ




                                                                                                                                 Page 38
                                                  y3                                              y3
         0 = −θk Φ x ( y1 , x, k )d τ − ∫                  yd Φ x ( y, x, k )d τ − (1 − τ ) y3φx ( y3 , x, k )
                                                                                                     dτ
                                            y1                                                    τ
                                                    y                                           y
               − [ y3 − (1 + r ) K ]φx ( y3 , x, k ) 3 d τ + [ y3 − (1 + r ) K ]φx ( y3 , x, k ) 3 d τ
                                                     τ                                           τ
                     ∞                                                  y3
               − ∫ yd Φ x ( y , x, k )d τ + (1 − τ ) y3φx ( y3 , x, k ) d τ
                    y3                                                  τ
                                                  y3                                         ∞
            = −θk Φ x ( y1 , x, k )d τ − ∫                 yd Φ x ( y , x, k )d τ − ∫             yd Φ x ( y , x, k )d τ ,
                                                 y1                                          y3


implying
                                                                              ∞
                                            θk Φ x ( y1 , x, k ) + ∫              yd Φ x ( y , x, k )
                                dτ
                                   =
                                                                             y3
                                                                                                          .
                                dτ                         −∫
                                                                  y3
                                                                       yd Φ x ( y , x, k )
                                                                 y1


By the FOC of the objective function on x, we have
                                                       ∞
                      θk Φ x ( y1 , x, k ) + ∫              [ y − ( K − k )] d Φ x ( y , x, k ) = c ′( x ),
                                                      y1


implying
                                        ∞
            θk Φ x ( y1 , x, k ) + ∫        yd Φ x ( y , x, k ) = −( K − k ) Φ x ( y1 , x, k ) + c ′( x ) > 0.
                                       y1


Hence, by (12), when τ = τ = τ
                             ˆ,
                                                                              ∞
                                            θk Φ x ( y1 , x, k ) + ∫              yd Φ x ( y , x, k )
                          dτ
                                       =                                                                  > 1.
                                                                             y3

                          dτ
                                                                  y3
                                τ =τ                       −∫          yd Φ x ( y , x, k )
                                                                y1


This means that the curve ϕ( τ , τ ) = 0 defined by (28) must cut the 45o line at τ = τ im-
                                                                                      ˆ,
plying that one part of the curve is in the zone with τ > τ. Hence, there must be a pair
(τ ∗ , τ ∗ ) with τ ∗ > τ ∗   and τ ∗ , τ ∗ ∈ (0, 1) that satisfies (28).

                                       τ


                                       1




                                                               ..(τˆ, τˆ )
                                                (τ ∗ , τ ∗ )


                                                  ϕ(τ , τ ) = 0

                                            0                                       1                 τ

                               Figure 12. Existence of (τ ∗ , τ ∗ ) with τ ∗ > τ ∗

                                                                                                                             Page 39
The exact values of τ ∗ and τ ∗ are determined by the bargaining power of the two parties. By
choosing this pair of conversion ratios, the manager and the investor will have enough
incentives to provide the first-best investments.


Proof of Proposition 7

    We have
                                                              ∞
             Π M + Π I = θk Φ( y1 , x, k ) + ∫                    [ y − ( K − k )] d Φ( y, x, k ) − k − c( x ) .
                                                          y1


Since this objective function has nothing to do with τ , q, τ and y , we choose ( x ∗ , k ∗` ) to
maximize it without the IC condition. We can then find τ , q, τ and y to satisfy the IC
condition. The IC condition is
               ∂Π M                                             y3
         0=         = (1 − τ )θk Φ x ( y1 , x, k ) + (1 − τ ) ∫ yd Φ x ( y , x, k ) − c ′( x )
                ∂x                                             y2
                                                                                                                                         (30)
                                    y3                                                     ∞
                             + ∫ [ y − (1 + r ) K ] d Φ x ( y, x, k ) + ∫                       (1 − τ ) yd Φ x ( y , x, k ).
                                   y3                                                      y3


In an extreme case with τ = τ , using (4), condition (30) becomes
                                                          ∞
    0 = (1 − τ )θk Φ x ( y1 , x, k ) + (1 − τ ) ∫               yd Φ x ( y , x, k ) − c ′( x )
                                                         y2
                                                                                                   ∞
      = (1 − τ )θk Φ x ( y1 , x, k ) − (1 − τ ) y2Φ x ( y2 , x, k ) − (1 − τ ) ∫                        Φ x ( y , x, k )d y − c ′( x )
                                                                                                  y2
                                                                                                   y2
      = (1 − τ )θk Φ x ( y1 , x, k ) − (1 − τ ) y2Φ x ( y2 , x, k ) + (1 − τ ) ∫                        Φ x ( y , x, k )d y
                                                                                                  y1
                         ∞
         − (1 − τ ) ∫        Φ x ( y , x , k ) d y − c ′( x )
                        y1

                  ⎡                                              y2                   ⎤
      = (1 − τ ) ⎢θk Φ x ( y1 , x, k ) − y2Φ x ( y2 , x, k ) + ∫ Φ x ( y , x, k )d y ⎥ − τ c ′( x )
                 ⎢⎣                                             y1                   ⎥⎦
                  ⎡                                    y2              ⎤
      = (1 − τ ) ⎢−( K − k ) Φ x ( y1 , x, k ) − ∫ yd Φ x ( y , x, k )⎥ − τ c ′( x ).
                  ⎢⎣                                  y1               ⎥⎦

The above equation implies a solution for τ :
                                                                                 y2
                                   ( K − k ∗ ) Φ x ( y1∗ , x∗ , k ∗ ) + ∫             yd Φ x ( y , x, k )
                   τ=
                                                                                y1
                   ˆ                                                     y2
                                                                                                                   .                     (31)
                             ( K − k ∗ ) Φ x ( y1∗ , x ∗ , k ∗ ) + ∫          yd Φ x ( y , x, k ) − c ′( x ∗ )
                                                                        y1


By (13), we have τ ∈ (0, 1) . By (30), taking a total differentiation for (τ , τ ) yields
                 ˆ




                                                                                                                                    Page 40
                                                 y3                                              y3
        0 = −θk Φ x ( y1 , x, k )d τ − ∫                  yd Φ x ( y, x, k )d τ − (1 − τ ) y3φx ( y3 , x, k )
                                                                                                    dτ
                                           y2                                                    τ
                                                   y                                           y
              − [ y3 − (1 + r ) K ]φx ( y3 , x, k ) 3 d τ + [ y3 − (1 + r ) K ]φx ( y3 , x, k ) 3 d τ
                                                    τ                                           τ
                    ∞                                                  y3
              − ∫ yd Φ x ( y , x, k )d τ + (1 − τ ) y3φx ( y3 , x, k ) d τ
                   y3                                                  τ
                                                 y3                                        ∞
           = −θk Φ x ( y1 , x, k )d τ − ∫                 yd Φ x ( y , x, k )d τ − ∫            yd Φ x ( y , x, k )d τ ,
                                                y2                                         y3


implying
                                                                            ∞
                                           θk Φ x ( y1 , x, k ) + ∫             yd Φ x ( y , x, k )
                              dτ
                                 =
                                                                           y3
                                                                                                      .
                              dτ                          −∫
                                                                y3
                                                                     yd Φ x ( y , x, k )
                                                               y2


The FOC of the objective function on x is
                                                      ∞
                     θk Φ x ( y1 , x, k ) + ∫              [ y − ( K − k )] d Φ x ( y , x, k ) = c ′( x ),
                                                     y1


which implies
                                       ∞
           θk Φ x ( y1 , x, k ) + ∫        yd Φ x ( y , x, k ) = −( K − k ) Φ x ( y1 , x, k ) + c ′( x ) > 0.
                                      y1


Then, by (13),
                                                 ∞
                    θk Φ x ( y1 , x, k ) + ∫              yd Φ x ( y , x, k )
                                                y2
                                                 ∞                                    y2
                 = θk Φ x ( y1 , x, k ) + ∫               yd Φ x ( y , x, k ) − ∫          yd Φ x ( y , x, k ) > 0.
                                                y1                                   y1


Hence, using (13) again, when τ = τ ,
                                                                            ∞
                                           θk Φ x ( y1 , x, k ) + ∫             yd Φ x ( y , x, k )
                         dτ
                                      =                                                                > 1.
                                                                           y3

                         dτ
                                                                y3
                              τ =τ                        −∫         yd Φ x ( y , x, k )
                                                               y2


This means that the curve ϕ( τ , τ ) = 0 defined by (30) must cut the 45o line at τ = τ as
                                                                                      ˆ,
shown in Figure 12. Hence, there must be a pair (τ ∗ , τ ∗ ) with τ ∗ > τ ∗ and τ ∗ , τ ∗ ∈ (0, 1)
that satisfies (30). The exact values of τ ∗ and τ ∗ are determined by the bargaining power of
the two parties. Hence, by choosing these conversion ratios, the manager and the investor
will have enough incentives to provide the first-best investments.




                                                                                                                           Page 41
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