If I>I *, either the entrepreneur is able to invest I - I * and the second-best outcome is feasible, that is, the value of the project is V* defined above, or one must solve the general program adding the constraint
Ave + Af >I *. (A55)
Replace AF and Ave by their value in (PCve) and (PCf), set aE = 0(whichis obviously optimal when equation (A55) holds) and use the fact that constraint (17) is binding to get
Ave + Af = -(b - Y2 + b(Ru - Rd)Y + Rd, (A56)
where Ystands for aARu - aARd. The determinant A is
A = (RU b2Rd)2 - 2((Ave + Af) - Rd) . (A57)
The solution is readily computed and gives
Y = y(R -2Rd) -ybpA. (A58)
Replacing Y by its value, and using equation (17) to find the expression of af gives, for Ave + AF>I*
u* Ru d* Rd y(R -Rd)-ybVA
A R a A R - o . a :
Check that the value of the project is then strictly lower than v* defined in equation (A54). When the entrepreneur is forced to raise an amount of outside capital strictly larger than I*, the value of the project decreases. Put differently, if I>I*, the entrepreneurs financial participation increases the value of the project.
Proof of eorollary 1: Use Lemma 1 and the optimal contract derived in the proof of Proposition 3 to compute the optimal levels of effort when Ave + AF>I*.Note that A decreases with Ave + AF. It follows immediately that e decreases with Ave + AFand a increases with Ave + AF. □
Proof of Proposition 4: See that A, defined in the proof of Proposition 3, is positive if and only if
Avc + Af < Rd + 2b-R) = I *- (A60)
Hence the maximal amount of outside financing is Imax. Simple comparison with the maximal level of initial investment defined in Section I yields the result of Proposition 4. □
Proof of Proposition 5: I first derive the conditions under which the investor acquires common stocks and the entrepreneur gets preferred stocks.
Preferred stocks ensure a minimum rate of return (dividend) to their owner before common stocks returns are paid. When the outcome of the project is sufficiently high, both types of stocks give the same rate of return. Define R as the minimum dividend pledged on each preferred stock, multiplied by the number of preferred stocks. Let a be the fraction of preferred stocks in the firms equity. The fraction of common stocks is (1 - a). To be able to distinguish between preferred and common stocks, assume that aRd<R<Rd and R<aRu. Hence, when the income is low, it is impossible to remunerate common stocks with the same dividend as preferred stocks. When the income is high, both types of stocks generate the same dividend. Under these assumptions, the optimal contract can be implemented by giving common stocks to the investor and preferred stocks to the entrepreneur if and only if
(1 - aE )Rd = Rd - R; (A61)
(1 - aE )Ru = (1 - a)Ru, (A62)
a 2 RRu; Rd-i (A63)
R < Rd. (A64)
When Avc < I * (A61) and (A62) write
R = I * - Avc , (A65)
j(Ru - Rd)-Avc + I *
It is easy to check that (A64) is satisfied if and only if Avc > I * - Rd. Besides, (A63) is satisied ifand only if
I next turn to the case where the investor acquires convertible bonds or preferred stocks.
In this stylized model, issuing convertible bonds or preferred stocks generates the same pattern of return for their owner: The face value of the bond corresponds to a minimum dividend pledged before common shareholders are remunerated. When the projects income is high, bonds are converted, and the return they generate is similar to the return of preferred (or common) stocks. Differences between these two types of claim usually concern the right to trigger bankruptcy, which is irrelevant in this setting. Convertible bonds are characterized by a face value D, and a fraction 1 - a of the firms equity, such that if (1 - a)Ry < D(6e[d; u}), the investor gets min[D; Ry]; if (1 - a)Ry>D, the investor gets (1 - a)Ry.
To be able to distinguish between convertible bonds and common stocks, assume (1 - a)Rd < D < (1 - a)Ru.
Consider convertible bonds with D < Rd. Such a contract implies Ave < I*, since the investors revenue must be lower than (or equal to) Rd in state d.The contract must verify
1 - auE)Ru = 1 - a)Ru;
Rd D Ru D
D < Rd.
Replacing aEd and aEu by their values, (A68) and (A69) become
D = Ave - I * + Rd;
Ru - Rd)+Ave - I* + Rd
Condition (A70) implies Ave> Av*e. It follows that issuing convertible bonds (as structured above) is possible if and only if Avee]Av*e,I*]. By the same reasoning, one can show that convertible bonds with D> Rd can be issued when
Ave> I*. □
Admati, Anat, and Paul Pfleiderer, 1994, Robust financial contracting and the role of venture capitalists, Journal of Finance 49, 371-402.
Allen, Franklin, and Andrew Winton, 1995, Corporate financial structure, incentives, and optimal contracting, in Robert A. Jarrow,Vojislav Maksimovic, William Ziemba, eds.: Handbooks in Operations Research & Management Science Vol. 9 (Elsevier Science, Amsterdam).
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