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Proof of Corollary 1: Use Lemma 1 and the optimal contract derived in the proof of
Proposition 3 to compute the optimal levels of e¡ort when AVC þ AF4I n. Note
that D decreases with AVC þ AF. It follows immediately that e decreases with
AVC þ AF and a increases with AVC þ AF . &
2082 The Journal of Finance

Proof of Proposition 4: See that D, de¢ned in the proof of Proposition 3, is posi-
tive if and only if
g °Ru À Rd Þ2
d
AVC þ AF Rþ  Imax : °A60Þ
2b 2g À b

Hence the maximal amount of outside ¢nancing is Imax. Simple comparison
with the maximal level of initial investment de¢ned in Section I yields the result
of Proposition 4. &

Proof of Proposition 5: I ¢rst derive the conditions under which the investor ac-
quires 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 su„-
ciently high, both types of stocks give the same rate of return. De¢ne 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 ¢rm™s equity.
The fraction of common stocks is (1 À a). To be able to distinguish between pre-
ferred and common stocks, assume that aRdoR Rd and RoaRu. 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 gen-
erate 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 À ad ÞRd ¼ Rd À R; °A61Þ
E

°1 À au ÞRu ¼ °1 À aÞRu ; °A62Þ
E

RR
a 2Š ; Š; °A63Þ
Ru Rd

Rd : °A64Þ
R

When AVC I n, (A61) and (A62) write
R ¼ I n À AVC ; °A65Þ
g u
À Rd Þ À AVC þ I n
bþg°R
a¼ °A66Þ
:
Ru

It is easy to check that (A64) is satis¢ed if and only if AVC ! I n À Rd. Besides,
(A63) is satis¢ed if and only if

g
Rd  AVC :
n
In À °A67Þ
AVC
bþg
Financing and Advising 2083

I next turn to the case where the investor acquires convertible bonds or pre-
ferred 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 corre-
sponds to a minimum dividend pledged before common shareholders are remun-
erated. When the project™s income is high, bonds are converted, and the return
they generate is similar to the return of preferred (or common) stocks. Di¡er-
ences between these two types of claim usually concern the right to trigger bank-
ruptcy, which is irrelevant in this setting. Convertible bonds are characterized by
a face value D, and a fraction 1 À a of the ¢rm™s equity, such that if
(1 À a)Ry D(yA{d; u}), the investor gets min[D; Ry]; if (1 À a)Ry4D, the investor
gets (1 À a)Ry.
To be able to distinguish between convertible bonds and common stocks, as-
sume (1 À a)RdoDo(1 À a)Ru.
Consider convertible bonds with D Rd. Such a contract implies AVC I n,
since the investor™s revenue must be lower than (or equal to) Rd in state d. The
contract must verify
°1 À ad ÞRd ¼ D; °A68Þ
E


°1 À au ÞRu ¼ °1 À aÞRu ; °A69Þ
E

d 
R À D Ru À D
a2 °A70Þ
; ;
Rd Ru

Rd : °A71Þ
D

d u
Replacing aE and aE by their values, (A68) and (A69) become
D ¼ AVC À I n þ Rd ; °A72Þ
 
1 b
°Ru À Rd Þ þ AVC À I n þ Rd :
1Àa¼ u °A73Þ
R bþg

Condition (A70) implies AVC4AVC. It follows that issuing convertible bonds (as
n

structured above) is possible if and only if AVCA]AVC,I n]. By the same reasoning,
n

one can show that convertible bonds with D4Rd can be issued when
AVC4I n. &


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