Continuous Time, Closed Form Solution
for the Case Without Learning and π(t) Non-Constant
(Addendum to: ”Valuation and Return Dynamics of New Ventures” by
Berk, Green and Naik, forthcoming Review of Financial Studies)
When the success intensity is not constant but a known function of the
stage alone, π(t) = πn , the Bellman-Hamiliton-Jacobi equation becomes
1 2 2 ∂2 n
∂
x σ
V (x) + µ̂x V n (x) − r̂V n (x)
2
∂x2
∂x h
i
+ u∗n (x) πn V n+1 (x) − V n (x) − a − bx = 0
where u∗n (x) denotes the optimal investment rule. This recursive system
of ordinary differential equations is solved by starting with the terminal
value and working backwards. At each stage the solution involves several
constant parameters, and a number of constants for stage n that are defined
recursively given the constants obtained solving the previous stage and the
boundary conditions in Corollary 2 in the main paper. The next proposition
states this solution.
Proposition 1 When the success intensity at each stage is known to be πn
the value of the R&D project, Vn (x), is
( P
N −1
Vn (x) =
i=n
An xβ
Fni xγi + Bn x + Gn
x ≥ x∗n
x < x∗n
(1)
where,
β =
γn =
Gn =
Bn =
Fni =
(σ 2 − 2µ̂) +
p
8r̂σ 2 + (σ 2 − 2µ̂)2
2σ 2
p
(σ 2 − 2µ̂) − 8(r̂ + πn )σ 2 + (σ 2 − 2µ̂)2
2σ 2
Gn+1 − πan
1+
r̂
πn
b
πn
r̂−µ̂
πn
Bn+1 −
1+

0






i=N
πn
i
πn −πi Fn+1






(x∗n )−γn
πn
n<i<N
(r̂ − µ̂)Bn x∗n + r̂Gn −
1
PN −1
i=n+1 πi
(x∗n )γi Fni
i=n

An =

N
−1
X
(x∗n )−β 
∗
(πn + (r̂ − µ̂)γn )Bn xn + r̂γn Gn +
(γi πn − γn πi ) (x∗n )γi Fni  .
βπn
i=n+1
and x∗n solves the following equation:

1 −
1
+
β
(r̂ − µ̂) 1 −
πn
γn
β


 Bn x∗ + 1 +
n
N
−1
X
r̂ 1 −
πn

γ
 i +
=
β
i=n+1
πi 1 −
πn
γn
β
γn
β

 Gn

− 1 (x∗n )γi Fni .
The constants are solved for recursively, subject to the following two boundary conditions:
GN
= 0
BN
=
1
.
r̂ − µ̂
The proof of this proposition follows exactly the same logic as the proof of
Proposition 7 in the main paper.
2