1. Unicorn Valuation
Jake Edison
December 22, 2015
1 Introduction
We assume a venture capital fund invests in the startup at time t = 0 and
that the firm is sold at time τ.
• V0 = initial nominal value of the firm. This is the value reported by
the press and V0 ≥ $1b is the condition required for the startup to be
labeled a “unicorn”.
• W = dollars invested in startup by venture capital fund.
• p ∈ (0, 1) is the percentage of the startup purchased by the venture
capital fund. It satisfies
W = pV0
so that p is the post investment percentage of the firm owned by the
fund. (p or equivalently N0 is negotiated and the entrepreneur is keen
to get a high value of N0 but of course given the other negotiating terms
and in particular the participation rate, it can be very misleading.)
• x = participation factor. This means that the venture fund will receive
xW upon liquidation of the fund before any other investors (including
the founders) get paid.
• Vt = true time t value of startup.
The firm is liquidated at time τ and the payoff, Pτ , to the venture capital
fund is given by
Pτ = min(xW, Vτ ) + p max(0, Vτ − xW) (1)
1

2. We assume Vτ is observed at time τ because the firm is sold / liquidated
for that value then. The first term on the rhs of (1) reflects the participation
rate whereby the venture fund gets paid before everyone else. The second
term on the rhs reflects the payoff to the fund in proportion to their owner-
ship, p, of the firm. The venture fund pays W for a time τ payoff given by
(1). From their perspective we therefore have
W = Value (Pτ )
= E0 e−rτ
(min(xW, Vτ ) + p max(0, Vτ − xW)) (2)
where r is an appropriate risk-adjusted discount rate and the expectation is
taken with respect to the true probability measure. By the same token the
value of the portfolio held by the founders is given by:
E0 e−rτ
(Vτ − min(xW, Vτ ) − p max(0, Vτ − xW)) (3)
While a risk-adjusted interest rate may be used for a single case it should
be noted that as the company is assumed to not be publicly traded our mar-
ket is not complete. Thus standard risk neutral pricing techniques are not
appropriate here and we must define a stochastic discount factor instead. Al-
ternatively, in adopting utility functions for our market participants we may
fix a single discount rate as the utility function itself defines the stochastic
discount factor.
In this paper we consider the exponential utility function defined below:
Ut(x) = −e−αer(T −t)x
(4)
Notice that instead of the discounting our cashflows we are instead pro-
jecting all cashflows to time T by means of the term er(T−t)
. To find the price
of the an asset in this market at time t we simply find the number P(t, Vt)
such that:
Ut(P(t, Vt)) = Et [Uτ (min(xW, Vτ ) + p max(0, Vτ − xW))] (5)
P(t, Vt) can be interpreted as the amount of money the investor would
require risk-free today to be indifferent to having the future risks payoff of the
portfolio. For this reason P is often referred to as the certainty equivalence
of the portfolio.
2

3. Recall that as the investors have already agreed to give the founders $
W the certainty equivalence of their investment must be W. Using this fact,
assuming a distribution for VT given V0, we can then solve for V0 numerically
by the initial valuation such that:
U0(W) = E0 [Uτ (min(xW, Vτ ) + p max(0, Vτ − xW))] (6)
2 Discrete Time Case
In discrete time we assume that Vt follows a standard binomial lattice model
such that:
Vt = V0uxt
dt−xt
(7)
xt ∼ Bin(t, p) (8)
p =
eµdt
− d
u − d
(9)
u =
1
d
= eσ
√
dt
(10)
dt =
T
N
(11)
(12)
Moreover, we impose that the firm can only be liquidated for:
τmin ≤ τ ≤ T (13)
Assuming the party determining liquidation time is perfectly rational
pricing the investor’s portfolio under these conditions is nearly equivalent
to standard American option pricing. If the investors have full control over
liquidation time then the only caveat that we must account for is the utility
function. Let:
Ut(P(t, Vt)) = Et [Uτ (Vτ − min(xW, Vτ ) − p max(0, Vτ − xW)))] (14)
= Et [Ut+1(P(t + 1, Vt+1))] (15)
3

4. where P(t, Vt) is by definition the certainty equivalence of the portfolio
at time t and current spot Vt. As the investors can liquidate prior to expiry
the following recursion arises:
Ut(P(t, Vt)) = max( Et [Ut+1(P(t + 1, Vt+1))] , (16)
Ut(Vt − min(xW, Vt) − p max(0, Vt − xW)) ) (17)
Thus the investors liquidate when the time adjusted utility of the intrinsic
portfolio value today is greater than the expected future utility of waiting
to liquidate. Solving repeatedly for P(t, V ) yields the corresponding price
surface.
If instead the founders are in control we need only to apply the same
recursion to their portfolio to obtain their liquidation decisions. We then
implement these decisions while pricing the founder’s portfolio to obtain the
price under these conditions.
3 Continuous Time Case
Let us now assume that Vt follows a geometric brownian motion and that:
dVt = µVtdt + σVtdwt (18)
As in discrete time we let:
Ut(P(t, Vt)) = Et [Uτ (Vτ − min(xW, Vτ ) − ρ max(0, Vτ − xW)))] (19)
(20)
By the tower property we have:
Ut(P(t, Vt)) = Et [Uτ (Vτ − min(xW, Vτ ) − p max(0, Vτ − xW)))] (21)
= Et [Es [Uτ (Vτ − min(xW, Vτ ) − p max(0, Vτ − xW))]] (22)
= Et [Us(P(s, Vs))] (23)
Now letting:
f(t, Vt) = Ut(P(t, Vt)) (24)
f(T, V ) = UT (min(xW, V ) + p max(0, V − xW)) (25)
4

5. this expression can be written as:
f(t, Vt) = Et [Es [f(τ, Vτ )]] = Et [f(s, Vs)] (26)
(27)
The martingale property yields the following PDE:
µxfx +
σ2
x2
2
fxx + ft = 0 (28)
(29)
A finite difference scheme can now be utilized to solve for f. As in discrete
time, if the investors are in control we add the following step during each
iteration of the finite difference method:
f(t, V ) = max(f(t, V ), Ut(min(xW, V ) + ρ max(0, V − xW)) (30)
Where f(t, V ) is initialized via the difference equation at time t and the
above line is executed for 0 ≤ V ≤ Vmax prior to calculating the solution for
the next time step.
Alternatively, if the founders are in control we once again price their port-
folio, ffounders, first to obtain the liquidation decisions and instead include
the step:
if ffounders(t, V ) == Ufounders
t (Vt − min(xW, V ) − ρ max(0, V − xW) : (31)
f(t, V ) = Uinvestors
t (min(xW, V ) + ρ max(0, V − xW)(32)
In either case we simply take U−1
t (f(t, V )) at the end to obtain the price
surface. Here I have labeled the respective utility functions of the founders
and investors to account for the possibility of using different risk aversion
parameters α for the different participants.
For implementation we utilized a fully implicit scheme with non-uniform
spacing. As you’ll see in the price surfaces in later sections, certain sections
of the (t, V ) grid require much finer resolution to obtain accurate results. For
boundary conditions we assumed:
fxx(t, 0) = fxx(t, Vmax) = 0 (33)
This reflects the idea the expected utility should converge to −1 for Vt
close to 0 and that the second derivative should be very small for large values
of Vt.
5

6. 4 Exercise Patterns
The liquidation decisions of either party follow the same usual pattern. Both
parties have respective exercise barriers, above which they will liquidate their
position. As the parties have different portfolios and possibly different risk
aversion parameters these barriers generally vary widely depending on the
parameters chosen.
For now assume:
x = 1 (34)
W = 100 (35)
r = .01 (36)
σ = .1 (37)
αfounders = αinvestors = .01 (38)
τmin = 0 (39)
T = 10 (40)
(41)
Heuristically, if µ is small then as a founder you’d likely choose to close
your position in the firm as soon as possible. On the other hand, notice that if
the firm is sold for less than W then you receive nothing. Thus we’d expect
that even with a small µ that the founders would be much more inclined
to hold out for a higher valuation before liquidating. Below is the exercise
barrier for the founders with µ = −.05.
6

7. The corresponding graph for the investors is entirely blank since they
would choose to liquidate at any price. This reflects the fact that the investors
are heavily protected on the downside whereas the founders generally receive
nothing in these cases. Raising µ to 0 we can see that the founders are now
more greedy with their sale price.
7

8. As µ surpasses r = .01 one can observe that the investors now have a non
trivial exercise barrier.
8

9. Increasing µ further predictably raises the barrier for both parties.
9

10. 5 The Price Surface
Once again assume the following parameters:
x = 1 (42)
W = 100 (43)
r = .01 (44)
σ = .1 (45)
αfounders = αinvestors = .01 (46)
τmin = 0 (47)
T = 10 (48)
(49)
The price surface then follows two general shapes. With negative µ and
the founders in control we observe a sharp jump in the surface along the
founder exercise barrier.
This characteristic is the result of the price surface being generated with
the founder’s exercise decisions. Typically we observe that American option
prices transition smoothly as the exercise region is entered. Here, however,
10

11. since the exercise decisions are stemming from an entirely different option
we observe the price transition sharply. This affect is amplified by the fact
that our utility functions makes the investors risk adverse.Thus they value
a certain payoff today much higher than an future risky payoff despite the
fact that their expected payoffs, without considering a utility function, are
nearly the same.
The other main shape shape we observe is a surface that closely follows
the intrinsic value. Here we see that when given control in this scenario,
the investors liquidate in any state. As we saw when evaluating the exercise
patterns the surface becomes more interesting in this case when µ > r.
Below is the case with µ = .05 and the founders in control
11

12. Above is the price surface when the founders have control and below is
the case when investors have control. When the investors have control notice
some subtle changes in our surface. As illustrated by the exercise boundaries
in the previous section we can see the the latter surface has an observable
kink as Vt crosses the investor’s much lower threshold for liquidation.
12

13. Apart from this nuance, however, it is important to note that as |µ|
gets large, from a root finding perspective the party in control is irrelevant.
If the company in question is obviously failing then all parties agree that
they should liquidate as soon as possible with the caveat that founders will
always refuse to sell for a price less than W. Observe that as µ becomes more
negative the portion of the surface outside of the exercise region approaches
zero. Thus the price becomes P(0, V ) = IV ≥W W + ρ max(V − W, 0) and
then clearly V = W is the solution. When the investors have control in such
cases we already observed that the price surface is exactly the intrinsic value
at all points. Thus once again V0 = 100.
13

14. Similarly, if the company is obviously growing then both parties are happy
to let their investments grow instead of liquidating now. In this case as µ
14

15. gets large τ −→ τmax and the portfolio prices are perfectly equivalent.
6 Value of Control
One question of interest is what is the value to the investor of having control
over the liquidation time. Numerically this can be determined by solving for
V0 in the case when the founders have control and comparing it to the value
obtained by letting the investors have control. Let us then define the value
of control to be the percentage change in V0 when we transfer control to the
investors. That is:
Vcontrol =
V founders
0 − V investors
0
V investors
0
(50)
Normalizing by V investors
0 is done so we can compare Vcontrol across a
range of parameters. It should be clear from this definition that Vcontrol is
nonnegative since the investors must value one startup higher than another
if they are willing to invest the same amount of money in the first without
the added advantage of control over liquidation time.
Unless otherwise stated the following graphics were produced with:
σ = .1 (51)
r = .01 (52)
τmin = 0 (53)
T = 10 (54)
N = 100 (55)
W = 100 (56)
x = 1 (57)
α = .01; (58)
Below observe Vcontrol as we vary µ and τmin. Notice that imposing τmin >
0 will merely truncate the exercise barriers previously observed. In short,
after τmin has passed both parties will follow the same liquidation strategies
as shown in prior sections. Intuitively we would then expect that as τmin → T
Vcontrol → 0 since the region in which the party in control can make decisions
15

16. is being made smaller and smaller. As discussed at the end of the Price
Surface section, as |µ| → ∞ Vcontrol → 0 which can be seen in the surface
below.
In the surface below notice that as σ increases the inflection point for µ
begins to increase. This reflects the idea our risk adverse investors desire a
higher rate of return for a higher rate of volatility. As before, as µ moves
away from this point on either side Vcontrol begins to decrease.
16

17. Now we consider what happens if we vary αinvestors
αfounders
with µ. In previous
cases this ratio was set to 1 for simplicity. Here we see that as the ratio
increases (keeping αfound = .01), Vcontrol increases since the investors are now
more willing to liquidate at a low price to avoid risk.
17

18. Finally, we examine the affect of varying the initial investment size.
Much as when we increased α, increasing W causes the inflection point
for µ to increase reflecting that our investors want a higher return to risk
more money. We also observe that Vcontrol seems to be generally decreas-
ing in W. This is a result of the higher initial investment implying a
higher V0. As V0 is pushed above both parties’ exercise barriers we expect
to see Vcontrol diminish as everyone agrees that they should liquidate now.
18

19. 19