Today’s Lecture
Rules of Thumb
Firm Valuation
TIP
If you do not understand
something,
ask me!
Rules of Thumb
We will look at two typically used rules of
thumb
Hurdle rates
Profitability Index
2
Fixing an arbitrary threshold
Consider the option to invest in a project
with no operating cost that costs I.
Assume the decision to invest is made
when the value of the project hits VA
Then the value of the option is
V
W (V ,VA ) (VA I )
VA
3
Effect of Mistakes
Since the shape of the curve is flat near
the optimal, small mistakes do not have
much effect.
Investing too early is more costly than
investing too late.
4
Hurdle Rate Rule
This rule says invest whenever the NPV
of the project is positive for discount rate
γ.
5
Profitability Index
Invest whenever the ratio of the NPV of
the project over the initial investment is
greater than Π
6
The Idea
Model the firm as
Collection of ongoing projects
Options to invest in new projects --- Growth
Stochastic Interest Rate
Vasicek model
7
The Firm
Consists of a continuously evolving portfolio of
projects.
Thus, the risk of the firm as a whole evolves as
projects evolve.
Projects require an initial investment, I, and
provide cash flows for a random amount of
time.
Projects disappear randomly and independently
of everything in the economy
8
Project Cashflow (while
alive)
1 2 ( t )
C2 i
Ci (t ) Ie i i
IID series of cashflows
It can be thought of as the optimal
investment level (that results from an
unmodeled optimization problem).
9
Expected Cash Flow of the
Firm
Let j(t) be an indicator function describing
whether the project taken on at time j is alive at
time t, then the expected cash flow of the firm as
a whole is
t t
E C (t 1) (t 1) e
t i i
C
I (t )
i
i 1 i 1
e b(t )
C
b(t) has a natural interpretation as the book
value of assets.
10
Growth
Each period the firm has an option to
undertake a one time investment in a
project (i.e., invest I and receive the
cashflow C(t) so long as the project is
alive.)
The firm only under takes the investment
if it has positive NPV.
The firm effectively owns a series of
options.
11
Pricing
The price at date t of any cashflow C(T) is given
z (T )
by
Et C (T )
z (t )
where
1 2 ( t 1)
r (t ) 2 z
z(t 1) z(t )e z
Evolution of the short rate r(t) is given by the
Vasicek model. 12
Vasicek Model (Discrete
Time)
The evolution of the short rate is
r (t 1) r (t ) (1 )r r (t 1)
with
zr r z cov( (t ), (t ))
13
Risk of an Individual
Project
The risk of a project's cash flows is given by
its “beta:”
i i z cov( (t ), i (t ))
is assumed to be drawn from a
distribution F independent of everything
else in the model.
14
Value of the jth Ongoing
Project
z ( s)
V j (t ) Et C j (s) j (s)
s t 1 z (t )
15
Consider a Particular Term
z ( s)
Et C j (s) j ( s)
z (t )
C j
s t
Ie B( s t , r (t ))
16
Summing over all terms
gives
C j s t
V j (t ) Ie B( s t , r (t ))
s t 1
C j
Ie D(r (t ))
where D(r(t)) is the price of a declining
perpetuity.
17
So what is the value of all
ongoing projects?
To do this we need to keep track of which
projects are alive.
If the project that arrives at time t is taken on
we set t(t)=1 otherwise we set t(t)= 0
If the project at time t is alive at some later
time we set t()= 1 otherwise we set t()= 0
Then the value of all ongoing projects is
just the sum over all projects that are alive
18
Value of all ongoing projects
t
V t Ie
C j
D (r (t )) j (t )
j 0
t I j (t )
b(t )e D (r (t ))
C j
e
j 0 b(t )
C (t )
b(t )e D(r (t ))
where
t I j (t )
(t ) ln
j
e
j 0 b(t ) 19
What affects the value of
ongoing projects
Value changes as old projects die or new
projects are added
20
Let’s keep track of the
important assumptions
Each project has constant risk
Each project as constant expected
cashflow --- no growth
So where is the growth coming from
addition of new projects
why does this make sense?
Risk changes as projects die or are added
21
Value of Growth
Opportunities
The firm will therefore invest whenever the
NPV of the investment is positive:
Vt t I Ie C t
D(r (t )) I
I e
C t
D(r (t )) 1 0
The initial investment merely determines the
scale of the project.
Whether the NPV is positive is determined by
the project's riskiness or beta.
22
The Intuition
If we condition on beta, then these options are
simply bond options, i.e.,
the strike is the interest rate for which the invest
opportunity has zero NPV
it is in the money of lower interest rates and out of the
money for higher interest rates
These options can be priced explicitly in the
Vasicek model.
So price em and add em up!
23
Value of Growth Options
Evaluating all such growth options
provides
V (t ) I (t )e J (r (t ))
* c *
where J*(r(t)) is the value of the portfolio of
bond options.
24
Value of the firm
Summing the value of ongoing projects and the
value of growth opportunities give the current
value of the firm:
P t b(t )e C (t )
D (r (t ))
I (t )e J ( r (t ))
c *
Note how interest rates affect value in two
distinct ways:
discount rate
set of positive NPV investments.
25
Expected Return (or
discount rate) of the firm
To compute the expected return we need
to compute the expected price and
cashflow one period hence.
For simplicity we will denote the
expectation of any function (one period
hence) with a subscript “e”.
For example, the expected price of the
default consol bond is denoted:
De (r (t )) Et D(r (t 1)) 26
Expected Return as a
function of
E 1 Rt 1
b (t )
I
1 De (r (t ))e
(t )
J (r (t ))
*
e
(t )
b (t )
I D ( r (t ))e J (r (t ))
*
27
Limits
1 De (r (t ))e
(t )
lim Et 1 Rt 1 (t )
b ( t ) D(r (t ))e
*
J (r (t ))
lim Et 1 Rt 1 e
*
b ( t ) 0 J (r (t ))
In general these two limits are not the same,
they vary in time.
This implies a physical size effect
Sign depends on r(t). 28
Testability
The above expression for the price is untestable
because it requires measuring the firms beta.
Beta is unmeasurable because:
The pricing kernel is unobservable
Even if the kernel was observable, the firm's beta is
the weighted average of all its projects and
individual projects are not observable.
Luckily an expression for the expected return
can be derived entirely in terms of observables.
29
Expected Return as a
function of Fundamentals
De (r (t )) C b(t )
E 1 Rt 1 e
D(r (t )) P(t )
* De (r (t )) I
e J e (r (t )) J (r (t ))
C *
P(t )
D(r (t ))
30
Expected Return with
constant interest rates
b(t ) 1 e 1 r
E Rt 1 1 e
C
K r
Constant
p(t ) 1 e p(t )
Capital
Constant x B/M Constant x 1/size
Depreciation
Current Projects Growth
This is the Fama-French regression
equation!
31
Growth vrs Value
One implication of this work is that the
current characterization of growth and
value stocks is misguided
B/M is a measure of current assets more
than a measure of growth
A better measure might be the coefficient
on 1/size.
32
Empirical Performance of
this model
The model is too new for a full empirical
study
However, since the model does predict a
relation that is already been documented,
we know that the must be some empirical
validity
This is also a problem --- how do we know
that the empirical effect is consistent with the
model
33
Numerical Assumptions
Distribution of : x *
*
F ( x) e
Parameters are set by satisfying the
following two conditions:
1 in 10 projects are taken on when r(t)=0
1 in 20 projects are taken on when r(t)=
r=7.4%
34
Parameter Values
35
Beta
Coef . Prob .
3.5
3
2.5
2
Fama- French
1.5
1
0.5
Beta
-0.05 0 0.05 0.1 0.15 0.2 0.25 36
MV
A: Coefficient
FF
- 0.2 - 0.1 0 0.1
37
MV and B/M
BM
-0.0359
0.028 0.383
MV 0.802
-0.042
1.22
-0.11 1.64
2.06
-0.18 2.48
-0.25
-0.32
-0.39
10
Coef. Prob.
5
Fama- French
0
38
MV and Beta
Beta
-0.375
0.11
-0.278
MV -0.181
0.049
-0.0838
0.0134
-0.011 0.111
0.208
-0.071
-0.13
-0.19
2
Coef. Prob.
1
Fama- French
0
39
Momentum
0.5 1.
0.25
F ra c t i o n o f P ro j e c t s S u rv i v i n g
H
0.75
L
0
P a y o ff $
- 0.25
0.5
- 0.5
0.25
- 0.75
HL
- 1
0
0 5 10 15 20 25 30
Horizon y ears
40