Today’s Lecture
Rules of Thumb Firm Valuation
TIP
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Rules of Thumb
We will look at two typically used rules of thumb
Hurdle rates Profitability Index
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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
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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.
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Hurdle Rate Rule
This rule says invest whenever the NPV of the project is positive for discount rate γ.
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Profitability Index
Invest whenever the ratio of the NPV of the project over the initial investment is greater than Π
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The Idea
Model the firm as
Collection of ongoing projects Options to invest in new projects --- Growth
Stochastic Interest Rate
Vasicek model
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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
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Project Cashflow (while alive)
Ci (t ) Ie
1 2 ( t ) C2 i i i
IID series of cashflows It can be thought of as the optimal investment level (that results from an unmodeled optimization problem).
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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 C t i i i i 1 i 1 b(t) has a natural interpretation as the book value of assets.
E C (t 1) (t 1) e
I (t )
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e b(t )
C
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.
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Pricing
The price at date t of any cashflow C(T) is given by
z (T ) Et C (T ) z (t )
where
z(t 1) z(t )e
1 2 ( t 1) r (t ) 2 z z
Evolution of the short rate r(t) is given by the Vasicek model.
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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 ))
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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.
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Value of the jth Ongoing Project
z ( s) V j (t ) Et C j (s) j (s) s t 1 z (t )
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Consider a Particular Term
z ( s) Et C j (s) j ( s) z (t )
s t
Ie
C j
B( s t , r (t ))
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Summing over all terms gives
V j (t ) Ie Ie
C j
s t 1 C j
s t
B( s t , r (t ))
D(r (t ))
where D(r(t)) is the price of a declining perpetuity.
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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
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Value of all ongoing projects
V t Ie
j 0
t
C j
D (r (t )) j (t )
t
b(t )e D (r (t ))
C j 0
I j (t ) b(t )
e
j
where
b(t )e
t j 0
C (t )
D(r (t ))
e
j
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(t ) ln
I j (t ) b(t )
What affects the value of ongoing projects
Value changes as old projects die or new projects are added
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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
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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 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.
I e
C t
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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!
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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.
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Value of the firm
Summing the value of ongoing projects and the value of growth opportunities give the current value of the firm: C (t )
P t b(t )e
D (r (t ))
c *
I (t )e J ( r (t ))
Note how interest rates affect value in two distinct ways:
discount rate set of positive NPV investments.
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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))
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Expected Return as a function of
E 1 Rt 1
b (t ) I
1 De (r (t ))e J (r (t )) b (t ) (t ) * J (r (t )) I D ( r (t ))e
* e
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(t )
Limits
lim Et 1 Rt 1
1 De (r (t ))e
D(r (t ))e
* e *
(t )
b ( t )
(t )
J (r (t )) lim Et 1 Rt 1 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).
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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.
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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 )) P(t ) D(r (t ))
C
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Expected Return with constant interest rates
1 e 1 b(t ) E Rt 1 1 e K r p(t ) 1 e p(t ) Constant
r C
Capital Depreciation Constant x B/M Current Projects Constant x 1/size Growth
This is the Fama-French regression equation!
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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.
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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
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Numerical Assumptions
Distribution of :
F ( x) e
x * *
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%
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Parameter Values
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Beta
Coef . Prob . 3.5 3
2.5 2 1.5
Fama- French
1 0.5 Beta
-0.05
0
0.05
0.1
0.15
0.2
0.25
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MV
A: Coefficient
FF
- 0.2
- 0.1
0
0.1
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MV and B/M
BM MV -0.0359 0.028 0.383 0.802 -0.042 -0.11 -0.18 -0.25 -0.32 -0.39 1.22 1.64 2.06 2.48
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Coef. Prob.
5
Fama- French
0
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MV and Beta
Beta 0.11 -0.375 -0.278 MV -0.181 0.049 -0.0838 0.0134 -0.011 0.111 -0.071 -0.13 -0.19 0.208
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Coef. Prob.
1
Fama- French 0
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Momentum
0.5 1.
0.75 0
- 0.25
0.5
- 0.5
- 0.75
0.25
- 1
0
5
10
15
Horizon y ears
H L
0 25 30
20
40
F ra c t i o n o f P ro j e c t s S u rv i v i n g
0.25
P a y o ff $
H L