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Chap 9
Estimating Volatility :
Consolidated Approach
Estimating Volatility : Consolidated Approach
 Volatility
of a project is not the same as the
volatility of any of the input variables, nor is it
equal to the volatility of the company’s equity.
 How to use a Monte Carlo approach to value a
project.
 Monte Carlo tools are fairly simple to use and
can model the cross correlations among various
inputs such as price and quantity, as well as time
series properties such as mean reversion.
 Two
approaches for estimating a consolidated
measure of the volatility of a value-based event
tree are discussed later in this chapter.
 We call them the historical and the subjective
approaches.
 We use the term “consolidated” because the
output is a single estimate of volatility, built up
from the many uncertainties that contribute to it .
 Estimates of these separate uncertainties are
taken either from historical data, or from the
subjective estimates of management.
Monte Carlo analysis for combining
uncertainties
 Each
sampling of a set of parameters generates
an estimate of the present value of a project ( or
a company), PVt
PVt  PV0 e
PVt
ln
 rt
PV0
rt
 We
start with a present value spreadsheet, model
the variable uncertainties, use the Monte Carlo
simulation to estimate the standard deviation of
rates of return, and then construct the event tree
binomial lattice.
 1. Price per unit
 2. Quantity of output
 3. Variable cost per unit
 If
the weighted average cost of capital is 12
percent, the present value is $1,507.63; and if the
investment outlay is $1,600, the net present
value is -$92.37.
 The expected price per unit starts out at $10,but
declines over time due to anticipated competitive
pressure.
 The expected quantity sold grows at 20 percent
in the year after the product is released, but
despite price cuts, growth shows to 5.8 percent
in the last year of the forecast.
 The
variable cost per unit is expected to decline
from $6.00 in the first year of operation to $3.56
per unit in the last year.
 They believe that their errors of estimation will
be highly positively correlated through time
(autocorrelation of 90%), implying that if they
underestimate the price that is achievable in one
year, they are highly likely to have
underestimated it the next year as well.
 This
time dependence is called autocorrelation,
and if it is negative, implying that a high value is
more likely to be followed by a low value (and
vice versa), then it is called mean reversion.
 1. Define Assumption.
 2. Set Autocorrelations.
“correlate icon”
Crystal Ball to “select an assumption”
 3. Define Forecast Variable.
 PV1  FCF1 

z  ln 
PV0


7
FCFt
PV1  
t 1
t  2 (1  WACC )
 4.
Run the Simulation.
“run” icon
“reset” “run preferences”
 The
mean return was 13 percent, and the annual
standard deviation was 21 percent.
 The standard deviation of prices was equal to 10
percent, but the standard deviation of the rate of
return on the project is 21 percent.
 The input variables that drive uncertainty is not
the same as the volatility of the project.
Building the event tree
= 1; therefore, u  e T  e 0.21  1.2337
and d = 1/u = 0.8106.
 The present value of the project, after the free
cash flows have been paid out at the end of the
first time period, is $1,569.12.
 By the end of the second time period, we can see
from Exhibit9.7 that the value can be either
$1,935.79 or $1,271.90.
T
 The
present value of the expected free cash
flow that period is $262 and the expected
present value at that point in time is
7
FCFt
PV2  
t
(1

WACC
)
t 3
 $261  $265  $253  $233  $209  $180
 $1401
$261/$1,401 = 0.1863
 Multiplying
this ratio times the up state value,
$1,935.79 yields the up state dividend, namely
$360.66.
 The down state dividend is the same ratio
multiplied by the down state value, namely
$236.95.
More on auto-correlation
X t 1  X t  b[ X t  E ( X )]   t
 bE ( X )  (1  b) X t   t
    X t  t
X t 1     X t   t 1

COV ( X t , X t 1 )
 t t 1
COV ( X t , X t 1 )

VAR( X t )
 2 ( X t )   2 ( X t 1 )
therefore  t   t 1
2
 Consequently  t  t 1   t  VAR( X t )
 This implies that the r-squared and beta
coefficient are identical, that is,    .
 Note also that beta is equal to one minus the
speed of adjustment, 1-b,
 And
 Positive
autocorrelation will result in greater
volatility than assuming independence across
time.
 Negative autocorrelation, with the simple mean
reversion that we have been describing, assumes
that positive error terms are followed by
negative errors.
 Note that positive autocorrelation increase the
standard deviation of project returns and
negative autocorrelation decreases it.
Using historical data
 Use
the r-squared of the time serious regression
to capture the autocorrelation effect.
 Cross-correlation between variables is captured
by the r-squared between them.
 The standard deviation for each of the variables
is the standard deviation of the residuals that
results from the time serious regression.
 1 x0  E ( x)2  2
sey
yˆ  t (n  2,0.95)  
2

n
S
(
x
)


 1  x0  E ( x)2  2
 1 21  10  2
 
 sey   
s
2
2  ey
n

S
(
x
)
20
S
(
x
)




 sey
S ( x )  20(11)
1
11
 sey

2
20S ( x )
20 S ( x 2 )
 sey
1
12

20 S ( x 2 )
2
 Therefore,
the confidence interval, out-of-sample,
increase approximately as  T
Subjective estimates provided by
management
 Geometric
Brownian motion
r  [rT  2 T , rT  2 T ]
upper[r ]  rT  2 T
lower[r ]  rT  2 T
upper[VT ]  V0erT  2
lower[VT ]  V0e
T
rT  2 T
T
RT   ri
i 1
rT  2

upper[VT ]  V0 e
T
rT  2

lower[VT ]  V0e
T
 VTupper  n
   ri
ln 
V0  i 1


2 T
Vt  Vt  t e
rt
 VTlower
ri  ln 

V0
i 1


2 T
n



A more complicated case : meanreverting procedures
 The
general model for mean reversion may be
written as :
Vt  Vt t   (V  Vt t )  dz
E(Vt t )  V


E  V  Vt t  0
E (dz )  0
E(Vt )  E(Vt t )  V
 (1   ) 
2
T
T  t
T t
t 2
2

T
T t

VT  V  2 t  (1   )
, V  2 t

t 2
V  VTlower
VTupper  V
T 


 (1   ) 
T
2

t 2
2
T t
t 2
 (1   ) 
T
2
 (1   ) 
T
t 2
Vt  Vt t   (V  Vt t )  dz
T t
2
2
T t


