# Topic Sensitivity and Breakeven Analysis and Simulation

Document Sample

```					      Fin650:Project Appraisal

Lecture 5

Project Appraisal Under Uncertainty and
Appraising Projects with Real Options
1
Project Analysis Under Risk

Incorporating risk into project
analysis through adjustments to the
discount rate, and by the certainty
equivalent factor.

2
Introduction: What is Risk?
 Risk is the variation of future
expectations around an expected
value.
 Risk is measured as the range of
variation around an expected value.
 Risk and uncertainty are
interchangeable words.

3
Where Does Risk Occur?
   In project analysis, risk is the
variation in predicted future cash
flows.
Forecast Estimates of
Varying Cash Flows
End of   End of       End of       End of
Year 0   Year 1       Year 2       Year 3
-\$760    ?   -\$876    ?   -\$546    ?
-\$235    ?   -\$231    ?   -\$231    ?
-\$1,257   \$127     ?    \$186    ?    \$190    ?
\$489     ?    \$875    ?    \$327    ?
\$945     ?    \$984    ?    \$454    ?   4
Handling Risk
There are several approaches to handling risk:
 Risk may be accounted for by (1) applying a
discount rate commensurate with the riskiness of
the cash flows, and (2), by using a certainty
equivalent factor
 Risk may be accounted for by evaluating the project
using sensitivity and breakeven analysis.
 Risk may be accounted for by evaluating the
project under simulated cash flow and discount
5
rate scenarios.
Using a Risk Adjusted Discount Rate
   The structure of the cash flow
discounting mechanism for risk is:-

 The \$ amount used for a ‘risky cash flow’ is the
expected dollar value for that time period.
 A ‘risk adjusted rate’ is a discount rate calculated to
include a risk premium. This rate is known as the
RADR, the Risk Adjusted Discount Rate.                   6
Defining a Risk Adjusted Discount
Rate

  Conceptually, a risk adjusted discount
rate, k, has three components:-
1. A risk-free rate (r), to account for the
time value of money
2. An average risk premium (u), to
account for the firm’s business risk
3. An additional risk factor (a) , with a
positive, zero, or negative value, to
account for the risk differential
between the project’s risk and the
7
Calculating a
Risk Adjusted Discount Rate
A risky discount rate is conceptually defined
as:
k=r+u+a
Unfortunately, k, is not easy to estimate.
Two approaches to this problem are:
1. Use the firm’s overall Weighted Average Cost of
Capital, after tax, as k . The WACC is the overall rate
of return required to satisfy all suppliers of capital.
2. A rate estimating (r + u) is obtained from the
Capital Asset Pricing Model, and then a is added.   8
Calculating the WACC
Assume a firm has a capital structure of:
50% common stock, 10% preferred stock,
40% long term debt.

Rates of return required by the holders of each are :
common, 10%; preferred, 8%; pre-tax debt, 7%.
The firm’s income tax rate is 30%.

WACC = (0.5 x 0.10) + (0.10 x 0.08) +
(0.40 x (0.07x (1-0.30)))
= 7.76% pa, after tax.
9
The Capital Asset Pricing Model
 This model establishes the covariance
between market returns and returns
on a single security.
 The covariance measure can be used
to establish the risky rate of return, r,
for a particular security, given
expected market returns and the
expected risk free rate.

10
Calculating r from the CAPM
   The equation to calculate r, for a
security with a calculated Beta is:

 Where : E ~  is the required rate of
r
return being calculated, R f is the risk free
rate:  is the Beta of the security, and Rm
is the expected return on the market.     11
Beta is the Slope of an Ordinary
Least Squares Regression Line

Share Returns Regressed On Market
Returns

0.12
0.10
Returns of Share, %

0.08
0.06
0.04
pa

0.02
0.00
-0.10       -0.05   -0.020.00   0.05   0.10   0.15   0.20

-0.04
12
Returns on Market, % pa
The Regression Process

The value of Beta can be estimated as the regression coefficient
of a simple regression model. The regression coefficient ‘a’
represents the intercept on the y-axis, and ‘b’ represents Beta,
the slope of the regression line.

rit  a  bi rmt  u it
Where,
= rate of return on individual firm i’s shares at time t
rmt = rate of return on market portfolio at time t
uit = random error term (as defined in regression
analysis)                                              13
The Certainty Equivalent Method: Adjusting the
cash flows to their ‘certain’ equivalents

The Certainty Equivalent method adjusts the
cash flows for risk, and then discounts these
‘certain’ cash flows at the risk free rate.
CF  b CF2  b
NPV     1                 etc  CO
1  r 1
1  r 2

Where: b is the ‘certainty coefficient’ (established by
management, and is between 0 and 1); and r is the
14
risk free rate.
Analysis Under Risk :Summary

   Risk is the variation in future cash flows
around a central expected value.
   Risk can be accounted for by adjusting the
NPV calculation discount rate: there are two
methods – either the WACC, or the CAPM
   Risk can also be accommodated via the
Certainty Equivalent Method.
   All methods require management judgment
and experience.

15
Appraising Projects with Real Options
•Critics of the DCF criteria argue that
cash flow analysis fails to account for
flexibility in business decisions.
•Real option models are more focused on
describing uncertainty and in particular
the managerial flexibility inherent in
many investments
•Real options give the firm the
opportunity but not the obligation to take
certain action
16
What is Options?
   In finance, an option is a derivative financial instrument that
specifies a contract between two parties for a future transaction
on an asset at a reference price. The buyer of the option gains the
right, but not the obligation, to engage in that transaction, while
the seller incurs the corresponding obligation to fulfill the
transaction. The price of an option derives from the difference
between the reference price and the value of the underlying asset
(commonly a stock, a bond, a currency or a futures contract) plus
a premium based on the time remaining until the expiration of the
option. Other types of options exist, and options can in principle
be created for any type of valuable asset.
   An option which conveys the right to buy something is called a
call; an option which conveys the right to sell is called a put. The
reference price at which the underlying may be traded is called
the strike price or exercise price. The process of activating an
option and thereby trading the underlying at the agreed-upon
price is referred to as exercising it it. Most options have an
expiration date. If the option is not exercised by the expiration
date, it becomes void and worthless.

17
What is Real Options?
 Application of financial options theory to
investment in a non-financial (real) asset
 Hence the name real options

18
Real Options: Link between
Investments and Black-Scholes Inputs

19
Real Options in Capital Projects
   Ten real options to:
   Invest in a future capital project
   Delay investing in a project
   Choose the project’s initial capacity
   Expand capacity of the project subsequent to the
original investment
   Change the project’s technology
   Change the use of project during its life
   Shutdown the project with the intention of restarting it
later
   Abandon or sell the project
   Extend the life of the project
   Invest in further projects contingent on investment in
the initial project
20
Real Options in Capital Projects
   Simply adjusting the discount rate for the risk
does not account for the full impact of
uncertainty
   Uncertainty affects investment in two ways
   Uncertainty about investment (I) required
   Uncertainty about the present value (PV) that the future
investment might generate
   Since the future values (FVs) of I and PV may
both be uncertain, we need to simplify by
combing them into a single variable:
Profitability index = Present value/Investment
PI = PV/I
21
Real Options in Capital Projects
   Real option and profitability index
   Exercise real option only if PI turns out to be
greater than of equal to 1.00
   Otherwise, keep the funds I invested in the
financial market where PI virtually always
equals 1.00

22
Uncertainty and Real Options Value
   In the year 2000 GROWTHCO had a prospective project
under development
   The decision to invest will not be made until 2003
   Investment in the project is contingent upon PI being
greater than 1
   Therefore, in 2000 the potential to invest in 2003 was a
real option for GROWTHCO
   Management expected to invest \$25 million in the project if
PI>1
   R&D budget to make the project ready is \$ 1 million per
year.
   The actual size of the investment is uncertain, it depended
on market information fully available until 2003
   Real options payoff histogram

23
Calculation of expected PI of payoff
Interval     Interval value   Probability   PI of payoff   Expected PI of payoff
(1)         (2)              (3)           (4)              (3x4)

0<x<=0.4          0.2             0.16          1.00              0.160

0.4<x<=0.8        0.6             0.21          1.00              0.210

0.8<x<=1.2         1              0.26          1.05              0.273

1.2<x<=1.6        1.4             0.21          1.40              0.294

1.6<x<=2          1.8             0.16          1.80              0.288

1.00                          1.225       24
Real Options in Capital Projects
Probability                     Real Option Payoff Histogram

0          0.4   0.8   1.0   1.4   1.8        2.0           25
PI
Calculation of the Expected PI of
Payoff
   The first column shows selected intervals of the PI used in
the histogram
   The second column is the average value of the PI for each
interval
   The third column gives the probability management
assigned to each interval
   The fourth column gives the value of the PI of the payoff
depending on whether or not management would exercise
the investment option
   The final column gives the product of the PI and its
probability for each interval
   The sum at the bottom of the column gives the expected PI
of the payoff

26
Calculation of the Expected PI of
Payoff
   For example, in the fourth row PI falls between 1.2 and 1.6
   The second column shows the average value of the interval, 1.4
   The third column shows the probability management assigned to
this interval, 0.21
   Because the average interval value of 1.4 is greater than 1,
management would intend to invest in this interval, gaining an
average PI value of 1.4 with probability of 0.21
   The final column gives the product of the PI value (1.4) and its
probability(0.21)i.e. 0.294
   The first two rows PI value is less than 1, management under
these circumstances would invest in financial market and get a
value of 1.00, as shown in the fourth column
   The third row has an interval value of 1. Therefore we use
weighted average 0.5x1.1+0.5x1=1.05
   The expected value of the investments PI with option payoff is
1.225 as shown at the bottom row

27
Risk Neutral Valuation of Real Options
  Management’s option to reject the unfavorable payoffs
alters distributions of PI
 Risk adjustment factor
F = Risk adjustment factor for PV/ Risk adjustment factor for
I
 Risk adjustment factor for PV= (1+RF)T/ (1+RPV)T
 Risk adjustment factor for I= (1+RF)T/ (1+RI)T
 Therefore, F = (1+RI)T/ (1+RPV)T , where
RI represents the discount rate for future investment
expenditure and RPV is the Project’s discount rate
 Assuming RI= 0.5 and RPV=0.10, we get F=0.870
 Multiply all the class intervals by the risk-adjustment
factor

28
Real Options in Capital Projects
Probability                       Risk adjusted Histogram

0          0.37   0.75         1.12   1.49   1.87        29
1.0
PI
Risk-neutral Valuation of the expected
PI with payoff
Interval     Risk-neutral   Interval   Probability   PI of payoff   Expected PI of payoff
interval       value

0<x<=0.4     0<x<=0.37      0.18          0.16         1.00               0.160

0.4<x<=0.8   0.37<x<=0.74   0.55          0.21         1.00               0.210

0.8<x<=1.2   0.74<x<=1.10   0.92          0.26         1.01               0.264

1.2<x<=1.6   1.10<x<=1.47   1.29          0.21         1.29               0.271

1.6<x<=2     1.47<x<=1.84   1.66          0.16         1.66               0.265

1.00                            1.169      30
Risk Neutral Valuation of Real Options
 Re-calculate other values using the risk-neutral
intervals. The result is a smaller expected PI payoff
1.169
 Present value of the option= Present value of the
expected investment expenditure * (Present value of the
PI-1)
= \$25/(1+0.07)3*(1.169-1)= \$3.449 million
 R&D budget to make the project ready is \$ 1 million per
year. The present value of this three year annuity
discounted at 5% is \$2.723 million
 Therefore, addition to shareholder value, due to
exercising this option, is \$3.449 - \$2.723 million =
0.726 million
 R&D should go ahead

31
What is a real option?
 Real options exist when managers can
influence the size and risk of a project’s
cash flows by taking different actions
during the project’s life in response to
changing market conditions.
 Alert managers always look for real options
in projects.
 Smarter managers try to create real
options.
What is the single most important
characteristic of an option?

   It does not obligate its owner to take
any action. It merely gives the
owner the right to buy or sell an
asset.
How are real options different from
financial options?

Financial options have an underlying
asset that is traded--usually a
security like a stock.
A real option has an underlying asset
that is not a security--for example a
project or a growth opportunity, and it

(More...)
How are real options different from
financial options?

The payoffs for financial options are
specified in the contract.
Real options are “found” or created
inside of projects. Their payoffs can
be varied.
What are some types of
real options?

 Investment timing options
 Growth options
   Expansion of existing product line
   New products
   New geographic markets
Types of real options (Continued)

   Abandonment options
   Contraction
   Temporary suspension
   Flexibility options
Five Procedures for Valuing
Real Options

1. DCF analysis of expected cash flows,
ignoring the option.
2. Qualitative assessment of the real
option’s value.
3. Decision tree analysis.
4. Standard model for a corresponding
financial option.
5. Financial engineering techniques.
Analysis of a Real Option: Basic
Project
Initial cost = \$70 million, Cost of
Capital = 10%, risk-free rate = 6%,
cash flows occur for 3 years.
Annual
Demand       Probability     Cash Flow
High           30%            \$45
Average        40%            \$30
Low            30%            \$15
Approach 1: DCF Analysis
   E(CF) =.3(\$45)+.4(\$30)+.3(\$15)
= \$30.
 PV of expected CFs = (\$30/1.1) +
(\$30/1.12) + (\$30/1.13) = \$74.61
million.
 Expected NPV = \$74.61 - \$70
= \$4.61 million
Investment Timing Option
    If we immediately proceed with the
project, its expected NPV is \$4.61 million.
   However, the project is very risky:
   If demand is high, NPV = \$41.91 million.
   If demand is low, NPV = -\$32.70 million.
Investment Timing (Continued)
 If we wait one year, we will gain
additional information regarding demand.
 If demand is low, we won’t implement
project.
 If we wait, the up-front cost and cash
flows will stay the same, except they will
be shifted ahead by a year.
Procedure 2: Qualitative
Assessment
   The value of any real option increases if:
   the underlying project is very risky
   there is a long time before you must exercise
the option
   This project is risky and has one year
before we must decide, so the option to
wait is probably valuable.
Decision Tree Analysis
(Implement only if demand is not low.)

Cost       Future Cash Flows            NPV this
a
0        Prob.         1        2         3         4      Scenario

-\$70      \$45        \$45      \$45         \$35.70
30%
\$0     40%           -\$70      \$30        \$30      \$30         \$1.79
30%
\$0       \$0         \$0       \$0          \$0.00
Discount the cost of the project at the risk-free rate, since the cost is
known. Discount the operating cash flows at the cost of capital.
Example: \$35.70 = -\$70/1.06 + \$45/1.12 + \$45/1.13 + \$45/1.13.
Use these scenarios, with their given probabilities, to
find the project’s expected NPV if we wait.

E(NPV) = [0.3(\$35.70)]+[0.4(\$1.79)]
+ [0.3 (\$0)]
E(NPV) = \$11.42.
Decision Tree with Option to
Wait vs. Original DCF Analysis
 Decision tree NPV is higher (\$11.42
million vs. \$4.61).
 In other words, the option to wait is
worth \$11.42 million. If we implement
project today, we gain \$4.61 million but
lose the option worth \$11.42 million.
 Therefore, we should wait and decide
next year whether to implement project,
based on demand.
The Option to Wait Changes Risk

   The cash flows are less risky under the
option to wait, since we can avoid the low
cash flows. Also, the cost to implement
may not be risk-free.
   Given the change in risk, perhaps we
should use different rates to discount the
cash flows.
   But finance theory doesn’t tell us how to
estimate the right discount rates, so we
normally do sensitivity analysis using a
range of different rates.
Use the existing model
of a financial option.
 The option to wait resembles a financial
call option-- we get to “buy” the project
for \$70 million in one year if value of
project in one year is greater than \$70
million.
 This is like a call option with an exercise
price of \$70 million and an expiration date
of one year.
Inputs to Black-Scholes Model for
Option to Wait
 X = exercise price = cost to implement
project = \$70 million.
 rRF = risk-free rate = 6%.
 t = time to maturity = 1 year.
 P = current stock price = Estimated on
following slides.
2 = variance of stock return = Estimated
on following slides.
Estimate of P
   For a financial option:
   P = current price of stock = PV of all of stock’s
expected future cash flows.
   Current price is unaffected by the exercise cost
of the option.
   For a real option:
   P = PV of all of project’s future expected cash
flows.
   P does not include the project’s cost.
Step 1: Find the PV of future CFs
at option’s exercise year.
Future Cash Flows PV at
0       Prob.        1          2       3      4    Year 1

\$45        \$45        \$45 \$111.91
30%
40%                     \$30        \$30        \$30 \$74.61
30%
\$15        \$15        \$15 \$37.30

Example: \$111.91 = \$45/1.1 + \$45/1.12 + \$45/1.13.
Step 2: Find the expected PV at
the current date, Year 0.
PVYear 0                   PVYear 1

\$111.91
High
\$67.82       Average      \$74.61
Low
\$37.30
PV2004=PV of Exp. PV2005 = [(0.3* \$111.91) +(0.4*\$74.61)
+(0.3*\$37.3)]/1.1 = \$67.82.
The Input for P in the Black-
Scholes Model
 The input for price is the present value of
the project’s expected future cash flows.
 Based on the previous slides,
P = \$67.82.
Estimating 2 for the Black-Scholes
Model
 For a financial option, 2 is the variance of
the stock’s rate of return.
 For a real option, 2 is the variance of the
project’s rate of return.
Three Ways to Estimate       2

 Judgment.
 The direct approach, using the results
from the scenarios.
 The indirect approach, using the expected
distribution of the project’s value.
Estimating   2   with Judgment

 The typical stock has 2 of about 12%.
 A project should be riskier than the firm
as a whole, since the firm is a portfolio of
projects.
 The company in this example has 2 =
10%, so we might expect the project to
have 2 between 12% and 19%.
Estimating 2 with the Direct
Approach

 Use the previous scenario analysis to
estimate the return from the present
until the option must be exercised. Do
this for each scenario
 Find the variance of these returns, given
the probability of each scenario.
Find Returns from the Present until
the Option Expires
PVYear 0                  PVYear 1 Return

\$111.91 65.0%
High
\$67.82       Average      \$74.61 10.0%
Low
\$37.30 -45.0%
Example: 65.0% = (\$111.91- \$67.82) / \$67.82.
Use these scenarios, with their given probabilities, to
find the expected return and variance of return.

E(Ret.)=0.3(0.65)+0.4(0.10)+0.3(-0.45)
E(Ret.)= 0.10 = 10%.

2 = 0.3(0.65-0.10)2 + 0.4(0.10-0.10)2
+ 0.3(-0.45-0.10)2
2 = 0.182 = 18.2%.
Estimating 2 with the Indirect
Approach
 From the scenario analysis, we know the
project’s expected value and the
variance of the project’s expected value
at the time the option expires.
 The questions is: “Given the current
value of the project, how risky must its
expected return be to generate the
observed variance of the project’s value
at the time the option expires?”
The Indirect Approach (Cont.)

 From option pricing for financial options,
we know the probability distribution for
returns (it is lognormal).
 This allows us to specify a variance of
the rate of return that gives the variance
of the project’s value at the time the
option expires.
Indirect Estimate of 2
   Here is a formula for the variance of a
stock’s return, if you know the
coefficient of variation of the expected
stock price at some time, t, in the
future:
ln[ CV  1] 2
 
2
t
We can apply this formula to the real
option.
From earlier slides, we know the value of the
project for each scenario at the expiration date.

PV Year 1

\$111.91
High
Average    \$74.61
Low
\$37.30
Use these scenarios, with their given probabilities, to
find the project’s expected PV and PV.

E(PV)=.3(\$111.91)+.4(\$74.61)+.3(\$37.3)
E(PV)= \$74.61.
PV = [.3(\$111.91-\$74.61)2
+ .4(\$74.61-\$74.61)2
+ .3(\$37.30-\$74.61)2]1/2
PV = \$28.90.
Find the project’s expected coefficient of
variation, CVPV, at the time the option expires.

CVPV = \$28.90 /\$74.61 = 0.39.
Now use the formula to estimate
2.

   From our previous scenario analysis, we
know the project’s CV, 0.39, at the time
it the option expires (t=1 year).

ln[ 0.39  1]
2
 
2
14 .2%
1
The Estimate of 2
   Subjective estimate:
   12% to 19%.
   Direct estimate:
   18.2%.
   Indirect estimate:
   14.2%
   For this example, we chose 14.2%, but
we recommend doing sensitivity
analysis over a range of 2.
Value of the Real Option


68
Use the Black-Scholes Model:
P = \$67.83; X = \$70; rRF = 6%;
t = 1 year: 2 = 0.142

V = \$67.83[N(d1)] - \$70e-(0.06)(1)[N(d2)].
ln(\$67.83/\$70)+[(0.06+0.142/2)](1)
d1 =
(0.142)0.5 (1).05
= 0.2641.
d2 = d1 - (0.142)0.5 (1).05= d1 - 0.3768
= 0.2641 - 0.3768 =- 0.1127.
N(d1) = N(0.2641) = 0.6041
N(d2) = N(- 0.1127) = 0.4551

V = \$67.83(0.6041) - \$70e-0.06(0.4551)
= \$40.98 - \$70(0.9418)(0.4551)
= \$10.98.

Note: Values of N(di) obtained from Excel using
NORMSDIST function.
Use financial engineering techniques.
 Although there are many existing models
for financial options, sometimes none
correspond to the project’s real option.
 In that case, you must use financial
engineering techniques, which are covered
in later finance courses.
 Alternatively, you could simply use
decision tree analysis.
Other Factors to Consider When Deciding
When to Invest

 Delaying the project means that cash
flows come later rather than sooner.
 It might make sense to proceed today if
there are important advantages to being
the first competitor to enter a market.
 Waiting may allow you to take
advantage of changing conditions.
A New Situation: Cost is \$75
Million, No Option to Wait
Cost                   Future Cash Flows    NPV this
Year 0 Prob.          Year 1 Year 2 Year 3 Scenario

\$45        \$45        \$45    \$36.91
30%
-\$75 40%                 \$30        \$30        \$30    -\$0.39
30%
\$15        \$15        \$15    -\$37.70

Example: \$36.91 = -\$75 + \$45/1.1 + \$45/1.1 + \$45/1.1.
Expected NPV of New Situation
   E(NPV) = [0.3(\$36.91)]+[0.4(-\$0.39)]
+ [0.3 (-\$37.70)]
   E(NPV) =   -\$0.39.

   The project now looks like a loser.
Growth Option: You can replicate the
original project after it ends in 3 years.

 NPV = NPV Original + NPV Replication
= -\$0.39 + -\$0.39/(1+0.10)3
= -\$0.39 + -\$0.30 = -\$0.69.
 Still a loser, but you would implement
Replication only if demand is high.

Note: the NPV would be even lower if we separately discounted
the \$75 million cost of Replication at the risk-free rate.
Decision Tree Analysis
Cost                         Future Cash Flows                NPV this
Year 0 Prob.       1       2    3     4     5             6   Scenario

\$45    \$45     -\$30    \$45     \$45    \$45 \$58.02
30%
-\$75 40%          \$30    \$30      \$30     \$0     \$0      \$0    -\$0.39
30%
\$15    \$15      \$15     \$0     \$0      \$0    -\$37.70

Notes: The Year 3 CF includes the cost of the project if it is optimal to
replicate. The cost is discounted at the risk-free rate, other cash
flows are discounted at the cost of capital.
Expected NPV of Decision Tree
   E(NPV) = [0.3(\$58.02)]+[0.4(-\$0.39)]
+ [0.3 (-\$37.70)]
   E(NPV) =   \$5.94.

   The growth option has turned a losing
project into a winner!
Financial Option Analysis: Inputs
 X = exercise price = cost of implement
project = \$75 million.
 rRF = risk-free rate = 6%.
 t = time to maturity = 3 years.
Estimating P: First, find the value
of future CFs at exercise year.
Cost                    Future Cash Flows                PV at  Prob.
Year 0Prob.     1      2    3     4    5        6        Year 3 x NPV

\$45   \$45 \$45 \$111.91 \$33.57
30%
40%                            \$30   \$30 \$30 \$74.61 \$29.84
30%
\$15   \$15 \$15 \$37.30 \$11.19

Example: \$111.91 = \$45/1.1 + \$45/1.12 + \$45/1.13.
Now find the expected PV at the
current date, Year 0.
PV Year 0                      Year 1 Year 2 PVYear 3

\$111.91
High
\$56.05       Average                                 \$74.61
Low
\$37.30

PVYear 0=PV of Exp. PVYear 3 = [(0.3* \$111.91) +(0.4*\$74.61)
+(0.3*\$37.3)]/1.13 = \$56.05.
The Input for P in the Black-
Scholes Model
 The input for price is the present value of
the project’s expected future cash flows.
 Based on the previous slides,
P = \$56.05.
Estimating 2: Find Returns from
the Present until the Option Expires
Annual
PVYear 0      Year 1 Year 2 PV Year 3Return

\$111.91 25.9%
High
\$56.05 Average                        \$74.61 10.0%
Low
\$37.30 -12.7%
Example: 25.9% = (\$111.91/\$56.05)(1/3) - 1.
Use these scenarios, with their given probabilities, to
find the expected return and variance of return.

E(Ret.)=0.3(0.259)+0.4(0.10)+0.3(-0.127)
E(Ret.)= 0.080 = 8.0%.

2 = 0.3(0.259-0.08)2 + 0.4(0.10-0.08)2
+ 0.3(-0.1275-0.08)2
2 = 0.023 = 2.3%.
Why is 2 so much lower than in the
investment timing example?

 2 has fallen, because the dispersion of
cash flows for replication is the same as
for the original project, even though it
begins three years later. This means the
rate of return for the replication is less
volatile.
 We will do sensitivity analysis later.
Estimating 2 with the Indirect
Method
   From earlier slides, we know the value
of the project for each scenario at the
expiration date.

PVYear 3

\$111.91
High
Average   \$74.61
Low
\$37.30
Use these scenarios, with their given probabilities, to
find the project’s expected PV and PV.

E(PV)=.3(\$111.91)+.4(\$74.61)+.3(\$37.3)
E(PV)= \$74.61.
PV = [.3(\$111.91-\$74.61)2
+ .4(\$74.61-\$74.61)2
+ .3(\$37.30-\$74.61)2]1/2
PV = \$28.90.
Now use the indirect formula to
estimate 2.

 CVPV = \$28.90 /\$74.61 = 0.39.
 The option expires in 3 years, t=3.

ln[ 0.39  1]
2
 
2
 4 .7%
3
Use the Black-Scholes Model:
P = \$56.06; X = \$75; rRF = 6%;
t = 3 years: 2 = 0.047

V = \$56.06[N(d1)] - \$75e-(0.06)(3)[N(d2)].
ln(\$56.06/\$75)+[(0.06 +0.047/2)](3)
d1 =             (0.047)0.5 (3).05
= -0.1085.
d2 = d1 - (0.047)0.5 (3).05= d1 - 0.3755
= -0.1085 - 0.3755 =- 0.4840.
N(d1) = N(0.2641) = 0.4568
N(d2) = N(- 0.1127) = 0.3142

V = \$56.06(0.4568) - \$75e(-0.06)(3)(0.3142)
= \$5.92.

Note: Values of N(di) obtained from Excel using
NORMSDIST function.
Total Value of Project with Growth
Opportunity
Total value = NPV of Original Project +
Value of growth option
=-\$0.39 + \$5.92
= \$5.5 million.
Sensitivity Analysis on the Impact of
Risk (using the Black-Scholes model)
   If risk, defined by 2, goes up, then value
of growth option goes up:
 2 = 4.7%, Option Value = \$5.92
 2 = 14.2%, Option Value = \$12.10
 2 = 50%, Option Value = \$24.08
Project Analysis Under Certainty: Recap

Discounted cash flow techniques
The ideal investment decision making
technique is Net Present Value.
N P V measures the equivalent present
wealth contributed by the investment.
NPV-- relates directly to the firm’s goal of
wealth maximization
-- employs the time value of money
-- can be used in all types of investments
-- can be adjusted to incorporate risk.
92
Other Project Evaluation Techniques
Internal Rate of Return – calculates
The discount rate that gives the
project an NPV of 0. If the IRR is
greater than the required rate, the
project is accepted. IRR
is given as % pa.

CF1            CF2
\$ 0 ( NPV )                             ......  IO
(1  IRR ) 1
(1  IRR ) 2

93
Other Project Evaluation Techniques
Modified Internal Rate of Return –
calculates the discount rate that gives
the project an NPV of \$0, when future
cash flows can be re-invested at the
Re-Investment Rate, a rate different
from the IRR. If the MIRR is greater
that the required rate, the project is
accepted. MIRR is given as % pa.
CF1  (1  RIR ) n CF2  (1  RIR ) ( n 1)
\$ 0( NPV )                                             ......  IO
(1  IRR ) 1
(1  IRR ) 2
Other Project Evaluation Techniques
Non-Discounted Cash Flow Techniques

Accounting Rate of Return- measures the ratio
of annual average accounting income to an
asset base value. ARR is given as % pa.

Payback Period – measures the length of time
required to retrieve the initial cash outlay.
Payback is given as number of years.

95
Selection of Techniques
NPV is the technique of choice; it satisfies the
requirements of: the firm’s goal, the time value of
money, and the absolute measure of investment.

IRR is useful in a single asset case, where the
Cash flow pattern is an outflow followed by all
positive inflows. In other situations the IRR may
not rank mutually exclusive assets properly, or
may have zero or many solutions.

96
Selection of Techniques
MIRR   is useful in the same situations as
the IRR, but requires the extra
prediction of a re-investment rate.
ARR allows many valuations of the asset
base, does not account for the time value
of money, and does not relate to the
firm’s goal. It is not a recommended
method.
PB does not allow for the time value of
money, and does not relate to the firm’s
goal. It is not a recommended method
except for situations of uncertainty.  97
The Notion of Certainty
   Certainty assumption
   Financial decision makers are rational, risk-averse,
wealth maximizers
   Financial markets are efficient and competitive
    Future is certain, outcome is known
   Certainty allows demonstration and evaluation
of the capital budgeting techniques, whilst
avoiding the complexities involved with risk.
   Certainty requires forecasting, but forecasts
which are certain.
   Certainty is useful for calculation practice.
   Risk is added as an adaption of an evaluation
model developed under certainty.
98
NPV Applications
•Asset retirement
•Asset replacement
•Correct ranking of mutually exclusive
projects.
•Where projects have different lives.
•Where projects have different outlays.

99
Class Exercise: Asset Replacement
Assume that SNU Ltd. has an asset with about three years of

Operating life remaining, today being the asset’s fifth year.
The net operating inflows are shown in the table below. When
should the asset be retired?

End of year             Net operating         Salvage value
inflow
5                       -                     22,000

6                       7,000                 17,500

7                       6,400                 14,375

8                       4,250                 8980
100
Net Present Value
THE model to use in all investment
evaluations.
Other criteria, such as IRR, MIRR,
ARR,and Payback may be used as
complementary measures.
Class Exercise
Consider the following three cash flow profiles:
Year ending
0      1        2         3       4       5
Project 1       -100    20       20        20      20      120
Project 2       -100    33.44    33.44     33.44   33.44   33.44
Project 3       -100    85.22    85.22     85.22   85.22   -300

Calculate IRR, NPV, and Payback periods for the projects

102
Class Exercise
Project   IRR(%)       NPV     Payback

1         20           37.9    5

2         20           26.8    3

3         19.9, 44.4   -16.4   1.2

103
Pitfalls in Project Appraisal
   Specifying project’s incremental cash flow
requires care
   Relevant expected after-tax cash flow associated with
two mutually exclusive scenarios, without and with the
project
   Allocation of overheads
   Expected versus most likely cash flows
   Mean versus mode
   Limited capacity
   The IRR is biased
   The IRR’s of projects with different cash flow profiles are
not comparable
   Projects with equal IRR can have different NPVs when
they have different payback periods
   IRR calculation uses IRR itself as the discount rate
104
Pitfalls in Project Appraisal
   The payback period is often ambiguous
   Does not reflect the time value of money
   Ignores cash flows after the payback period
   Unsuitable for projects requiring investment over a period of
years
   Discount rates are frequently wrong
   Fallacy of single discount rate, projects have widely differing
risks
   Rising inflation rates are dangerous
   Use of a nominal rate to discount nominal cash flows and use
of a real rate to discount real cash flows
   All cash flows do not change equally with the rate of inflation
   Inflation increases the required investment in nominal working
capital
   Inflation increases corporate tax rate

105
Pitfalls in Project Appraisal
   The precise timing of cash flows is important
   Cash flows occur at the end of the year assumption
   Two methods for precise discounting
   Use monthly discount rates
   For example 1.5 –year discount factor
   Forecasting is often untruthful
   Increase the hurdle rate by the average forecasting bias
   Subsidiary forecast
   Risk adds value to real options
   Real options affect the NPV rule

106
Critique of DCF
   Ignores risks inherent in capital projects
   Uses the same discount rate to cash flows with
different risks
   Uses the same discounts rates throughout the
life of the project
   Considers investment one-time
irreversible decision

107

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 3 posted: 9/30/2012 language: Unknown pages: 107