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Topic Sensitivity and Breakeven Analysis and Simulation

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					      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
   firms’ business risk.
                                         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
 isn’t traded.

                                   (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

				
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