Real options by dfhdhdhdhjr

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									              Real options
Part of material is borrowed from
  Aswath Damodaran’s website
   There are three areas in particular where
    traditional DCF, most widely articulated as the
    net present value rule (NPV), comes up short
    versus options theory:
        Flexibility. Flexibility is the ability to defer, abandon,
        expand, or contract an investment.
        Contingency. This is a situation when future
        investments are contingent on the success of today’s
        investment. Managers may make investments
        today—even those deemed to be NPV negative—to
        access future investment opportunities.
        Volatility. In options theory, higher volatility —
        because of asymmetric payoff schemes—leads to
        higher option value.
Some examples of business situations
that can be modeled as real options:
   Waiting to invest options, as in the case of a tradeoff
    between immediate plant expansion (and possible
    losses from decreased demand) and delayed
    expansion (and possible lost revenues)
   Growth options, as in the decision to invest in entry
    into a new market
   Flexibility options, as in the choice between building
    a single centrally located facility or building two
    facilities in different locations
   Exit options, as in the decision to develop a new
    product in an uncertain market
   Learning options, as in a staged investment in
    advertising.
Advantages of using real options

   The real options method applies financial options
    theory to quantify the value of management
    flexibility in a world of uncertainty.
   Real options capture the value of managerial
    flexibility to adapt decisions in response to
    unexpected market developments.
   The real option method enables corporate decision-
    makers to leverage uncertainty and limit downside
    risk.
   Companies create shareholder value by identifying,
    managing and exercising real options associated
    with their investment portfolio.
Real Options and Strategic Planning
(Michael J. Mauboussin, CREDIT SUISSE FIRST
BOSTON CORPORATION, 1999)
    Invest/Grow Options
   Scale up. The initial investments scale up to future value-
    creating opportunities. Scale-up options require prerequisite
    investments. For example, a distribution company may have
    valuable scale-up options if the served market grows.
   Switch up. A switch( flexibility) option values an opportunity
    to switch products, process, or plants given a shift in the
    underlying price or demand of inputs or outputs. A utility
    company that has the choice between three boilers: natural
    gas, fuel oil, and dual-fuel. Although the dual-fuel boiler may
    cost the most, it may be the most valuable, as it allows the
    company to always use the cheapest fuel.
   Scope up. This option values the opportunity to leverage an
    investment made in one industry into another, related
    industry. This is also known as link-and-leverage. A company
    that dominates one sector of e-commerce and leverages that
    success into a neighboring sector is exercising a scope-up
    option.
Defer/Learn Options

   Study/start. This is a case where
    management has an opportunity to invest
    in a particular project, but can wait some
    period before investing. The ability to wait
    allows for a reduction in uncertainty, and
    can hence be valuable. For example, a
    real estate investor may acquire an option
    on a parcel of land and exercise it only if
    the contiguous area is developed.
    Disinvest/Shrink Options
   Scale down. Here, a company can shrink or
    downsize a project in midstream as new
    information changes the payoff scheme. An
    example would be an airline’s option to abandon a
    non-profitable route.
   Switch down. This option places value on a
    company’s ability to switch to more cost-effective
    and flexible assets as it receives new information.
   Scope down. A scope-down option is valuable
    when operations in a related industry can be
    limited or abandoned based on poor market
    conditions and some value salvaged. A
    conglomerate exiting a sector is an example.
Case Study 1: Cable’s “Stealth Tier”—A
Scale-Up Option
 Of the 750 MHz available in an upgraded cable system, approximately
 648 MHz are being used for four visible revenue streams (analog
 video, digital video, high-speed data, and telephone). Figure below
 includes a diagram of the typical uses for 750 MHz cable plant. We
 refer to the remaining 102 MHz as the “Stealth Tier.” It is the tier of
 future interactive services that do not exist today.
A Scale-Up Option
   Lack of visibility does not mean a lack of value. The
    Stealth Tier could include services such as video
    telephone, interactive ecommerce, interactive
    games, and any other application that requires
    enormous amounts of bandwidth.
   The present value of the four visible revenue
    streams equals the current public trading value per
    home passed by cable wire. Accordingly, investors
    are attributing no value to the 17 empty 6 MHz
    channels on the interactive tier.
   Embedded in the upgrade of the cable plant is a
    growth option—or scale-up option —that is being
    overlooked. We know that the additional 102 MHz
    will be used, we just do not know when or how.
    Real options valuation
   We consider five potential NPV outcomes in our
    analysis.
   To minimize analytical complexity, we hold four
    of the five option inputs constant, making the
    valuation impact of the various NPV
    assumptions transparent.
   We hold volatility (s2) constant at 45% per year
    (the midpoint of the volatility range), time (t)
    constant at ten years (cable plant’s life), the risk-
    free rate (Rf) constant at 5.2%, and the marginal
    cost (X) per proposed project at 50% of the
    project’s value.
Real options valuation
Using these variables for each 6 MHz channel in the
Stealth Tier, we can determine a range of values. Using
just the 17 empty 6 MHz channels available today implies
a call option value per home passed of $197-1,979 for the
Stealth Tier. The midpoint of this range is $1,088,
representing approximately 50% of today’s trading value
per home passed.
    Case Study 2: Enron—Flexibility Options

    In 1998, the electricity prices briefly surged from $40 to
    an unprecedented $7,000 per megawatt hour in parts of
    the Midwest. Although the magnitude of this jump was
    unusual, a combination of capital intensity, transmission
    constraints, a lack of storage capability, deregulation,
    and always-uncertain weather has led to a secular
    increase in electricity price volatility.
   Enron learned from the events of 1998. Management
    realized that its diverse skills and meaningful resources
    made it uniquely positioned to capitalize on this volatility
    and immediately began work on a “peaker” plant
    strategy.
   This summer, Enron is slated to open three “peaker”
    plants—gas-fired electricity generating facilities that have
    production costs 50-70% higher than the industry’s finest.
    The plants, situated at strategic intersections between
    gas pipelines and the electric grid, are licensed to run
    only 1,200 hours per year but are much cheaper to build
    than a normal facility.
    In effect, they serve as the equivalent of underground
    storage in the gas business: they start up when electricity
    prices reach peak prices.
   Real options analysis demonstrated that the flexibility of
    the peakers is more valuable than their relative
    inefficiency, given ENE’s wholesale businesses and risk
    management capabilities.
Case Study 3: Merck and Biogen
Contingent Options
   In late 1997, Biogen announced that it had signed
    an agreement with Merck to help it develop and
    bring to market an asthma drug. Merck paid Biogen
    $15 million up front, plus the potential of $130
    million of milestone payments over several years.
   Before the drug becomes commercially viable,
    Biogen has to shepherd it through the development
    process. Along the way, Biogen could face
    expanded tests, a changing asthma drug market,
    and the risk of abandonment for safety reasons.
    Contingent Options
   In this case, Merck purchased a stream of options,
    including scale-up and scale-down (abandonment)
    options. Drug development represents “options on
    options,” or a series of contingent options. And Merck’s
    abandonment option must also be considered. The
    result is that Merck’s upside is unlimited, while its
    downside is capped by the payments.
   Real options analysis revealed that the deal was worth
    more than the $145 million of up-front and milestone
    payments that Merck pledged.
   From Biogen’s perspective, the value of the joint
    venture is the up-front payment plus the expected value
    of the milestone payments. In effect, Biogen transferred
    options to Merck that cannot be valued using traditional
    methods.
    Case Study 4: Amazon.com
    An Options Smorgasbord
   Scope-up options. Amazon has leveraged its position in
    key markets to launch into similar businesses. For
    example, it used it market-leading bookselling platform to
    move into the music business. These can be considered
    contingency options.
   Scale-up options. Flexibility options are part of Amazon’s
    announced growth in distribution capabilities. The
    company is adding capacity that will support significantly
    higher sales volumes in current businesses as well as
    capacity in potential new ventures. Management believes
    the cost of this option is attractive when weighed against
    the potential of disappointing a customer.
   Learning options. The company has made a number of
    acquisitions that may provide the platform for
    meaningful value creation in the future. The recently
    acquired business Alexa is an example. Alexa offers
    Web users a valuable service, suggesting useful
    alternative Web sites. It also tracks user patterns.
    Amazon may be able to use this information to better
    serve its customers in the future.
   Equity stakes. Amazon has taken equity stakes in a
    number of promising businesses, including
    drugstore.com and pets.com. These new ventures are
    best valued using options models. Figure below shows
    a conceptual diagram of how value has been created
    at Amazon.
   The company started by selling books. So there was a
    DCF value for the book business plus out-of-the-money
    contingent options on other offerings.
   As the book business proved successful, the contingent
    option on music went from out-of-the-money to in-the-
    money, spurring the music investment.
   As the music business thrived, the company exercised
    an option to get into videos.
   As time has passed, Amazon’s real options portfolio
    has become more valuable. For example, the recent
    foray into the auction business, unimaginable one year
    ago, was contingent on a large base of qualified users.
      APPLICATIONS OF
OPTION PRICING THEORY
  TO EQUITY VALUATION
A few caveats on applying
option pricing models
   1. The underlying asset is not traded.
   2. The price of the asset follows a continuous
    process.
   3. The variance is known and does not
    change over the life of the option
   4. Exercise is instantaneous
Valuing Equity as an option
Payoff Diagram for Equity as a Call Option. Equity can thus be
viewed as a call option the firm, where exercising the option
requires that the firm be liquidated and the face value of the
debt (which corresponds to the exercise price) paid off.
Application to valuation: A simple example

   Assume that a firm whose assets are currently
    valued at $100 million and that the standard
    deviation in this asset value is 40%.
   Further, assume that the face value of debt is
    $80 million (It is zero coupon debt with 10
    years left to maturity).
   If the ten-year treasury bond rate is 10%,
   how much is the equity worth? What should the
    interest rate on debt be?
Model Parameters
   Value of the underlying asset = S = Value of the firm
    = $ 100 million
   Exercise price = K = Face Value of outstanding debt
    = $ 80 million
   Life of the option = t = Life of zero-coupon debt = 10
    years
   Variance of the underlying asset = Variance in firm
    value = 0.16
   Riskless rate = r = Treasury bond rate corresponds
    to option life = 10%
Valuing Equity as a Call Option
                   rf T
 C  SN ( d1 )  e kN ( d 2 )
         S
      ln( )  ( rf  0.5 )T
                         2

 d1     k
                T
 d 2  d1   T
d1 = 1.5994 N(d1) = 0.9451
d2 = 0.3345 N(d2) = 0.6310
Value of the call = 100 (0.9451) - 80 exp(-0.10)(10)(0.6310) =
$75.94M
Value of the outstanding debt = $100 - $75.94 = $24.06 million
Interest rate on debt = ($ 80 / $24.06)**1/10 -1 = 12.77%
    Valuing equity in a troubled firm:

   The parameters of equity as a call option are as follows:
   Value of the underlying asset = S = Value of the firm =
    $ 50 million
   Exercise price = K = Face Value of outstanding debt =
    $ 80 million
   Life of the option = t = Life of zero-coupon debt = 10
    years
   Variance in the value of the underlying asset = 0.16
   Riskless rate = r = Treasury bond rate corresponding to
    option life = 10%
   d1 = 1.0515 N(d1) = 0.8534
    d2 = -0.2135 N(d2) = 0.4155
   Value of the call = 50 (0.8534) - 80 exp(-0.10) (10)
    (0.4155) = $30.44
   Value of the bond= $50 - $30.44 = $19.56 million
   The equity in this firm has substantial value,
    because of the option characteristics of equity.
   This might explain why stock in firms, which are in
    Chapter 11 and essentially bankrupt, still has
    value.
    The Conflict between bondholders
    and stockholders
   Stockholders and bondholders have different objective
    functions, and this can lead to agency problems, where
    stockholders can expropriate wealth from bondholders.
   Stockholders have an incentive to take riskier projects,
    and to pay more out in dividends than bondholders
    would like them to.
   An increase in the variance in the firm value, other
    things remaining equal, will lead to an increase in the
    value of equity.
   Stockholders can take risky projects with negative net
    present values, which while making them better off,
    may make the bondholders and the firm less valuable.
Illustration: Effect on value of the conflict
between stockholders and bondholders
   Consider again the firm with a value of assets of $100
    million, a face value of zero-coupon ten-year debt of $80
    million, a standard deviation in the value of the firm of
    40%.
   Value of Equity = $75.94 million
   Value of Debt = $24.06 million
   Value of Firm == $100 million
   Now assume that the stockholders take a project with a
    negative NPV of -$2 million, but assume that this project
    is a very risky project that will push up the standard
    deviation in firm value to 50%.
    Valuing Equity after the Project
   Value of the firm = S = $ 100 million - $2 million = $ 98 million
   Exercise price = K = Face Value of debt = $ 80 million
   Life of the option = t = Life of zero-coupon debt = 10 years
   Variance in the value of the firm value= 0.25
   Riskless rate = r = Treasury bond rate = 10%
     Value of Equity = $77.71
     Value of Debt = $20.29
     Value of Firm = $98.00


   The value of equity rises from $75.94 million to $ 77.71 million,
    even though the firm value declines by $2 million. The
    increase in equity value comes at the expense of bondholders,
    who find their wealth decline from $24.06 million to $20.19
    million.
Illustration: Effects on equity of a
conglomerate merger

                         Firm A                    Firm B
 Value of the firm       $100 million        $ 150 million
 Face Value of Debt      $ 80 million        $ 50 million
 (Zero-coupon debt)
 Maturity of debt        10 years        10 years
 Std. Dev. in firm value       40 %      50 %
 Correlation between firm cash flows 0.4
 The ten-year bond rate is 10%.

 Variance in combined firm value = (0.4)**2 (0.16) +
 (0.6)**2 (0.25) + 2 (0.4) (0.6) (0.4) (0.4) (0.5) = 0.154
    Valuing the Combined Firm
   The values of equity and debt in the individual firms and
    the combined firm can then be estimated using the
    option pricing model:

                      Firm A        Firm B        Combined
    Value of equity $75.94          $134.47       $ 207.43
    Value of debt     $24.06        $ 15.53       $ 42.57
    Value of the firm $100.00       $150.00       $ 250.00

    The combined value of the equity prior to the merger is $
    210.41 million and it declines to $207.43 million after. The
    wealth of the bondholders increases by an equal amount.
    There is a transfer of wealth from stockholders to
    bondholders, as a consequence of the merger. Thus,
    conglomerate mergers that are not followed by increases
    in leverage are likely to see this redistribution of wealth
    occur across claim holders in the firm
Obtaining option pricing inputs -
Some real world problems
   There were only two claim holders in the
    firm - debt and equity.
   There is only one issue of debt outstanding
    and it can be retired at face value.
   The debt has a zero coupon and no special
    features (convertibility, put clauses etc.)
   The value of the firm and the variance in
    that value can be estimated.
       Input                                      Estimation Process

                       Cumulate market values of equity and debt (or)
Value of the Firm      Value the firm using FCFF and WACC (or)
                       Use cumulated market value of assets, if traded.

                       If stocks and bonds are traded,
                         2firm = w e 2 e 2 + w d 2 d 2 + 2 w e w d e d ed
                       Where  e = variance in the stock price
                                    2



Variance               w e = MV weight of Equity
in Firm Value           d 2 = the variance in the bond price
                       w d = MV weight of debt
                       If not traded, use variances of similarly rated bonds.
                       Use average firm value variance from the industry in
                       which company operates.


                       Face value weighted duration of bonds outstanding (or)
Maturity of the Debt
                       If not available, use weighted maturity
    Valuing Equity as an option - The
    example of an airline
   The airline owns routes in North America, Europe and South America,
    with the following estimates of current market value:

    North America        $ 400 million
    Europe               $ 500 million
    South America        $ 100 million

   It has four debt issues outstanding with the following characteristics:

    Maturity             Face Value      Coupon          Duration
    20 year debt         $ 100 mil       11%             14.1 years
    15 year debt         $ 100 mil       12%             10.2 years
    10 year debt         $ 200 mil       12%             7.5 years
    1 year debt          $ 800 mil       12.5%           1 year
    Valuing Equity as an option - The
    example of an airline

   The annualized standard deviation in ln(stock
    prices) has been 25%, and the firm's debt has
    been approximately 90% of the firm value.
   bonds have had an annualized standard deviation
    of 10% (in ln(bond prices)). The correlation
    between B rated bonds and stock price is 0.3.
   The firm pays no dividends. The current T. Bond
    rate is 8%.
    Valuing Equity as an option - The
    example of an airline
   Step 1: Value of the firm = 400 + 500 + 100 = 1,000 million
   Step 2: Average duration = (100/1200) * 14.1 + (100/1200) *
    10.2 + (200/1200) * 7.5 + (800/1200) * 1 = 3.9417 years
   Step 3: Value of debt = 100 + 100 + 200 + 800 = 1,200 million
   Step 4: Estimate the variance in the value of the firm =
    (.1) 2 (.25) 2 + (.9) 2 (.10) 2 + 2 (.1)(.9)(.3) (.25)(.10) = 0.010075

   Step 5: Value equity as an option
    d1 = 0.7671      N(d1) = 0.7784
    d2 = 0.5678      N(d2) = 0.7148
    C=1000(0.7784)-1200exp(-0.08)(3.9417)(0.7148)= $ 152.63M
    Illustration: Valuing Equity as an
    option - Cablevision Systems
   Cablevision Systems was a firm in trouble in March 1995.
   The book value of equity in March 1995 was negative: - 1820
    million
   It had $ 3000 million in face value debt outstanding. The
    weighted average duration of this debt was 4.62 years
   The value of the firm estimated using projected cash flows to
    the firm, discounted at the weighted average cost of capital
    was $2,871 million.
   The stock is traded; its (monthly prices) variance is 0.0133.
   Cablevision bonds are traded; its (monthly prices) variance is
    0.0012.
   The correlation between stock and bond price changes has
    been 0.25. The debt proportion during the period was 70%.
    Illustration: Valuing Equity as an
    option - Cablevision Systems
   The stock and bond price variance are first
    annualized:
    Annual var. in stock = 0.0133 * 12 = 0.16
    Standard deviation = 0.40 Annual var. in bond =
    0.0012 * 12 = 0.0144 Standard deviation = 0.12
    Annualized variance in firm value = (0.30)**2
    (0.16) + (0.70)**2 (0.0.0144) + 2 (0.3)
    (0.7)(0.25)(0.40)(0.12)= 0.02637668
   The five-year bond rate (correspond to the
    weighted average duration of 4.62 years) is 7%.
    Illustration: Valuing Equity as an
    option - Cablevision Systems
   The parameters of equity as a call option are as follows:
    Value of the firm = S = Value of the firm = $ 2871 million
    Exercise price = K = Face Value of debt = $ 3000 million
    Life of the option = t = Weighted average duration of debt =
    4.62 years
    Variance in the value of the underlying asset = 0.0264
    Riskless rate = r = Treasury bond rate corresponding to option
    life = 7%
   d1 = 0.9910 N(d1) = 0.8391
    d2 = 0.6419 N(d2) = 0.7391
   C= 2871(0.8391) - 3000 exp(-0.07)(4.62) (0.7395) = $ 817
    million
   Cablevision's equity was trading at $1100 million in March
    1995.
Valuing Natural Resource
          Options/ Firms
    Valuing Natural Resource Options/ Firms
   In a natural resource investment, the underlying asset
    is the resource and the value of the asset is based
    upon two variables - the quantity of the resource that is
    available in the investment and the price of the
    resource.
   In most such investments, there is a cost associated
    with developing the resource, and the difference
    between the value of the asset extracted and the cost
    of the development is the profit to the owner of the
    resource.
   Defining the cost of development as X, and the
    estimated value of the resource as V, the potential
    payoffs on a natural resource option can be written as
    follows: = V - X if V > X   = 0 if V<X
Input                      Estimation Process
1. Value of Available      Expert estimates; The present value of the after-
Reserves of the            tax cash flows from the resource are then
Resource                   estimated.
2. Cost of Developing      Past costs and the specifics of the investment
Reserve (Strike Price)

3. Time to Expiration      Relinquishment Period: if asset has to be
                           relinquished at a point in time.
                           Time to exhaust inventory - based upon
                           inventory and capacity output.

4. Variance in value of    Based upon variability of the price of the
underlying asset           resources and variability of available reserves

5. Net Production          Net production revenue every year as percent of
Revenue (Dividend Yield)   market value.

6. Development Lag         Calculate present value of reserve based upon
                           the lag.
Valuing Natural Resource Options/ Firms

                 rf t*                      rf t
C  (S  De               ) N (d1 )  Xe             N (d 2 ), to account for dividend payout


           div*t                   rf t
 C  Se             N (d1 )  Xe            N (d 2 ), to account for natural resources
 lost for each year of waiting to develop
Illustration: Application to
valuation: A gold mine
   Consider a gold mine with an estimated inventory of 1
    million ounces, and a capacity output rate of 50,000
    ounces per year. The price of gold is expected to grow
    3% a year. The firm owns the rights to this mine for the
    next twenty years.
   The present value of the cost of opening the mine is
    $40 million, and the average production cost of $250
    per ounce. This production cost, once initiated, is
    expected to grow 4% a year.
   The standard deviation in gold prices is 20%, and the
    current price of gold is $350 per ounce. The riskless
    rate is 9%, and the cost of capital for operating the
    mine is 10%.
    Inputs for the Option Pricing Model:
   Value of the underlying asset = Present Value of expected
    gold sales (@ 50,000 ounces a year) = (50,000 * 350) * (1-
    (1.0320/1.1020))/(.10-.03) - (50,000*250)* (1- (1.0420/1.1020))
    /(.10-.04) = $ 42.40 million
   Exercise price = PV of Cost of opening mine = $40 million
   Variance in ln(gold price) = 0.04
   Time to expiration on the option = 20 years
   Riskless interest rate = 9%
   Dividend Yield = Loss in production for each year delay = 1 /
    20 = 5%
   d1 = 1.4069 N(d1) = 0.9202
   d2 = 0.5124 N(d2) = 0.6958
   C = 42.40exp(-0.05)(20)(0.9202)-40exp(-0.09)(20)(0.6958) =
    $ 9.75 M                  div*t              rf t
                      C  Se         N (d1 )  Xe N (d 2 ),
Illustration 10: Valuing an oil reserve

   Consider an offshore oil property with an estimated
    oil reserve of 50 million barrels of oil, where the
    present value of the development cost is $12 per
    barrel and the development lag is two years.
   The firm has the rights to exploit this reserve for the
    next twenty years and the marginal value per barrel
    of oil is $12 per barrel currently (Price per barrel -
    marginal cost per barrel).
   Once developed, the net production revenue each
    year will be 5% of the value of the reserves. The
    riskless rate is 8% and the variance in ln(oil prices)
    is 0.03.
     Valuing an oil reserve
   S = Value of the developed reserve discounted back the length of the
    development lag at the dividend yield = $12 * 50 / (1.05)**2= $ 544.22
   (The oil will not be available for sale until two years from now. The
    estimated opportunity cost of this delay is the lost revenue over the
    delay period. Hence, the discounting of the reserve back at the
    dividend yield)
   Exercise Price = PV of development cost = $12 * 50 = $600 million
   Time to expiration on the option = 20 years
   Variance in the value of the underlying asset = 0.03
   Riskless rate =8%
   Dividend Yield = Net production revenue / Value of reserve = 5%
   d1 = 1.0359 N(d1) = 0.8498
   d2 = 0.2613 N(d2) = 0.6030
   C =544 .22exp(-0.05)(20)(0.8498)-600exp(-0.08)(20)(0.6030)=97.08 M
   This oil reserve, though not viable at current prices, still is a valuable
    property because of its potential to create value if oil prices go up.
    Valuing product patents as options
   A product patent provides the firm with the right to
    develop the product and market it.
   It will do so only if the present value of the expected
    cash flows from the product sales exceed the cost of
    development.
   If this does not occur, the firm can shelve the patent
    and not incur any further costs.
   If I is the present value of the costs of developing
    the product, and V is the present value of the
    expected cash flows from development, the payoffs
    from owning a product patent can be written as:
   Payoff from owning a product patent = V - I if V> I
                                             = 0 if V<I
  Obtaining the inputs for option valuation
Input                      Estimation Process

Value of the Underlying    PV of Cash flows from taking
Asset                      project now
Variance in value of       Variance in cash flows
underlying asset           Variance in present value from
                           capital budgeting simulation.
Exercise Price on Option   Cost of investment on the project.

Expiration of the Option   Life of the patent
Dividend Yield             Cost of delay, each year of delay
                           translates into one less year of
                           value-creating cashflows
                           Annual cost of delay = 1/n
Illustration: Valuing a product option

   Assume that a firm has the patent rights, for
    the next twenty years, to a product which
    requires an initial investment of $ 1.5 billion to
    develop, and a present value, right now, of
    cash inflows of only $1 billion.
   Assume that a simulation of the project under
    a variety of technological and competitive
    scenarios yields a variance in the present
    value of inflows of 0.03.
   The current riskless twenty-year bond rate is
    10%.
    Valuing the Option
   Value of the underlying asset = Present value of inflows
    (current) = $1,000 million
   Exercise price = PV of cost of developing product = $1,500 M
   Time to expiration = Life of the patent = 20 years
   Variance in underlying asset = Variance in PV of inflows = 0.03
   Riskless rate = 10%
   d1 = 1.1548 N(d1) = 0.8759
   d2 = 0.3802 N(d2) = 0.6481
   Call Value= 1000 exp(-0.05)(20) (0.8759) -1500 (exp(-0.10)(20)
    (0.6481)= $ 190.66 million
   This suggests that though this product has a negative net present value
    currently, it is a valuable product when viewed as an option. This value
    can then be added to the value of the other assets that the firm
    possesses, and provides a useful framework for incorporating the value
    of product options and patents.
    Illustration: Valuing a firm with only product options

    Consider a bio-technology firm, which has no cash flow-
     producing assets currently, but has one product in the
     pipeline that has much promise in providing a treatment for
     diabetes. The product has not been approved by the FDA,
     and, even if approved, it could be faced with competition
     from similar products being worked on by other firms.
    The firm, however, would hold the patent rights to this
     product for the next 25 years. After a series of simulations,
     the expected present value of the cash inflows is
     estimated to be $500 million, with a variance of 0.20
     (signifying the uncertainty of the process).
    The expected present value of the cost of developing the
     product is estimated to be $400 million. The annual cash
     flows, once developed, are expected to be 4% of the
     present value of the inflows. The twenty-five year bond
     rate is 7%.
    Valuing the Option
   Value of underlying asset = PV of expected cash flows = $ 500
    million
   Exercise price = PV of cost of commercial use = $400 mil
   Time to expiration on patent rights = 25 years
   Variance in value of underlying asset = 0.20
   Riskless rate = 7%
   Dividend yield = Expected annual cash flow / PV of cash
    inflows = 4%
   d1 = 1.5532 N(d1) = 0.9398
   d2 = -0.6828 N(d2) = 0.2474
   C= 500 exp(-0.04)(25)(0.9398)-400exp(-0.07) (25) (0.2474)
    =155.66 M
   The estimated value of this firm is $155.66 million. This is a
    more realistic measure of value than traditional discounted cash
    flow valuation (that would have provided a value of $100 million)
    because it reflects the underlying uncertainty in the technology
    and in competition.
    Valuing Biogen
   The firm is receiving royalties from Biogen discoveries
    (Hepatitis B and Intron) at pharmaceutical companies.
    These account for FCFE per share of $1.00 and are
    expected to grow 10% a year until the patent expires (in
    15 years). Using a beta of 1.1 to value these cash flows
    (leading to a cost of equity of 13.05%), we arrive at a
    present value per share: Value of Existing Products = $
    12.14
   The firm also has a patent on Avonex, a drug to treat
    multiple sclerosis, for the next 17 years, and it plans to
    produce and sell the drug by itself. The key inputs on the
    drug are as follows:
    Valuing Biogen
   PV from Introducing the Drug Now = S = $ 3.422 billion
   Present Value of Cost of Developing Drug = K = $ 2.875 billion
   Patent Life = t = 17 years
   Riskless Rate = r = 6.7% (17-year T.Bond rate)
   Variance in Expected Present Values = 0.224 (Industry average
    firm variance for bio-tech firms)
   Expected Cost of Delay = y = 1/17 = 5.89%
   d1 = 1.1362               N(d1) = 0.8720
   d2 = -0.8512              N(d2) = 0.2076
   Call = 3,422 exp(-0.0589)(17) (0.8720) - 2,875 (exp(-0.067)(17)
    (0.2076)= $ 907 million
   Call from Avonex = $ 907 million/35.5 million = $ 25.55
   Biogen Value Per Share = Value of Existing Assets + Value of
    Patent = $ 12.14 + $ 25.55 = $37.69
    Valuing an Option to Abandon; An Illustration

   Assume that a firm is considering taking a 10-year
    project that requires an initial investment of $ 100
    million in a real estate partnership, where the PV of
    expected cash flows is $110 million. While the NPV of
    $ 10 million is small, assume that the firm has the
    option to abandon this project anytime in the next 10
    years, by selling its share of the ownership to the
    other partners for $ 50 million. The variance in the
    present value of the cash flows from being in the
    partnership is 0.09.
   The value of the abandonment option can be
    estimated by determining the characteristics of the put
    option:
   Value of the Underlying Asset (S) = $ 110 million
   Strike Price (K) = Salvage Value from Abandonment = $ 50
    million
   Variance in Underlying Asset’s Value = 0.06
   Time to expiration = Period for abandonment option = 10
    years
   Assume that the ten-year riskless rate is 6%, and that the
    property is not expected to lose value over the next 10 years.
    The value of the put option can be estimated as follows:
   Call Value = 110 (0.9737) -50 (exp(-0.06)(10) (0.8387) = $
    84.09 million
   Put Value= $ 84.09 - 110 + 50 exp(-0.06)(10) = $ 1.53 million
   The value of this abandonment option has to be added on to
    the net present value of the project of $ 10 million, yielding a
    total net present value with the abandonment option of $11.53
    million.
    Valuing an Option to Expand: The
    Home Depot
   Assume that The Home Depot is considering opening a
    small store in France. The store will cost 100 million FF to
    build, and the present value of the expected cash flows from
    the store is 80 million FF. Thus, by itself, the store has a
    NPV of -20 million FF.
   By opening this store, the Home Depot acquires the option
    to expand into a much larger store any time over the next 5
    years. The cost of expansion will be 200 million FF, and it
    will be undertaken only if PV of the expected cash flows
    exceeds 200 million FF. At the moment, PV of the expected
    cash flows from the expansion is only 150 million FF. If it
    were not, the Home Depot would have opened to larger
    store right away. The Home Depot still does not know much
    about the market for home improvement products in France,
    and there is considerable uncertainty about this estimate.
    The variance is 0.08.
   S = PV of Cash Flows from Expansion, if done now=150
    million FF
   Strike Price (K) =Cost of Expansion = 200 million FF
   Variance in Underlying Asset’s Value = 0.08
   Time to expiration = Period for which expansion option = 5
    years
   Assume that the five-year riskless rate is 6%.
   C = 150 exp(-0.06)(5)(0.6314) -200exp(-0.06)(20) (0.3833)=
    37.91M
   NPV of Store = 80 million FF - 100 million FF = -20 million
   Value of Option to Expand = 37.91 million FF
   NPV of store with option to expand = 17.91 mil FF
   The Home Depot should open the smaller store, even though
    it has a negative net present value, because it acquires an
    option of much greater value, as a consequence.

								
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