Chapter 4 Comparing NPV, Decision Trees, and Real Options

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Chapter 4Comparing NPV, Decision Trees, and Real
Options
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Comparison of financial options (FO) and real
options (RO)

• Underlying
• FO a traded security, e.g. stock, bond,
  interest rates
• It is easier to estimate the parameters from
  traded security prices.
• RO a tangible asset non-tradable, e.g. a
  business unit or a project
• Thus, we make the Marketed Asset Disclaimer
  assumption that we can estimate the PV of the
  underlying without flexibility by using
  traditional NPV.
• Management controls
• FO side bets, option traders have no influence
  over the actions of the company and no control
  over the companys share price.
• RO management controls the underlying real
  assets on which they are written. E.g. to defer a
  project.
• Thus, ROA can enhance the value the underlying
  real asset, and also enhances the value of the
  option.
• The uncertainty of the underlying
• FO ?S is assumed to be exogenous.
• ?The rate of return on the underlying stock is
  beyond option traders control .
• RO the actions of a company that owns a real
  option (e.g. to expand production) may affect the
  actions of competitors, and thus affect the
  nature of uncertainty that the company faces.
  (Chapter 9 volatility estimation)

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Cash flows of a project and a twin security

• The purpose of introducing the idea of twin
  security
• A twin security has cash flows that are
  perfectly correlated with those of our project
  and therefore have the same beta.
• 170/345 65/135
• The twin security has cash payoffs that are
  exactly one fifth of the payoffs of our project,
  therefore, they are perfectly correlated.
• Use CAPM of this twin security to find the WACC
  of this project.
• WACC of this project ECAPM(Rtwin security)

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Decision Tree Analysis

• This is a long-standing method for attempting to
  capture the value of flexibility.

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• ????????
• NPV0(0.51700.565)/1.175-115 -15 lt0
• ??????
• NPV0(0.51700.565)/1.175-115/1.08 -6.48 lt0
• ??????????
• DTA method
• NPV0(0.5550.50)/1.175 23.4 gt0
• The current value of option to defer 23.4 -
  (-6.48) 29.88
• ROA method
• u34/201.7, d13/200.65
• p(1rf)-d/(u-d)(1.08-0.65)/(1.7-0.65)0.43/1.
  050.41?(1-p) 0.59
• C0(0.41550.590)/1.08 20.88 gt0
• The current value of option to defer
  C0-NPV0,???????? 20.88-(-15) 35.88
• The current value of option to defer
  C0-NPV0,????? 20.88-(-6.48) 27.36
• Replication method
• 55m?34B0?(18) 0m?13B0?(18) ?
  m2.62, B0?31.53
• V0 m?P0B0 2.62?20(?31.53) 20.87 gt0
• The current value of option to defer V0 ?
  NPV0,????? 20.87?(?6.48) 27.35

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• At first glance, the DTA seems to be a good
  approach, but on close reflection the DTA method
  is wrong. Why?
• Because the DTA approach violates the law of one
  price.
• The risk-adjusted discount rate (i.e. WACC) of
  17.5 is appropriate for a 50-50 chance of either
  170 or 65, and for any pattern of cash flows
  that are perfectly correlated (i.e., that are a
  constant multiple) with it.
• But the cash flows (55 or 0) of the deferral
  option (i.e.,???????????) are very different.
• The cash flows of (55 or 0) are not perfectly
  correlated with the net cash flows of the project
  (55,-50).
• To value the cash flows (55,0) provided by the
  deferral option, we need to use the replicating
  portfolio approach, or equivalently ROA.

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• ?????????????????deferral option??????????,?DTA???
  ???????.
• ???????????
• PV0 given by ROA q?maxVu?I1,0(1-q)?maxVd
  ? I1,0 / (1k)
• ?? 20.87 0.5?550.5?0 / (1k) ? k31.9
  
• ??,DTA????????,??WACC (I7.5)???.
  ??,?????deferral option?,???????????????,?????WACC
  ???????????????????.
• In general, the DTA approach will give the wrong
  answer because it assumes a constant discount
  rate throughout a decision tree, when the
  riskiness of the cash flows outcomes changes
  based on where we actually are located in the
  tree.

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Valuing the deferral flexibilityi.e. the current
value of option to defer?

• Way 1
• the current value of option to defer
  C0,project?NPV0,????? 27.36
• Way 2
• the current value of option to defer V0,project
  by Replication?NPV0,????? 27.35
• Way 3
• The deferral option allows the decision maker to
  avoid negative outcomes in the down state of
  nature.
• the current value of option to defer C0,option
  to defer by Replication 27.34
• See below.

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• Cu,option to deferm?Putwin security B0?(1rf)
• Cd,option to deferm?Pdtwin security B0?(1rf)
• ?m(Cu,option to defer-Cd,option to
  defer)/(Putwin security-Pdtwin security) delta
• 0m?34 B0?(18)
• 50m?13 B0?(18)
• ? m ? 2.38, B074.93
• ? PV of the deferral option m ? P0 B0 ?
  2.38 ? 20 74.93 27.34

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The Marketed Asset Disclaimer (MAD)

• However, the twin security approach is that it is
  practically impossible to find a priced security
  whose cash payouts in every state of nature over
  the life of the project are perfectly correlated
  with those of the project.
• It is nearly impossible to find market-priced
  underlying risky assets.
• Also, the volatility of the gold price (as the
  proxy of twin security of a gold mine project) is
  different from the volatility of the value of a
  gold mine that had the right to defer opening.
• Solution setting the MAD (Marketed Asset
  Disclaimer) assumption
• Instead of searching in financial markets, we
  recommend that you use the present value of the
  project itself, without flexibility, (i.e. the
  traditional NPV) as the underlying risky asset
  the twin security.
• i.e. the traditional NPV is the best unbiased
  estimate of the market value of the project were
  it a traded asset.
• E.g.
• VU1,with flexibilitym ?Vu1,no flexibility B0 ?
  (1rf )
• Vd1,with flexibilitym ?Vd1,no flexibility B0 ?
  (1rf )
• ? V0,with flexibility m ? V0,without
  flexibility B0
• ?55 m ? 170 B0 ? (18)
• 0 m ? 65 B0 ? (18)1
• ? m 0.524, B0 ?31.54
• V0,without flexibility (0.5 ? 170 0.5 ?
  65)/(117.5) 100
• ? V0,with flexibility m ? V0,without
  flexibility B0 0.524 ? 100 ?31.54
  20.86 gt 0

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Risk-Neutral Probability Approach

• It starts out with a hedge portfolio that is
  composed of one share of the underlying risky
  asset and a short position in m shares of the
  option that is being priced in our example this
  is a call option.
• form a perfect hedged portfolio (risk free over
  the next short interval of time)
• ?m?CE0,project with flexibility
  (XI1115,?1)?S0B0
• Making the MAD assumption
• Since the hedged portfolio is riskfree, at t1
• ?m?CE,u1,project with flexibility ?Vu1
  B0?(1rf) ?m?CE,d1,project with flexibility?Vd1
• ? m (Vu1 ? Vd1)/(CE,u1,project with flexibility
  ? CE,d1,project with flexibility )
  (171/100-65/100) ? 100 /(55-0) 1.91
• The PV of the hedged portfolio
• ?m?CE0,project with flexibility ?V0B0
  ?1.91?CE0,project with flexibility ?100
• The PV of the hedged portfolio multiplied (1rf)
  will equal its value at time 1 in either up or
  down state
• (?m?CE0,project with flexibility ?V0)?(1rf) B0
  ?(1rf) (?m?CE,u1,project with flexibility
  ?Vu1)
• ? (?1.91?CE0,project with flexibility
  ?100)?(18) B0 ?(1rf) (?1.91?55 ?170)
• ? CE0,project with flexibility 20.86 gt 0

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• (?m?CE0,project with flexibility ?V0)?(1rf) B0
  ?(1rf) (?m?CE,u1,project with flexibility
  ?Vu1)
• m (Vu1 ? Vd1)/(CE,u1,project with
  flexibility ? CE,d1,project with flexibility )
• ? CE0,project with flexibility 1/(1rf) ?
  CE,u1,project with flexibility? (1?rf)?d /
  u?d ? CE,d1,project with flexibility?
  u?(1?rf) / u?d 1/(1rf) ?
  CE,u1,project with flexibility?p ? CE,d1,project
  with flexibility?(1?p)
• where p (1?rf)?d / u?d
• (1-p) u?(1?rf) / u?d

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More on the Risk-Adjusted and Risk-Neutral
Approaches
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• Suppose we have a CtA(X95,?2 periods), with
  objective probability q0.5, rf8, and the tree
  of its underlying risky project shown as follows,

u1.2 , d0.8333
Vu2u2?V0144
p(1rf?d)/(u?d)0.537
Vu1u?V0120
E0(V2)p?Vu2(1?p)Vm2106.1
Vm2ud?V0100
V0100
E0(V1)p?Vu1(1?p)Vd1103
Vd1d?V083.33
Vd2d2?V069.44
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Option valuation objective probabilities

• Replicating portfolio approach

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F m?Vu1 ? (1rf )?B27.77 m?Vd1 ? (1rf
)?B2.75 ?m0.6823, B?52.53 ?C0m?V0 ?B15.70
q0.5 u1.2 , d0.8333
p(1rf?d)/(u?d)0.537
A Cu2max0,144-9549
D maxVu1?X25, Cu1m?Vu1?B27.7727.77
E0(V2)p?Vu2(1?p)Vm2106.1
E0(V1)p?Vu1(1?p)Vd1103
B Cm2max0,100-955
F C0 maxV0?X,C0 15.70
E maxVd1?X ?11.67, Cd1m?Vd1?B2.752.75
C Cd2max0,69.44-950
D m?Vu2 ? (1rf )?B49 m?Vm2 ? (1rf
)?B5 ?m1, B?92.23 ?Cu1m?Vu1?B27.77
E m?Vm2 ? (1rf )?B5 m?Vd2 ? (1rf
)?B0 ?m0.1636, B?10.88 ?Cd1m?Vd1 ?B2.75
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• We have also calculated the risk-adjusted rate of
  return at each node by finding the rate that
  equates the PV of the option with its expected
  cash flows, discounted at the risk-adjusted rate
  (RAR). For example, at node D
• Cu q?Cuu(1?q)?Cud / (1RAR)
• 27.77 0.6?49(1?0.6)?5 / (1RAR)
• ?RAR13.07
• The risk-adjusted return rate hedge portfolios
  (m and B) change from node to node reflecting the
  changing risk of the payoffs.
• However, the risk-neutral probabilities remain
  constant from node to node.
• Note that the risk-neutral probability does not
  depend on the state of nature (node). It is a
  function of only the risk-free rate, and the up
  and down movements, u and d.
• These up and down movements are related to the
  volatility of the underlying asset.

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Option valuation risk-neutral probabilities
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q0.5 u1.2 , d0.8333
p(1rf?d)/(u?d)0.537
A Cu2max0,144-9549
D maxVu1?X25, Cu127.7727.77
E0(V2)p?Vu2(1?p)Vm2106.1
B Cm2max0,100-955
E0(V1)p?Vu1(1?p)Vd1103
F C0 maxV0?X,C0 15.70
E maxVd1?X ?11.67, Cd12.752.75
C Cd2max0,69.44-950
D Cu1 p?Cu2 ?(1?p)?Cm2 / (1rf )27.77
E Cd1 p?Cm2 ?(1?p)?Cd2 / (1rf )2.75
F C0 p?Cu1 ?(1?p)?Cd1 / (1rf )15.75
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Comparing real options to the Black-Scholes
approach
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• The replicating portfolio approach
• m?V0?B0C0
• The basic idea is to find the right number of
  units (i.e. m) of the undelrying risky assets, V0
  , plus some bonds, B0 , so that the portfolio has
  exactly the same payout in each state of nature
  as the call.
• The BS formula,
• S0?N(d1) ? Xe?rTN(d2) C0
• N(d1) is the number of units of the underlying
  necessary to form a mimicking portfolio,
• The second term is the number of bonds each
  paying 1 at expiration.
• N(d2) is the probability that the option will
  finish in-the-money (i.e., with the stock price
  greater than the exercise price), and
• Xe-rT is the exercise price at maturity
  discounted back to the present at the risk-free
  rate for T units of time.

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• Thus, the idea behind the BS formula and the
  replicating portfolio is the same.
• The main difference is that BS starts from Ito
  calculus (the calculus of stochastic differential
  equations), while the replicating portfolio
  concept is an algebraic approximation over the
  next short interval of time that approaches the
  BS equation in the limit as the number of
  discrete subintervals per unit of time becomes
  large.

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the end