Real Options Volatility Estimation with Correlated Assumptions

1
Real Options Volatility Estimation with
Correlated Assumptions

• Barry Cobb
• MST Seminar
• October 4, 2002
• Co-author John Charnes

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Real Options Analysis

• Provides a framework for valuing proactive and
  reactive managerial flexibility in investment
  decisions.
• Extends Black and Scholes (1973) model for
  valuing financial options to evaluation of real
  asset investments.
• Requires different approaches to estimating input
  parameters than those commonly used for financial
  option models.

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Parameters in Option Pricing Models
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Parameters in Option Pricing Models (cont)
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Black-Scholes/Merton Model

• Price of European call option (at time t)
• Price of European put option (at time t)
• where

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Black-Scholes/Merton Model (cont.)

• ParametersN(x) cumulative distribution
  function for standard normal random variableVt
  present value of expected cash flows (or stock
  price)K exercise price of optionD
  dividend yieldT time of expiration andr
  risk-free interest rate? volatility of cash
  flows

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Estimating Financial Options Volatility

• Estimate from historical market asset prices
• Stock price represents present value of future
  cash flows of the firm.
• Historical series of prices represents value of
  firm in various periods.
• Change in value between successive periods forms
  a historical rate of return distribution
• Implied volatility can be calculated from a
  current option price by inserting known
  parameters and solving for volatility.

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Estimating Real Options Volatility

• Estimate using perfectly correlated security
• Rarely exist in practice
• the volatility of gold is not the same as the
  volatility of a gold mine. (Copeland and
  Antikarov, 2001).
• Subjective estimate
• Monte Carlo simulation of future project cash
  flows (real asset value evolution)
• Requires a model of future cash flows
• Multiple simulation trials create a rate of
  return distribution for future cash flows

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Simulated Project Cash Flows

• Define the following (Herath and Park 2002
  Copeland and Antikarov 2001)
• Present Worth (PWn) present value of all cash
  flows from periods n to T.
• Market Value (MVn) present value of all cash
  flows from periods n1 to T.
• If An equals current period cash flow, PWn MVn
  An (assume PW0 MV0)
• Let k be the required rate of return

OR
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Simulated Project Cash Flows

• Solving for k in the PWn equation
• To create an estimate of k, we can use simulation
  to obtain the distribution of
• PW1 and MV0 are independent random variables.

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Simulation Example/Demo

• Unit contribution margin is assumed to have a
  normal distribution, and annual demand is assumed
  to have a triangular distribution.

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Simulation Example/Demo

• Generate a series of random variates for X1,,X5
  and D1,,D5 and insert in the following formula
• Generate another series of random variates for
  X1,,X5 and D1,,D5 and insert in the following
  formula
• For each trial, calculate , building a rate
  of return distribution.

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Simulation Example/Demo

• Expected value of the rate of return distribution
  will be equal to the required rate of return.
• Standard deviation of the rate of return
  distribution can be used to estimate the
  volatility parameterwhere ? time interval
  (assume 1 year for this example)

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Simulation Example/Demo
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Simulation Example/Demo

• Rate of Return distribution
• The standard deviation of the sample is 35.35,
  which will be used as the volatility estimate.

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Correlation of Assumptions

• Serial Price Correlation
• Positive Correlationprices that are higher
  (lower) than expected may be followed by
  additional periods of higher (lower) than
  expected prices.
• Negative Correlationprices that are higher
  (lower) than expected are followed by prices that
  are lower (higher) than expected.
• In this example, X1 is correlated with X2, X2
  correlated with X3, etc.
• Experiment test correlation factors between -0.9
  and 0.9 in increments of 0.1 (10,000 trials each)

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Serial Price Correlation
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Correlation of Assumptions

• Price-Demand Cross-Correlation
• Prices that are higher (lower) than expected
  correlated with lower (higher) than expected
  demand.
• Price inelasticity of demand implies little or no
  negative correlation price elasticity of demand
  implies large negative correlation coefficients.
• In this example, X1 negatively correlated with
  D1, X2 negatively correlated with D2, etc.
• Experiment test correlation factors between -0.9
  and 0.0 in increments of 0.1 (10,000 trials each)

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Price-Demand Cross-Correlation
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Experiment Results

• Serial Price Correlation
• Mean returns remain close to expected value
  throughout range of correlation coefficients.
• Volatility increases only slightly as serial
  price correlation increases from -0.9 to 0
• Volatility increases steadily as serial
  correlation increases from 0.0 to 0.9
• Price-Demand Cross-Correlation
• Volatility increases significantly as negative
  correlation becomes smaller price inelasticity
  implies lower volatility.

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Correlation of Assumptions

• Combined serial price correlation and
  price-demand cross-correlation
• Create separate model for ten levels of
  price-demand cross-correlation between -0.9 and
  0.0.
• Experiment test serial price correlation factors
  between -0.9 and 0.9 in increments of 0.1 (10,000
  trials each) for each price-demand
  cross-correlation model

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Correlation of Assumptions

• Combined serial price correlation and
  price-demand cross-correlation

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Experiment Results

• Lowest levels of volatility are associated with
  large negative price-demand cross correlation and
  large negative serial price correlation
• Highest levels of volatility are associated with
  zero price-demand cross correlation and high
  serial price correlation.
• An interaction effect exists the lines on the
  graph become closer as serial price correlation
  increases.

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Application of Volatility Parameters

• Using assumptions from the previous example,
  consider the following
• An initial investment of 4,950 is required.
• PV of project cash flows at Year 0 are 4,398, so
  Expected NPV is -552 (project would be rejected)
• Suppose the project can be abandoned after Year 1
  for salvage value of 4,508 (the PV at Year 1 of
  its year 2-5 expected cash flows)
• Risk-free interest rate of 7.
• Valuing using Black-Scholes at all combinations
  of serial price correlation and price-demand
  cross-correlation.

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Application of Volatility Parameters

• Total project values with flexibility to abandon
  after Year 1

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Application of Volatility Parameters

• Uncorrelated project value with flexibility to
  abandon is negative.
• Certain combinations of serial price correlation
  and price-demand cross-correlation lead to
  positive project values with flexibility.
• Managers can improve firm value by seeking
  projects with flexibility where demand is
  inelastic and price is highly serially
  correlated.
• On such projects, significant upside potential
  exists, while flexibility to abandon or
  scale-back limits downside risk.

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Conclusions

• Correlation assumptions are important in any
  model that uses probability distributions to
  represent uncertainty.
• Certain combinations of serial price correlation
  and price-demand correlation can increase project
  values where flexibility exists.
• Managers must understand how correlation affects
  volatility if subjective estimates are to be used
  for model parameters.
• Questions/Comments?