PONTIFCIA UNIVERSIDADE CATLICA DO RIO DE JANEIRO PUCRIO

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• PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE
  JANEIRO / PUC-RIO
• DEPARTAMENTO DE ENGENHARIA INDUSTRIAL
• Using Omega Measure for Performance Assessment of
  a Real Options Portfolio
• Javier Gutiérrez Castro Tara Keshar Nanda Baidya
• Fernando Antonio Lucena Aiube
• javiergc_at_aluno.puc-rio.br
• baidya_at_puc-rio.br
• aiube_at_ind.puc-rio.br

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ABSTRACT

• This work aims to establish a method for
  performing an optimized composition of a
  portfolio of real assets (investment projects),
  determining which of them will incorporate the
  portfolio, given a set of constraints and
  considering to exercise real options.
• This optimization is done maximizing the Omega
  measure (O), using Monte Carlo simulation
  approach.
• Omega (O) is a measure of performance
  (risk-return) which takes into account all the
  moments of the distribution of gains and losses,
  beyond that simply mean and variance (Markowitz,
  1952).
• The study of portfolio composition of investment
  projects with options and optimization of its
  performance by Omega measure (O), are the main
  contributions of this work.

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RISK MEASURES
VaR and Expected Shorfall
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RISK MEASURES
Properties of coherent measures of risk

• Translation Invariance The allocation of fixed
  income to the portfolio reduces the risk in the
  same amount.
• Risk(Xc)Risk(X)-c
• Subadditivity The risk of the sum of
  sub-portfolios is less than or equal to the sum
  of individual risk of sub-portfolios.
• Risk(X1X2) Risk(X1)Risk(X2)
• Positive Homogeneity By increasing the size of
  each portfolio's position increases the risk of
  the portfolio in the same proportion.
• Risk(cX)cRisk(X) ?c gt 0
• Monotonicity If gains in portfolio X are lower
  than those of portfolio Y for all possible
  scenarios, then the risk in portfolio X is
  greater than risk in portfolio Y.
• XY Risk(Y) Risk(X)

• Measures of risk like Standard Deviation and VaR,
  not always satisfy these properties, being the
  subadditivity one of the most important of them.
• The Expected Shorfall, in turn, satisfies the
  properties of coherence.

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Performance Assessment (Risk Return) of the
Portfolio

• Sharpe Ratio
• Sortino Ratio

(Downside Risk)
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A New Performance MeasureOmega (?)

• The Omega measure (O) incorporate all the moments
  of the distribution (regardless of the shape of
  this), providing a complete description of
  risk-return characteristics.
• Omega takes into account a level of return
  defined exogenously, the threshold L, which is
  the border between what is considered as gain and
  loss.

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Representation of Omega Function
Numerator (N(L)) and Denominator (D(L)) of Omega
Measure.
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Example
Omega ?(L1,4)
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Optimization with Omega Measure
where
Expected Shortfall
Excess Chance
Return of Portfolio P in period i
rij return of asset j in period i (there are
n assets) wj Part of the portfolio invested
in asset j.
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Optimization with Omega Measure

• Example

Statistics Properties of Historical Data of
Returns
Correlations Matrix
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Optimization with Omega Measure
Probability Distributions of four assets
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Optimization with Omega Measure
Portfolio Composition
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Optimization with Omega Measure
Probability Distribution of Optimized Portfolio
P. (a) Mean-Variance Optimization, (b) (c) e
(d) Optimization by Omega (L0, L3, L15)
(b)
(a)
(d)
(c)
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Optimization with Omega Measure
Main Statistics of Otimized Portfolio
Omega with L0
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Optimization with Omega Measure
Efficient frontiers in the scale ES vs. EC
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Optimization with Omega Measure
L vs. Omega
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Optimization with Omega Measure
Efficient frontiers for three thresholds (L)
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Methodology for Portfolio Optimization of Real
Assets with Options
Stage I Modeling Information
First step Identification of Risk Factors in
Projects
a) Economic Uncertainty ? Market Risk of Projects
b) Technical Uncertainty ? Private Risk of
Projects
c) Strategic Uncertainty ? Risk of acts of other
companies on the market
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Methodology for Portfolio Optimization of Real
Assets with Options
Stage I Modelling Information
Second step Modelling Uncertainty Variables
a) Econometrics Models
b) Modeling by Stochastic Processes
1) Geometric Brownian Motion (GBM)
2) Mean Reversion Processes - Arithmetic
- Geometric
3) Processes with Jumps
Third step Determination of correlations between
risk variables of Projects from Portfolio
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Methodology for Portfolio Optimization of Real
Assets with Options
Stage II Portfolio Optimization without Real
Options

• Following the spirit of MAD assumption (Marketed
  Asset Disclaimer, Copeland and Antikarov (2001))
  the market value of the project is its expected
  value without options, we do a parallel extending
  that assumption at the level of a projects
  investment portfolio.
• For selection of this portfolio, it is done an
  optimization by Omega (O), identifying the
  periods in which each project should be
  initiated.
• At literature, the Omega measure is always
  applied at the level of distributions of returns.
  The proposed methodology evaluates Omega O (L)
  using the distribution of NPV's.

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Methodology for Portfolio Optimization of Real
Assets with Options
Stage II Portfolio Optimization without Real
Options
Otimization Model
If P is a set of projects to be selected for
the portfolio Where Expected
Shortfall Excess Chance
NPV at t(0) of porfolio P in simulation
i
s.t.
(A project can only be initiated once)
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Methodology for Portfolio Optimization of Real
Assets with Options
Stage III Portfolio Optimization with Real
Options
First Step Determining the Market Value of
Projects and their Volatility
Market Value
Volatility
(Standard Deviation of z)
Or
(extract s)
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Methodology for Portfolio Optimization of Real
Assets with Options
Stage III Portfolio Optimization with Real
Options
Second Step Determining Correlations between
NPVs Projects Distributios
Third step Determining the Market Value of
Projects with Options
(Market Value with Options)
Fouth step Determining NPV0?s for each project j
(NPVs with RO distribution)
(NPV0 at simulation i, with RO distribution)
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Methodology for Portfolio Optimization of Real
Assets with Options
Stage III Portfolio Optimization with Real
Options
Fifth step Otimization Model with Options
If P is a set of projects to be selected for
the portfolio where Expected
Shortfall Excess Chance
NPV at t(0) of Portfolio P in simulação
i
wjt(s) is a binary variable 0 ou 1 , 1 means
that the project must be considered in portfolio
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Methodology for Portfolio Optimization of Real
Assets with Options
Stage III Portfolio Optimization with Real
Options
Constraints
If P is a set of projects to be selected for
the portfolio
(a project may or may not be selected)
(minimum and maximum number of projects)
(budget constraint K)
(obligatory project)
(mutually related projects)
(mutually exclusive projects)
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Final Considerations

• This paper shows a methodology for portfolio
  composition of investment projects including real
  options, with the extension of MAD assumption
  (duly corrected the volatility).
• Also proposes to use the Omega performance
  measure (O), which takes into consideration the
  entire format of the distribution of NPV's (and
  not simply the Mean and Variance).
• The approach using Monte Carlo simulation is
  designed to facilitate modeling of risk variables
  and calculating the value of options. Also it
  generates the distributions of NPV's in which
  Omega (O) would be assessed.
• Omega is a flexible and adaptable measure to
  different preference levels of investors,
  stipulating the threshold "L".

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• THANK YOU!