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Risk Analysis for Portfolios


Risk Analysis for Portfolios Analytica Users Group Modeling Uncertainty Webinar Series, #5 3 June 2010 Lonnie Chrisman, Ph.D. Lumina Decision Systems – PowerPoint PPT presentation

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Title: Risk Analysis for Portfolios

Risk Analysis for Portfolios
  • Analytica Users Group Modeling Uncertainty
    Webinar Series, 5
  • 3 June 2010
  • Lonnie Chrisman, Ph.D.
  • Lumina Decision Systems

Course Syllabus (tentative)
  • Over the coming weeks
  • What is uncertainty? Probability.
  • Probability Distributions
  • Monte Carlo Sampling
  • Measures of Risk and Utility
  • Risk analysis for portfolios (Today)
  • Common parametric distributions
  • Assessment of Uncertainty
  • Hypothesis testing

Todays Outline
  • Review Risk Metrics (VaR, Eshortfall)
  • Build a portfolio model.
  • Graph reward vs. risk for portfolios.
  • Efficient Frontier
  • Covariance
  • Continuous portfolio allocations.
  • Duration 90 Minutes

Risk in Portfolios
  • Portfolio Theory asserts that
  • You can lower risk substantially with only minor
    impact to potential benefit by assembling
    combinations of assets.
  • Diversification
  • Reducing exposure to individual factors by
    holding many assets.
  • Hedging
  • Pairing assets that react to factors in opposite

Portfolios of...
  • Financial assets
  • Equipment (e.g., airplanes, machines, vehicles,
  • Products or technologies
  • Projects
  • Personel with varying skill sets
  • Inventory of supplies or suppliers

Review of Risk Measures
  • Measures of risk
  • Value-at-risk
  • Expected Shortfall
  • State Transition Model exercise
  • (See power point slides from last session)

Prelude to a Modeling Exercise
  • Were going to build a model of five potential
    investments with uncertainty.
  • Each is impacted to varying extents by
  • Changes in fuel price
  • Financial crises
  • One future point in time (i.e., one year).
  • Afterwards, well compute risk-return for
    combinations of investments (portfolios).

Exercise The Potential Assets
  • Let
  • FPC Fuel price change Normal(0,4)
  • Crisis Financial crisis occurs Bernoulli(5)

Inv. Base Mean FPC impact CR impact Std. dev.
A 2 0 0 0
B 3 0.5 -1 1
C 4 0 -2 3
D 5 -1 -1 5
E 6 0 1 7
E.g., Asset_B Normal(30.5fpc-1crisis, 1)
Exercise Explore individual investments
  • Collect the returns along an index named Asset
    (having 5 elements)
  • Plot the CDF of all 5 investments.
  • Use Sample Size 1000
  • In separate variables, compute
  • Mean return
  • Value-at-risk
  • Expected shortfall
  • Standard Deviation
  • Create a risk-reward scatter plot
  • Will have 5 dots

Combinations of Portfolios
  • How many possible portfolios (i.e., combinations
    of assets) do we have?

  • Create and define a variable Portfolio_return
  • It should be the average (equally weighted) of
    all assets in each portfolio.
  • View its
  • mean result
  • CDF (slicing 1 portfolio at a time)

Exercise Plot all portfolios
  • Create result variables for
  • Portfolio Value-at-risk
  • Portfolio Expected Shortfall
  • Portfolio Risk/Return scatter plot
  • Explore the scatter plot.
  • Identify the Efficient Frontier
  • Find each one-asset portfolio.
  • For each, can you decrease risk without damaging

Exercise Scatter Plot Color
  • Define a variable Portfolio_size
  • The number of assets in portfolio
  • 1 thru 5
  • Use this as the color in your scatter plot.

The Efficient Frontier
Capital Market Line Market Portfolio
Capital Market Line
Market Portfolio (maximal reward/risk ratio)
Risk-free asset
Exercise Parametric Analysis
  • How sensitive is the risk-reward relation to the
    probability of a financial crisis?
  • Define Index P_crisis Sequence(5,40,5)

Exercise Insurance Asset (Put Option)
  • Add a sixth asset
  • A put-option (i.e., insurance contract) on
    asset E.
  • Pays for any loss in asset E (even if you dont
    own it)
  • Does not pay out when E profits
  • You always pay a 1 premium for the contract.
  • Explore the risk/return scatter plot.
  • Should you buy the insurance? (hedge)

Comparison to Markowitz Portfolio Theory
  • Harry Markowitz (1952)
  • Statitionary Gaussian distributions
  • Mean covariance matrix
  • RewardMean
  • RiskStandard Devation
  • Continuous allocations
  • Todays presentation
  • Structured models, arbitary distributions
  • Reward, Risk Any measure.
  • Binary (yes,no) allocations.

  • Measures a connection between two inter-related
  • Definition
  • Computed by Analytica function Covariance(x,y)
  • Note Covariance(x,x) Variance(x)

Exercise Compute Covariance
  • Compute the covariance between assets B and D.
  • Compute the full covariance matrix.
  • Hint Youll need a copy of the Investment index.
  • Use the Gaussian function (in Multivariate
    Distribution library), and this covariance
    matrix, to create a Markowitz model of returns.

Continuous Allocation
  • Exercise Consider all portfolios with some
    continuous proportion of asset B and asset D
  • 0w2,w41, w2w41
  • rw w2rB w4rD
  • Exercise Graph Mean vs. SDeviation for this set
    of continuous portfolios
  • A continuous allocation w w1,..,wN is a
    vector with ? wi 1.

Identifying the Entire Efficient Frontier
  • Theorem (Black 1972)
  • In a continuous allocation, the set of all
    portfolios on the efficient frontier can be
    written as
  • z c x (1-c) y
  • where x and y are any two distinct efficient
    portfolios and 8ltclt8 is a constant.
  • Note assumes portfolios may short sell assets.

  • Find (approximately) all efficient continuous
    allocations for our 6 investments.
  • Use the scatter plot to manually identify two
    portfolios that appear to be efficient.
  • Plot Mean vs. SDeviation for all convex
  • Why is this not entirely correct?

  • Asset allocation is the practice of selecting
    mixes of assets to reduce risk while continuing
    to maximize return.
  • The efficient frontier characterizes the
    portfolios that cannot be improved upon without
    increasing risk.
  • Markowitz Portfolio Theory makes lots of
    parametric assumptions for analytical
    tractability. With Monte Carlo, most assumptions
    arent required.
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