Copyright © 2010 Lumina Decision Systems, Inc. Risk Analysis for Portfolios Analytica Users Group Modeling Uncertainty Webinar Series, #5 3 June 2010 Lonnie.

Copyright © 2010 Lumina Decision Systems, Inc.

Risk Analysis for PortfoliosAnalytica Users Group

Modeling Uncertainty Webinar Series, #53 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


Today’s Outline

• Review: Risk Metrics (VaR, E[shortfall])

• 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.

HedgingPairing assets that react to factors in opposite ways.


Portfolios of...

• Financial assets • Equipment (e.g., airplanes,

machines, vehicles, factories)• Products or technologies• Projects• Personel with varying skill sets• Inventory of supplies or suppliers


Review of Risk Measures

• Measures of risk:Value-at-riskExpected Shortfall

• State Transition Model exercise

(See power point slides from last session)


Prelude to aModeling Exercise

• We’re going to build a model of five potential investments with uncertainty.

• Each is impacted to varying extents by:Changes in fuel priceFinancial crises

• One future point in time (i.e., one year).• Afterwards, we’ll compute risk-return for

combinations of investments (portfolios).


Exercise:The Potential Assets

• Let:o FPC = Fuel price change: Normal(0,4%)o 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(3%+0.5*fpc-1%*crisis, 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 returnValue-at-riskExpected shortfallStandard Deviation

• Create a risk-reward scatter plotWill have 5 dots


Combinations of Portfolios

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


Exercise

• 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-riskPortfolio Expected ShortfallPortfolio Risk/Return scatter plot

• Explore the scatter plot.Identify the Efficient FrontierFind each one-asset portfolio.

For each, can you decrease risk without damaging return?


Exercise:Scatter Plot Color

• Define a variable: Portfolio_sizeThe 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

Risk-free asset

Capita

l Mar

ket L

ine

Market Portfolio(maximal reward/risk ratio)


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 don’t own it)Does not pay out when E profitsYou 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

Reward=MeanRisk=Standard DevationContinuous allocations

• Today’s presentationStructured models, arbitary distributionsReward, Risk = Any measure.Binary (yes,no) allocations.


Covariation

• Measures a connection between two inter-related quantities.

• Definition:

• Computed by Analytica function:Covariance(x,y)

• Note: Covariance(x,x) = Variance(x)

)])([(),( yyxxEyxCov


Exercise:Compute Covariance

• Compute the covariance between assets B and D.

• Compute the full covariance matrix.Hint: You’ll 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:

0≤w2,w4≤1, w2+w4=1

rw = w2*rB + w4*rD

• 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) ywhere x and y are any two distinct efficient portfolios and –∞<c<∞ is a constant.

Note: assumes portfolios may “short sell” assets.


Exercise

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 combinations

• Why is this not entirely correct?


Summary

• 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 aren’t required.

Copyright © 2010 Lumina Decision Systems, Inc. Risk Analysis for Portfolios Analytica Users Group Modeling Uncertainty Webinar Series, #5 3 June 2010 Lonnie.

Documents

lumina decision systems

risk portfolio

compute risk

lower risk

time slide

combinations of portfolios

suppliers slide

session slide