Risk Analysis for Portfolios
Analytica Users Group
Modeling Uncertainty Webinar Series, #5
3 June 2010
Lonnie Chrisman, Ph.D.
Lumina Decision Systems
Copyright © 2010 Lumina Decision Systems, Inc.
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
Copyright © 2010 Lumina Decision Systems, Inc.
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
Copyright © 2010 Lumina Decision Systems, Inc.
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
ways.
Copyright © 2010 Lumina Decision Systems, Inc.
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
Copyright © 2010 Lumina Decision Systems, Inc.
Review of Risk Measures
• Measures of risk:
Value-at-risk
Expected Shortfall
• State Transition Model exercise
(See power point slides from last session)
Copyright © 2010 Lumina Decision Systems, Inc.
Prelude to a
Modeling Exercise
•
•
We’re 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, we’ll compute risk-return for
combinations of investments (portfolios).
Copyright © 2010 Lumina Decision Systems, Inc.
Exercise:
The Potential Assets
Let:
•
o
o
Inv.
FPC = Fuel price change: Normal(0,4%)
Crisis = Financial crisis occurs: Bernoulli(5%)
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%)
Copyright © 2010 Lumina Decision Systems, Inc.
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
Copyright © 2010 Lumina Decision Systems, Inc.
Combinations of Portfolios
• How many possible portfolios (i.e.,
combinations of assets) do we have?
Copyright © 2010 Lumina Decision Systems, Inc.
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)
Copyright © 2010 Lumina Decision Systems, Inc.
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 return?
Copyright © 2010 Lumina Decision Systems, Inc.
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.
Copyright © 2010 Lumina Decision Systems, Inc.
The Efficient Frontier
Copyright © 2010 Lumina Decision Systems, Inc.
Capital Market Line
& Market Portfolio
Market Portfolio
(maximal reward/risk ratio)
Risk-free asset
Copyright © 2010 Lumina Decision Systems, Inc.
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%)
Copyright © 2010 Lumina Decision Systems, Inc.
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 profits
You always pay a 1% premium for the
contract.
• Explore the risk/return scatter plot.
Should you buy the insurance? (“hedge”)
Copyright © 2010 Lumina Decision Systems, Inc.
Comparison to
Markowitz Portfolio Theory
• Harry Markowitz (1952)
Statitionary Gaussian distributions
Mean & covariance matrix
Reward=Mean
Risk=Standard Devation
Continuous allocations
• Today’s presentation
Structured models, arbitary distributions
Reward, Risk = Any measure.
Binary (yes,no) allocations.
Copyright © 2010 Lumina Decision Systems, Inc.
Covariation
• Measures a connection between two interrelated quantities.
• Definition:
Cov( x, y)  E[(x  x )( y  y )]
• Computed by Analytica function:
Covariance(x,y)
• Note: Covariance(x,x) = Variance(x)
Copyright © 2010 Lumina Decision Systems, Inc.
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.
Copyright © 2010 Lumina Decision Systems, Inc.
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.
Copyright © 2010 Lumina Decision Systems, Inc.
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 –∞<c<∞ is a
constant.
Note: assumes portfolios may “short sell” assets.
Copyright © 2010 Lumina Decision Systems, Inc.
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?
Copyright © 2010 Lumina Decision Systems, Inc.
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.