Risk Analysis for Portfolios: Understanding Uncertainty & Efficient Frontier

1. Risk Analysis for Portfolios Analytica Users GroupModeling Uncertainty Webinar Series, #5 3 June 2010 Lonnie Chrisman, Ph.D. Lumina Decision Systems

2. 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

3. 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

4. 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.

5. 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

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

7. 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 price • Financial crises • One future point in time (i.e., one year). • Afterwards, we’ll compute risk-return for combinations of investments (portfolios).

8. Exercise:The Potential Assets • Let: • FPC = Fuel price change: Normal(0,4%) • Crisis = Financial crisis occurs: Bernoulli(5%) E.g., Asset_B :=Normal(3%+0.5*fpc-1%*crisis, 1%)

9. 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

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

11. 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)

12. 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?

13. 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.

14. The Efficient Frontier

15. Capital Market Line & Market Portfolio Capital Market Line Market Portfolio(maximal reward/risk ratio) Risk-free asset

16. 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%)

17. 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”)

18. 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.

19. Covariation • Measures a connection between two inter-related quantities. • Definition: • Computed by Analytica function:Covariance(x,y) • Note: Covariance(x,x) = Variance(x)

20. 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.

21. 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.

22. 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.

23. 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?

24. 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.