Transcript Scenario Optimization, part 2 Contents CVAR portfolio optimization  Demo of VAR and CVAR optimization  Put-call efficient frontiers  Professor Alexei A.

Scenario Optimization,
part 2
Contents
CVAR portfolio optimization
 Demo of VAR and CVAR optimization
 Put-call efficient frontiers

Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management
Value at Risk in portfolio optimization
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Loss function
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Probability that loss does not exceed some threshold
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Probability of losses strictly greater than some threshold
Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management
Value at Risk in portfolio optimization
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Relation between different quantities
Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management
Value at Risk in portfolio optimization
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Conditional Value at Risk
Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management
CVAR portfolio models
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Loss function
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CVAR for z being 100%a VAR:
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Or for discrete scenarios
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Assuming denominator equal 1-a
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We need to simplify this keeping in mind our objective of using CVAR in portfolio risk
management models
Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management
CVAR portfolio models
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General CVAR portfolio model
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Two possible advantages of this model
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It takes into account the losses incurred if abnormal scenarios materialize
 CVAR is convex function of portfolio as opposed to VAR and for this reason it is
easier to compute
 In order to take advantage of this it is necessary to look more carefully into CVAR
formulation
Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management
VaR and CVaR: comparison
VaR/CVaR
10
CVaR may give very misleading ideas about VaR
9.5
9
8.5
8
7.5
7
6.5
6
0
0.2
0.4
0.6
0.8
1
fraction of portfolio 2
Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management
LP formulation of CVAR portfolio model
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Introduce auxilliary variables
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Or in case of discrete scenarios
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Averaging these with respect to scenarios
Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management
LP formulation of CVAR portfolio model
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Which gives
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Dividing this by 1-a and rearranging we get
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And recalling expression for CVAR we get finally
Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management
LP formulation of CVAR portfolio model
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LP CVAR portfolio model
This is linear model
if losses are linear
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Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management
Portfolio optimization with CVAR constraints
Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management
Put-call efficient frontiers
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Portfolio performance is measured against random target g:
liabilities, benchmark, index, competition, etc
Reward: portfolio exceeds target, risk: portfolio is below target
Integrated view of financial management process
Upside potential: payoff of a call option on the future portfolio value
relative to target
Downside potential: short position in a European put option on the
future portfolio value relative to the target
Portfolio call value: expected upside, put value: expected downside
Put/call efficient portfolios and put/call efficient frontiers
Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management
Put-call efficient frontiers
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Tracing put/call efficient frontier
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Start with LP without constraints
Portfolio value
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Target portfolio value
Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management
Put-call efficient frontiers
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Constraint which connects portfolio value with upside and downside
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Put/Call efficient portfolio
Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management
Dual problem
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Helps to obtain insight into the nature of solution
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Solution does not depend on w!!
This means that efficient frontier is the straight line
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Professor Alexei A. Gaivoronski
Financial Optimization and Risk Management