Presentasjon i TI挪7 Financial Optimization and Risk Management by yurtgc548

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									Scenario Optimization,
part 2
Contents
n CVAR portfolio optimization
n Demo of VAR and CVAR optimization
n Put-call efficient frontiers
Value at Risk in portfolio optimization
n   Loss function



n   Probability that loss does not exceed some threshold




n   Probability of losses strictly greater than some threshold
Value at Risk in portfolio optimization
n   Relation between different quantities
Value at Risk in portfolio optimization
n   Conditional Value at Risk
CVAR portfolio models
n   Loss function



n   CVAR for z being 100%a VAR:



n   Or for discrete scenarios




n   Assuming denominator equal 1-a




n   We need to simplify this keeping in mind our objective of using CVAR in portfolio risk
    management models
CVAR portfolio models
n   General CVAR portfolio model




n   Two possible advantages of this model
     ¨ 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
 VaR and CVaR: comparison
           CVaR may give very misleading ideas about VaR
VaR/CVaR




                                            fraction of portfolio 2
LP formulation of CVAR portfolio model
n   Introduce auxilliary variables



n   Or in case of discrete scenarios



n   Averaging these with respect to scenarios
LP formulation of CVAR portfolio model
n   Which gives




n   Dividing this by 1-a and rearranging we get




n   And recalling expression for CVAR we get finally
LP formulation of CVAR portfolio model
n   LP CVAR portfolio model




n    This is linear model
if losses are linear
Portfolio optimization with CVAR constraints
Put-call efficient frontiers
n   Portfolio performance is measured against random target g:
    liabilities, benchmark, index, competition, etc
n   Reward: portfolio exceeds target, risk: portfolio is below target
n   Integrated view of financial management process
n   Upside potential: payoff of a call option on the future portfolio value
    relative to target
n   Downside potential: short position in a European put option on the
    future portfolio value relative to the target
n   Portfolio call value: expected upside, put value: expected downside
n   Put/call efficient portfolios and put/call efficient frontiers
Put-call efficient frontiers
n   Tracing put/call efficient frontier




n   Start with LP without constraints
n   Portfolio value


n   Target portfolio value
Put-call efficient frontiers
n   Constraint which connects portfolio value with upside and downside




n   Put/Call efficient portfolio
Dual problem
n   Helps to obtain insight into the nature of solution




n   Solution does not depend on w!!
n   This means that efficient frontier is the straight line

								
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