# 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
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   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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