Scenario Optimization, part 2

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

Contents CVAR portfolio optimization Demo of VAR and CVAR optimization Put-call efficient frontiers

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

Value at Risk in portfolio optimization Loss function

Probability that loss does not exceed some threshold

Probability of losses strictly greater than some threshold

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

Value at Risk in portfolio optimization Relation between different quantities

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

Value at Risk in portfolio optimization Conditional Value at Risk

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

CVAR portfolio models Loss function

CVAR for being 100% VAR:

Or for discrete scenarios

Assuming denominator equal 1-

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

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

CVAR portfolio models General CVAR portfolio model

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

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

VaR and CVaR: comparison

0 0.2 0.4 0.6 0.8 16

6.5

7

7.5

8

8.5

9

9.5

10CVaR may give very misleading ideas about VaR

VaR/CVaR

fraction of portfolio 2

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

LP formulation of CVAR portfolio model Introduce auxilliary variables

Or in case of discrete scenarios

Averaging these with respect to scenarios

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

LP formulation of CVAR portfolio model Which gives

Dividing this by 1- and rearranging we get

And recalling expression for CVAR we get finally

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

LP formulation of CVAR portfolio model LP CVAR portfolio model

This is linear model if losses are linear

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

Portfolio optimization with CVAR constraints

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

Put-call efficient frontiers 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

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

Put-call efficient frontiers Tracing put/call efficient frontier

Start with LP without constraints Portfolio value

Target portfolio value

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

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

Put/Call efficient portfolio

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

Dual problem Helps to obtain insight into the nature of solution

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