@ A.C.F. Vorst - Rotterdam 1 Optimal Risk Taking under VaR Restrictions Ton Vorst Erasmus Center for Financial Research January 2000 Paris, Lunteren, Odense, Berlin

@ A.C.F. Vorst - Rotterdam 2 Maximizing expected return under Value at Risk restriction (e.g. in accepting projects) For normally distributed variables: VAR is 2.31  standard deviation. Hence equivalent with maximizing expected return under standard deviation restriction (Markowitz). Due to nonlinear instruments such as derivatives normality is questionable.

@ A.C.F. Vorst - Rotterdam 3 Banks worry about distributions VAR

@ A.C.F. Vorst - Rotterdam 4 Focus on portfolio manager who invests in index related derivatives S is the value of the index Problem: What kind of portfolio of the index, bonds and options can be built with a limited Value at Risk and a high expected return?

@ A.C.F. Vorst - Rotterdam 5 Binomial Model, 7 periods S Sd Su Suu Su 7 Su 6 d Su 5 du. Sd 7 Sud Sdu Probabilities are.5 and 1 + r = 1 i.e. r = 0,, the equity risk premium Sd

@ A.C.F. Vorst - Rotterdam 6 Arrow-Debreu security for a path: One receives unit payment if one exactly follows that path and zero otherwise. These securities can be created through a dynamic portfolio strategy ? 1 0 Su Sd S Buy  stock and invest B riskless  Su + B = 1  Sd + B = 0  = 1/S(u-d), B = -d/(u-d)

@ A.C.F. Vorst - Rotterdam 7 These are called Arrows-Debreu security prices. The system is called a stochastic discount factor. Lowest prices for the 7 upstate case. Highest prices for 7 downstates. Intuition: One is prepared to pay a higher price for financial protection in cases where the economy is in distress. Cost of portfolio:  S + B = (1-d)/(u-d) = p < ½ Hence state price for every state with 4 ups and 3 downs: p 4 (1-p) 3. General formula: p j (1-p) 7-j.

@ A.C.F. Vorst - Rotterdam 8 Solution: The lowest states have the highest prices. Do invest nothing in the lowest state (or you might even go short). Invest in all other states in 98 Arrow-Debreu securities for that state. Costs are (1 - (1 - p) 7 )  98. Buy additional [100 - (1 - (1 - p) 7 )  98]/p 7 Arrow- Debreu securities for highest state. Problem: Invest 100 such that the expected return is maximized and 1%-VaR < 2.

@ A.C.F. Vorst - Rotterdam 9 Distribution with 6% equity risk premium and ten day horizon

@ A.C.F. Vorst - Rotterdam 10 This looks very much like: VAR

@ A.C.F. Vorst - Rotterdam 11 Non recombining tree approach is not essential. Continuous time models: with X T standard normal distribution Pricing contingent claims Put pay off in highest states. Problem, there is no highest state and hence the solution has infinite expected return. Price per unit of probability

@ A.C.F. Vorst - Rotterdam 12 Arrow-Debreu securities are not traded, especially not the path dependent ones Digital options are traded, but illiquid With calls and puts one might approximate them

@ A.C.F. Vorst - Rotterdam 13 Calls: pay off profile increases for the cheaper states Puts: pay off profile increases for the expensive states Hence in an optimization model with options one goes long the highest strike calls and short the lowest strike puts, especially those with strikes below the VaR- boundary. Even if one does not follow a straightforward optimization but only compares the expected returns of different portfolios, the above drives the results

@ A.C.F. Vorst - Rotterdam 14 Volatility smiles and smirks Spot priceexercise price smirk smile Smiles in stock market before 1987, since then more smirks In currency markets more smiles Smirk: price increase for out-of-the-money puts. Low states become even more expensive. High states get cheaper. Problems will be more pronounced.

@ A.C.F. Vorst - Rotterdam 15 Utility functions do not improve the results if there is the equity premium puzzle Put a restriction on the expected loss below VaR E(Loss / Loss > VaR) For 7 steps case this gives a lower bound on the investment in the lowest state security Take an extra step (i.e. 8). 8 down steps 7 down steps 1 up · · less weight more weight

@ A.C.F. Vorst - Rotterdam 16 Other problems with VaR are signalled by Artzner, Delbaen, Eber and Heath “Definition of Coherent Risk Measures” If one has different portfolios with all reasonable VaR’s, the total might have a very large VaR 1%

@ A.C.F. Vorst - Rotterdam 17 ADEH suggest to specify a number of scenarios. Make portfolios that on these scenarios have limited downfall Methodology shows that this does not work if one specifies scenarios before optimization is allowed.

@ A.C.F. Vorst - Rotterdam 18 Ahn, Boudoukh, Richardson and Whitelaw, JoF February 1999 –Firm holds the underlying asset, which is governed by a geometric Brownian motion. Maturity T. – .05 is the 5% cut off point of lognormal distribution –Minimize VaR by buying puts. –Constraints are on costs of puts and total number of puts should be smaller than 1. –Linear programming solution –Costs can be reduced by writing calls –The restriction on number of puts is not necessary

@ A.C.F. Vorst - Rotterdam 19 Other Example Quantile Hedging: Hedge a contingent claim in such a way that the hedge portfolio exceeds the claim in at least 95% of the cases. – For exotic options – Regular claims with transaction costs – Cost reduction roughly 50%

@ A.C.F. Vorst - Rotterdam 20 Conclusion: Maximizing expected return with a VaR-restriction is dangerous. If markets get more securitized, dangers will increase Put other restrictions on portfolio composition