Optimal Risk Taking under VaR Restrictions

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Optimal Risk Taking under VaR Restrictions

• Ton Vorst
• Erasmus Center for Financial Research
• January 2000
• Paris, Lunteren, Odense, Berlin

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• 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.

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Banks worry about distributions
VAR
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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?
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Binomial Model, 7 periods
Su7 Su6d Su5du . . . . Sd7
Suu
Su
Sud
S
Sdu
Sd
Sd
Probabilities are .5 and 1 r 1 i.e. r 0,
, the equity risk premium
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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
Su
?
S
Sd
0
Buy ? stock and invest B riskless ?Su B 1 ?Sd
B 0 ? 1/S(u-d), B -d/(u-d)
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Cost of portfolio ?S B (1-d)/(u-d) p lt ½
Hence state price for every state with 4 ups and
3 downs p4(1-p)3. General formula pj(1-p)7-j.
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.
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Problem Invest 100 such that the expected return
is maximized and 1-VaR lt 2.

• 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/p7
  Arrow-Debreu securities for highest state.

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Distribution with 6 equity risk premium and ten
day horizon
0
98
493
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This looks very much like
VAR
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Non recombining tree approach is not
essential. Continuous time models with XT
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
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• 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

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• 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

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Volatility smiles and smirks
smile
smirk
Spot price
exercise price

• 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.

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• 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 gt 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

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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 VaRs, the total might have a
very large VaR
1
1
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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.
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• 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

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• 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

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Conclusion Maximizing expected return with a
VaR-restriction is dangerous. If markets get
more securitized, dangers will increase Put
other restrictions on portfolio composition