Priced Risk and Asymmetric Volatility in the Cross Section of Skewness

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Priced Risk and Asymmetric Volatility in the
Cross Section of Skewness

• Robert Engle and Abhishek Mistry
• NYU and JP Morgan Chase

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SKEWNESS IN STOCK RETURNS

• ARE STOCK INDICES SKEWED?
• ARE INDIVIDUAL STOCK RETURNS SKEWED?
• WHAT ECONOMIC MODEL GENERATES THESE OBSERVATIONS?
• SHOULD WE CARE?

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BRIEF LITERATURE REVIEW

• Skewness preference and pricing
• Kraus and Litzenberger(1976)
• Harvey and Siddique(2000)
• Dittmar(2002)
• Smith(2000)
• Asymmetric Volatility
• French Schwert Stambaugh(1987)
• Engle and Ng(1992)
• Glosten Jaganathan and Runkle(1992)
• Campbell and Hentschel(1992)
• Baekert and Wu(2000)
• Bae Kim and Nelson(2007)
• Berd Engle Voronov (2006)
• Skewness in Options Prices
• Bakshi Kapadia and Madan(2003)
• Dennis and Mayhew(2002)
• Duan and Wei(2006)
• Economic Models of Skewness
• Hong Wang and Yu(2007)

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SKEWNESS IN A MEAN VARIANCE SETTING?

• When expected market volatility increases,
  risk-averse individuals demand a higher expected
  return going forward
• This causes price to drop now.
• Thus asymmetric volatility is a consequence of
  risk aversion.
• volatility feedback effect
• Asymmetric volatility models imply negative
  skewness of time aggregated systematic returns.
• Of course, with a sufficiently long horizon CLT
  implies zero skewness

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ICAPM

• In the ICAPM all assets are priced by a pricing
  kernel linear in a set of state variables
• Assuming bs and betas are time invariant, then
  increasing the variance on any risk factor should
  increase the risk premium and lower the price.
• Changes in conditional variance and return should
  be negatively correlated.

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TIME AGGREGATION

• For a conditional volatility model
• One step conditional skewness is simply the
  skewness of epsilon
• But for time aggregated returns, the skewness can
  be negative if volatility is asymmetric

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ANALYTICALLY TARCH WITH SYMMETRIC INNOVATIONS
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Time Aggregation of TARCH
For TARCH with Gaussian shocks we can compute
closed form solutions for the skew and kurtosis
of aggregated returns
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Time Series Model

• The asymmetric GARCH model can generate this
  pattern of skewness
• Generatedpattern

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TEST RISK FACTORS

• We can test whether a factor is a priced risk
  factor by testing whether it has asymmetric
  volatility of the usual sign.
• This is similar to the test of Charoenrook and
  Conrad but should be more powerful
• Testing one at a time ignores some covariances
• Other news such as cash flow news may reduce the
  power of the test.

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SP 500 DAILY RETURNS
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TRIMMING .001 IN EACH TAIL (8 DAYS)
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SKEWNESS OF MULTIPERIOD RETURNS
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Fama-French TARCH Estimation 1988-2005
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Market
Size
B/M
Momentum
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INDIVIDUAL STOCK SKEWNESS

• Decompose skewness
• To get
• Across stocks the R3 and the idiosyncratic
  skewness may vary.
• We only use the market factor.

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Skewness of Individual Stocks 108,520 year x name
observations

• For each firm and year from 1988-2004
• Calculate Skewness using
• Daily returns
• Monthly returns starting on every day of year
• Quarterly returns
• Source
• CRSP, Computstat, IBES and Optionmetrics
• Beta, R2, volatility, Amihud illiquidity from
  prior year, lead and lag of market return.

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CALCULATE RISK NEUTRAL SKEWNESS

• From OptionMetrics calculate skewness of risk
  neutral distribution.
• Smooth implied vols on each date
• Compute BKM risk neutral skewness
• Average over the year
• 21,146 year x name observations
• 1996-2005

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SKEWNESS BY SIZE DECILE
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SKEWNESS BY R3 DECILE
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SUMMARY

• The more correlated a stock is with the market,
  the more negative is its skewness
• Skewness is increasingly negative for more
  aggregated returns
• On a daily frequency, skewness is generally
  positive and for small stocks it is generally
  positive.
• Risk neutral skewness is more negative than
  historical skewness.

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IMPLICATIONS

• Stocks and diversified portfolios over longer
  holding periods will generally have negative
  skewness.
• A mean variance investor will not care, but if
  she has preference for positive skewness, then a
  higher rate of return will be needed or perhaps
  diversified portfolios will not be optimal.
• Dynamic portfolio strategies could offer
  improvements.

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CREDIT RISK

• Default is associated with extreme declines in
  equity prices
• The correlation of defaults depends on the lower
  tail dependence of the joint distribution
• In a one factor model with Asymmetric GARCH
  factor, multiperiod returns will have lower tail
  dependence and higher default correlations than a
  Gaussian copula.
• This makes even senior tranches of a CDO more
  risky and makes them sensitive to changing
  volatilities.

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WHAT IS A CDO?

• Collateralized Debt Obligation a portfolio of
  bonds, residential mortgages, subprime mortgages,
  loans, and other types of credit.
• Investors can buy tranches of this portfolio that
  have more risk or less risk.
• How does this work?

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SAND OR OIL?

• An analogy mix sand, water and oil
• Tranches
• Senior and Super Senior Tranche
• Mezzanine Tranche
• Equity Tranche
• Under what circumstances are the senior tranches
  risky? Rising volatility and correlation.

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MODELING CDO TRANCHES

• Berd, Engle and Voronov(2007) propose a one
  factor model where the factor is an asymmetric
  garch model interpreted as the market return.
• The distribution of defaults depends upon the
  probability of large market declines. Because of
  the skewness of multiperiod returns, there is
  strong extreme correlation or tail dependence

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INDEX SPREADS (Default Premium)
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Tranche spreads of CDX.NA.IG8
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EFFECTS OF RISING VOLATILITY
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EFFECTS OF RISING CORRELATION
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UNDERSTANDING THE CREDIT CRISIS

• Most subprime mortgages and many other forms of
  risky debt were purchased in CDOs.
• Senior tranches were rated AAA and were
  considered to have virtually no risk, yet paid
  slightly higher interest than comparable
  investments.
• Credit Default Swaps (CDS) written by
  (inadequately funded) counterparties further
  guaranteed these investments

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• In the low interest environment of 2003-2007
  global investors demanded vast quantities of
  these products often with borrowed money as they
  were perceived to have low risk.

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WERE WE PREPARED?
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WHAT IS NEXT?
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