Title: Priced Risk and Asymmetric Volatility in the Cross Section of Skewness
1Priced Risk and Asymmetric Volatility in the
Cross Section of Skewness
- Robert Engle and Abhishek Mistry
- NYU and JP Morgan Chase
2SKEWNESS IN STOCK RETURNS
- ARE STOCK INDICES SKEWED?
- ARE INDIVIDUAL STOCK RETURNS SKEWED?
- WHAT ECONOMIC MODEL GENERATES THESE OBSERVATIONS?
- SHOULD WE CARE?
3BRIEF 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)
4SKEWNESS 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
5ICAPM
- 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.
6TIME 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
7ANALYTICALLY TARCH WITH SYMMETRIC INNOVATIONS
8Time Aggregation of TARCH
For TARCH with Gaussian shocks we can compute
closed form solutions for the skew and kurtosis
of aggregated returns
9Time Series Model
- The asymmetric GARCH model can generate this
pattern of skewness - Generatedpattern
10TEST 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.
11SP 500 DAILY RETURNS
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13TRIMMING .001 IN EACH TAIL (8 DAYS)
14SKEWNESS OF MULTIPERIOD RETURNS
15Fama-French TARCH Estimation 1988-2005
16Market
Size
B/M
Momentum
17INDIVIDUAL STOCK SKEWNESS
- Decompose skewness
- To get
- Across stocks the R3 and the idiosyncratic
skewness may vary. - We only use the market factor.
18Skewness 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.
19CALCULATE 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
20SKEWNESS BY SIZE DECILE
21SKEWNESS BY R3 DECILE
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23SUMMARY
- 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.
24IMPLICATIONS
- 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.
25CREDIT 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.
26WHAT 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?
27SAND 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.
28MODELING 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
29INDEX SPREADS (Default Premium)
30Tranche spreads of CDX.NA.IG8
31EFFECTS OF RISING VOLATILITY
32EFFECTS OF RISING CORRELATION
33UNDERSTANDING 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
34- 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.
35WERE WE PREPARED?
36WHAT IS NEXT?
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