Downside Risk: Downside Risk: Implications for Implications for
Financial Management Financial Management Robert EngleRobert Engle
NYU Stern School of BusinessNYU Stern School of BusinessCarlos III, May 24,2004Carlos III, May 24,2004
WHAT IS ARCH?WHAT IS ARCH?
Autoregressive Conditional HeteroskedasticityAutoregressive Conditional Heteroskedasticity
Predictive (conditional)Predictive (conditional) Uncertainty (heteroskedasticity)Uncertainty (heteroskedasticity) That fluctuates over time (autoregressive)That fluctuates over time (autoregressive)
THE SIMPLEST PROBLEM THE SIMPLEST PROBLEM ––WHAT IS VOLATILITY WHAT IS VOLATILITY NOWNOW??
One answer is the standard deviation over the One answer is the standard deviation over the last 5 yearslast 5 years But this will include lots of old information that But this will include lots of old information that
may not be relevant for short term forecastingmay not be relevant for short term forecasting Another answer is the standard deviation over Another answer is the standard deviation over
the last 5 daysthe last 5 days But this will be highly variable because there is so But this will be highly variable because there is so
little informationlittle information
THE ARCH ANSWERTHE ARCH ANSWER
Use a weighted average of the volatility over a long Use a weighted average of the volatility over a long period with higher weights on the recent past and period with higher weights on the recent past and small but nonsmall but non--zero weights on the distant past.zero weights on the distant past.
Choose these weights by looking at the past data; Choose these weights by looking at the past data; what forecasting model would have been best what forecasting model would have been best historically? This is a statistical estimation problem.historically? This is a statistical estimation problem.
FINANCIAL ECONOMETRICSFINANCIAL ECONOMETRICS
THIS MAY ALSO BE THE BIRTH OF THIS MAY ALSO BE THE BIRTH OF FINANCIAL ECONOMETRICSFINANCIAL ECONOMETRICS
STATISTICAL MODELS DEVELOPED STATISTICAL MODELS DEVELOPED SPECIFICALLY FOR FINANCIAL SPECIFICALLY FOR FINANCIAL APPLICATIONSAPPLICATIONS
TODAY THIS IS A VERY POPULAR AND TODAY THIS IS A VERY POPULAR AND ACTIVE RESEARCH AREA WITH MANY ACTIVE RESEARCH AREA WITH MANY APPLICATIONSAPPLICATIONS
FROM THE SIMPLE ARCH FROM THE SIMPLE ARCH GREW:GREW:
GENERALIZED ARCH (Bollerslev) a most GENERALIZED ARCH (Bollerslev) a most important extensionimportant extension
Tomorrow’s variance is predicted to be a Tomorrow’s variance is predicted to be a weighted average of the weighted average of the Long run average varianceLong run average variance Today’s variance forecastToday’s variance forecast The news (today’s squared return)The news (today’s squared return)
ANDAND
EGARCH (Nelson) very important as it EGARCH (Nelson) very important as it introduced introduced asymmetryasymmetry Weights are different for positive and negative Weights are different for positive and negative
returnsreturns
NEW ARCH MODELSNEW ARCH MODELS GJRGJR--GARCHGARCH TARCHTARCH STARCHSTARCH AARCHAARCH NARCHNARCH MARCHMARCH SWARCHSWARCH SNPARCHSNPARCH APARCHAPARCH TAYLORTAYLOR--SCHWERTSCHWERT
FIGARCHFIGARCH FIEGARCHFIEGARCH Component Component Asymmetric ComponentAsymmetric Component SQGARCHSQGARCH CESGARCHCESGARCH Student tStudent t GEDGED SPARCHSPARCH
ROLLING WINDOW ROLLING WINDOW VOLATILITIESVOLATILITIES
NUMBER OF DAYS=5, 260, 1300NUMBER OF DAYS=5, 260, 1300
.0
.2
.4
.6
64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 00 02
V5 V260 V1300
ARCH/GARCH ARCH/GARCH VOLATILITIESVOLATILITIES
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
65 70 75 80 85 90 95 00
GARCHVOL
CONFIDENCE INTERVALSCONFIDENCE INTERVALS
-.10
-.05
.00
.05
.10
1990 1992 1994 1996 1998 2000 2002
3*GARCHSTD SPRETURNS -3*GARCHSTD
FINANCIAL FINANCIAL APPLICATIONSAPPLICATIONS
VALUE AT RISKVALUE AT RISK
Future losses are uncertain. Find a LOSS that Future losses are uncertain. Find a LOSS that you are 99% sure is worse than whatever will you are 99% sure is worse than whatever will occur. This is the occur. This is the Value at RiskValue at Risk.. One day in advanceOne day in advance Many days in advanceMany days in advance
This single number (a quantile) is used to This single number (a quantile) is used to represent a full distribution. It can be represent a full distribution. It can be misleading.misleading.
CALCULATING VaRCALCULATING VaR
Forecast the one day standard deviationForecast the one day standard deviation––GARCH style models are widely used. Then:GARCH style models are widely used. Then: Assuming normality, multiply by Assuming normality, multiply by 2.332.33 Without assuming normality, multiply by the Without assuming normality, multiply by the
quantile of the standardized residuals. quantile of the standardized residuals. For the example, multiplier = For the example, multiplier = 2.652.65
MULTIMULTI--DAY HORIZONSDAY HORIZONS
If volatility were constant, then the multiIf volatility were constant, then the multi--day day volatility would simply require multiplying by volatility would simply require multiplying by the square root of the days.the square root of the days.
Because volatility is dynamic and asymmetric, Because volatility is dynamic and asymmetric, the lower tail is more extreme and the VaR the lower tail is more extreme and the VaR should be greater. should be greater.
TWO PERIOD RETURNSTWO PERIOD RETURNS
Two period return is the Two period return is the sum of two one period sum of two one period continuously continuously compounded returnscompounded returns
Look at binomial tree Look at binomial tree versionversion
Asymmetry gives Asymmetry gives negative negative skewnessskewness
High variance
Low variance
MULTIPLIER FOR 10 DAYSMULTIPLIER FOR 10 DAYS
For a 10 day 99% value at risk, conventional For a 10 day 99% value at risk, conventional practice multiplies the daily standard deviation practice multiplies the daily standard deviation by by 7.367.36
For the same multiplier with asymmetric For the same multiplier with asymmetric GARCH it is simulated from the example to beGARCH it is simulated from the example to be7.887.88
Bootstrapping from the residuals the multiplier Bootstrapping from the residuals the multiplier becomes becomes 8.528.52
OPTIONSOPTIONS Traded options always have multiple days to Traded options always have multiple days to
expiration. expiration. Hence the distribution of future price levels is Hence the distribution of future price levels is
negatively skewed.negatively skewed. Thus the Black Scholes implied volatility should Thus the Black Scholes implied volatility should
depend on strike if options are priced by GARCH.depend on strike if options are priced by GARCH. A skew in implied volatility will result from A skew in implied volatility will result from
Asymmetric GARCH, at least for short maturities. Asymmetric GARCH, at least for short maturities.
IMPLIED VOLATILITY SKEW IMPLIED VOLATILITY SKEW FOR 10 DAY OPTIONFOR 10 DAY OPTION
From simulated From simulated (risk neutral)(risk neutral) final values, find final values, find average put option payoff for each strike.average put option payoff for each strike.
Calculate Black Scholes implied volatilities Calculate Black Scholes implied volatilities and plot against strike.and plot against strike.
Notice the clear downward slope. This would Notice the clear downward slope. This would be zero for constant volatility.be zero for constant volatility.
PUT PRICESPUT PRICES
0
10
20
30
40
50
60
920 960 1000 1040 1080
K
PU
T
PUT IMPLIED VOLATILITIESPUT IMPLIED VOLATILITIES
.144
.148
.152
.156
.160
.164
.168
920 960 1000 1040 1080
K
PU
TIM
P
PRICING KERNELPRICING KERNEL
The observed skew is even steeper than this. The observed skew is even steeper than this. Engle and Rosenberg(2002) explain the Engle and Rosenberg(2002) explain the
difference by a risk premiumdifference by a risk premium Investors are especially willing to pay to avoid Investors are especially willing to pay to avoid
a big market drop. a big market drop. Others describe this in terms of jumps and risk Others describe this in terms of jumps and risk
premia on the jumpspremia on the jumps
WHAT IS NEXT?WHAT IS NEXT?MULTIVARIATE MODELSMULTIVARIATE MODELS--
DCC or Dynamic Conditional DCC or Dynamic Conditional CorrelationCorrelation
HIGH FREQUENCY MODELSHIGH FREQUENCY MODELS--Market MicrostructureMarket Microstructure
THE MULTIVARIATE PROBLEMTHE MULTIVARIATE PROBLEM
Asset Allocation and Risk Management Asset Allocation and Risk Management problems require large covariance matricesproblems require large covariance matrices
Credit Risk now also requires big correlation Credit Risk now also requires big correlation matrices to accurately model loss or default matrices to accurately model loss or default correlationscorrelations
Multivariate GARCH has never been widely Multivariate GARCH has never been widely used used –– it is too difficult to specify and estimateit is too difficult to specify and estimate
DDynamic ynamic CConditional onditional CCorrelationorrelation
DCC is a new type of multivariate DCC is a new type of multivariate GARCH model that is particularly GARCH model that is particularly convenient for big systems. See convenient for big systems. See Engle(2002) or Engle(2004).Engle(2002) or Engle(2004).
DCCDCC
1.1. Estimate volatilities for each asset and compute the Estimate volatilities for each asset and compute the standardized residualsstandardized residuals or or volatility adjusted volatility adjusted returnsreturns..
2.2. Estimate the time varying covariances between Estimate the time varying covariances between these using a maximum likelihood criterion and one these using a maximum likelihood criterion and one of several models for the correlations.of several models for the correlations.
3.3. Form the correlation matrix and covariance matrix. Form the correlation matrix and covariance matrix. They are guaranteed to be positive definite.They are guaranteed to be positive definite.
HOW IT WORKSHOW IT WORKS
When two assets move in the same direction, When two assets move in the same direction, the correlation is increased slightly.the correlation is increased slightly.
When they move in the opposite direction it is When they move in the opposite direction it is decreased.decreased.
This effect may be stronger in down markets. This effect may be stronger in down markets. The correlations often are assumed to only The correlations often are assumed to only
temporarily deviate from a long run meantemporarily deviate from a long run mean
Two period Joint ReturnsTwo period Joint Returns
If returns are both If returns are both negative in the first negative in the first period, then correlations period, then correlations are higher.are higher.
This leads to lower tail This leads to lower tail dependence dependence
Up Market
Down Market
DCC and the CopulaDCC and the Copula
A symmetric DCC model gives higher tail A symmetric DCC model gives higher tail dependence for both upper and lower tails of dependence for both upper and lower tails of the multithe multi--period joint density.period joint density.
An asymmetric DCC or ASYAn asymmetric DCC or ASY--DCC gives DCC gives higher tail dependence in the lower tail of the higher tail dependence in the lower tail of the multimulti--period density.period density.
TestingandValuingDynamicCorrelationsfor
AssetAllocation
Robert Engle and Robert Engle and RiccardoRiccardo ColacitoColacitoNYU SternNYU Stern
A Model for Stocks and BondsA Model for Stocks and Bonds
Daily returns on S&P500 FuturesDaily returns on S&P500 Futures
Daily returns on 10Daily returns on 10--year Treasury Note year Treasury Note FuturesFutures
Both from DataStream from Jan 1 1990 to Dec Both from DataStream from Jan 1 1990 to Dec 18 200218 2002
SUMMARY STATISTICSSUMMARY STATISTICS
5.15.18.18.1KurtosisKurtosis
.06.06CorrelationCorrelation
6.2%6.2%17.2%17.2%Annual Annual VolVol
2.0%2.0%8.6%8.6%Annual Annual MeanMean
10 Yr 10 Yr TreasTreasFut.Fut.
S&P 500S&P 500
200
400
600
800
1000
1200
1400
1600
1800
80
85
90
95
100
105
110
115
120
1990 1992 1994 1996 1998 2000 2002
S&P500 INDEX 10 YR TREASURY NOTE FUTURES
Volatilities and CorrelationsVolatilities and Correlations
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
500 1000 1500 2000 2500 3000
Volatility of Bonds Volatility of SP500
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
500 1000 1500 2000 2500 3000
Asy-DCC of SP500 and Bonds
THE FORMULATIONTHE FORMULATION
Solve a series of portfolio problems with a Solve a series of portfolio problems with a riskless assetriskless asset
Where Where rr00 is the required is the required excessexcess return and return and µµ is is a vector of a vector of excessexcess expected returnsexpected returns
With the true covariance matrix you can With the true covariance matrix you can achieve lower volatility or higher required achieve lower volatility or higher required returns than with the incorrect one.returns than with the incorrect one.
0
min '
. . 'tw
w H w
s t w rµ >
INTERPRETING RESULTSINTERPRETING RESULTS
A number such as 105 means required excess A number such as 105 means required excess returns can be 5% greater with correct returns can be 5% greater with correct correlations without increasing volatility.correlations without increasing volatility.
E.g. a 4% excess return with incorrect correlation E.g. a 4% excess return with incorrect correlation would be a 4.2% return with correct correlations.would be a 4.2% return with correct correlations.
With 10% required return, the value of such With 10% required return, the value of such correlations is 50 basis points.correlations is 50 basis points.
AN EXPERIMENTAN EXPERIMENT
Simulate 10,000 days of the DCC model Simulate 10,000 days of the DCC model documented above. documented above.
One investor knows the volatilities and One investor knows the volatilities and correlations every day, correlations every day, ΩΩ..
The other only knows the unconditional The other only knows the unconditional volatilities and correlations, volatilities and correlations, HH
What is the gain to the informed investor?What is the gain to the informed investor?
VALUE GAINSVALUE GAINSStocks Stocks vsvs Bonds, Simulated Data, Full CovarianceBonds, Simulated Data, Full Covariance
100
101
102
103
104
105
106
107
108
109
0.00% 0.16% 0.31% 0.45% 0.59% 0.71% 0.81% 0.89% 0.95% 0.99% 1.00%
Extreme CorrelationsExtreme Correlations(simulated data, full covariance)(simulated data, full covariance)
90
95
100
105
110
115
120
125
0 0.16 0.31 0.45 0.59 0.71 0.81 0.89 0.95 0.99 1
Bottom 5%Top 5%
Volatility ratiosVolatility ratiosStocks Stocks vsvs Bonds ( actual data with estimated DCC)Bonds ( actual data with estimated DCC)
100
102
104
106
108
110
112
0.00%0.16%0.31%0.45%0.59%0.71%0.81%0.89%0.95%0.99%1.00%
SP500 vs. DOW JONESSP500 vs. DOW JONES
Correlation and return structure of equity Correlation and return structure of equity indices is very differentindices is very different
Unconditional correlations are about .9Unconditional correlations are about .9
Asymmetry is greaterAsymmetry is greater
Expected returns are probably nearly equalExpected returns are probably nearly equal RESULTS ARE ABOUT THE SAMERESULTS ARE ABOUT THE SAME
VALUE GAINSVALUE GAINSSP500 SP500 vsvs DOW, Simulated Data, Full CovarianceDOW, Simulated Data, Full Covariance
100
102
104
106
108
110
112
114
0.00% 0.16% 0.31% 0.45% 0.59% 0.71% 0.81% 0.89% 0.95% 0.99% 1.00%
MONTHLY REBALANCINGMONTHLY REBALANCING
Monthly rebalancing lies between rebalancing Monthly rebalancing lies between rebalancing every day and never rebalancing.every day and never rebalancing.
The monthly joint distribution is asymmetric The monthly joint distribution is asymmetric with important lower tail dependence.with important lower tail dependence.
Daily myopic rebalancing takes account of this Daily myopic rebalancing takes account of this asymmetryasymmetry
Additional gains are possible with daily multiAdditional gains are possible with daily multi--period optimizationperiod optimization
INTEGRATING RISK MANAGEMENT INTEGRATING RISK MANAGEMENT AND ASSET ALLOCATIONAND ASSET ALLOCATION
Asset Allocation is considered monthly because only Asset Allocation is considered monthly because only at this frequency can expected returns be updatedat this frequency can expected returns be updated
Within the month, volatilities can be updatedWithin the month, volatilities can be updated Rebalancing can be done with futures Rebalancing can be done with futures –– portfolio portfolio
volatility can be reducedvolatility can be reduced Risk management can be done with futures or other Risk management can be done with futures or other
derivativesderivatives In this way, firms can integrate risk management and In this way, firms can integrate risk management and
asset allocationasset allocation
LONG RUN RISKSLONG RUN RISKS
A position may have little risk in the short run A position may have little risk in the short run but depending on some outcome, will have but depending on some outcome, will have much more in the long runmuch more in the long run For example credit riskFor example credit risk For example energy sectorFor example energy sector For example downside correlationsFor example downside correlations
Invest today taking account of the possibility Invest today taking account of the possibility that the risk will be higher in the future. that the risk will be higher in the future.
CONCLUSIONSCONCLUSIONS
The value of accurate daily correlations is The value of accurate daily correlations is moderate moderate –– maybe 5% of the required return. maybe 5% of the required return. Possibly why asset allocation is done monthly Possibly why asset allocation is done monthly and ignores covariances.and ignores covariances.
On some days, the value is much greater. On some days, the value is much greater. Possibly why risk management is done daily.Possibly why risk management is done daily.
Additional value may flow from coordinating Additional value may flow from coordinating these decisions.these decisions.
