Transcript Correlation Risk

Correlation Risk and Interest Rate
Swaps Spreads
BY
JAHANGIR SULTAN
Department of Finance
BENTLEY COLLEGE
WALTHAM, MA 02254
781-891-2518
[email protected]
Correlation Risk
“The rocket scientists of the financial world have
become obsessed with a phenomenon that they
cannot price or hedge directly but which they see
as the final piece in the jigsaw of risks they must
understand and control if they are to manage
portfolios effectively. That phenomenon is
correlation. Identifying and quantifying
correlation risk has become the derivative markets'
holy grail." (Parsley, Euromoney,1993)
Correlation Risk: Definition
• Correlation risk refers to the
change in the payoff/marked to
market value of an asset when
the correlation between the
underlying assets changes over
time.
Correlation Dependent Products
• Typical equity derivatives book contains volatility
risk as well correlation risk –two are correlated—
e.g., index arbitrage desk
• Products that pay off as a function of max-min of
two random processes/underlying stochastics
• Quanto options-requires estimating expected
correlation between exchange rate and the equity
index
• Interest Rate Swaps/Currency Swaps/Diff swaps
• Spreads
Sources of Correlation Risk
•Interdependence among risk factors –co-movement
•Contagion –structural breaks in data generating process
• Opposite of interdependence
•Deviation from a no-arbitrage relationship
•Time varying distribution
• Bivariate joint distribution of the underlying assets is
time varying (non-parallel yield curve shifts)
• Market breaks–extreme events –regime shifts
• Up market/down market
Correlation Risk: Interdependence
• Sensitivity of portfolio value to one risk factor is
dependent of the level of another risk factor.
Change in one risk factor affects the price
sensitivity of other risk factors
• Example
•CF (X1, X2) = CF(X1) * CF (X2)
•CF (cash flows)
•X1, X2 (risk factors)
Correlation Risk: Multi-factors
• Return generating process includes effects of
multi-factors
• Variance of the portfolio for example could
include own factor variances of assets as well as
correlations with other factors
hs , f  ( hs,t * h f,t ) *  t ( s, f )
Correlation Risk: Contagion
• Opposite of interdependence
• Increase in variance of one firm’s returns is
caused by economy wide relationship between
firm and its lending partners, suppliers, and
other related entities –wide spread credit
problem
• Excess correlation –during periods of market
breakdowns correlation is greater than under
normal market conditions
Correlation Risk:
Time Varying Distribution
• If the bivariate joint distribution of the underlying
assets is time varying then the covariance matrix is
changing over time;
 s 
 | t -1 ~ D(0, H t )
 f 
• So the correlation is also changing over time.
Estimating Measures of Correlation Risk
•Methods of estimating correlation risk:
•Implied correlation from options
•Error-correction term
• (represents short-term uncorrelatedness in a longterm steady-state equilibrium model)
•Time Varying correlations
• (multivariate GARCH models of two underlying
assets)
Error Correction Model (ECM)
•
ECM is derived from a cointegrating relationship between
two fundamentally related variables.
•
ECM represents the amount of short-term deviation of
variables from a no-arbitrage relationship between them. It
represents the amount of correction needed to bring the
system back to equilibrium.
ECM t = S t -1 -  F t -1
Multivariate ECM-GARCH Model
 ln S t =  0   1(ln St  1  ln Ft  1) +  s
 ln F t =  0 +  1(ln St  1  ln Ft  1)   f
 s 
   t -1 ~ D(0, H t )
 f 
 hs,t    s   a11 a12 a13    2s,t -1  b11 b12 b13   hs,t -1 

 
    
 


H t =hsf,t  =  sf  + a21 a22 a23  s,t -1  f,t -1 + b21 b22 b23 hsf,t -1
 h f,t   f  a31 a32 a33   2  b31 b32 b33  h f,t -1 
   
 
 f,t -1 
Multivariate GARCH Model
• To avoid convergence problems the H matrix is assumed to be
block-diagonal. This intuitively plausible assumption says that
volatilities are determined by the lagged squared residuals as
well as past volatilities much the same way as an ARMA model
is defined.
 hs,t    s   a11 0 0    2s,t -1  b11 0 0   hs,t -1 

 
    




H t =hsf,t  =  sf  + 0 a 22 0   s,t -1  f,t -1 + 0 b22 0  hsf,t -1
 h f,t   f  0 0 a33   2  0 0 b33  h f,t -1 
   


 f,t -1 
Time Varying Covariance Matrix
• Time Varying Hedge Ratio:
• Time Varying Correlations:
ht =
h sf,t
h f,t
hs , f
 t ( s, f ) =
( hs,t * h f,t )
Usefulness of Multivariate GARCH Models
• Volatility Spillover - off diagonal coefficients in H can help
trace the impact of one market on another.
• Dynamic Hedge Ratio • Arbitrage Trigger - ECM term can be used as the trigger
mechanism to design trading filters (index arb/program
trading).
• VaR - the efficient estimation of time varying covariance
matrix also can be incorporated in VaR analysis.
Pricing Correlation Risk: ECM
• Estimate a univariate GARCH model by including the ECM
term in the mean and the variance equations
SSt =  0 +  1 ECMt +  t
2
=
+

ht ,ss 0  1  t -1 +  2 ht -1 +  3 ECM
t
Pricing Correlation Risk:
GARCH Correlations
•
Estimate a univariate GARCH model by including time
varying correlations in the mean and the variance equations:
SSt =  0 +  1  t,sf +  t
ht ,ss =  0 +  + 2 ht -1 + 3  t,sf
2
1 t -1
Hedging Correlation Risk
• Estimate time varying hedge ratio
• Dynamic hedging
• Rebalance when expected correlation changes
• Measure correlation over time and see if it is priced in the
payoff/volatility
• Examine correlation behavior during market drops and upturns
–regime shifts—significant?
• Measure correlation between volatilities
• Measure sensitivity to changes in correlation and use linear
hedging strategies –delta hedging
Hedging Correlation Risk: Instruments
•
•
•
•
•
Traditional risk management products
Spread options
Diversification strategy
Stress testing
Selecting flexible hedging instruments
Empirical Modeling of Correlation Risk
EXAMPLE
Correlation Risk in Interest Rate Swaps
• Fixed-floating swap rates
• Swap rates keyed off U.S.Treasury benchmark rates
• Floating side is the 6m USD Libor
• Swap rate reflects
•
future sequences of fixed and floating rates
•
uncertainty in changes in underlying interest rates
•
uncertainty regarding marked to market of swaps
Determinants of Swaps Spreads
• Greenblatt (1995), Brown, et al (1994), Duffie and Singleton (1997),
Cooper and Mello (1991), Brown, et al (1994), Evans and Parente-Bales
(1991), Sun, Sundaresan and Wang (1993), and Liu, Longstaff and
Mundell (2000)
• swaps spread is a reward for bearing both liquidity and default risk
• default risk is the largest component of the swap spread
• liquidity of Treasury bonds have the greatest impact on changes in
the swap spread
• Minton (1997)
• swaps spread as a portfolio of series of forward contracts and a
portfolio of non-callable corporate bonds
Determinants of Swaps Spreads: Volatility
ion Underlying Yield Curves
• Brown, et al (1994)
• increased uncertainty regarding the underlying yield curve
increases the expected probability of default on existing swap
contracts, which in turn leads to changes in the swap spread
because changes in the interest rates may alter the option-like
features embedded in the fixed side of the swap contract
• the crucial link between swap spread and the volatility of the
interest rates
• Soresnon and Bolier (1994)
• swap spreads respond to shape and the volatility of the yield
curve because these factors affect the option value embedded
in the replacement cost of the swap.
Determinants of Swaps Spreads: Volatility
of the Underlying Yield Curve
• Subramahnyam, Eom, and Uno (1998), Minton (1997), Brown
et al (1994), Lekkos and Milas (2001), Duffie and Singleton
(1997)
• currency swap spreads are directly linked to the curvature
(volatility) of the interest rate
• interest rate swaps spreads respond to the slope of the yield
curve, curvature of the yield curve.
Determinants of Swaps Spreads: Correlation
•
Since correlation is an important
component of the joint volatility structure
of two underlying interest rates, changes in
the correlation offers information on the
amount of uncertainty in the markets and
therefore should affect the volatility of the
swap spreads.
Determinants of Swaps Spreads: Parallel
Shifts in Yield Curve
•
An unexpected increase in the fixed rate increases the
market value of the swap for the fixed rate payer
•
An unexpected decline in the fixed interest rate
increases the floating rate payer's savings
•
Parallel shift: If the rates are highly correlated, the
economic value of swap contracts rises because there
is less uncertainty.
Determinants of Swaps Spreads : Parallel
Shifts in Yield Curve
• There is more uncertainty in the swap market
if the correlation is low because yield curve
shifts would not be parallel.
• Changes in the Treasury rate reflect
macroeconomic conditions; changes in
Libor reflect changes in the credit
conditions; so these changes are not
always synchronized.
Determinants of Swaps Spreads:
Unmatched Swaps
• Correlation risk also increases the dealer’s cost of hedging
the book
• dealers hedge their unmatched swaps using interest rate
futures contracts
• high correlation between interest rates implies better risk
reduction by using these futures contracts; low
correlation imply a reduction in the hedging
effectiveness of these futures contracts
• high correlation also reduces the uncertainty regarding
marked-to-market of existing swaps
Empirical Results
• Three swaps: 2yr, 5yr, and 10yr { 4/1/87 - 11/20/00}
• Rates are non-stationary in the levels
• Substantial leptokurtosis
• Second moments are time varying
• 3 multivariate GARCH models to derive time varying
correlations
• Plot
Empirical Results
(contemporaneous correlation)
•Estimate three univariate GARCH models with time varying
correlation in both mean and conditional variance equations:
Model
Mean Equation
Variance Equation
Sign
Significance
Sign
Significance
2yr
-
NS
-
1%
5yr
-
NS
-
1%
10yr
-
NS
-
5%
Robust (Bollerslev-Wooldridge heteroskedasticity consistent)
estimates. Convergence level=.0005. NS: Not significant at 5%.
Empirical Results
(lagged correlation)
•Estimate three univariate GARCH models with time varying
correlation in both mean and conditional variance equations:
Model
Mean Equation
Variance Equation
Sign
Significance
Sign
Significance
2yr
+
NS
-
1%
5yr
+
NS
-
1%
10yr
-
NS
-
5%
Robust (Bollerslev-Wooldridge heteroskedasticity consistent)
estimates. Convergence level=.0005. NS: Not significant at 5%.
Empirical Results
(change in correlation)
•Estimate three univariate GARCH models with time varying
correlation in both mean and conditional variance equations:
Model
Mean Equation
Variance Equation
Sign
Significance
Sign
Significance
2yr
-
1%
-
1%
5yr
-
1%
+
1%
10yr
-
1%
+
1%
Robust (Bollerslev-Wooldridge heteroskedasticity consistent)
estimates. Convergence level=.0005. NS: Not significant at 5%.
Conclusions
• Adds another dimension to understanding the role
of information in pricing and hedging financial
assets.
• Multivariate GARCH model allows us to examine
the nature of the correlation over time and also
factors that contribute to such time variations in
the correlation.
Conclusions
• Adds another dimension to understanding the role
of information in pricing and hedging financial
assets.
• Multivariate GARCH model allows us to examine
the nature of the correlation over time and also
factors that contribute to such time variations in
the correlation.
Conclusions
• Adds another dimension to understanding the role
of information in pricing and hedging financial
assets.
• Multivariate GARCH model allows us to examine
the nature of the correlation over time and also
factors that contribute to such time variations in
the correlation.