Commodity Betas - Recent Advances in Commodity Markets

Report
ROBERT ENGLE
DIRECTOR: VOLATILITY INSTITUTE AT NYU STERN
RECENT ADVANCES IN COMMODITY MARKETS
QUEEN MARY, NOV,8,2013
 Asset
prices change over time as new
information becomes available.
 Both public and private information will
move asset prices through trades.
 Volatility is therefore a measure of the
information flow.
 Volatility is important for many economic
decisions such as portfolio construction on
the demand side and plant and equipment
investments on the supply side.
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 Investors
with short time horizons will be
interested in short term volatility and its
implications for the risk of portfolios of
assets.
 Investors with long horizons such as
commodity suppliers will be interested in
much longer horizon measures of risk.
 The difference between short term risk and
long term risk is an additional risk – “The risk
that the risk will change”
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 The
commodity market has moved swiftly
from a marketplace linking suppliers and
end-users to a market which also includes a
full range of investors who are speculating,
hedging and taking complex positions.
 What are the statistical consequences?
 Commodity producers must choose
investments based on long run measures of
risk and reward.
 In this paper I will try to assess the long run
risk in these markets.
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 The
most widely used set of commodities
prices is the GSCI data base which was
originally constructed by Goldman Sachs and
is now managed by Standard and Poors.
 I will use their approximation to spot
commodity price returns which is generally
the daily movement in the price of near term
futures. The index and its components are
designed to be investible.
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 Using
daily data from 1996 to July, 2012,
annualized measures of means and
volatilities are constructed for 21 different
commodities. These are roughly divided into
agricultural, industrial, precious metals and
energy products.
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 What
annual return from today will be worse
than the actual return 99 out of 100 times?
 What is the 1% quantile for the annual
percentage change in the price of an asset?
 Assuming
constant volatility and a normal
distribution, it just depends upon the
volatility. Here is the result. Here also is
the actual 1% quantile of overlapping annual
returns for each series since 1996.
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1%
$ GAINS
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0.0
ALUMINUM
BIOFUEL
BRENT_CRUDE
COCOA
COFFEE
COPPER
CORN
COTTON
GOLD
HEATING_OIL
LEAD
LIGHT_ENERGY
LIVE_CATTLE
NATURAL_GAS
NICKEL
ORANGE_JUICE
PLATINUM
SILVER
SOYBEANS
SUGAR
UNLEADED_GAS
WHEAT
80.0
70.0
60.0
50.0
40.0
30.0
20.0
Normal 1% VaR
1%Realized
10.0
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 Like
most financial assets, volatilities change
over time.
 Vlab.stern.nyu.edu is a web site at the
Volatility Institute that estimates and
updates volatility forecasts every day for
several thousand assets. It includes these
and other GSCI assets.
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 We
would like a forward looking measure of
VaR that takes into account the possibility
that the risk will change and that the shocks
will not be normal.
 LRRISK calculated in VLAB does this
computation every day.
 Using an estimated volatility model and the
empirical distribution of shocks, it simulates
10,000 sample paths of commodity prices.
The 1% and 5% quantiles at both a month and
a year are reported.
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 Some
commodities are more closely
connected to the global economy and
consequently, they will find their long run
VaR depends upon the probability of global
decline.
 We can ask a related question, how much
will commodity prices fall if the
macroeconomy falls dramtically?
 Or, how much will commodity prices fall if
stock prices fall.
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 Estimate
the model
yt     xt   t
 Where y is the logarithmic return on a
commodity price and x is the logarithmic
return on an equity index.
 If beta is time invariant and epsilon has
conditional mean zero, then MES and LRMES
can be computed from the Expected Shortfall
of x.
 But is beta really constant?
 Is epsilon serially uncorrelated?
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This is a new method for estimating betas that
are not constant over time and is particularly
useful for financial data. See Engle(2012).
 It has been used to determine the expected
capital that a financial institution will need to
raise if there is another financial crisis and here
we will use this to estimate the fall in
commodity prices if there is another global
financial crisis.
 It has also been used in Bali and
Engle(2010,2012) to test the CAPM and ICAPM.

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 ROLLING
REGRESSION
 INTERACTING VARIABLES WITH TRENDS,
SPLINES OR OTHER OBSERVABLES
 TIME VARYING PARAMETER MODELS BASED ON
KALMAN FILTER
 STRUCTURAL BREAK AND REGIME SWITCHING
MODELS
 EACH OF THESE SPECIFIES CLASSES OF
PARAMETER EVOLUTION THAT MAY NOT BE
CONSISTENT WITH ECONOMIC THINKING OR
DATA.
 IF
 yt , xt  ,
t  1,..., T
is a collection of
k+1 random variables that are distributed as
   y ,t   H yy ,t
 yt 
  Ft 1 ~ N  t , H t   N     ,  H
 xt 
  x ,t   xy ,t
H yx ,t  

H xx ,t  
 Then

yt xt , Ft 1 ~ N  y ,t  H yx,t H xx1,t  xt  x,t  , H yy ,t  H yx,t H xx1,t H xy ,t
 Hence:
t  H xx1,t H xy,t

 We
require an estimate of the conditional
covariance matrix and possibly the
conditional means in order to express the
betas.
 In regressions such as one factor or multifactor beta models or money manager style
models or risk factor models, the means are
insignificant and the covariances are
important and can be easily estimated.
 In one factor models this has been used since
h
Bollerslev, Engle and Wooldridge(1988) as t  yx,t
hxx.t
 Econometricians
have developed a wide
range of approaches to estimating large
covariance matrices. These include






Multivariate GARCH models such as VEC and BEKK
Constant Conditional Correlation models
Dynamic Conditional Correlation models
Dynamic Equicorrelation models
Multivariate Stochastic Volatility Models
Many many more
 Exponential
Smoothing with prespecified
smoothing parameter.

For none of these methods will beta appear
constant.

In the one regressor case this requires the ratio
of hyx ,t / hxx ,t to be constant.

This is a non-nested hypothesis

More precisely it is a partially nested model.
The point at which these models are nested is
when there is no heteroskedasticity and hence
they are identical. Pretest for
heteroskedasticity.
 Create
a model that nests both hypotheses.
 Test the nesting parameters
 Four possible outcomes




Reject f
Reject g
Reject both
Reject neither
 Consider
the model:
yt   ' xt   t  ' xt  vt
 If
f = 0, the parameters are constant
 If q = 0 , the parameters are time varying.
 If both are non-zero, the nested model may
be entertained.
 Notice that with several regressors there are
many possible outcomes.
 SUGGESTION: Nested Model is the MODEL
 Estimate
regression of commodity returns on
SP 500 returns. There is substantial
heteroskedasticity in residuals.
 Sample is daily 1996 – July 2012
 Do




Rolling Regression Model
Estimate beta from sample of t-n-1 to t-1
Using this estimated beta calculate residual at t
Compute sum of squared residuals for all t
Minimize over n
 Note:
no correction for heteroskedasticity
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BETA_ALUMINUM
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12
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The daily decay rate of the correlations is .998.
This is very slow moving but is indeed mean
reverting. It has a half life of about one and a
half years.
 The GJR-GARCH model for aluminum has a
persistence of .993 for a half life of about 100
days.
 The GJR-GARCH model for SP_500 is highly
asymmetric and has a persistence of .9868 for a
half life of 50 days.
 The beta is the correlation times the ratio of
these two volatilities, it is not clear how
persistent it really is.

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BETA_ALUMINUM
BETA_BIOFUEL
1.6
1.6
1.2
1.2
0.8
0.8
0.4
0.4
0.0
0.0
BETA_BRENT_CRUDE
2
1
0
-0.4
-1
-0.4
96
98
00
02
04
06
08
10
12
-2
96
98
00
BETA_COCOA
02
04
06
08
10
12
96
98
00
BETA_COFFEE
1.6
1.5
1.2
02
04
06
08
10
12
08
10
12
08
10
12
08
10
12
08
10
12
08
10
12
08
10
12
BETA_COPPER
1.5
1.0
1.0
0.8
0.5
0.5
0.4
0.0
0.0
0.0
-0.4
-0.5
-0.5
96
98
00
02
04
06
08
10
12
-1.0
96
98
00
BETA_CORN
02
04
06
08
10
12
96
98
00
BETA_COTTON
1.0
1.6
0.8
1.2
02
04
06
BETA_GOLD
.8
.4
0.6
0.8
0.4
.0
0.4
0.2
-.4
0.0
0.0
-0.2
-0.4
96
98
00
02
04
06
08
10
12
-.8
96
98
00
BETA_HEATING_OIL
02
04
06
08
10
12
96
98
BETA_LEAD
2
2.0
02
04
06
1.5
1.5
1
00
BETA_LIGHT_ENERGY
1.0
1.0
0
0.5
0.5
-1
0.0
0.0
-2
-0.5
96
98
00
02
04
06
08
10
12
-0.5
96
98
BETA_LIVE_CATTLE
00
02
04
06
08
10
12
96
98
00
BETA_NATURAL_GAS
.4
02
04
06
BETA_NICKEL
1.2
2.0
.3
1.5
0.8
.2
1.0
.1
0.4
0.5
.0
0.0
0.0
-.1
-.2
-0.4
96
98
00
02
04
06
08
10
12
-0.5
96
98
00
BETA_PLATINUM
02
04
06
08
10
12
96
98
BETA_SILVER
1.2
2.0
0.8
1.5
0.4
1.0
0.0
0.5
00
02
04
06
BETA_SOY BEANS
1.2
0.8
0.4
-0.4
0.0
-0.8
-0.5
96
98
00
02
04
06
08
10
12
0.0
-0.4
96
98
BETA_SUGAR
00
02
04
06
08
10
12
96
2
1.2
1.0
1
0.8
0.5
0
0.4
0.0
-1
0.0
-0.5
-2
-0.4
98
00
02
04
06
00
08
10
12
96
98
00
02
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06
08
02
04
06
BETA_WHEAT
1.5
96
98
BETA_UNLEADED_GAS
10
12
96
98
00
02
04
06
36
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 DCB
has smaller (better) SIC for all 21
commodities.
 This is not the case for other data sets. For
FF data on industry sectors, about half favor
constant beta and half favor time variation.
 Two of the constant betas are insignificant at
5% value.
 One of the dynamic betas fails to have a
t-stat greater than two.
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 Set
s=t
 Grid
search yields
(g = .031, c = (2 / 05 / 09))
 Schwarz
Criterion STR
3.220
 Schwarz Criterion Constant beta 3.247
 Schwarz Criterion DCB
3.216
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1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
96
97
98
99
00
01
02
03
04
05
06
07
08
09
10
BETANEST_ALUMINUM
BETALSTR_ALUMINUM
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11
12
 Are
these changes in beta permanent?
 Will resources become decoupled from broad
equity indices?
 Today the stock market is rising while
commodities are tanking.
 STR model cannot adjust to a return very
quickly because it is difficult to see regime
changes until there is sufficient data after
the change.
 What does DCB show?
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 The
one year VaR changes over time as the
volatility changes.
 The betas on equity markets have risen
dramatically since the financial crisis.
 The value of commodities for diversification
has been reduced but not eliminated.
 There is evidence post sample that
correlations for some commodities are mean
reverting while others are not.
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