For over a decade, academic and industry economists argued that the negative correlation between returns on stocks and commodity futures was evidence that institutional investors should add commodity futures index funds as an asset class in their portfolio management strategies. Does this negative correlation give rise to the possibility of unexploited profit opportunity in the financial markets? Using a rational asset-pricing model, we argue that such a negative correlation could arise as a no-arbitrage equilibrium phenomenon and reflects traders’ perceptions about the fundamental processes driving the economy and commodity prices.Stocks - Rate of return ; Commodity futures

Parantap Basu

William T. Gavin

English

Research Papers in Economics

      Research Division 
          Federal Reserve Bank of St. Louis 
                   Working Paper Series 
 
 
 
 
Negative Correlation between Stock and Futures Returns: 
An Unexploited Hedging Opportunity? 
 
 
 
 
Parantap Basu 
and 
William T. Gavin 
 
 
 
 
Working Paper 2011-005C 
http://research.stlouisfed.org/wp/2011/2011-005.pdf 
 
 
 
January 2011 
Revised May 2012 
 
 
 
FEDERAL RESERVE BANK OF ST. LOUIS 
Research Division 
P.O. Box 442  
St. Louis, MO 63166 
 
______________________________________________________________________________________ 
The views expressed are those of the individual authors and do not necessarily reflect official positions of 
the Federal Reserve Bank of St. Louis, the Federal Reserve System, or the Board of Governors. 
Federal Reserve Bank of St. Louis Working Papers are preliminary materials circulated to stimulate 
discussion and critical comment. References in publications to Federal Reserve Bank of St. Louis Working 
Papers (other than an acknowledgment that the writer has had access to unpublished material) should be 
cleared with the author or authors. Negative Correlation between Stock and Futures
Returns: An Unexploited Hedging Opportunity?￿
Parantap Basu
Durham Business School, Durham University, UK
William T. Gavin
Federal Reserve Bank of St. Louis, USA
January 2011
Revised May 2012
Abstract
The negative correlation between equity and commodity futures re-
turns is widely perceived by investors as an unexploited hedging oppor-
tunity. A Lucas (1982) two-country asset-pricing model is adapted to
analyze the fundamentals driving equity and commodity futures returns.
Using the model we argue that such a negative correlation could arise as
a no-arbitrage equilibrium phenomenon and re￿ ect traders￿perceptions
about the growth fundamentals in oil and GDP and does not necessarily
indicate an arbitrage opportunity.
￿Without implicating we acknowledge John Donaldson and Bart Taub for very useful
comments.
11 Introduction
The Commodity Futures Modernization Act in 2000 gave large ￿nancial ￿rms
wide latitude in trading commodity derivatives.1 The institutional fund man-
agers shifted out of equities into commodity futures partly in the belief that
it represents a previously unrecognized hedge for business cycle risk. Greer
(2000) argues that commodity index funds as an asset class is underused while
the index returns are negatively correlated with stocks and bonds over the pe-
riod 1970-99. Gorton and Rowenhorst (2006) also found that the returns on
long positions in commodity futures are negatively correlated with the returns
from comparable bond and equity portfolios. Erb and Harvey (2006) report
a similar historical record but caution against using historical correlations to
make prospective portfolio allocations. Boyuksahin, Haigh and Robe (2010)
provide detailed evidence of the correlation between equity and commodity re-
turns and ￿nd that commodities did not provide enough diversi￿cation when it
was needed. Likewise Daskalaki and Skiapoulos (2011) provide out-of-sample
evidence that commodities as an asset class do not improve returns over port-
folios which include only traditional asset classes.
A common question that arises in all these extant studies is: Does the nega-
tive correlation between commodity and equity returns provide an unexploited
hedging opportunity? This question cannot be e⁄ectively answered without
an asset pricing model that identi￿es the common macroeconomic fundamen-
tals driving both commodity and equity returns. To the best of our knowledge,
there is no theoretical treatment of the common macroeconomic fundamentals
driving returns to both equities and commodity futures using general equilib-
rium principles.
In this paper, we adapt a Lucas (1982) international asset-pricing model to
analyze the fundamentals driving equity and commodity futures returns. We
show that in a frictionless complete market setting, even though households are
fully hedged, a negative correlation could arise as a no-arbitrage equilibrium.
Such a negative correlation by itself cannot be used as a hedging motive. In
the model, the home country is exposed to two types of endowment risks. The
￿rst is the business cycle risk of its own output. The second is the commodity
supply risk arising from the foreign endowment. We explicitly model the home
resident￿ s investments in commodity futures and equity. The model is kept quite
simple and stylized where oil, used as a stand-in for commodities generally, is
1See, Basu and Gavin (2011) for a documentation of the rise commodity trading.
2treated as a consumption item, and all returns are real. We demonstrate that
the correlation between equity and futures real returns depends crucially on the
variance-covariance matrix of these two economic fundamentals, oil and home
output .
A central implication of our two-country asset pricing models is that the
equity return is positively related to the growth rate of home output while the oil
futures return is determined by the growth rate of GDP and the news about the
future oil output. If home output and oil supply are positively correlated, then
a greater supply of home output signals better news about the oil output. Thus
during a business cycle boom (recession) equity return would be higher (lower)
while futures return would be lower (higher). The immediate implication is
that the equity and futures return negatively correlate as long as home output
and foreign oil production positively correlate, i.e., if the systematic risk of oil
and home output show positive comovement.
2 A Lucas Tree Model
There are two countries, home and foreign. At date t, the home country is
endowed with ya
t units of its own good and the foreign country owns yb
t units
of oil: The growth rates of these endowments evolve stochastically as a Markov
process with a stationary distribution. Agents receive direct utility from the
consumption of oil. At date t ,there are two consumables, home good (ca
t) and
imported oil (cb
t) respectively. Individuals in both countries are identical in
terms of preferences. The instantaneous utility function of households is: u(ca
t)+
v(cb
t): In view of the complete market nature of the ￿nancial environment, all
conceivable Arrow-Debreu securities can be traded. However, we will focus
only on four ￿nancial instruments which traders hold in equilibrium: (i) equity
claims (za
t+1) to future ￿ ows of home output which sell at the price qa
t today,
(ii) equity claims (zb
t+1) to future ￿ ows of oil which sell at the price qb
t today,
(iii) claims to future delivery of oil at a price of oil contracted today which we
call futures, (iv) a discount bond (bt+1) held domestically which pays a risk free
return rt+1 in the following period. Let f
j
t be the date t price for delivery of
one barrel of oil at date t+j and n
j
t be the number of barrels of oils contracted
at date t for delivery at date t+j; k be the number of such traded futures which
means j = 1;2:::k and st be the spot price of oil (or the real exchange rate)
which by de￿nition is equal to f0
t . By de￿nition, n0
t is the spot purchase of oil
3which is the same as cb
t:
The ￿ ow budget constraint facing the home country is:
ca
t + stcb
t + qa
t (za
t+1 ￿ za
t ) + qb
t(zb
t+1 ￿ zb
t) +
k X
j=1
f
j
t n
j
t + bt+1
= za
t ya
t + zb
tstyb
t +
k￿1 X
j=0
f
j
t n
j+1
t￿1 + (1 + rt)bt (1)
The home household receives direct utility from home goods, ca
t and oil cb
t:
In other words, the household maximizes the discounted stream of utilities :
E0
1 X
t=0
￿
t[u(ca
t) + v(cb
t)]
subject to (1), where 0 < ￿ < 1.
Foreign country￿ s problem is symmetric. It holds claims to home country￿ s
output and takes short position in issuing oil futures. In equilibrium, n
j
t add up
to zero across home and foreign because if the home takes a long position, the
foreign must take a corresponding short position. Hereafter, we will specialize
to the home country￿ s problem. The ￿rst order conditions are:
Home equity:
za
t+1 : u0(ca
t)qa
t = ￿Etu0(ca
t+1)fqa
t+1 + ya
t+1g (2)
Foreign equity:
zb
t+1 : u0(ca
t)qb
t = ￿Etu0(ca
t+1)fqb
t+1 + st+1yb
t+1g (3)
Futures:
n
j
t : f
j
t u0(ca
t) = ￿f
j￿1
t+1 u0(ca
t+1); j = 1;2;:::k (4)
Bond:
bt+1 : 1 + rt+1 = ￿
￿1Et
u0(ca
t)
u0(ca
t+1)
(5)
Spot :
f0
t = st =
v0(cb
t)
u0(ca
t)
(6)
4The equity price equations (2), (3) are standard. The futures price equation
(4) basically means that if a trader buys a j period futures today at the price
f
j
t ; he has the option to sell the same futures tomorrow at the price f
j￿1
t+1 and
make a notional capital gain or loss which explains the right hand term. Spot
price (6) is given by the intratemporal marginal rate of substitution between ca
and cb.
2.1 Calculating Futures Price
Using (4) and (6), we can write the futures price with one period prior to
maturity as:
f1
t+j￿1u0(ca
t+j￿1) = ￿Et+j￿1f0
t+j:u0(ca
t+j) (7)
Next note from (6) that
f0
t+j = st+j =
v0(cb
t+j)
u0(ca
t+j)
(8)
which upon substitution in (7) gives:
f1
t+j￿1u0(ca
t+j￿1) = ￿Et+j￿1v0(cb
t+j) (9)
Using (9 recursively backward and also noting that at each date, the spot
price is given by (6) one obtains,
f
j
t u0(ca
t) = ￿
jEtv0(cb
t+j) (10)
Since both countries have identical preferences and start with the same initial
wealth positions, international asset markets will be used to pool risks. We price
the assets assuming, as in Lucas, that perfect risk pooling occurs. This means
that home country will hold half of its own output claims and half of oil claims.
In equilibrium, ca
t = :5ya
t and cb
t = :5yb
t:
Assume log utilities u(ca
t) + v(cb
t) = lnca
t +lncb
t: It is easy to verify that the
futures price is given by:
f
j
t = ￿
jEt
"
ya
t
yb
t+j
#
(11)
52.2 Correlation between returns on equities and futures
The equilibrium equity prices are proportional to home production as follows:
qa
t = qb
t =
￿
1 ￿ ￿
ya
t
Note that the foreign equity price is also proportional to home production
because the real exchange rate st is ya
t =yb
t which means styb
t = ya
t :
This means that ex post returns on home and foreign stocks are equal. Call
this equity return RE
t+1: We have:
RE
t+1 = ￿
￿1ya
t+1
ya
t
(12)
which means
lnRE
t+1 = ￿ln￿ + ln(
ya
t+1
ya
t
) (13)
In other words, the ex post equity return is proportional to the growth rate
of home output.
The ex post return on futures (call it RF
t+1) is: f
j￿1
t+1 =f
j
t . Using (11), it
follows that
RF
t+1 =
￿
ya
t+1
ya
t
￿
:
Et+1
h
1
yb
t+j
i
Et
h
1
yb
t+j
i (14)
For the sake of tractability, hereafter we specialize to a one period futures
(which means that the futures horizon j = 1): Rewrite (14) as follows:
RF
t+1 =
￿
ya
t+1
ya
t
￿
:
Et+1
h
y
b
t
yb
t+1
i
Et
h
yb
t
yb
t+1
i
= > lnRF
t+1 = ln
ya
t+1
ya
t
+ ln
￿
yb
t
yb
t+1
￿
￿ lnEt
￿
yb
t
yb
t+1
￿
Last equality comes from the fact that
y
b
t+1
yb
t
is already realized at date t+1
which means that lnEt+1
h
y
b
t
yb
t+1
i
= ln
h
y
b
t
yb
t+1
i
:
6Next assuming a lognormal distribution for the growth rate of oil output
rewrite the above as2:
lnRF
t+1 = ln
ya
t+1
ya
t
￿
￿
ln
￿
yb
t+1
yb
t
￿
￿ lnEt
￿
yb
t+1
yb
t
￿￿
￿ 0:5vart ln
￿
yb
t+1
yb
t
￿
(15)
The ex post one period futures return depends on the growth rate of home
output (the ￿rst square bracket term) and the news about future production of
oil (the second square bracket term). Everything else equal, better news about
future oil production will depress the expected return to oil futures because
the signal that oil production will rise also signals a lower future spot price.
Likewise, a greater perceived uncertainty about oil production also depresses
future spot price.
The correlation between equity and oil futures returns is ambiguous. It
depends on the correlation between home production and oil production. To
see this clearly, assume an iid process for oil production which means that ln
(yb
t+1=yb
t)~N(￿;￿2): One can rewrite (14) as:
RF
t+1 =
￿
ya
t+1
ya
t
￿￿
yb
t
yb
t+1
￿
exp(￿ ￿ :5￿2)
which means
lnRF
t+1 = (￿ ￿ :5￿2) + ln
ya
t+1
ya
t
￿ ln
yb
t+1
yb
t
(16)
Using (13) and (16) one obtains:
covt(lnRE
t+1;lnRF
t+1) = vart(ln
ya
t+1
ya
t
) ￿ covt(ln
ya
t+1
ya
t
;ln"b
t+1) (17)
where covt(:) and vart(:) are conditional covariance and variance respectively.
If cov(ln
y
a
t+1
ya
t ;ln
y
b
t+1
yb
t
) > 0 and it exceeds var(ya
t+1); futures and equity returns
will be negatively correlated. This will happen in a no arbitrage equilibrium.
2For any lognormal random variable x, following property holds: lnE(xk) = kE(lnx) +
0:5k2var(lnx) where k is a constant.
72.3 A beta based intuition
The negative correlation between equity and futures returns can be understood
as an inverse association between the systematic risks of futures and oil. Suppose
a commodity investor wants to ascertain whether there is any predictable nega-
tive relationship between futures return and equity return. He runs a regression
of futures return (lnRF
t+1) on the contemporaneous equity return (lnRE
t+1).
This regression coe¢ cient is simply cov(lnRF
t+1;lnRE
t+1)=var(lnRE
t+1). Call
this regression coe¢ cient ￿
F which can be interpreted as the beta of the futures
given that RE
t+1 is the market portfolio.
Based on (12), (16) and (17), notice that this regression coe¢ cient is
1 ￿
n
cov(ln
y
a
t+1
ya
t ;lnyb
t+1=yb
t)=var(ln
y
a
t+1
ya
t )
o
: Next note that bracketed term is
simply the regression coe¢ cient of the log of oil growth on the log of home out-
put growth. Alternatively, this regression coe¢ cient can be interpreted as the
beta of oil (referred as betaoil) given that the home output captures all aggre-
gate risk. Thus equation (17) basically means the following tight relationship
between these two betas:
betaF = 1 ￿ betaOil (18)
Note that betaF on the left hand side of (18) represents the systematic risk in
the oil futures market. betaOil on the right hand side summarizes the systematic
risk in the real oil sector. The model predicts an inverse relation between betaF
and betaOil: If the systematic risk of oil is quite substantial (betaOil > 1), a
predictable relationship (a negative correlation) emerges between oil futures
return and equity returns which means a negative betaF. However, such a
negative relationship cannot be exploited by investors because it arises as a
no-arbitrage condition.
3 Conclusion
The negative correlation between equity and future returns is often interpreted
as a potential hedging opportunity for investors. In this short paper, we es-
tablish that such a negative correlation can easily arise in equilibrium as a no
arbitrage condition. We illustrate this point using a variant of Lucas (1982) in-
ternational consumption CAPM model. The model shows that the correlation
8between equity and oil futures returns stems from the variance and covariance
properties of GDP and oil production.
Our model uses general equilibrium perspective to understand the impli-
cations for correlation between equity and futures returns. The lesson that we
learn from such a general equilibrium exercise is that commodity and equity
markets are integrated and should not be studied in isolation. Thus a negative
correlation between these two returns should not necessarily be misconstrued
as a hedging opportunity to common macroeconomic shocks. Rather, it re￿ ects
the equilibrium response of equity and futures markets to fundamental shocks
driving the economy.
References
[1] Basu, Parantap, and William T. Gavin. ￿What Explains the Growth in
Commodity Derivatives?￿Federal Reserve Bank of St. Louis Review 93(1)
(2011).
[2] B￿y￿k‚ sahin, Bahattin, M. Haigh, and Michel Robe. "Commodities and Eq-
uities: Ever a "Market of One"?" Journal of Alternative Investments 12
(2010) 76-95.
[3] Daskalaki, Charoula and George Skiadopoulos. "Should Investors include
commodities in their portfolios after all? New Evidence." Journal of Banking
and Finance 35(2011) 2606-2626.
[4] Erb, Claude B. and Harvey, Campbell R. ￿The Strategic and Tactical Value
of Commodity Futures.￿Financial Analysts Journal, 62(2) (2006), 69-97.
[5] Gorton, Gary, and K. Geert Rouwenhorst. ￿Facts and Fantasies About Com-
modity Futures,￿Financial Analysts Journal 62(2) (2006), 47-68.
[6] Greer, Robert J. ￿The Nature of Commodity Index Returns.￿ Journal of
Alternative Investments, 3(1) (2000), 45-53.
[7] Lucas, Robert E. Jr. ￿Interest Rates and Currency Prices in a Two-Country
World,￿Journal of Monetary Economics 10 (1982) 335-359.
9

Negative correlation between stock and futures returns: an unexploited dedging opportunity?

http://research.stlouisfed.org/wp/2011/2011-005.pdf

Negative correlation between stock and futures returns: an unexploited dedging opportunity?

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics