LIBRARY
OF THE
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
Ucwc
NOV \4
'7S
my'
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
VARIANCE MINIMIZATION AND THE THEORY OF
INFLATION HEDGING
Zvi Bodie
WP 806-75
September 1975
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
VARIANCE MINIMIZATION AND THE THEORY OF
INFLATION HEDGING
Zvi Bodie
WP 806-75
September 1975
M.I.T.
LIBRARIES
NOV 171975
RECEIVED
-1-
Var lan ce MinlmizaLion aud the Theory
of Inflation Hedging
I.
Introduction
One of the casualties of the unprecedented infla-
tionary experience in this country in the past several
years has been the so-called riskless asset of portfolio
theory.
The widespread practice of dismissing short-run
inflation-risk as being negligibly small and treating
short term government securities as risk-free has become
questionable, to say the least.
Although in principle it is possible to eliminate
inflation uncertainty either by creating futures markets
or by linking deferred payments to price indices,
no
such option is currently available in the developed
capital markets of the world.
^
In other words, at least
in these capital markets there is no perfect inflation
hedge.
In
the absence of a perfect inflation hedge it is
Important to determine how and to what extent one can
hedge against
inflation with the financial Instruments
currently available, and that is the subject of this
paper.
0725868
2-
Tho lerm liudglng against inflation as used in this
paper, means reducing the specific kind of risk, which
stems from nncnrtainty about the future level of the
prices of consumption goods.
In other words, we use
the term in exactly the same way one would use it to
describe the forward purchase or sale of commodities or
foreign currencies for the purpose of eliminating the
risk of unanticipated changes in spot prices or exchange
rates.
Just as futures contracts for commodities and
foreign currencies are perfect hedges against unantlcipattul
fluctuations in spot prices and exchange rates,
so futures contracts for the specific basket of consumer
goods used to define the "real" value of money (i.e.,
the "purchasing power" of money) would be a perfect
hedge against inflation risk.
Recently there has been a revival in this country
Df the proposal to link deferred payments to some index
of the cost-of-living like the CPl.^
Effectively an
index-linked bond is equivalent to an ordinary nominal
For example,
bond plus a luLiires contract on the CPI.
Imagine an
i
lulcx-iinked Trrasury Mill wJ
t.li
a
maturity of
-3-
one year and a face value of $1000 of today's purchasing
power.
If an inve&tor were to purchase such a bond he
would in effect be buying forward $1000 of purchasing
power, but instead of contracting to pay for it in the
future, which is the practice in a standard futures contract, he would be paying for it now.
The example of an index-linked Treasury Bill can
also be used to clarify a very important point about
hedging against inflation which is often overlooked.
If the Treasury were in fact to issue the 1-year Bill
described above iL might under current market conditions
be able to selJ
it at a price in excess of $1000.
In other Trords the riskless real rate of interest on
such a bond could very well be negative.
Now many people would claim that if it has a negative
real rate of return then the security is not an inflation
hedge.
In their
view an inflation hedge is an asset
which "keeps up" with inflation, i.e., has a nominal rate
of return at least as great as the rate of Inflation.
It does not take extensive research to discover
that no such security currently exists in
tlie
financial
markets of this country and that there Is no way for an
investor to create a portfolio having that feature
-4-
(l.e., a guaranteed non-negative real rate of return).
Furthei-raore,
it
should be added that the proponents of
issuance by the Federal Government of index-linked
bonds have not stipulated that the real rate of interest
on these bonds be non-negative, although it would probably
be fair to say that they have assumed that the rate at
least on longer maturities would be positive.
The point is that there is nothing in the definition
of an inflation hedge as used in this study which would
imply that it has a non-negative real rate of return or
that an investor would want to hold it.
Although it is
natural to assume that risk-averse consumer-investors
would want to hedge some of their inflation risks, how
much to hedge would depend among other things on the
cost of hedging.
In order to deal with the question of hedging against
inflation
tlie
issues must be more sharply defined.
Since
there is one type of security whose real return is
certain but for inflation risk, namely single-period,
riskless-in-te.rms-of default money-fixed bonds, it seems
natural to identify inflation risk with the stochastic
component of the real return on such a bond.
Therefore
in this paper the conceptual criterion for deciding the
.
-5-
extent to wliich a particular security or class of
securities
L.s
an inflation hedge is the extent to
which it can be used to reduce the uncertainty of
the real return on a nominal bond.
We adopt the convention of identifying the
riskiness of a probability distribution with its
variance.
Accordingly, we measure the effectiveness
of a security as an inflation hedge as the proportional
reduction
in
the variance of the real return on a one-
period, risk-free-in-terms-of-default
,
nominal bond
attainable by combining the hedge security and the bond
in their variance minimizing •proportions
It is worthwhile to indicate at
this point the
relationship between this view of hedging against inflation and the investor's ultimate objective of optimal
portfolio selection.
This can best be done in the frame-
work of the Markowitz-Tobin mean-variance model of
portfolio choice.^
-6-
In that model the process of portfolio selection
is divided into two separate stages:
(1)
of the efficiunt portfolio frontier and
the optimal portfolio on that frontier.
identification
(2)
choosing
This paper
focuses on one particular point on the efficient
frontier
—
the minimum variance portfolio.
From this
perspective hedging against inflation is essentially the
process of taking a risk-free-in-terms-of-default nominal
bond as the starting point and using other securities to
eliminate as much of the variance of its real return as
possible.
We define the difference between the mean real
return on a nominal bond and the mean real return on
the minimum variance portfolio as the "cost" of hedging
against inflation.
Since the unhedged nominal bond does
not in general lie on the efficient portfolio frontier,
the cost of hedging may be either positive or negative.
The approach to hedging against inflation adopted
in this paper can perhaps be further clarified with the
aid of figure 1, in which mean real return is measured
along the vertical axis and standard deviation of real
return along the horizontal.
Point A is the hypothetical
location of an unhedged nominal bond.
The minimum
variance porL folio Ilea Homewliere to the left of A
-7-
Figure 1
U
'
-a
o
B
Standard deviation of
real return
-8-
(e.g., points B or C)
the farther to the left it is,
;
the more effective the inflation hedge.
The cost of
hedging is represented in the diagram by the vertical
distance between point A and the point representing
the minimum variance portfolio.
Thus if point B were
the minimum variance portfolio the cost of hedging
would be positive; If point C, the cost of hedging
wouJ d be
neg.'il
ivc.
The mln Lmnm-variance portfolio is in principle
composed of all available securities.
there are n +
1
Assume that
such securities, with the n + Ist
being nominal bonds.
In part II of the paper we
view the minimum variance portfolio as being composed
of just two securities:
the nominal bond and the
portfolio of the other n securities, which we refer
to as the optimal inflation-hedge security, or
simply the hedge, security.
Part II tries to relate
the effectiveness and the cost of hedging to the
parameters of Lhc joint probability distribution of
the real returns on the two component securities.
-9-
ParL III dc;als with the composition of the optimal
inflation-hedge portfolio, while part IV tries to
indicate how uncertainty about the parameters of the
joint probability distribution of security returns
affects one's ability to hedge against inflation.
In part V we try to relate the theory of inflation
hedging developed in the previous parts of the paper
to
modem portfolio
theory, and specifically, to the
capital asset pricing model.
Finally, in part VI we
summarize the results of the paper and draw its
major conclusions.
.
-lo-
ll..
The
Lip.CeniiiiiaiUs
LpL
l'(t)
nominal value at
Let
of lledginj; ErfecLivencs.s
b(»
l\\v
LiinM
t
price level at time
I.e.,
t,
the
of some standard basket of commodities.
r^(t) be the continuously compounded onc-pciriod nominal
rate of return on security
i
during period
compounded real rate of return
R.(t)
r.(t) -
=
The continuously
t.
on security
then given by:
is
i
„(t)
where n(t) is the rate of inflation during period
t
defined by:
Tr(t)
^
and tildes
C)
Suppose
matures at
nominal
Its
real
r(t+_i)
=
I
are used to denote random variables.
he scMuiriiy under consid(,^rat:ion
tinu'
I-
I
rate n\
1
irlniii,
1' (
H
i
I
)
from Its mean, p(t).
of inflation.
of relurn on
r
n
<ir
K
(i),
is
known
v^^lLli
risk,
lerlainly at time
since it
I,
71
(l)
into its mean, n(t), and deviation
p(t) represents the "unani
ri.sk-froe nominal bond is sim[)ly:
a
--
II
- p
'
n
so that its
J
c
J
pated" rate
Suppressing the time subscripts, the real rate
n
\{
n
free of default
bond that
a
.
Let us di'(om|.ose
K
|-
Is
leLuni, Jiowcver, is uncertain at time
depends on
(I)
and
is
|)
t,
.
-u
Now consldi'i-
securities
he sccoiul security,
I
tlie
portfolio of risky
to hedge the nominal bond against
uscul
and hereafter referred to as the hedge security.
inflation
I.et
us also
decompose its real rate of return into its mean nnd deviation
from
(lie
mean:
(2)
\
'
K,^
^
^
The deviation from the mean,
c
can be further decomposed
,
into two orthoj'.onnl components as follows:
)i
cov(c,p)
-._.».i_
,
where
+
u^^p
^
f:
a^^
(2)
_
then becomes:
vnr(p)
(3)
.
R,^
+
^^
+
.,^P
M
can be called the "inflation-risk" component of the real
a p
return on the hedjv' security while
ii
is
the "non-
ii\ f
latlon risk"
conipontMiL
slioLild
li
(Iclin
about
i
I
ional ly
Llie
that since tlicse reJ at lomdij ps are
;.u-('.sscd
h(^
Liuc,
they assume nothing and
link or
cnus.i.l
tlie
tlioy
Imply nothing
time series relat ionsii
i
p
the return on the security and the rate of inflnllon.
between
Neither do
they assume anytliing about the specil'ic probabillly distributions
of p and p.
Now if we roplnce
with
(3')
tlie
nominal
r,
r
r,
n
where
it
rate
ntp
I
li
-
i»
I-
1
I
i^f
I-
real rate of return on the hedge security
lie
return then equation
\i
('3)
becomes:
:
-12-
Our objective is to combine nominal bonds and the hedge
security so
iis
Letting w be the proportion of (he hedge
of real return.
security
Ihe inlninnim variance portfolio,
In
rate of ret urn,
K
H
(4)
R
R
.=
mm
js
f-
w(K,
i
tut
R
n
R
-
h
n
)
+
(w
(3):
mm the
.
+
u-i)p
2
wii
?
Let Oi
a^
Irilter's real
)
from (1) and
ini;
+ w(R,
n
-
h
n
the
approximation:^
to a very close
,
mln
= K
.
mit)
or by subsL
create the portfolio with minimiim variance
Ici
bi>
t
lie
the varJanre of
variance of p, 02
\i ,
and
variance of R
..
)i\ln
From (A) we get
0^
(5)
-
min
(w a-l.)^'
'
'
^
+
w^o^
ai
2
The value of w wliich minimizes
o''
is:
nti.n
w =
(C)
-of
Substi tut
variance
(7)
Ls
^?
ii)(;
from (b) into (5),
given by;
the vaLue of the ininimtzed
-13-
Our measure
(if
effectiveness of the
Llic
lii'd;'/'
security
as an inflation hedge is the proportional reduction In the
variance of the real return on the nominal bond attainable
by combining the two securities in the variance minimizing
proportions.
llu'
fonmila for this measure is:
1
--.--— --^-
,,
(8)
.1
It
p^,
+
2
is n simplf inatLer to show that S
the coeLEicieut ol
Is
determination between
none olhor than
and
r
p.
Thus one can say that the more highly correlated (either
negatively or positively) the nominal rate of return on
security is with unanticipated inflation,
it
Js as an inflation hedge.
cannot sefl
hedge
if
tiic
Note that
hedge security short
and only
if
i'a
nominal
tlien
return
lated witii unani lei paled inflation.
If
is
more effective
tlie
the
it
a
Investor
is an
ju.)S
It
J
Inflation
vivl^
corre-
-14-
FinaJ ly as a measure of the "cost" of hcdRini; we take
the difference between the expected values of the real return
on the unhedged nominal bond and the real return on the mini-
mum variance portfolio, which from
C H i;(R
(9)
-
R
n
p
Equations (6) nml (8) revt;al that
functions of O-./o^ and
measure of the
rioii-
i
nf
a.
1
is:
w('r" - R, )
n
h
E
)
(4)
.
^^J^l
^^
bot)i
l^'^'-
w and n^ are
'"'iti')
t'lo
''f
at Lon rj.sk in the real return on
the liedge secuiily to the measure of inflation risk, while
a is
tlie
regression coefficient of the nominal rale of return
on the hedge security on the unanticipated rate oT inflation.
Now let us explore the implications of equal ions (6)
and (8) with
tlu-
aid of figure 2, in which o->/0|
alonj; the horizontal
is
axis and a along the vertical.
let us direct our attention to
(8),
the formula
for
measured
I'^irst
"^
f)
or S.
-15-
Flgure. 2
a
02
o7
-16-
Each "iso-cf feet Lveiiess" curve in figure
;'
each
i.c^.
,
locus of paramoLcr vaJ.ues yielding the same vaJue of p^
a
seL of
2
lines which
sLrnij'Jit
.symmetric with
at
tlie
is
the origin and are
aL
to the hori^.onLal axis.
iKs|)erl.
important exceptions
niiiet
,
Tliere are
two
The iso-efl ccti veness
extremes.
curve corresiHinding to an p^ value of zero is the horizontal
axis itself (excluding the origin), and the
curve corresponding to an
D
value of one is
axis (excluding the origin).
i
so-e f I'ectiveness
I
vertical
lie
The closer the iso-ef fectiveness
lines are to the vertical axis (i.e., the greater the angle
they make with the horizontal axis),
the higher the value of p^.
Along the horii'.ontal axis the nominal rate of return on
uncorrelated with unanticipated inflation
the hedge security
is
so no reduction
variance is possible.
In
The reason for this
can most easiiy he seen by comparing the expressions for
l^
R,
and
in this case:
R
=
n
=
K,^
R
n
- p
|-^^-p +
M
Because the regression coefficients on
any amount of
tiie
hedjic
p
are the same, combining
security, positive or nc>)vitive, witli the
nominal boml would jusl add non-inflation risk lo
without olimlnat
Ini',
any of the inflation risk.
tlic
portfolio
Ivpiation
(5)
-17ciMifirnis
rain
sijue wlien a =
UliL;;
it reduces
to:
2
1
Similarly equal ion
(6)
indicates that when a = 0, w is zero.
Along the vertical axis, on the other hand, p^
02 = 0,
i.e.,
tlie
is
1
because
only source of uncertainty in the return on
the hedge securtty is unanticipated inflation.
that case
In
the rates of return ou the nominal bond and the hedge security
are perfect] y correlated, and they can therefore he combined to
create a real risk-free composite security.
For a given v.ilue of
(different from zero)
a
monotonically as 02/^1 fises indicating that
tlie
,
p^-
falls
greater the
degree of non-JnllatJda risk in the return on the hedge
security relalivc to
tlie
degree of uncertainty nbouL inflation,
the less effocti-vc the security is as an inflation hedge.
a
as
given value of
(i.e.,
|u|
tlie
Conseiiurnl
curves there
tliat
I
lie
"p/i'j,
I
is
y
,
a
on
tlie
other
Iiand
p''
,
(Hsi.ince from the liorizontaL
.ihui);
1
For
rises monotonically
.ixi.":)
increases.
all but the two extreme iso-ef fectiveness
rado-of f between
|<x
|
and
";'./iM
In the sense
loss of hedging effectiveness resulting from a rise in
o^/oj can be (•(impcnsal cd by a rise in
|
a
|
the
"marginal
rate
?.
of substitution" betwi^eu
being
a
constant.
|
a
|
and
i^p./O]
for a given value of
p
-18
I'ir.mi'^
of equal Ion
'
also he used Lo
'III
((i)
Li
the lOrraula for w.
,
lusLraLc
ImpllcaLions
Llic
As already RLni.od above,
no reduction in variance can be achieved through diver-
if a =
sification, and there fore w equals zero all along the horizontal
axis (except at the origin where it is undefined),
same sign as
indicating that the hedger must
en
position if the
w has the
tal<p
a
long
rate of return is positively correlated
ncMiiinnl
with unanticipated inflation and a short position
if
negatively
correlated.
An iso-w curve
figure
in
2
is defined as
combinations of paranicLer values wliich yield
Uicus of all
tlie
mi ninium-variance
portfolios containing the same proportion of the hoiigo security.
The iso-w curves for positive values of w form a set of
c'.'ci
-w idening semL-cin- Les whicli fan out from the origin in an
Although they all converge at
upward direction.
the origin itself is not on any of them.
tlie
set
Tiie
ol
origin,
iso-w curves
for negative values of w is symmetric to the positive set with
respect to the iKnizonLiL axis, i.e.
Lso-w curve
f;iiniiii);
identical uegal
upward from
out
i.vi'-va
I
i
I
or every
lie
orl(.',in
por.
I
Live-valued
tluMc Is an
iso-w curve with the same absoluLe
iicd
value fanning downward from the origin.
The only iso-w curve shown in figure
ponding to
lying on
a
value of
tlii s
1
Is
the one corres-
For any combination of |)arameter values
i.
simtii-c ire
2
e
,
the minimum-variance |H>rtfolio will
:udso Security;
will convain no nocinal
ko:u:i-.t
::«>Nl\
bonds.
The siiaded arta witliin the unit iso-w curve contains all
v' (
the parameter comb
;
lu'
inai- ions'
it
at which nominal
bonds must be sold
short in order to creale the minimum variance poit folio.
-19
The two sti-aigliL
J
Lnes emanating from the oriyl.n and
making A5 degree angles with the horizontal axis have special
significance.
=
p''
at all points along tliese lines
t-'irstly,
But more importantly, if we hold 02/ai constant and
.5.
examine the way w varies as a function of a, we find that w
reaches its maximum wliere a = 02/0^ and its minimum where
a.
= - a-^/O].
I'igiiro
2
Ji\
throuj'.h
pomlLui; to
siuir:
other words if we drew a vcrlicaJ
1
lie
value
[xjlnt on
ol
the horizontal axif; corres-
the standard deviation ratio, w
would be at Its lulnLmum where the line Internerts
43 degree
line and at
Lts
"1
_
20 2
11
llie
lower
maximum where the line hiLersects
the upper A5 degree line.
w =
line in
II
Along the upper 45 degrees line
and along the lower one w = -
"1
2a2
-20-
Perliaps
fuLLlicr light can be sliod on the
p" and
between
in which
a and between w and
Each curve in figure 3a shows the
functional relal ionsliip between w and a
O2/01.
The efroc-L of a decrease in
in
oj/^i, to curve
Curve
2.
fndicatinj', Miat
sensitive
Is
small, the
w to
In
rij;ur('
the value
curvc to
steeper
o!
I
tlian
iiulJ cat inj;
I
i
lie
ve
Hi
sign of
Llie
I'H
both
<'"'e
curv
Mi.;
to
a.
we see the graphs of
1
and
2
tlie
two
in figure 3a.
p
-^
Tlie
functions
smaller
die "closer" the two branches of the p2-
vertical axis.
p'
\n\
<iri<'
the more
must be either very long or very short the hedge se-
aj/d]
tliat
Is
Thus if 02/01
.
correspcjnding to curves
i
is much steeper than curve 1
2
the smaller the value of a^/o^
|ft|
hed};'''-
curity dcpcndiuf,
;;ens
02/01 is Illustrated by
vicinity nf the origin (i.e., for smaLl vaJ ues of
thr-
|a|)
tor a given value of
from curve 1, which corresponds to the higher value
the shift
of
by examining figure 3
along the horizontal axis and w
« is iiRvisured
along the vert Leal.
a
reJ jitionships
1
the vicinity of
In
smaller
u
As in figure 3a, curve
.
tlie
value of
tlie
2
is much
tuigin thus
>T;,/(1i
Mie more
-2]
a.
w
.
-22
Ln
Llie.
so that p^
]
=-
tmir.
,
wIk^h
Idr
1
£iJ
i
o^ = 0» I'erfect hedging
values of a
in figure 3a shows how w varies with
except zero.
and 2, curve
asymptotic to
3
vertical axis.
tlic
a^i
= 0, a
This is because
curves
in
the
siiab Ic
from
the stochastic component of Jts real rate
of return and would eiLher dominate (if
dominated by
clo
cannot be equal to zero or
else Lhe hedge security would be indistinguJ
In
3
splits into two parts, both of which are
special case where
nominal bonds
Curve
in this limiting case.
a.
Note that instead of passing through the origin as
1
possible
is
nomlii.il
bonds (if R
>
R,
)
R,
h
>
1^
n
)
or be
-23
Mil's
111
I'll
pftiiit
I
wLLh v.iriaiu-e icilurL inn
of equnticjn (9),
I'.quation
of two cases.
the
I
discussion has
lie
dc-i
Let us now cx.iniiao
.
lumula for
(9)
indicates
riu>
first
tlie
tliat
1
cx'-
1
1
ns vnly
Jm|i]
Llic
i
J
cations
cost of hedging.
C will be positive in either
is if both the mean real
return on
the nominal bond exceeds that of the hedge security and the
hedger must take a long position in the hedge security
a^
(i.e.,
).
i'iie
real return on
I
will
bond,
l)e
K
(2)
R
>
R,
n
h
<
™
aiul
"
h
meaning
tliat
the mean
tiian
an unliedged nominal
and w is negative.
positive.
Ls
h
The following table summarizes these relationships.
Cost of h edging
R
•
R,
R
<
n
a
-
n
<
is
the remaining two possible cases, namely:
and w
R,
n
K,
the minimum varianci?. portfolio
i.e.,
nf;gativc,
in either of
(1)
<
return on the unhedged nomluaL bond.
greater mean real return
a
n
he ininLmum variance portfolio would be less
than the mean leai
will have
K
In either case C is positive,
negative.
C
second case is if
R,
h
-Ik
Ihe Optimal
III.
I
at] nLion-lIedge Portfolio.
Up to this point we have treated the creation of the
minimum variance portfolio as a problem involving just two
securities.
but
there are n +
fact
in
st
the n + 1
to choose from,
1
risky securities
being nominal bonds.
ri\e
pur-
pose of this section is to examine the composition of the
portfolio of the first n securities, which 1b combined
with nominal bonds to itreate the minimum variance portfolio.
be the continuously compounded real rate of
Let R
return on security
As in section II decompose it into
1.
its mean and deviation
R.
c
+
R.
1
--^
L
and decompose
from the mean:
e.
i
into an inflation-risk component and a
.
non-inflation risk component:
~
Since security
a|;^^
so that.
"
hot
n
.\
llir
_L
-^
'
+
^1
"i''
I
1
nominal bonds we know thai:
l.s
and
-I
be
~
R~
.
'
0.
H
„^^_^^
covariance lietween
of
I
and "2
p.
.
.
I
In-
O
<-ii
va
Ihe vari
I
c
itV'^
Is
mcc
v'.ivrn
r
lance between
K
.
.
:
'^^
s^-cu!
and K.,
W.
L.I
ol
bv
(l
R R
a O
.
L
.
J
(V
and
li
the vnrJanco
1
.1
n
.
.
Then:
'•
1
^
+
9,.
'^ij
.
the real return on any portfolio composed of these
:
-25
„^
or
and
-26
The values ul
R
a
R
,,,,.1
"
for this portfolio
,Mre
nivon by
K
i=l
^-
o?
^
^
n
n
i=i
j=l
It is shown in the
1 J
^ij
appendix that the composition of the optimal
inflation-hedge portfolio (i.e., the set of
weichts w'
is independent of
of,
i=l, n)
d.e variance of unanticipated
inflation.
This is a kind of sc.parabi] ity or
mutual fund tlioorem for
lu'clging
against
iuflaiion because it implies that
.-,
hedger can
create the mini mum variance portfolio in
two soparaio stages.
In
the first s(age
l.n
selects a portfolio out
ol'
all
securities oth^r U.an the risk-free nominal
bond so as to maximize
the cocfficiouL of deu-ni.lnation
between the uominnl
resultant portfolio aud unauticipated
Inflation.
return on the
The relative pro-
portions of the various securities in this
optimal inflation-hedge
portfolio depend only on
«^.
matrix of
tiie
non
-
1
= 1..., n, and
tlie
variance-covarlance
inf lat on stochastic components of
i
rates of return, but nol on
In the second stage,
o"^
the securities'
.
a judgment is made about
lli.^
variance of
unanticipaK'd inflation and the optimal inf
f ation-iudge portfolio
is
combined
wiili
nominal
bonds to create the minimu,,, variance
portfolio as described above in section 11.
Tluis
even
I
Uouj.li
variance of unant
f<
Invosl ors may have diverKO
Ipal .d
inflation,
If
J
udgmcaitB about
they nf;n>e about
thti
the vn.lues of
all the other parameters then they will
agree about the composition of
the optimal inflation-hedge portfolio to
be combined with nominal bonds
to form the minimum-variance portfolio.
:
-27
JV.
'Hie
I^f
L^cL
Para iiiGter UnccirLainty on
i»^r
Ileclylnj',
Up to this point we have assumed tliat a, a'
recLiveness
r.r
and
o''-
are
2
1
Generally, however, they will themselves be subject to
known.
uncertainty.
chosen for
that case the portfolio proportion actually
In
security will not necessarily be equal to
hcdj^c
tlic
the true varJance-minimizing proportion.
actuaJJy chosen
w
,ind
Llie
Let w be the proportion
"correct" proportion.
Define:
w - w
From (5)
(13)
where
liave
we.
(T'^
-
c"
is
actually
11
c'^
2wifn''
--
+
12
+
w'^(a''a''-
a'')
the v.iriance of the real return on
By substituting; w +.c
sc^.Jeelcil.
I
for w
he portfolio
(13) we
in
i;et:
=
a''-
which by
(14)
11
0''
-
sul)st
i
02
tui
=
in^;
0-'
12
+
2vi'o)'^
w''(a'^a''-
lor
-
v;
(
1
,
+ a')
- 2f;aa^
+
(i
from (b)
reduces to:
L
0^
+
+ ^_?^
'
I-
12
2wi;)(a^a^ + o^)
1
)o^1
f;2(a2(T?
+ e^)
]
,?
1
The first
nominal bond;
attainable by
I
I
(Mm
iti
(14)
is
he .'UH-oml term is
lietlj;
i
n)',
1
the variance of the unhedged
tlie
reduction
he bond against
in
t
iiat
variance
Inflation assuming that
:
-28-
one knows the currecl variance-minimizing
prdporlion for the
hedge seciirlLy; and the third term is the increase in variance
due to usinR a proportion for the hedge security which differs
by
c
from
correct variance-minimizing proportion.
tiie
third term is greater than the absolute value of
t
lie
Investor
is
tlie
If the
second,
really increasing the variance of his real
return by tryinj; to hcilge against inflation.
To gain fuilher Insight into
the effect of parameter un-
certainty let us assume that the source of the error in the
portfolio proportion
is
a
where
ex
is
an error in estimating
2
Tlius
let:
the deviation of a
from
a.
a + n
our (Estimate of a and
n
Is
the "true" value.
In
tion
that ca.se the deviation of our clioson portfolio propor-
for the
lu'djte
security from the correct
v.ir
lance-mi nimizing
proporl ion will be
tio^
(15)
r.
h
let
"n
(n"^
+
2ai])<T^-
-^-}
0^02 +
^
where we
- a
a?(--5i
=
h
(n2 + 2an)'T2l
1
stand for the variance of the nominal return
on the hedge security and make use of the identity:
a
-.
u'^o^
+
o'-
-29-
Now suppose
Lh.it
'i
= 0,
the hedge security is
i.e.,
uii-
correiated with unanticipated inflation, but we mistakenly
assume that
lIk>
regression coefficient is not zero.
case (15) rechjccs
In this
to:
no 2
L
(16)
2
which is also
funnula for w.
tlie
we get:
into (14)
Substitvitinj.',
-
.r
(1.7)
1
11
+
^-2
(n^o^ + a?)2
I
(17)
fitiuation
leaving well
Lni[>Lios
(Mioiigh
;.
]
»
?.
that the
not
clone by
"dain.igf?"
aJone (i.e., by combinJng some of the mis-
taken hedge security with the risk-free nominal 1)ond)is a mono-
tonically inci easing ful^ction of n^
>
o'^
and
if
a
Liic
error in the estimate of «
•larger the value of
(i.e.,
ly
that
a
,
will be larger the
the residual variance).
security whose nominal return has a
JnflatLon risk
hedge even
If
in
ii
t(ui,
lot
A
therefore,
of non-
should probably not be used as an inflation
the point
ferent from zero.
then n^.
techiiif|ues
reasonabJe ctmc] usion to draw from this discuss
is
practice
?
is estimated using standard regression
the square of
In
a''.
1
estimate of its u
value
Is
quite dif-
-3(1
V.
Inn., ion-risk, M.n kcL-rlsk and
Hqu
la nKHl..i-a
J,
1
ibr
i
K..,,„ms.
un,
.a^LLal market Lheory a basic
disLinctioa is
draw,, between syste.mu
i.r.
and unsystematic risk.
the
rn
SharlK-Lintner-^k,ssiM version of the
capitai asset pricing
model (CArM)O
,,,.
two most important theorems
are:
The eqni Librium expected excess
return on a s.M-.urity
(1)
:.s
proportlouai to its systematic risk as
measured by
Lho rejirossion roe
"T
(2)
tlic
th..
propositions
l.cicnt of
porilolios have no
absen.e
tlic
ol
s.iJI
is
with refercnr.. ,o
instead of
i:
its rcti.n. on the return
"mark.'t porliolio."
Kllielcnt
In
1
i
a
n
eiiiat
I
c
risk.
rLskiess asset, the firsi
,orrect
he mi
luisysl
i
i:
of these
one n.easures ex.ess returns
imnm-var ianee-/.ero-bet
a
i-ort
folio
rljikJess asset. ^0
Since a large proportion of the
wealth of
•sector is in the
form
ol
a negative correlation
money-fixed assets there
portfolio.
^
predict a posiliv risk premium (above
y.vro
is
undoubtedly
between unanticipated inflation and
the return on the markei
minimum varJanr.-
private
tiie
so that the
tlie
I
h,-ory
rcl.nn on
beta iiortfolLo) on any
s.'e,n
i
1
tiie
y
negatively eor.elaled
wi,|,
unanticipated inflation (i.e..
I'osUively ...rrelaled
will,
tlie
assets)
.
real return on money-llxed
would
-31-
"""1<" 'I- lina
noL s.nviv..
asset.
.r:-nsl,ion to a
.1,,.
have some and i-orhaps
I-c^t
portlollo
in |..Ti,Kl
deviation Crnm
i
will,
no riskicss
j„ general
unsystematic risk.
he mean:
-
"^n,(t)
+ m(t)
can then be decomposed into its
i
sum of Lwu orthogonal stochastic
components:
tlie
1^(L)
K.(t) +
:
''ovCR
^
P.
surond docs
the real return on the market
The real return on se<-urlty
where i.^u;
wnere
(t)
lu-
i
and decompose It Into Its mc^an and
t
'^m^'^^
mean plus
CAJ'M,
capUal markoL
,;reat deal of
.,
r.|,rrs.„t
K^^^Cij
the.
market olflciciit portfolios will
.such a
Iti
tlu-cnm of
.
.
^i
(l
.
)
,m(t)
n^(t)m(t) + n^(t)
'
)
,
„
The ^.
first, 3j(t)m(t),
„,.
.
var(m(t))
is the systematic risk of security
'
^
and
1,
n
(t)
Is
Its un-
systematic risk.
J^y
the first
tiieorem of the CAl'M:
E(Hj(L)) = K(Ky(t)) + B.(t) E(K,/t)
where E(
of return on
'^'^''
''(^)'
a
i
""'
then the va hn-
implying
m iniiiuiiii-variance zero-beta portfolio.
tiic
seems Jik<Hy,
l<^^(t))
the expectation operator and R^ is the
real rate
is
)
-
|r.
niaikcl
iiiiant
(^l
as
portfolio is negatively cc.rrelated
i.ipated change
lyi)
If,
the
In
rate
tor an unhedged nominal
positive lisk premium for K
.
n
nl
l.nnd
inriation,
is
positive
-32-
Sinco
n>Lurns on nil eFficicnl
ic\)l
llir
positively c.orro
L:itc(l
Llic
'
,
variaacc port.ldj io also
iiiLnimuiu
has a posiLivc risk pn-mium.
imrt fdl ios are
Which security has
liigher
a
return, and therefore whether the "cost" of hedjvinf; is positive
or negative, cannoL be clatermLned on the basis of juirely
l.heuret icai cons i.tlurat ions.
Note, liowever,
will have
a
en
the
expe.cLi-d
cost of
the market porlTolio
or |)osiLively rorreiated with
ri'lateil
rt'luni
if
Mial:
nmi-pii:;
in
i
ii
i
nmiii
Li'lnni en
hedj'.inj',
i
I
v.i r
Lve
I
In
l)el.a.
ani'e.
jiorl
infLatioii
an iiiihcdgeil noiiiinal bond
will
LhcMi
that case
folio will
uncc^r-
is
i
a
expected
in'
the
exe<'e(l
^nul
I
nominal
liere.fore.
the
be negative.
Although these theoretical considerations dn not tell
us whetlier the
c-.osi;
of liedging is positive or negative they
do impose an upper bound on
bet.i
pertfOlie.
the n)ean real
^;pl'(il
tlie
iealJy,
mean real retui'n on
if
t:he
a
zero-
theory is ceirect then
leinrn on a zero-beta portfolio
iimis(
be less
than the expecled iiMuni on the minimum variance portfolio.
bond
-31'
VI.
SniiMH.i
I
me
ii
(
pniiii
Till-
inflation
.111(1
s'
heel)', inj;
I
ir;
i
fiii'j
urc for this paper was the view Lhat
()(
il.'iMi
is
t-;;»;onLially
1
process of crfatiug the
tlie
minimum variance pniilolio by tal<ing
(lofault moni-y-f
r
isk-f roi^- n-Lorms-of
i
and using other securilios to eliminate
hcnul
lx(.'(l
a
as much of Jts variance as possibJ.e.
In part
I
was shown that the extent to wiiich this
it
1
variance, wiiLch was Identified with inflation-risk, can be reduced
depends on
return on
liigluM"
1
i
lie
variance.
Lliat
a
noinin.il
cuerricieat of determination between the rate of
Llie
he
value
Ihii::
..eciii
i
rate
I
y
el
negatively -
this parameter,
oi
as'-.um
i
un an
is
I
i
to the exleiii
-
eLther pos
variance
lai io
ol
variance ratio,
The second
I
i
vi'
!
its
y or
lie
paiaiii','1 t>r
smaller the coef f
wa;
tlie
i
i- i
cnt
1
n\
ation stoch-
secnrtty to
hedj',i'
The
first of
Tiie
of the variance of the non- inf
unanticipated inflation.
t
i
lliat
icient of determination was then broken down
astic component of the rate of return on the
tiie
in
inflc'ition.
and analyzed in terms of two other parameters.
these was the
leduction
sales are possible mie can say
correlated
el pa ted
The
inflai.lon.
the larger the
inllatii>n iu'dge
return
I
shoi^l
iij',
an
ii.
villi
coc^f
Tlie
hsecurity and unanticipated
liedjM-
larj'.er
diM c
coefficient of unanl
i
i
this
nil
nal Ion
c Ipatec.!
inflation in the regression equation lor the nominal
rate of
.
-34i^'lurn
1
(Ml
r);ri;s'!
sriiiriiy.
lic(lc,c
111,'
I
((n.'
(.(Ml
T
I
i
c
i
cnl
'Ilio
gii.\UiM-
the {greater
,
Llic
al-so
l.lu'
idc-
1
I
i
r
i
ii
I
n
<
I
i
value of this
c>
o
|
determination.
paj
fn
theoix'in for
I:
I
I
IkmIj',
w
1
i
nj-,
I'ldvod a
a)',aLnst
separability
iaiiation.
oi'
iiiiiLii.il
Fund
sluiwn
that
It was
even though iiivosiors may have diverse judgenienLs about the
variance of unauL
inflation, if they agree about the
iciiiattui
values of all other parameters then they will agree about the
composition of the
combined w
port
i
()
1
i
would-be
n<iiuinal
li
pajt
ol
rcttiin;;
I
inflation-hedge portfolio to he
bonds to form the m Lnimnm-vav ance
i
o.
In
iiicters
i
o|)tiiiial
I
lie
sli(>w(!d
juiiil
probability disLrLbution
icid
ini;',lil
how uncc!rLainty
IV wi'
licdg(-|-
i
lo
I
he pa rach)X ca
I
1
aixnil.
oi
sitnal. ion
i
lie
piira-
srMnrity
\%lii|-(.'
Llie
ai-inaily increasing ratlu'f tlian decreasing
::
the variance of his real return.
Finally,
pail
in
V we sliowed that although |)ortfolio
theory does have some implications for the cost of iic^dglng
against inflation, it
of
theoretici
imsLtive or
I
cons
nc^j'.-'l^
'
is
i(h.'rat
ve.
impossible to determine on the basis
ions alone whetlier
thai,
cost is
,
-35-
Appendix
Proof that w'
1
^
= 1,...,
n do not depend on
a''-.
1
By assumption we hold a
and
^2-1
fixed for the first n
•
securities and consider the effect of changing the value of o^.
Let w'
represent the weight of security
in the optimal
i
inflation-hedge portfolio corresponding to a value for a^ of
Oi
and wV
where
be the weight corresponding to a value of o^
oj ^ oj
From equation
.
o^
min
f
'2
"
O'
O
(7)
we have:
and 0^
1
.
%
"
.
™i"
i^
tt
A
A
2
?
1
1
.
-2
2
2
«^:
^A
1
where:
n
>;
2
n
A
w'a
i=l
^ ^
n
n
I
V
i=l
Ann
i=l j=i
From the fact that
»
a
a^ ^
min
i J
'iJ
2
i=i j=i
is a minimum for
1
have:
-^2
=
<
°
^-
'
o^-
a2
1
we
= o
o
1
-36-
which implies that;
(1)
Similarly,
1
1
<
— + 4-
T- + T-
rain
-r
which implies that!
0(2
>
0(2
(2)
From (1)
.-iiul
wo have:
(2)
A
(3)
.
"
;
Since (3) musL hold for all possible values of a. and
a?., it must be that:
11
w!
1 =
= wV
Q.E.D.
l,...,n
-
-By-
Footnotes
This paper ijjnores the ambigii Ll ies and difficulties
involved in defining and measuring thi.' general price
level.
It assumes that the Consumer I'rice Index or
some such index is an appropriate measure.
2
Although there are futures markets for many Iiasic
commodities, these do not generally correspond to
the components of a basket of final consumption goods.
The idea of establishing futures markets for the al
item CPI and its components lias been suj'.gestetl by
I
LovoU and Voge
3
4
I
(1^71).
MtcraLure.
for a review of this
See Hliatia (197A)
See the articles by Friedman (1974), Modif.llani
and Tobin (]'J()i) for example.
(1974),
Originally formulated by Markowitz (1952), this model
was thought to be consistent with expected utility
maximization only under very restrictive issumptions
about either the stochastic si)ecif icat on or tiie utility
function.
Hecent work by Merton, (]9r)9, 1971) however,
has shi)wn that tiie mean-variance model lias vaJidiLy for
a broad class of stochastic specif icai ions and utility
functions when the trading interval is suflicitmlly
small.
i
I'rool
I
liaL
vari.inic
is
p
are orthogonal,
idiul ically zero:
cov(p,
.Mid
|i
cov(p,
|i)
cov(i>,
I
a var(p)
:i
-
I
)
i.e.,
up)
- a vat (p)
- a
var(p)
that
(heir
it)-
—
-38-
This approximation becomes an equality in the limit
as the length of the holding period approaches zero
since the instantaneous rate of return is by its
nature a continuously compounded rate.
8
Proof that
o
p"^
^
S:
cov(rj^, p)'
P^-
=
a''v.]r (r.
)
(aa2)2
1
o^Ca^a^ + a2)
2
1
1
1
7^'
1
+
a'
2
oP-a'^
1
9
For a thorough review of the capital asset pricing
model and the literature on that model see Jensen (1972)
See Black (1972)
Tor a proof of this proposition.
There is a great deal of evidence that the real, return on conunon stocks is also negatively correlated with
unanticipated inflation.
See, for example. Body (1974).
12
See Merton (1972) for a proof of this proposition.
-39-
Ref erences
K.
Bhatia, "Index-LinkLiig of Financial Contracts,"
Research
Report 7412, 1974, Dept. of Economics University of Western
Ontario.
F.
Black, "Capital Market Equilibrium with Restricted Borrowing,"
J.
Z.
of Business
,
1972, 45, 444-454.
Body, "Common Stocks as a Hedge Against Inflation," Boston
University School of Management Working Paper, 1974.
M.
Friedman, Column, N ewsweek , January 21, 1974, p. 80.
M.
Jensen, "Capital Markets:
Theory and Evidence," Bell Journal
of Economics and Ma nagement Science , Autumn 1974,
M.
Lovell and R. Vogel
,
357-98.
"A CPI-Futures Market," JPh^ 1971,
1009-1012.
11.
Markowitz, "Portfolio Selection," Journal of F inance
7,
R.
C.
1952,
77-91.
Merton, "Lifetime Portfolio Selection under Uncertainty:
The Continuous-Time Case," Rev. Econ
,
,
.
Statist.
,
1969,
51,
247-257,
"Optimum Consumption and Portfolio Rules in a
Continuous-Time Model,"
J_-
of Economic Theor y, 1971,
3,
37j-4J3.
"An AnaJytic Derivation of the Efficient Portfolio
Frontier,"
F.
.
J^.
of F inancial and Q uant
.
Anal ., 1972, 1851-1871,
Modigliani, "Some Economic Implications of the Indexing
of Financial Assets with Special Reference
to
Mortgages,"
Proceedings of Lhe Conference on The New Inflation, 1974.
J.
Tobin, "An Essay on Principles of Debt Management," Fiscal and
Debt Management Policies
,
Englewood Cliffs, New Jersey:
Prentice-Hall, 1963, 143-218.
Date Due
BASEMENT
-^4
Ferr^^
*<?*
25 19
9&M
8ff
1
ul^
FEB25'S2
DEC
V
Lib-26-67
T-J5
w
143
Agmon
75
no.804-
B/lnternational
Tamir
72516^
diversiti
000198""
n^Pt^S
Hi
111
Hi
nV
ODD b4b
TOflQ
3
T-J5 143 w no.805- 75
Gordon, Judith/Descnbinq the external
725175
.
niB*vs
.
ODD bMb 07E
TDflD
3
ni)oiiH/8
H028M414 no806-75
Bodie,
/Variance minimization a
Zvi.
725868
P.^BkS
.
.
TDfiO
3
T-J5
143
.
72587R
nnoi984F
ODD bMS 371
w no807-
Rosoff, Nina.
75
/Educational
p»BKS
interventio
00019850
DDQ bMS 470
TOflO
3
.
143 w no.SOB- 75
Marsten, Roy E/Parametric integer prog
000198 47
D»BKS
72586.1
T-J5
3
ODD bMS 413
TDfiO
T-J5 143 w no.809- 75
Agmon, Tamir B/Trade effects
D»BKS
725863
the
of
oi
mn.m.
TOaO ODD b4S 7T3
HD28.M414 no.810- 75
Agmon Tamir B/Financial intermediatio
D»BKS „„ „ „,(jO0 19867
D»
22
725822
3
iiiiii:
TOflO
UlT
IIIIII
I
IIIIII
III
ODD b45
6
Lll
III
'g
3
TDflO
DQD
4
=110
03