common stocks as a hedge against inflation

8/7/2019 Common stocks as a hedge against inflation

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THE JOURNAL OF FINANCE VOL. XXXI, NO. 2 MAY 1976

COMMON STOCKS AS A HEDGE AGAINST INFLATIONZVI BODIE*

I. INTRODUCTION

THIS PAPER DEFINES the effectiveness of common stocks as an inflation hedge as theextent to which they can be used to reduce the risk of an investor's real returnwhich stems from uncertainty about the future level of the prices of consumptiongoods.' Since there is one type of security whose real return is certain but forinflation risk, namely single-period^, riskless-in-terms-of-default nominal bonds, itseems appropriate to identify inflation risk with the variance of the real return onsuch a bond. Accordingly, we measure the effectiveness of common stocks as ahinflation hedge as the proportional reduction in that variance attainable by com-bining a "representative" well-diversified portfolio of common stocks and thenominal bond in their variance minimizing proportions.It is worthwhile to indicate the relationship between this view of hedging againstinflation and the investor's ultimate objective of optimal portfolio selection. Thiscan best be done in the framework of the Markowitz-Tobin-mean-variance modelof portfolio choice.^In that model the process of portfolio selection is divided into two separatestages: (1) identification of the efficient portfolio frontier and (2) choosing theoptimal portfolio on that frontier. This paper focuses on one particular point on theefficient frontierthe minimum variance portfolio. From this perspective hedgingagainst inflation is essentially the process of taking a risk-free-in-terms-of-defaultnominal bond as the starting point and using other securities to eliminate as muchof the variance of its real return as possible.We define the difference between the mean real return on a nominal bond andthe mean real return on the minimum variance portfolio as the "cost" of hedging

Assistant Professor of Economics and Finance, Sloan School of Management, MassachusettsInstitute of Technology. I am grateful to Stanley Fischer and Robert Merton for their advice and manyhelpful suggestions and to Boston University for financial support. An earlier draft of this paper wasgreatly improved by the comments of John Lintner and Steven Shavell. Since writing this paper I havebecome aware of the papers by Nelson (1975) and Jaffee and Mandelker (1975), which deal with one ofthe issues discussed in this paper.1. This paper ignores the ambiguities and difficulties involved in defining and measuring the generalprice level. It assumes that the Bureau of Labor Statistics Consumer Price Index is an appropriatemeasure.2. The nominal bond is assumed to be a pure discount bond with a maturity equal to the length of theholding period.


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460 The Journal of Financeagainst inflation. Since the unhedged nominal bond does not in general lie on theefficient portfolio frontier, the cost of hedging may be either positive or negative.Of course, ours is not the only possible definition of the term hedging againstinflation or inflation hedge. Indeed, in the literature on common stocks andinflation the term has been employed in two distinct ways, both of which differfrom the definition given above.According to the first of these alternative definitions a security is an inflationhedge if it offers "protection" against inflation, which in turn means the elimina-tion or at least the reduction of the possibility that the real rate of return on thesecurity will fall below some specified "floor" value such as zero. This definitionhas been made most explicit in the two papers by Reilly, Johnson and Smith (1970,1971), and it is implicit in Cagan (1974).In effect in order for common stocks to qualify as an inflation hedge under theReilly et al. definition they must be free of "downside" risk stemming from allsources, not just from inflation. Considering the large variance associated withequity returns, it is not at all surprising that these studies find that the nominal rateof return on common stocks is often less than the rate of inflation.The second alternative definition of the term inflation hedge as apphed tocommon stocks can be expressed as follows: a security is an inflation hedge if andonly if its real return is independent of the rate of inflation. This is the definitionemployed either explicitly or implicitly by Branch (1974), Fama and MacBeth(1974), and Oudet (1973).

Economic theorists have long considered common stocks an inflation hedge inthis sense because stocks represent ownership of physical capital whose real valueis assumed to be independent of the rate of inflation."* This independence impliesthat a ceteris paribus change in the rate of inflation should be accompanied by anequal change in the nominal rate of return on equity. Indeed this view is mostcommonly expressed in somewhat looser terms as a positive correlation betweenthe nominal rate of return on equity and the rate of inflation.Strictly speaking, a zero correlation between the real rate of return on equity andthe rate of inflation implies that in a regression with the nominal rate of return onequity as the left-hand variable, the coefficient of the rate of inflation should be 1.It should be pointed out that the three definitions of inflation hedge dis-tinguished above are not mutually exclusive. It is quite possible for a security toqualify as a perfect inflation hedge in more than one sense, although the onlysecurity which could conform to all three definitions would be one with acompletely riskless, non-negative real rate of return.The primary purpose of this paper is to determine to what extent common stocksare an inflation hedge in the first sense explained above. More specifically thequestion dealt with here is to what extent an investor can reduce the uncertainty ofthe real return on a nominal bond by combining it with a "representative"well-diversified portfolio of common stocks. As a necessary by-product, the paperalso addresses the question of whether the real return on common stocks is or is


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Common Stocks as a Hedge against I/Ration 461derives from it a quantitative measure of hedging effectiveness. Section III de-scribes the empirical tests performed and reports the findings, and Section IVsummarizes the study and draws the major conclusions.

II. HE DG ING NOMINAL BONDS AGAINST INFLATIONLet F(t) be the price level at time /, i.e., the nominal value at time / of somestandard basket of commodities. Let 1 -I- /-,(?) be the nominal one-period return perdollar invested in security / at time t. Since the real end-of-period value of thismoney return is 1 -I- ri(t)/F(t+ 1) baskets, the real return on security /, 1 + Ri(t), is

p(t)\ +RM[\ + (t)Wwhere tildes (~) are used to denote random variables.Suppose the security under consideration is a bond that m atures at time t + 1 andis free of default risk, so that its nominal return 1 -I- /(/), is known with certainty attime t. Its real return, however, is uncertain at time /, since jt depends on F(t+l).Let D(t) stand for F(t)/P(t_+ l),_and let us decompose D(t) into its mean, D(t),and deviation from its mean d(t). d(t) is the "unanticipated" proportional changein the purchasing power of money, which with a negative sign in front of it wouldbe called unanticipated inflation. Suppressing the time subscripts, the real return ona risk-free nominal bond can now be written as

or more simply^ J (1)

Note that the risk-free nominal interest rate has no subscript in (1); this conventionis adopted throughout the remainder of the paper.Now consider a second security, a representative portfolio of common stocks.Let us also decompose its real return into its mean and deviation from the mean:

\ +R^=\ +R^-^l (2)The deviation from the mean, I, can be further decomposed into two orthogonalcomponents as follows:

1=ad+ p ,where a = cov(e.,d)/va.T(d). (2) then becomes:

(3 )ad can be called the "inflation-risk" component of the real return on equity while ju


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46 2 The Journal of Financevariance, and the question therefore becomes what is the maximum reduction inthe variance of the real return on nominal bonds attainable by combining themwith the equity portfolio.

Letting w be the proportion of equities in the composite inflation-hedgedportfolio, the real return on the portfolio is:

or by substituting from (1) and (3):\ + R = \ + R + W{R^ -R) + [\ + r-w(\ + r-a)]d+ w^i. (4)

Let a] be the variance of d, al the variance of jii, and a^ the variance of the realreturn on the resulting inflation-hedged portfolio. From (4) we get:

al=[\ + r-w(\ + r-a)'fa]+w^(jl (5)The value of w which minimizes a is:

(l + r)(l-hr-)(l + ) V ( | / ? )

If a is negative, then the real return on equity is negatively correlated with theunanticipated rate of change in purchasing power, or in more familiar terms,positively correlated with unanticipated inflation. Equation (6) implies that w isthen positive, indicating that the hedger will be long in equity.Even if a is positive, meaning that equity as well as nominal bonds is negativelycorrelated with unanticipated infaltion, the hedger should take a long position inequity if 1 H- /-> a, i.e., if equity has the smaller coefficient for the d term. However,if 1 -f-r


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Common Stocks as a Hedge against Inflation 463expected value of the real return on the nominal bond, \ + R^, and the expectedvalue of the real return on the minimum variance portfolio, \ + Rfj, which is:

E^w{R-R,) (8)The sign of E can be either positive or negative. Assuming that R^ > R, the costof hedging will be positive if a > ( l -l-r) because in that event the hedger must sellequity short. On the other hand, if R e> Rn and a


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464 The Journal of FinanceA serious practical difficulty which arises in the estimation of (9a) and (9c) istjiat in addition to the regression parameters, the historical values of the variableD{t), the expected value of the index of purchasing power, are not observable. The

way I have chosen to deal with this problem is to assume that D evolves accordingto the model of "adaptive expectations," which says thatD{t) =D{t-\)+ e[D{t-\)-D{t-\)]

making the current D a function of past realized values of D{t).I have constructed a set of ten artificial series to serve as D{t), corresponding tovalues of of .1 , .2, .3, .4, .5, .6, .7, .8, .9, and 1.0, starting each series by settingD{t) equal toD{t) in January 1926. I then used D{t)-D{t) as the proxy for theunanticipated change in purchasing power, d{t).All equations were estimated by ordinary least squares regression, and allregressions and other calculations were done using holding periods of threedifferent lengths: 1 month, 3 months and 1year. Unless otherwise stated, thesample period used was January, 1953 toDecember, 1972.The definitions of variables and data sources used are as follows:

1. D{t), the index of purchasing power, is the value of the Consumer Price index(CPI) at the beginning of the holding period divided by its value at the end.Source: Bureau of Labor Statistics2. 1 + r{t), the risk-free nominal return, is the terminal value per dollar investedin a Treasury Bill with a maturity equal to the length of the holding period.Source: Salomon Bros., "Analytical Record of Yields & Sp reads."3. 1 + r^{t) is the terminal nominal return per dollar invested at time / which aninvestor would have received had he followed a policy of buying and reinvesting allreceipts in equal dollar amounts of all NYSE common stocks at the beginning ofeach month during the holding period.Source: CRSP tape.4. 1 + ^^(0 is \ + r^{t) multiplied by D{t). All returns are before taxes and

commissions.A . Estimates Based on (9a)Specification (9a) is:

Table 1 presents some regression results for a one year holding period for fivedifferent values of 9, the speed of adjustment parameter used in generating the timeseries for d{t).These results strongly suggest that a, is positive. Since d is the unanticipatedchange in purchasing power, a positive value of a, indicates that the real return on


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Common Stocks as a Hedge against InflationTABLE 1

R E G R E S S IO N S OF THE R E A L R E T U R N ON E Q U IT Y , 1 + R^, ON THE U N A N T IC IP A T E DP R O P O R T IO N A L C H A N G E IN P U R C H A S I N G P O W E R , dSample period: 1953-1972 1 Year holding period

46 5

0

.2

.4

.6

.8

1.0

Coefficients(Standard error of

coefficient)t statistics

0 1

1.1285(0,0482)

1.1393(0.0482)

1,1415(0.0475)

1.1431(0.04672)

1.1446(0.04646)

6.4665(2.9319)2.20568.0644

(3.4374)2.32619,3041

(3.7176)2.5027

10.2862(3.8657)2.660910.6683(3,9157)2.7240

N u m b e rof

observations

20

20

20

20

20

R^

0.2128

0.2342

0,2582

0.2824

0,2920

StandardError of

Regression

0.2175

0.2145

0.2111

0.2076

0.2062

DurbinWatsonStatistic

2.301

2.151

2.036

1.962

1.940

measurement. But measurement errors in d would only bias our estimates of a,towards z ero.'We can take the standard error of the regression as an estimate of Oj.Regressions were also run using the sample period February, 1926 to December,1952. In almost all of them the hypothesis that a,=0 could not be rejected. Thusthe result that common stock returns and the rate of inflation are negativelycorrelated is restricted to the post-Korean War period.

B. Estimates Based on (9b)Specification (9b) is:

5. For a discussion of the bias caused by measurement errors see Johnston (1972, p, 281), To check


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466 The Journal of FinanceSince D{t) = D{t) + d{t) this specification is equivalent to

\ + R,{t) = Po+'^iD{t) + a,d{t) + ^{t)Note that this specification does not require us to know D{t) in order to estimatei- Tables 2a, 2b, and 2c contain the regression results for 1 year, 3 month and 1month holding periods, respectively. In interpreting these results one should keep inmind that an inflation rate of zero corresponds to a value of 1 for D{t).

T A B L E 2REGRESSIONS OF THE REA L RETUR N ON EQUITY, 1 + R^ , ON THE INDEX OF PURCHASING

P O W E R , D

Sample period

Coefficients(standard errorsof coefficients) N o, of

t statistics observationsStandard Durbin-Erro r of W atson

Regression Statistic

a. 1 Year Holding Period1953-1972 -5 ,68 28 6.9730

(2,8598) (2.9270)-1,9871 2.3823

b. 3 Month Holding Period"1953-1972 -4 ,9 92 6 6,0575(1,7610) (1,7713)

-2.8351 3,4197c. 1 Month Holding Period"

1953-1972 -4 ,8 18 2 5.8390(1.1910) (1.1934)

-4 ,0455 4 .89305-year subperiods:1953-1957 -5,4449 6,4606

(1,5057) (1,5075)-3,6162 4,2857

1958-1962

1963-1967

1968-1972

-7 . 1 3 7 3 8.1574(2.9059) (2.9089)

-2 .4561 2.8043

- 2 , 5 6 5 0 3,5871(2.8733) (2.8787)

-0,89 2 7 1.2461

-7 . 7 36 9 8,7719(4.3760) (4.3924)

-1.7681 1.9971

20

80

240

60

60

60

60

0,2391 0.2137

0,1312 0,0843

0,0953 0.0415

0,2449 0.0305

0,1224 0,0410

0.0309 0,0377

0,0661 0.0537

2.371

1,906

1.665

1.768

1.478

1,823

1.547


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Common Stocks as a Hedge against Inflation 467These results seem to confirm the belief that a, is positive. Moreover, the resultsfor a one month holding period indicate that even when the twenty year sampleperiod is broken down into four 5-year subperiods a, is still significantly positive

with a magnitude of around 6.*C. Estimates Based on (9c)

Specification (9c) is:

This specification is more flexible than (9b) because it allows D{t) and d{t) to havedifferent coefficients.The coefficient ^8, measures the effect of an anticipated change in purchasingpower on the real return on equity, while a, measures the effect of the unantici-pated rate of change. Even those who are willing to believe that a, is positive wouldexpect fi^ to almost surely be insignificantly different from 0.But the evidence presented in Table 3 seems to warrant the rejection of thathypothesis. The table shows regression results for 1 year, 3 month and 1 monthholding periods and for three different values of 0, the speed of adjustmentparameter used in computing the .0(0 series. Although the t statistics in theregressions for the 1 year holding period a re rather low, they are based on arelatively small sample of 20 observations, which leads to large standard errors ofthe coefficients. As we go from a 1 year to a 3 month and then to a 1 monthholding period, the num ber of observations increases and so do the t statistics.

These results are even more disturbing than the earlier finding that a, is positive,for they seem to imply that not only is the real return on equity negativelycorrelated with unanticipated inflation, but it is also inversely related to anticipatedinflation. Even the lowest estimate of ;8, reported in Table 3 implies that a ceterisparibus increase of 1 percentage point in the expected rate of inflation is associatedwith a decline of 4 percentage points in the real return on equity.To check for the stability of the parameters reported in Table 3 I divided thetwenty year sample period into four 5-year subperiods and reestimated (9c) for the1-month holding period. Although the resulting estimates of ^, and a, varied quitea bit from subperiod to subperiod they were all positive and in most casessignificantly greater than zero. F-tests were performed to test the hypothesis thatthe regression coefficients were the same during all four subperiods and thathypothesis could not be rejected at the .05 level of significance.What do these results imply about the effectiveness of common stocks as aninflation hedge? Our regressions indicate that for a one-year holding period, a isabout 7 and a^ about 0.21.In order to calculate 5 one must also set the values of 1 + r and a,. In thecalculations presented below 1 + r is arbitrarily set at 1.08. During the 1953-1972


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46 8 The Journal of FinanceT A B L E 3

REGR ESSION S O F 1 -i- RJ^t) ON D(t) A N D d{t)Sample period : 1953-1972

Coefficients(standard errorsof coefficients)

d t statistics/So )3, a,

a. I Year Holding Period,2 -4 ,5 18 2 5.7807 7.0712

(7.0716) (7,2393) (3,0572)-0 .6 38 9 0.7985 2.3130

N o . ofObservat ions R ^

20 0.2413

StandardError of

Regression

0.2197

Durbin-WatsonStatistic

2.356

.6 -2 .9 22 7 4,1560 10,3190(3,4586) (3.5364) (3.7789)

- 1 , 2 9 6 1 1,1752 2.73071.0 - 3 , 6 3 1 0 4 , 8 85 0 1 3 , 1 92 7

( 2 . 8 0 1 6 ) ( 2 . 86 5 4 ) ( 4 , 0 0 7 3 )- 1 . 2 9 6 1 1 .7 04 8 3 , 2 9 2 2

b. 3 Month Holding Period,2 -3 ,5 68 0 4.6251 6,8953

(2.7293) (2,7447) (2,1571)- 1 , 3 0 7 3 1,6851 3.1967

.6 -5 ,5 59 4 6,6274 5,5475(2.2150) (2,3377) (2,1464)

-2.5099 2.9751 2,58461,0 -6 ,3 1 1 9 7,3841 4.7322

(2.0016) (2,0232) (2.0231)-3.1378 3,6496 2,3392

c, 1 Month Holding Period,2 -5 .5 85 3 6.6075 5,4782

(1,9078) (1.9114) (1,3853)-2,9276 3,4568 3.9546,6 -6 ,3 30 7 7.3543 4,9940

(1,5931) (1,5962) (1,3306)-3,9737 4.6073 3.7534

1,0 -5 .8 4 28 6.8655 5.2527(1.4365) (1,4393) (1,2781)

-4,0673 4,7699 4,1097

20 0,3139 0,2089 2,144

20 0.3954 0.1961 2.0316

80 0.1364 0.0845 1.936

80 0.1332 0.0847 1.888

80 0,1508 0,0838 1,871

240 0,0962 0.0416 1.666

240 0.1029 0.0414 1.678

240 0,1013 0,0414 1,675

period, the ten estimated standard deviations of d (one for each value of 9) for a


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Common Stocks as a Hedge against Inflation 469nominal bonds a hedger could eliminate roughly 18 per cent of the variance of thereal return on those bonds. If 02/0, = 17.8 then 5' = 0.10 and H '= .02.To compute the cost of hedging we must assume values for R^ and R^. The meanvalues of these real rates of return for the 1953-1972 period are 1.46 per cent p.a.and 12.9 per cent p.a., respectively. Thus, if 02 /0 , = 12.6 the cost of the 18 per centreduction in the variance of the real return on the nominal bond would be areduction in expected return of 0.34 per cent p.a.

IV. SUMMARY AND CONC LUSIONSThis paper has attempted to address the question of how effectively an investor canhedge against inflation with a "representative" well-diversified portfolio of com-mon stocks. Since hedging against inflation means reducing the uncertainty of realreturns which stems from uncertainty about the future price level, the measure ofhedging effectiveness used is the proportional reduction in the variance of the realreturn on a nominal bond attainable by combining it with the equity portfolio.The formula for this measure reveals that the effectiveness of common stocks asan inflation hedge depends on two parameters. The first of these is the ratio of thevariance of the non-inflation stochastic component of the real return on commonstocks to the variance of unanticipated inflation. The larger this variance ratio is,the less effective is equity as an inflation hedge.The second parameter is the difference between the nominal return on thenominal bond and the coefficient of unanticipated inflation in the equation for thereal return on equity. The greater the absolute value of this difference, the moreeffective is equity as an inflation hedge.Using annual, quarterly and monthly data for the twenty year period 1953 to1972, these parameters were estimated under a number of different assumptionsabout the stochastic process generating the data. The regression results obtained inderiving the estimates seem to indicate that, contrary to a commonly held beliefamong economists, the real return on equity is negatively related to both antici-pated and unanticipated inflation, at least in the short run. This negative correla-tion leads to the surprising and somewhat disturbing conclusion that to usecommon stocks as a hedge against inflation one must sell them short.

R E F E R E N C E S1. B. Branch, "Common Stock Performance and Inf lat ion: An Internat ional Compar ison," J.

Business, 1974, 47, 48-52.2. P. Cagan, "C om mon Stock Values and Inf lat ionThe H istor ical Record of M any Countr ies,"

Nationat Bureau Report Supptement, New York, 1974.3. E. F. Fama and J. D. MacBeth, "Tests of the Mult iper iod Two-parameter Model ," J. Financiat

Economics, 1974, 1, 43-66.4. J. Jaffee and G. Mandelker, "The 'Fisher Effect ' for Risky Assets: An Empirical Test ," workingpaper, 1975.


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470 The Journal of Finance8. R. C. Merton, "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," Rev.Econ. Statist; 1969, 51, 247-257.9. . "Optimum Consumption and Portfolio Rules in a Continuous-Time Model," Journal ofEconomic Theory, 1971, 3, 373-413.10. C. R. Nelson, "Inflation and Rates of Return on Common Stocks," working paper. University ofChicago, February, 1975.11. B. Oudet, "The Variation of the Return on Stocks in Periods of Inflation," J. Financial an dQuantitative Analysis, 1973, 8, 247-258.12. F. K. Reilly, G. L. Johnson, and R. E. Smith, "Inflation, Inflation Hedges, and Common Stocks,"Financial Analysts Journal, 1970, 28, 104-110.

13. F. K. Reilly et al, "Individual Common Stocks as Inflation Hedges," /. Financial and QuantitativeAnalysis, 1971, 6, 1015-1024.


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common stocks as a hedge against inflation

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