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This article appeared in the Spring 1999 issue of the Journal of Portfolio Management.

Note that the tables and figures may not format properly in some browsers.


Money Illusion Revisited:
Linking Inflation to Asset Return Correlations


 Ross M. Miller
Miller Risk Advisors
2255 Algonquin Road
Niskayuna, NY 12309


Evan Schulman
Upstream Technologies
69 Mount Vernon Street
Boston, MA  02108



In 1979, in a pioneering work in behavioral finance, Franco Modigliani and Richard Cohn showed that investors tend to undervalue firms in inflationary times if they do not properly account for the effects of inflation on a company’s income statement. They termed this effect “money illusion”. This paper examines a corollary of their result: in the presence of “money illusion” the correlation between stock and bond returns will be abnormally high during periods of high inflation. For the U.S., it is shown that inflation had exactly this effect on the stock/bond correlation during the postwar era. Asset allocation strategies that rely on the high correlation coefficients generated over the last 20 years can be expected to generate inefficient portfolios in regimes of low inflation.


The gyrations of the U.S. equity market, particularly as reflected in Standard and Poor’s 500 Stock Index, have received unremitting attention from academicians and practitioners alike over the past twenty years. An early, and very controversial, contribution to this literature was the Modigliani and Cohn (1979) article in the Financial Analysts Journal. This article was one of several appearing at the end of the 1970s to fashion an explanation that linked the two scourges of that period: high inflation and depressed equity markets. In a pioneering work in the nascent field of behavioral finance, Modigliani and Cohn demonstrated how a simple behavioral quirk, “money illusion” could generate a direct link between high inflation and undervalued equity.

The form that the money illusion took was straightforward: an upward movement in inflation would generate a one-time permanent reduction in nominal accounting net income without changing the real value of the firm, and this illusory drop in earnings would trigger a decline in equity values. The Modigliani-Cohn work ignited a firestorm of research that looked for purely “rational” explanations for the behavior of the US equity market. However, for either rational or irrational reasons, the decline in equity markets ended with the taming of inflation in the early 1980s and the great rise in valuations since then has coincided with a continuing long-term decline in inflation.

Equity returns were not the only asset returns that were affected by inflation. The return on debt instruments, such as government bonds, was affected in a more direct and rational manner. As inflation increased beyond the expected levels, the present value of both the bond coupons and principal declined, causing the bond to depreciate, frequently more than offsetting any accrued interest. As inflation declined back to its “normal” level, the value of the bonds rebounded, generating returns substantially in excess of the interest payments. Because the decline in bonds was nearly simultaneous with the decline in stocks and the subsequent rise in the bonds mirrored a similar rise in stocks, the correlation of returns between these two assets classes has been quite high by historical standards since the 1970s. In contrast, during the 1950s it was not uncommon for the returns on stock and bonds to be somewhat negatively correlated.

The ability to forecast the future correlation between stock and bond returns is critical to the practice of asset allocation—the division of assets among different asset classes based on their anticipated returns and their correlation. Quantitative approaches to asset allocation have their roots in the pioneering (and Nobel-prize winning) work of Harry Markowitz on portfolio optimization. While Markowitz’s model can be applied directly to the problem of allocating assets between stocks and bonds, making it a  staple of investments textbooks (see Bodie, Kane, and Marcus, 1996), practitioners dating back to Grauer and Hakansson (1982) have refined the basic model in a variety of ways. [1] Regardless of the approach that one takes to asset allocation, it is generally the case the better one is able to forecast future correlations between the returns in the asset classes, the greater will be the performance of the portfolio of assets that one constructs.

This paper will show that the inflation rate can be an important determinant of the correlation between stock and bond returns. We refer back to Modigliani and Cohn’s original model on money illusion as inspiration for a simple explanation for the linkage between the inflation rate and the correlation between these returns. It is an important consequence of their model, not mentioned in the original paper, that any real or behavioral influence that tends to depress the market’s perception of equity values will also tend to increase its perception of financial leverage, which in turn will increase the sensitivity of equity values to changes in real interest rates. The aggregate equity market should then be more tightly linked to debt markets during inflationary times than in noninflationary times.

This paper begins by reviewing the accounting basis for money illusion and the behavioral link to the improper valuation of equity. Then, we look at the empirical evidence that inflation exerts an influence on the correlation between equity and debt. Finally, we examine both the limitations and potential for future refinement and application of the analysis.

The Money Illusion

To illustrate the illusion let us imagine a firm, the demand for whose product is real, impervious to inflation and in zero growth steady state. While the reader may have such a firm in mind, we have selected Sewer Security Ltd. as our example. It is based just outside the City of New York, has been in business for some 30 years and, by regulation, can service only the island of Manhattan. That regulation also stipulates that Sewer Security is the only firm allowed to so service Manhattan. Its product is designed to prevent rodents and other vermin from gaining access to buildings through plumbing connections. The device is guaranteed for five years, and disintegrates in five years and 60 days. Virtually all of their installations are now replacements. Installation of the replacement takes but minutes, payment is COD, and prices are set to maintain the firm's operating profit at 50% of sales. Also, the real value of the physical plant equals the real value of sales, and the plant depreciates at a constant and continuous rate of 10% per annum.

The owner is fiscally conservative. He limits the amount of debt to the book value of Sewer Security's physical plant. Also, he has observed that upward sloping yield curves are the norm and, to reduce interest costs, finances his operations only with floating rate paper. [2] The rate on the paper is tied to the rate on government treasury bills: it changes daily. In a presentation to the students at a prestigious east coast business school, he noted that his firm was a real business in that any changes in costs, positive or negative, are immediately passed on to his customers. Further, his debt management policy–constant real debt at a floating rate–together with the firm’s natural advantages, should leave the real wealth of his debt and equity holders, unaffected by changes in inflation.

The corporate tax rate is 50% and the real rate of interest is 4%. Both remain so throughout the period we examine.

For ease of exposition, the income statement below starts with sales of 100.

Sewer Security


(Steady State, No Inflation)


Sales                                                   100

Expenses                                             50

Operating Profit                                  50 [by definition above]


Interest                                                   4

Plant Replacement                            10 [10% of plant (100)]


Net Before Taxes                               36

Let us now assume, at the beginning of the next year, unanticipated inflation at a continuous annual rate of 20% that ceases, again unexpectedly, after a year. When we examine this operation after a year of inflation we find the following:

Sewer Security

Income Statement

(After 1 Year of Steady State, Unanticipated Inflation at an Annual Rate of 20%)


                                    Sales                                                   110

                                    Expenses                                             55

                                    Operating Profit                                  55


                                    Interest                                                 15.2

                                    Plant Replacement                            11

                                    Net Before Taxes                               28.8

To explain: we grew the sales and expenses in a lagged response to the unanticipated inflation. Even though sales and expenses are running at a rate 20% above last year’s by year-end, the average for the year is only half that. Clearly the ratio of operating profit to sales is unaffected by inflation.

However, interest expenses jumped dramatically. Remember, the rate changes daily as a function of the Treasury bill rate. We assume that rate encapsulates realized inflation, with a one-day lag, in such a way as to protect the debt holder from the ravages of inflation. Thus by the end of the year, lenders will be asking for the 4% real rate plus the 20% inflation to date. The average for the year can be calculated as the compounding of the 4% real rate plus the 20% continuously compounded inflation. Furthermore, any changes in the risk premium will now have a greater relative impact on earnings. Hence, not only will the value of the firm be depressed if investors base valuation on earnings, which was Modigliani and Cohn’s original point, but the apparent increase in the leverage of the firm, as reflect in its debt/equity ratio, will make it more interest-rate sensitive. As a result its returns will correlate more closely with those of debt instruments, which are also interest-rate sensitive.

Finally, returning to the income statement, depreciation is 10% of the physical plant, which is now on the books for 110. We had to continuously replace 10% of the plant over the year, at an average cost 10% higher than the plant's book value. The reader will note that earnings (and stock prices, to the extent that stock prices reflect earnings) decline along with bond prices as nominal interest rates rise.

We excerpted comments from a security analyst's report on Sewer Security:

“Sewer Security’s earnings plummeted 20% in a most disappointing year. Over the years Sewer has become the prototypical yield stock on which those with income needs could depend. The past stability of Sewer’s operations has allowed management to pay out virtually 100% of earnings as dividends year in and year out. We are at a loss to understand how management can continue to maintain the dividend at its current rate.


The main culprit in this tragedy appears to be management’s naive and shortsighted debt management policy. Interest payments almost quadrupled with devastating impact on earnings. Unless management gets a better handle on the principles of corporate finance, this company is in for some very rocky times.”

Some years later, after inflation had subsided, management presented the following annual report:

Sewer Security

Income Statement

(Post the 20% Inflationary Shock, Steady State, No Inflation)


                                    Sales                                                    120

                                    Expenses                                              60

                                    Operating Profit                                   60


                                    Interest                                                     4.8

                                    Plant Replacement                              12

                                    Net Before Taxes                                 43.2

The reader can confirm that the Net Before Taxes of 43.2 is 20% more than the original Net of 36. As expected, on a point-to-point basis, our firm appears impervious to the ravages of inflation: however, there is a one to one relationship between the inflation rate and Sewer Security’s earnings.

We managed to find a security analyst's comment on this report, which is reprinted below; we were unable to determine whether it was from the same analyst quoted above.

“Sewer Security's management pulls off a dramatic turnaround! Earning skyrocketed 51% over the low of a couple of years ago. Showing confidence in the strength and quality of their renewed earnings, management raised the firm's dividend, which is now a full 20% above historic levels.

The focus of this turnaround was the firm’s debt management policy. Interest payments fell by some two-thirds from their high with a most favorable impact on earnings. Management took a strong bet on the level of rates, and it paid off handsomely!”

To recap, we designed a firm to be impervious to inflation: we protected it in terms of demand for its product, its costs and its financial structure. However, this protection failed, and failed dramatically, during periods with changing levels of inflation.

Accounting Theory

Before we look at the empirical evidence, we should take a moment to explain what happens to the “earnings”, and to note a theme in the literature that argues that accounting, appropriately practiced, should not fall prey to money illusion.

In correspondence with the second author (reprinted here in the Appendix), André Perold at the Graduate School of Business at Harvard University proves that the observed loss of earnings appears in the properly calculated value of the plant, exactly as Modigliani and Cohn suggested. But we note that most accountants do not practice calculus, and it is unlikely even if they did, that FASB would accept such adjustments.

Fischer Black (1980) and Jack Treynor (1993) among others, argue that accountants should adjust earnings to reflect the firm’s underlying value. To quote from Fischer Black (1980, p. 20), “Security analysts are clearest in their thinking about earnings, . . . They would like the accounting process to give an earnings figure they can simply multiply by 10 to get an estimate of value.” Black then notes that the ratio of price to earnings is more stable than the ratio of price to book. He argues that this proves that accountants do attempt to make earnings a statement of value as opposed to a statement of change in value.

We accept his argument and evidence. However, price earnings ratios are not constant either across firms or through time. Accountants may attempt to make earnings a statement of value but, according to investors, they fail.

The Empirical Evidence: Stock/Bond Correlations and Inflation

Although the procedure for estimating the variance of a financial time series has received enormous attention, especially once listed options appeared in the 1970s, the estimation of covariance or correlation, which is needed to compute the overall variance and risk profile of a portfolio of assets, is only beginning to receive serious attention. [3] It is common practice to use from 36 to 60 monthly observations (possibly weighted by a time-decaying factor) to compute the correlation between two assets classes–in our case the U.S. stock and long-term government bond returns series that are reported the Stocks, Bonds, Bills, and Inflation database prepared by Ibbotson Associates [4] . Figure 1 gives a graph of the 36-month historical correlation between U.S. stocks and bonds during the post-war period beginning in 1952. Following the lead of Modigliani, Cohn, and countless other researchers, we will restrict our analysis to data taken from this period; to the extent that the analysis has been done going back to 1926 the additional data tends to “noise up” the results but not change the basic conclusions. The volatility of this series indicates, as we shall demonstrate statistically, that past correlation by itself is of very limited use for forecasting future correlation.


A natural way to proceed would be to divide the period from 1952 to 1995 into non-overlapping 36-month periods and then determine the extent to which the correlation during a period depends on the information available at the beginning of the period; in particular, the observed correlation for the past period and the rate of inflation [5] . An obvious problem with this method is there are only 14 distinct 36-month periods available during the postwar era, too few from which to draw meaningful statistical conclusions. To get around this problem we will use two approaches. First, we will use a shorter 12-month window, which will expand the number of observations to a more comfortable, but still low, 42. Second, we will greatly expand the number of observations to 400 or more by using monthly overlapping windows. While this method makes the fullest use of the information available, it also violates the independence assumptions that underlie regression analysis, which limits the applicability of the results. Nonetheless, these two approaches taken together provide some insight into the correlation between stocks and bonds.

Although the introduction of inflation-indexed Treasury Bonds in 1996 may ultimately lead to meaningful time series that give the market expectation of future inflation, for this analysis we are forced to make do with much less. For the sake of completeness, we use two separate proxies for inflation. The first proxy is the one-month Treasury Bill rate reported by Ibbotson. We convert it from a monthly rate to an annual rate for comparability and ease of interpretation. As a measure of inflation, the T-Bill rate has the advantage that it is forward-looking, if only for a month. However, it has the disadvantage that if the real rate of interest is nonconstant, which is especially likely during times of active monetary intervention, it provides a noisy and potentially biased measure of inflation. The second proxy is the consumer inflation rate for the previous year (the change in CPI) also as reported by Ibbotson. (Unlike the T-Bill rate, a single month’s number cannot be used because of its lack of precision with which it is reported by the government.) While this proxy is a backward-looking and subject to substantial measurement error, it does not directly conflate other macroeconomic variables as does the T-Bill rate. Figure 2 gives a graph of these two proxies, which closely track one another much of the time as evidenced by a correlation of 0.7472.

In the absence of any fundamental insight as to how inflation and historical correlation affect the future correlation between stocks and bonds, we use a simple linear specification. We recognize that because correlation is limited to the range of –1.0 to 1.0 that this simplification may lead to misspecification of the relationship for extreme values of inflation. We accept this limitation in light of the value of viewing the linear specification as a convenient first-order approximation to the real relationship.

The regression results for the 12-month windows are given in Table 1. Each of the 42 windows is constructed to span a single, complete calendar year. The first regression fits the 12-month stock-bond correlation to the annual T-Bill rate (for the previous December) and a constant term. This proxy of inflation by itself accounts for nearly 36% of the variance in the 12-month stock/bond correlation. From the equation we see that a 1% increase in the annualized T-Bill rate leads to rough a 0.07% (or a 7 basis point) increase in the correlation between stocks and bonds. Looking at the second and third regressions, the historical correlation (from the previous 12 months) taken alone explains only 19% of the variance in correlation and adds only a slight amount to the explanatory value of the T-Bill rate. The use of the CPI-based inflation measure in place of the T-Bill in the final two regressions generates results that are substantially the same with a bit less precision. In all cases, the t-statistics for the coefficients of the two inflation proxies, which appear in parentheses under the coefficients, are significant at well beyond the 99% level. A regression with both inflation proxies included is presented neither here nor later in this paper because the high degree of collinearity between the two proxies, as noted above, makes the output of this regression of little additional value.

                           T-Bill             Change in         Prior Year’s
Constant             rate                   CPI               Correlation                      d                   R2

 -0.1790            7.0924                                                                        1.7970             0.3596
(-1.8381)         (4.7391)

   0.1244                                                              0.4395                     2.3307             0.1921
 (2.0022)                                                            (3.0843)

  -0.1582           6.0193                                       0.1765                     2.2459             0.3823
(-1.6077)         (3.4654)                                     (1.1987)

  -0.0563                                   6.6334                                               1.5732             0.3306
(-0.7077)                                 (4.4443)

 -0.0590                                   5.4656               0.2288                      2.2217             0.3724
(-0.7567)                                 (3.3470)            (1.6126)

Table 1: Regressions for predicting the monthly stock/bond correlation for non-overlapping 12-month annual windows from 1953 to 1995. (T-statistics are in parentheses, d is the Durbin-Watson statistic, and N=42.)

Because the three time series under consideration—stock/bond correlation, T-Bill rates, and consumer inflation—are all time series that tend to trend upward over the time period under consideration, one must be alert to the possibility of spurious regression, the tendency for regressions between macroeconomic time series to show significant relationships caused by the violation of the assumptions of the regression model rather than any true relationship between the variables. Fortunately, there is no evidence of spurious regression any of the regressions in Table 1: the Durbin-Watson statistics show no undue correlation of the residuals (for spurious regressions the Durbin-Watson statistic is often less than the R2) and the dependent variable, stock/bond correlation, as the second regression equation indicates, does not have a unit root. [6]

The second approach to estimating stock/bond correlations, which uses overlapping windows, exhibits low Durbin-Watson statistics, one of the symptoms of spurious regression. This is simply an artifact of the overlap in the data. The inputs used to compute the correlation for two successive months will be the same except for the latter will drop the oldest stock and bond return observation and replace it with the stock and bond returns for the current period. This will automatically generate high autocorrelation in the stock/bond correlation time series and the degree of autocorrelation will increase with the size of the window.

Table 2 repeats the five regressions done for the non-overlapping windows in Table 1 for 12-month, 36-month, and 60-month correlations. The top set of five regressions, which cover the 12-month correlations over 504 monthly observations show little change in results except that the t-statistics are all much larger and the Durbin-Watson, which is not included in the table, is near 0.2 for the 12-month regressions and near 0.1 for the others. Comparing the results in Tables 1 and 2 we find that for a 12-month window the additional information contained in monthly observations may generate somewhat more accurate estimates of the parameter coefficients; however, it is difficult to gauge that accuracy because the autocorrelation in the errors leads to an overstatement of the t-statistics. [7] Indeed, had we not run the nonoverlapping regression first, these overlapping regressions would at first glance appear to be spurious.

                                      T-Bill               Change in         Prior Year’s
Constant            rate                    CPI                Correlation                 N         R2

  12      -0.1646              6.8761                                                                      504      0.3180
           (-5.8068)         (15.2998)

  12      0.1175                                                               0.4614                      504      0.2118
           (6.8276)                                                           (11.6140)

  12      -0.1365              5.4371                                     0.2487                      504      0.3656
           (-4.9191)         (11.0226)                                  (6.1326)

  12      -0.0421                                      6.0002                                              504      0.2524
           (-1.7231)                                 (13.0197)

  12      -0.0389                                      4.4723             0.3019                      504      0.3267
           (-1.6765)                                  (9.24715)         (7.4341)

  36      -0.0897              5.8388                                                                      456      0.4271
           (-4.2989)          (18.3966)

  36       0.1248                                                              0.6526                      456      0.4767
          (11.1179)                                                          (20.3367)

  36      -0.0401              3.5134                                     0.4489                     456      0.5849
          (-2.2068)          (10.8665)                                 (13.1239)

  36        0.0224                                    5.0369                                              456      0.3460
             (1.2456)                               (15.4988)

  36        0.0243                                    2.8750             0.5016                      456      0.5639
             (1.6551)                                 (9.5187)         (15.0454)

  60        0.0274            4.2565                                                                      408      0.4428
             (1.6891)       (17.9624)

  60        0.1832                                                            0.5669                      408      0.5645
(23.1364)                                                       (22.9407)

  60        0.0914           1.9579                                      0.4166                      408      0.6185
             (6.4321)        (7.5717)                                  (13.6578)

  60        0.1144                                    3.6097                                              408      0.3478
             (8.0791)                               (14.7132)

  60        0.1328                                   1.4504              0.4644                      408      0.6022
           (11.9308)                                (6.1956)          (16.0951)

 Table 2: Regressions for predicting the monthly stock/bond correlation for overlapping windows in the postwar period (t-statistics in parentheses).

The results in Table 2 for the 36-month and 60-month correlations are qualitatively similar to the 12-month correlation, only the overall fit and the precision of the coefficient estimates is better. This is to be expected given that the longer windows make the measurement of correlation less noisy. Also, the correlation for the prior window now predicts future correlation better than does either of the inflation proxies. As before, there is little difference in explanation power between the two measures of inflation, the T-Bill performs minimally better than the CPI. The important thing is that inflation, either alone or in combination with the past correlation, significantly aids in the prediction of future correlation between stocks and bonds over the sample period (and there is no basis in received theory for this to happen).

Interpreting and Extending the Results

 Methodologically, this paper should be considered as a point of departure for further investigation into the nature of stock/bond (and other asset) correlations. We hope to have opened Pandora’s Box just a bit and to have avoided the temptations of “data mining” in that we did not try to find the “best” estimate of stock/bond correlation. Clearly, inflation is not the only factor to effect the real or perceived aggregate leverage of S&P 500 firms. Growth in the interest-sensitive financial services sector, from its inclusion in the S&P 500 in 1976 to the present, and the fact that major S&P 500 companies such as General Electric, General Motors, and Ford have significant financial subsidiaries, may underlie an apparent secular uptrend in stock/bond correlations that this analysis does not directly take into account. [8]  

However, starting with the insight that money illusion would affect not only the value of equity but also the correlation of its returns with debt returns we found empirical evidence to support the expected linkage between inflation and stock/bond correlations. While we have not derived a precise measurement of stock/bond correlation and inflation, the linkage is robust to alternative methods for measuring both. The straightforward analysis performed in this paper is not put forth as definitive proof that money illusion drives equity valuation; instead, we demonstrate that a simple behavioral model can provide a fuller understanding of a neglected area—the correlation between stock and bond returns.

The main empirical message of this paper is that asset allocators, whose performance ultimately depends on their ability to forecast asset correlations accurately, need to look beyond historical returns time series to macroeconomic and other variables in making their projections. The effect of inflation on equity values, through money illusion or a related mechanism, is enough to justify its incorporation into any estimate of stock/bond correlation. It almost goes without saying that similar effects are likely to affect the correlations between other pairs of asset classes.


Definitions and Assumptions:

Pt = plant at time “t”

St = sales running rate = Pt by assumption

Bt = bonds outstanding = Pt by assumption

Ct = cumulative cash flow from 0 to “t”, C0= 0

i = inflation rate (constant)

g = profit margin (gross, before interest and depreciation
      [depreciation = plant ‘maintenance’])

y = real rate of interest

r = nominal rate of interest = i+y

m = maintenance rate


Plant grows with inflation:  dP/dt = i  => Pt = P0eit = St = Bt                                (1)

Cash Flow:  dC/dt = gSt - mPt - rBt + iPt +rCt                                                      (2)

            Where:  gSt = gross profit

                          mPt = maintenance expenditures

                          rBt = interest on debt

                          iPt = gain in nominal value of plant

                                  = increased borrowings which are ‘dividended’ out

                          rCt = reinvestment of cash flow at rate r (in a money market fund)

therefore dC/dt = (g - m + i - r) * Pt + rCt                                                                (3)

                                    substitute r = i + y and,

               dC/dt            = (g - m - y) * P0eit + (i + y) * Ct                                           (4)

Note the inflation gain in the plant is canceled by the inflation component of the nominal rate.

The solution to the differential equation (4) is

                        Ct = (g - m - y) * P0eit * ((eyt - 1)/y)                                                  (5)

To calculate the present value of this cash flow, note that $1 growing at rate r grows to ert by time “t”

Therefore the Value of the Firm =

e-rt * Ct = e-(i+y)t * Ct = (g - m - y) * P0 *((1 - e-yt )/y)         (6)


1.)        Inflation i affects only the plant = P0eit and hence the cash flow Ct  in (5)

            However, i disappears from value in (6)

2.)        Reinvestment of cash flow at rate r is needed only to calculate the present value of the cash flow.  If the cash is dividended out, the accounting numbers are as follows (for the period 0 to t):

Sales =
òoT S(t)/dt = òoT P0eit dt = P0 * ((eit - 1)/i) = P0*a   where a = ((eit - 1)/i)  (7)

Gross Profit = g * P0*a

Maintenance = m * P0*a

Interest Expense = (i + y) * P0*a

Inflation Gain in Plant = i * P0*a

Accounting Net Income = (g - m - i - y) * P0*a

Economic Net Income = (g - m - i - y + i) *
P0a = (g - m - y) * P0*a


Black, Fischer , 1980. “The Magic in Earnings: Economic Earnings versus Accounting Earnings.” Financial Analysts Journal, vol. 36, no. 6 (November/December):19-24.

Bodie, Zvi, Alex Kane, and Alan J. Marcus, 1996. Investments. 3rd ed. Chicago: Richard D. Irwin.

John Y. Campbell, Andrew W. Lo, and A. Craig MacKinlay, 1997. The Econometrics of Financial Markets. Princeton, N. J.: Princeton University Press.

Engle, Robert F., and Joseph J. Mezrich, 1996. “Correlating with GARCH.” RISK, vol. 9, no. 8 (August):36-40

Grauer, Robert R., and Nils H. Hakansson, 1982. “Higher Return, Lower Risk: Historical Returns on Long-Run, Actively Managed Portfolios of Stocks, Bonds, and Bills, 1936-1978. Financial Analysts Journal, vol. 38, no. 2 (March/April):39-53.

Koskosidis, Yiannis A., and Antonio M. Duarte, 1997. “A Scenario-Based Approach to Active Asset Allocation.” Journal of Portfolio Management, vol. 24, no. 2 (Winter):74-85.

Modigliani, Franco, and Richard A. Cohn, 1979. “Inflation, Rational Valuation and the Market.” Financial Analysts Journal, vol. 35, no. 2, (March/April):24-44.

Treynor, Jack, 1993. “Feathered Feast: A Case.” Financial Analysts Journal, vol. 19, no. 6 (November/December):9-12.

Wainscott, Craig B., 1990. “The Stock-Bond Correlation and Its Implications for Asset Allocation.” Financial Analysts Journal, vol. 46, no. 4, (July/August):55-60, 79.

* An earlier version of this article was presented at the 1996 First Quadrant Advisory Board Meeting. The authors thank the participants for insights and helpful comments, in particular, André Perold and Jack Treynor. Jarrod Wilcox also offered some telling comments.

[1] Koskosidis and Duarte (1997) provide a brief survey of the asset allocation literature.

[2] Use of longer term debt will lead to the same conclusions as the stylized analysis presented here; however, the effect of inflation on the firm’s income statement will take more time to fully manifest itself.

[3] Wainscott (1990) noticed that the correlation between stock and bond returns could be influenced by macroeconomic variables, but not did provide a mechanism to link the two. More recently, Engle and Mezrich (1996) address the issue of “conditional correlations” in high-frequency data that parallels the conditional variance models, such as GARCH, that apply to a single time series.. Standard textbooks in financial econometrics, such as Campbell, Lo, and MacKinlay (1997) do not yet deal the estimation of correlations between assets.

[4] The return series for stocks is the total return (pre-tax dividends and capital appreciation) for the S&P 500 Stock Index since its inception in 1957 and appropriate proxies prior to then. The (long-term) bond return series attempted to approximate the return of an “ideal” 20-year Treasury bond.

[5] To gain better comparability over long time periods we use the correlation between stock and bond returns rather than their covariance. The two measures of association are related as follows:  Correlation(Stocks, Bonds) =

[6] In general, the regression of the stock/bond correlation on its previous value will generate a coefficient somewhat under 0.5. This is to be expected given that the correlation, which is bounded between
 –1 and 1, cannot easily behave as if it would eventually wander off to plus or minus infinity.

[7] The effect of raising the t-statistics uniformly without changing the R2 can be achieved with an arbitrary set of data simply by repeating each observation twelve times which, to a first approximation, is what we are doing here.

[8] Some of the effect attributed to inflation may be attributable to trending behavior; however, the inflation effect loses some of its significance after simple linear detrending, largely due to the introduction of collinearity. Inclusion of such a trend in the analysis is a clear misspecification of the regression equation because it implies that the stock/bond correlation would, given enough time, exceed any positive number. This is in contrast to the desirable steady-state properties of the current specification. Of course collinearity issues complicate the search for the proper “missing variable.”

[9] This appendix is taken from correspondence with André Perold.


Copyright © 1999 Institutional Investors, Inc. and electronically posted with permission of the publisher.