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TRADING RULES AND EXCESS VOLATILITY by George Bulkley Department of Economics, University of Exeter, Exeter, EX4 4RJ, England and Ian Tonks Department of Accounting and Finance, London School of Economics, London WC2A 2AE, England. Tel: 071-955-7230 June 1991 We would like to thank James Poterba for kindly providing the data series.

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TRADING RULES AND EXCESS VOLATILITY ABSTRACT A number of recent papers have reported evidence that stock prices are more volatile than is consistent with efficient markets. We argue that the excess volatility tests address a definition of efficient markets which makes an extreme information assumption. We go on to test a weaker definition of efficient markets, due to Jensen (1978). We show the existence of a profitable trading rule that earns a significantly higher rate of return than a buy-and-hold strategy, and so conclude that stock prices are too volatile, even when judged by this weaker definition.

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I Volatility Tests

The hypothesis that stock prices are determined by rational expectations of the

discounted present value of an infinite stream of dividends, the efficient markets model, has been

examined in a number of recent papers using so-called variance bounds tests. These tests entail

calculating an upper limit to the volatility that such rational expectations prices should exhibit,

and then testing whether actual stock market prices are more volatile than this bound. Shiller

(1981) and LeRoy and Porter (1981) were the first to apply such volatility tests, and they

reported that the volatility of US stock prices grossly exceeded the appropriate bound. Although

the particular specification of the variance bound in these two papers has since been subjected to

considerable criticism [Flavin (1983), Kleidon (1986) and Marsh and Merton (1986)], a number

of second generation tests derived by other authors, for example Mankiew, Romer and Shapiro

(1985), Campbell and Shiller (1987) and West (1988), have nevertheless confirmed the original

excess volatility result.

If actual stock prices deviate from rational expectations prices, then it should be possible

for an agent to construct a trading rule which exploits these excess fluctuations in the stock

market and so yield greater profits than a simple buy and hold policy. As Shiller (1981)

proposed:

"If real stock prices are too volatile,....., then there may be well be a source of real profit opportunity. Time variation in expected real interest rates does not itself imply that any trading rule dominates a buy-and-hold strategy, but really large variations in expected returns might seem to suggest that such a trading rule exists." p 423

The objective of this paper is to test this implication using the same data set that has been

shown to exhibit excess volatility, Standard and Poors Index of Stock Prices, 1871-1985.

Prior to the development of the excess volatility literature a widely used test for efficient

stock prices was to look for the existence of trading strategies which beat the market. For

example Jensen (1978) stated that

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"a market is efficient with respect to information set θ, if it is impossible to make economic profits by trading on the basis of θ. By economic profits we mean the risk adjusted rate of return, net of all costs." (Jensen, 1978, p. 98)

After the initial work of Alexander (1961), who claimed to uncover a profitable trading

rule, a consensus emerged in later work that, after allowing for transactions costs, no systematic

rule could consistently beat a simple "buy-and-hold" strategy.1 However trading rules studied in

this literature were filter rules based on very short term price movements (say daily) of individual

securities around a slightly longer time period (say monthly) trend, even though these rules were

sometimes applied over long periods. For example Alexander studied a filter rule, where x is

the filter: if the daily closing price of a particular security moves up by at least x per cent buy

and hold the security until its price moves down at least x per cent from a subsequent high, at

which time simultaneously sell and go short.

The trading rule implied by the excess volatility literature is of a very different form to

these rules. It suggests buying a broad based market portfolio, whenever the stock market index

is low relative to its mean, and holding until it is high relative to its mean. It is essentially a long

run rule with, as we shall see below, an average period of five years between transactions. Rules

of this structure, which exploited the long run stationarity of the data, were not studied in the

earlier literature. A rule of this form avoids four problems which plagued short term filter rules.

First, transactions costs are now of a second order of significance, given the infrequency

of trades, as we shall discuss below. Secondly, since securities or bonds are always held, the

question of how to measure earnings at times when no profitable opportunities are present does

not arise. Thirdly since the holding period is always at least one year, it is straightforward to

include the appropriate dividends in returns, whereas this caused considerable data problems

with short term trading rules. Fourthly, the opportunity of taking a short position in a security

1 See Fama and Blume (1966) for a survey of this evidence, and for more recent work Sweeney (1988), who shows that a floor trader could have identified excess profit opportunities using a filter rule.

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does not arise since we may safely assume that the transactions cost of maintaining a long term

short contract is prohibitively costly.

Another trading rule approach stimulated in part by the excess volatility literature, is due

to De Bondt and Thaler (1985, 1987), who suggest that an explanation of excess volatility may

be investor overreaction to new information. This leads to a trading rule that is rather different to

the one studied here. They examine returns to individual shares and construct portfolios of shares

which have recently underperformed the market. They show that such "loser" portfolios produce

significant excess returns relative to the market index, over one to five years.

If we wish to test Jensen's definition of an efficient market we must first specify the

information set available to agents, before we see whether agents made efficient use of it. In

constructing our trading rule, we ensure that all trading decisions are strictly conditioned on

current information. This means that the agents' expectations of future dividends should be

derived from a model whose parameters have been estimated using only data that available at the

time of the forecast. This contrasts with empirical measures of rational expectations prices

employed in volatility tests.

The methodology of both first and second generation volatility tests was to define

rational expectations of future dividends as those derived form the "true" model of dividends.

This was implemented empirically by estimating the proposed model over the whole data set,

and deriving expectations at each date within the sample from this estimated model. If one

assumes that the sample is sufficiently large so that the difference between the estimated model

and the true model can be neglected, then the information set assumed available to agents at each

date is the true model of the dividend process. That is the volatility tests assume that

uncertainty/forecast errors are solely due to the errors/innovations in a given true model:

uncertainty about the model itself is not allowed for. For example Shiller writes

"uncertainty about future dividends is measured by the sample standard deviation of real dividends around their long term exponential growth path". (Shiller, 1981, p 434).

It is additional uncertainty about this long run exponential growth path itself which may

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account for the observed volatility of actual prices as new observations result in revised estimates

of the long run growth rate itself. Small changes in an estimated growth rate of dividends

translate into substantial changes in the present value of all future dividends.

The strict interpretation of the volatility tests to date is that actual prices are excessively

volatile compared to those that should be set if agents had rational expectations and knew the

true model of the dividend process. We will now investigate the profitability of a trading rule

which requires that agents forecast dividends from a model estimated using only information

available at the date of the forecast.

II A Trading rule: Methodology

The trading rule which we examine in this section is based on the assumption that the

true model which generates dividends is stationary after detrending.2 Of course we now have a

joint test of inefficient markets and stationarity, so that the existence of a profitable rule would

lend support to both assumptions whereas negative results need only imply rejection of one of

these hypotheses.

We consider a risk averse agent who in 1881 had $100 to invest in either bonds or

equities. We assume that there is a constant risk premium over the whole period, which leaves

him indifferent between bonds and equities: this is measured by the average excess return of

equities over bonds in the full sample. The objective of the trading rule is to identify periods

when the yield on equities is relatively high or low which, in conjunction with the assumption of

a constant risk premium, implies excess returns on either bonds or equities. By excess returns we

mean returns adjusted for risk by the average risk premium of the whole sample. This can be

measured by the X-statistic introduced by Sweeney (1986), and defined as

X = RF - (1-f)RBH (1)

2 See Poterba and Summers (1988) for recent evidence in support of this assumption.

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where RF is the sample average annual return from following the trading rule over the sample

period, RBH is the sample average annual return from the buy and hold strategy and (1-f) is the

proportion of years in which the investor's wealth is placed in the market index. In this way the

X-statistic, makes an allowance for the lower risk attributable to the trading rule strategy which

dictates f years when the investor is holding a risk free asset. The test statistic X has an expected

value of zero, and a standard deviation given by

σX = σ[f(1-f)/N]1/2 (2) where σ is the square root of the sample variance of the annual returns from the buy and hold

strategy, and N is the total number of years in the sample.

The rule used for identifying excess returns is based on the results from the volatility

literature: we estimate the long run average stock price, around which stock prices have been

shown to be excessively volatile, and then compute a k% filter to exploit this volatility. To

tackle the first problem and obtain an empirical measure of the long run average stock price we

fitted an exponential trend to prices using the data set from 1871 up to the year for which the

parameter is to be estimated. In Table I we report the estimated exponential growth rate of prices,

for each year, based on data only available up to that time. It can be seen that the growth rates are

much higher in the early part of the sample, which means that agents will incorrectly estimate the

growth rate early in the sample so that we might expect poor trading rule performance.

The second problem is to choose the value of k to incorporate in a rule of the form: Buy

when actual prices are k% below their long run average and sell when prices are k% above their

long run average. The optimal value for k will be determined by the following considerations.

Suppose the exponential growth were certain, and the density of the error process known. Then

optimal k would reflect a trade-off between the frequency with which some excess profits are

made (increased by small k) and the size of excess profits on the occasions when they are made

(increased by large k). When the exponential trend itself has to be estimated, there is in addition

the uncertainty surrounding the true trend, given its estimated value. Thus if k is small, the

chance of estimation error resulting in buying when actual prices are above their true trend,

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possibly resulting in eventual losses, is increased. Calculation of the optimal value for k in terms

of the estimated structural model is a forbidding problem. We propose the following empirical

rule. At any date t, the value kt is chosen which would have maximised profits over the period

(0,t-1), when incorporated in a trading rule where the forecast price at each data within this sub

period was estimated as described above. With the benefit of hindsight we estimate at each date

the value of kt which would have maximised the return from following the trading rule up to that

date, and then apply that estimated value kt in the current period. In this way the trading rule

strictly conforms to the requirement that in all respects it is based only on currently available

information.3

In principle trading rules should allow both long and short positions to be taken, as

appropriate. We exclude the latter however, on the grounds that the periodicity of the cycle of

actual prices around their efficient market valuation is so long relative to the time horizon of

short sale contracts that a long sequence of short sale contracts might be required before any

profits were realised. This could result in substantial transactions costs, as well as short term

losses which would require financing at the rate of interest appropriate to borrowers undertaking

this kind of speculation.

III Trading Rule: Results

In Table II we report the evolution of wealth and the annual return from following the

optimal trading rule from 1881-1985. Decisions on whether to switch in or out of the Standard

and Poor's Composite Real Stock Price Index are made on December 31st each year.4 Annual

3 From table II it can be seen that although the value of k is volatile initially, it settles down to 6% until 1934, and then remains at 8%. This constancy is due to the stationary model under consideration. As we move through time the number of time periods used to estimate k gets larger, which means that the early stock prices always feature in the later k estimation procedure.

4 Shiller (1981) describes the Index data in detail. In 1881 it would not have been possible to purchase a stock index as such, though it would have been possible to replicate the positions taken by the Index. Holding a well diversified homemade index would involve automatic rebalancing, though we do not allow for firms entering and leaving the Index. The sensitivity of the trading rule to transactions costs as well as buying at the ask and selling at the bid, is discussed below.

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real wealth and returns as a result of buying $100 of shares on December 31st 1881 and holding

thereafter are also shown in Table II. Each year bonds are held the yield is the nominal annual

interest rate on prime commercial paper minus the actual inflation rate,5 and this and all

dividends are reinvested.

The figures in table II do provide some support for the view that an individual who

formed rational expectations and accepted the efficient markets model could have successfully

identified times when excess returns were available on a representative market portfolio. The

geometric average annual return from adopting the trading rule is 7.494%, which is an excess

return of 1.18% over the buy-and-hold strategy. The excess returns were sufficiently large to

more than compensate for the fact that bonds, which on average had a lower yield than equities,

had to be held when there were no profitable opportunities available on the stock market. From

the returns columns in table II we can see that the trading rule correctly identified the poor

performance of equities in 1882-84, and the slumps in 1887, 1899, 1902/03, 1906/07, 1931 and

1937. Though the rule predicted the Great Depression, it was a little too early, and failed to take

advantage of the high returns on equities in 1928. Up to 1954 the trading rule had an average

return of 9% which outperformed buy and hold by 2.27%. However the rule failed to identify the

growth of equities throughout the fifties and sixties, incorrectly viewing the stock market as

overvalued, and consequently remained in bonds over this period, until the slump of 1973/74.

Since then the rule correctly anticipated the subsequent bull market.

The excess returns described in the previous paragraph are calculated without any

allowance for risk. In fact the trading rule is less risky than the buy and hold strategy since, under

the trading rule, for a fraction of the period wealth is held in risk free bonds. Sweeney's X-

statistic is designed to make allowance for the reduced risk of the trading rule. We computed the

value of X in equation (1) as 0.03289 over the whole sample, with a standard deviation of

0.00835, yielding a Z-ratio of 3.94. By comparing this value with the standard normal

distribution tables, we can state that the trading rule's returns are significantly greater from the

buy and hold returns at the 99% per cent confidence level.

5 Taken from Friedman and Schwartz (1982) and updated.

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The sample variance in equation (2) will be biased if RBH exhibits autocovariance

[Anderson (1971), p. 463], with the bias equalling

N

(-1/N) Σ σ(t-s) s,t=1

where σ(r) is the autocovariance at lag r. We tested the RBH series for autocorrelation of all lag

lengths between one and twelve, and for no lag length could we reject the null hypothesis of zero

autocorrelation at the 95 per cent confidence level.6

Note that in the computation of the X-statistic, it is assumed that the risk associated with

equities, and hence the risk premium is constant over the whole sample. We checked this by

calculating the variance of market returns during years when the agent was holding the market

index as 0.0298, and the variance of the market returns when the agent was out of the market as

0.0326. A simple F test with (34,68) degrees of freedom could not reject the null hypothesis that

the two variance were the same.

Further we can also calculate Sweeney's X-statistic as we move forward throughout the

sample, in order to determine the importance of the start date, in the calculation of the excess

returns. In the first column of table III we compute the X-statistic from initiating the trading rule

at that date and following the rule for the remainder of the sample. We can see that in all cases

the point estimate shows a positive excess return and, from the Z-values in the second column of

table III, that in 83.7 per cent of the start years the excess return is significantly different from

zero at 90 per cent confidence limits.

We can also see whether an investor would have been persuaded to follow the trading

rule strategy at any date on the basis of the historical performance of the rule. To this end we

computed the X-statistic from the returns in the initial year up to the year under consideration,

and the results are reported in the third and fourth columns of table III. In all cases throughout the

6 Furthermore the sum of the point estimates of the autocovariances is -0.0108 for 12 lag lengths, so that if there is any bias in the sample variance it will be positive, which will only serve to strengthen the significance of the results reported below.

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entire sample the historical performance always generates a significant excess return.

The above trading rule neglected transactions costs. It was not possible to obtain a full

set of historical data on transactions costs, but we have calculated that transactions costs would

have had to have been 8.2 per cent in order to reduce the gross return, before any risk adjustment,

from our trading rule to the same value as a buy and hold strategy. Sweeney (1988) states that

recent transactions costs on the New York Stock Exchange are much less than one per cent. We

repeated the calculation of the X-statistic after taking account of one per cent and two per cent

transactions costs, to reflect the commission charges on trades and the bid ask spread. As can be

seen from tables IV and V, incorporating transactions costs whenever the investor moves from

the market index into bonds or vice versa reduces the returns from the trading rule but the effect

is small, probably because the long term nature of the rule suggests switching on only fifteen

occasions throughout the entire sample.

III Conclusions

First and second generation excess volatility tests, whether or not they require

stationarity, are built on the assumption that agents have strong form rational expectations i.e. the

agents knew the parameters of the true model which generated dividends. The only "news" was

about the realisations of the disturbances in the stationary model or innovations in the

non-stationary case. In practice agents' forecasts have to be obtained from an estimated model,

which will be revised over time, and weak form rational expectations only requires such

revisions themselves be not forecastable. An empirical test of weak form rational expectations

should not require model innovations to be exactly zero each period, as imposed by the excess

volatility tests.

In order to test whether actual stock prices exhibited excess volatility when this was

allowed for, we examined whether agents, constrained to estimate the structural model, and

choose parameters of a trading rule using only currently available data, could earn a return in

excess of that from buying and holding equities for the whole period. We found that risk neutral

agents could have earned an annual excess return of 1.18% from following our trading rule. For a

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risk averse agent, the excess returns are statistically significant over the whole period.

Furthermore if agents had considered such a rule at any date within the period it would have

appeared profitable on past data and in expectation, would have been profitable from any starting

date within the period up to the terminal date.

Given the scale of excess volatility reported in earlier work it may seem surprising that

this abnormal return is not larger. In this paper we required that agents estimate a trading rule

from historical information about the trend. As we saw in table I, these estimates change over

time, so that in the fifties, agents with the benefit of history believed equities were overvalued,

and remained in bonds missing out on the spectacular growth in equities over the next two

decades. However even allowing for these mistakes, we can still conclude from the existence of a

profitable trading rule, that stock market prices appear to be sufficiently volatile to reject Jensen's

relatively weak definition of an efficient market.

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Table I - Estimated long term growth rate of equity prices 1882 0.043 1935 0.013 1883 0.043 1936 0.015 1884 0.040 1937 0.016 1885 0.034 1938 0.014 1886 0.040 1939 0.015 1887 0.042 1940 0.014 1888 0.039 1941 0.013 1889 0.038 1942 0.011 1890 0.038 1943 0.011 1891 0.035 1944 0.012 1892 0.037 1945 0.012 1893 0.035 1946 0.013 1894 0.032 1947 0.011 1895 0.030 1948 0.010 1896 0.028 1949 0.010 1897 0.027 1950 0.010 1898 0.027 1951 0.010 1899 0.029 1952 0.010 1900 0.028 1953 0.011 1901 0.029 1954 0.010 1902 0.030 1955 0.012 1903 0.030 1956 0.013 1904 0.028 1957 0.013 1905 0.029 1958 0.012 1906 0.030 1959 0.014 1907 0.030 1960 0.014 1908 0.027 1961 0.014 1909 0.028 1962 0.015 1910 0.028 1963 0.015 1911 0.027 1964 0.016 1912 0.027 1965 0.017 1913 0.026 1966 0.017 1914 0.025 1967 0.017 1915 0.023 1968 0.018 1916 0.023 1969 0.018 1917 0.020 1970 0.018 1918 0.012 1971 0.018 1919 0.010 1972 0.018 1920 0.007 1973 0.018 1921 0.008 1974 0.017 1922 0.011 1975 0.016 1923 0.011 1976 0.016 1924 0.011 1977 0.016 1925 0.012 1978 0.016 1926 0.013 1979 0.015 1927 0.014 1980 0.015 1928 0.017 1981 0.015 1929 0.021 1982 0.015 1930 0.020 1983 0.015 1931 0.018 1984 0.015 1932 0.014 1985 0.015 1933 0.014 1934 0.015 The table shows the progression in the estimated growth rate coefficients of stock prices, obtained from an exponential time trend over the period 1870-t, where t progresses from 1881 to 1985. The stock price series is the Standard and Poor's Composite Real Stock Price series.

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Table II - Annual wealth levels and returns from trading rule and buy-and-hold strategies Dec Buy and Hold Trading Rule B/S k 31st Wealth Return Wealth Return 1882 105.52 0.055 100.24 0.002 B 0.14 1883 108.03 0.023 112.47 0.122 B 0.14 1884 105.18 -0.026 127.64 0.134 B 0.14 1885 140.23 0.333 170.18 0.333 S 0.05 1886 156.21 0.114 189.57 0.114 S 0.07 1887 148.62 -0.049 191.38 0.010 B 0.06 1888 159.96 0.076 200.47 0.048 B 0.06 1889 178.92 0.118 224.22 0.118 S 0.04 1890 164.58 -0.080 206.25 -0.080 S 0.06 1891 206.81 0.257 259.17 0.257 S 0.06 1892 205.72 -0.005 257.81 -0.005 S 0.06 1893 190.39 -0.074 238.60 -0.074 S 0.06 1894 205.96 0.082 258.11 0.082 S 0.06 1895 212.65 0.032 266.49 0.032 S 0.06 1896 225.79 0.062 282.96 0.062 S 0.06 1897 263.36 0.166 330.05 0.166 S 0.06 1898 333.63 0.267 418.11 0.267 S 0.06 1899 297.34 -0.109 402.30 -0.037 B 0.06 1900 370.11 0.245 500.77 0.245 S 0.06 1901 429.65 0.161 581.33 0.161 S 0.06 1902 423.07 -0.015 567.46 -0.024 B 0.06 1903 368.93 -0.128 589.77 0.039 B 0.06 1904 474.47 0.286 758.48 0.286 S 0.06 1905 571.62 0.205 913.79 0.205 S 0.06 1906 550.47 -0.037 924.52 0.011 B 0.06 1907 429.10 -0.221 930.80 0.006 B 0.06 1908 573.93 0.338 1244.97 0.338 S 0.06 1909 604.90 0.054 1312.15 0.054 S 0.06 1910 628.34 0.039 1363.00 0.039 S 0.06 1911 650.99 0.036 1412.13 0.036 S 0.06 1912 656.19 0.008 1423.41 0.008 S 0.06 1913 639.38 -0.026 1386.94 -0.026 S 0.06 1914 607.68 -0.050 1318.18 -0.050 S 0.06 1915 707.55 0.164 1534.83 0.164 S 0.06 1916 583.70 -0.175 1266.17 -0.175 S 0.06 1917 396.60 -0.321 860.30 -0.321 S 0.06 1918 431.37 0.088 935.73 0.088 S 0.06 1919 441.23 0.023 957.11 0.023 S 0.06 1920 517.59 0.173 1122.76 0.173 S 0.06 1921 702.63 0.357 1524.16 0.358 S 0.06 1922 813.76 0.158 1765.22 0.158 S 0.06 1923 876.24 0.077 1900.75 0.077 S 0.06 1924 1071.11 0.222 2323.46 0.222 S 0.06 1925 1339.88 0.251 2906.49 0.251 S 0.06 1926 1610.07 0.202 3492.57 0.201 S 0.06 1927 2177.45 0.352 4723.33 0.352 S 0.06 1928 3205.35 0.472 4846.14 0.026 B 0.06 1929 3030.92 -0.054 5167.92 0.066 B 0.06 1930 2768.94 -0.080 5940.27 0.149 B 0.06 1931 1829.59 -0.344 7083.48 0.192 B 0.06 1932 1846.25 0.009 7147.97 0.009 S 0.06 1933 2422.52 0.312 9379.08 0.312 S 0.06 1934 2049.77 -0.154 7935.92 -0.154 S 0.06 1935 3074.93 0.500 11904.96 0.500 S 0.06 1936 3851.75 0.253 14912.53 0.253 S 0.08 1937 2799.11 -0.273 14022.99 -0.060 B 0.08 1938 3387.53 0.210 16970.90 0.210 S 0.08 1939 3403.13 0.005 17049.05 0.005 S 0.08

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1940 3050.15 -0.104 15280.69 -0.104 S 0.08 1941 2364.76 -0.225 11846.98 -0.225 S 0.08 1942 2671.20 0.130 13382.19 0.130 S 0.08 1943 3249.11 0.216 16277.44 0.216 S 0.08 1944 3818.58 0.175 19130.37 0.175 S 0.08 1945 5814.34 0.357 25972.57 0.357 S 0.08 1946 3480.85 -0.329 17438.41 -0.329 S 0.08

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Dec Buy and Hold Trading Rule B/S k 31st Wealth Return Wealth Return 1947 3180.72 -0.086 15934.79 -0.086 S 0.08 1948 3546.58 0.115 17767.70 0.115 S 0.08 1949 4371.37 0.233 21899.75 0.233 S 0.08 1950 5034.75 0.152 25223.16 0.152 S 0.08 1951 6173.22 0.226 30926.68 0.226 S 0.08 1952 7236.91 0.172 36255.53 0.172 S 0.08 1953 7373.80 0.019 36941.35 0.019 S 0.08 1954 10829.42 0.469 54253.32 0.469 S 0.08 1955 13714.61 0.266 54924.98 0.012 B 0.08 1956 14047.69 0.024 54372.98 -0.010 B 0.08 1957 13049.17 -0.071 54613.86 0.004 B 0.08 1958 18111.15 0.388 55933.88 0.024 B 0.08 1959 19511.74 0.077 57191.83 0.022 B 0.08 1960 20628.79 0.057 59402.29 0.039 B 0.08 1961 24611.68 0.193 61939.36 0.042 B 0.08 1962 24016.82 -0.024 63582.62 0.027 B 0.08 1963 28916.64 0.204 65856.34 0.036 B 0.08 1964 33524.86 0.159 68054.63 0.033 B 0.08 1965 36118.50 0.077 69391.22 0.020 B 0.08 1966 33293.11 -0.078 70131.62 0.011 B 0.08 1967 38249.77 0.149 73883.66 0.053 B 0.08 1968 41010.09 0.072 75804.64 0.026 B 0.08 1969 35893.25 -0.125 77319.21 0.020 B 0.08 1970 37561.79 0.046 80541.88 0.042 B 0.08 1971 41106.94 0.094 84125.18 0.044 B 0.08 1972 45236.62 0.100 84660.22 0.006 B 0.08 1973 32378.36 -0.284 77541.99 -0.084 B 0.08 1974 21969.57 -0.321 69239.57 -0.107 B 0.08 1975 29179.16 0.328 91961.39 0.328 S 0.08 1976 31019.61 0.063 97761.76 0.063 S 0.08 1977 26704.66 -0.139 84162.73 -0.139 S 0.08 1978 28172.11 0.055 88787.55 0.055 S 0.08 1979 28630.71 0.016 90232.88 0.016 S 0.08 1980 32241.24 0.126 101611.88 0.126 S 0.08 1981 28710.39 -0.110 90484.01 -0.110 S 0.08 1982 36805.34 0.282 115996.16 0.282 S 0.08 1983 43121.80 0.172 135903.16 0.172 S 0.08 1984 46196.08 0.071 145592.09 0.071 S 0.08 1985 58264.93 0.261 183629.44 0.261 S 0.08 B denotes bonds held, S denotes shares held up to, but not including, December 31st of the current year. The table shows the annual terminal wealth levels obtained from $100 invested in 1881 up to 1985, for both the buy-and-hold and the trading rule strategies. The stock price series is the Standard and Poor's Composite Real Stock Price Index, 1871-1985, and the dividend series is the associated real dividends payable on the Index. The real interest rate series is the prime commercial paper rate, taken from M. Friedman and A. Schwartz, Monetary Trends in the US and the UK, (1982), p.128.

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Table III - Forward and Backward X-Statistics Forward Z-value Backward Z-value X-stat. X-Stat. 1882 0.033 3.94 1883 0.033 3.93 1884 0.032 3.77 1885 0.030 3.55 0.124 4.10 1886 0.030 3.45 0.101 3.69 1887 0.030 3.46 0.094 3.85 1888 0.029 3.37 0.088 4.36 1889 0.029 3.34 0.080 4.21 1890 0.029 3.32 0.061 3.18 1891 0.030 3.37 0.065 3.29 1892 0.030 3.30 0.055 3.03 1893 0.030 3.34 0.045 2.59 1894 0.031 3.40 0.043 2.66 1895 0.031 3.40 0.039 2.64 1896 0.032 3.41 0.036 2.68 1897 0.032 3.41 0.036 2.81 1898 0.032 3.37 0.037 2.89 1899 0.032 3.28 0.040 2.97 1900 0.031 3.22 0.041 3.08 1901 0.031 3.16 0.040 3.19 1902 0.031 3.13 0.040 3.21 1903 0.031 3.11 0.046 3.56 1904 0.029 2.91 0.047 3.64 1905 0.029 2.84 0.047 3.74 1906 0.029 2.80 0.049 3.90 1907 0.028 2.71 0.054 4.06 1908 0.025 2.48 0.056 4.10 1909 0.025 2.40 0.053 4.11 1910 0.025 2.44 0.051 4.11 1911 0.026 2.45 0.049 4.10 1912 0.026 2.46 0.047 4.06 1913 0.027 2.48 0.046 3.98 1914 0.028 2.52 0.042 3.87 1915 0.029 2.58 0.042 3.95 1916 0.029 2.54 0.039 3.63 1917 0.031 2.66 0.035 3.07 1918 0.033 2.87 0.034 3.12 1919 0.034 2.96 0.033 3.12 1920 0.035 2.99 0.033 3.20 1921 0.035 2.94 0.034 3.25 1922 0.034 2.81 0.036 3.32 1923 0.034 2.80 0.033 3.35 1924 0.035 2.81 0.033 3.42 1925 0.034 2.73 0.033 3.49 1926 0.034 2.64 0.033 3.56 1927 0.033 2.58 0.034 3.60 1928 0.032 2.43 0.027 2.71 1929 0.036 2.75 0.030 2.99 1930 0.034 2.67 0.035 3.44 1931 0.030 2.35 0.044 4.09 1932 0.023 1.73 0.043 4.08 1933 0.024 1.85 0.044 4.14 1934 0.022 1.71 0.042 3.99 1935 0.025 1.88 0.043 3.99 1936 0.022 1.64 0.043 4.06 1937 0.021 1.60 0.046 4.19 1938 0.018 1.39 0.046 4.25 1939 0.017 1.36 0.044 4.24

18

1940 0.018 1.41 0.043 4.15 1941 0.021 1.55 0.041 3.96 1942 0.024 1.80 0.040 4.00 1943 0.024 1.83 0.040 4.06 1944 0.024 1.72 0.040 4.11 1945 0.023 1.66 0.041 4.16 1946 0.020 1.42 0.038 3.89 1947 0.026 1.84 0.037 3.83 1948 0.029 2.17 0.037 3.87 1949 0.030 2.18 0.037 3.92 1950 0.028 2.03 0.037 3.97

19

Forward Z-value Backward Z-value X-stat. X-Stat. 1951 0.028 1.99 0.037 4.02 1952 0.026 1.84 0.036 4.06 1953 0.026 1.76 0.056 4.06 1954 0.028 1.86 0.036 4.08 1955 0.020 1.35 0.034 3.79 1956 0.023 1.60 0.034 3.79 1957 0.024 1.63 0.035 3.88 1958 0.025 1.48 0.032 3.47 1959 0.027 1.74 0.032 3.47 1960 0.028 1.84 0.032 3.51 1961 0.027 1.73 0.032 3.42 1962 0.029 1.78 0.033 3.52 1963 0.028 1.64 0.032 3.42 1964 0.031 1.75 0.031 3.37 1965 0.037 1.86 0.031 3.38 1966 0.035 1.89 0.032 3.50 1967 0.033 1.69 0.032 3.49 1968 0.036 1.76 0.032 3.51 1969 0.037 1.80 0.033 3.66 1970 0.031 1.47 0.034 3.73 1971 0.031 1.46 0.034 3.76 1972 0.032 1.54 0.033 3.73 1973 0.038 1.91 0.035 3.81 1974 0.023 1.42 0.036 3.87 1975 0.036 3.95 1976 0.036 3.95 1977 0.035 3.84 1978 0.034 3.84 1979 0.034 3.82 1980 0.034 3.85 1981 0.033 3.76 1982 0.033 3.83 1983 0.033 3.87 1984 0.033 3.88 1985 0.033 3.94 The 1882 forward X-statistic and the 1985 backward X-statistic are both calculated over the entire sample and have the same values. There are no forward X-stats over the last eleven years since the rule dictated holding the market index, and so X is by definition zero over this period. Similarly there are no backward X-stats over the first three years because σX is zero.

20

Table IV - Forward and Backward X-Statistics with 1% transactions costs Forward Z-value Backward Z-value X-stat. X-Stat. 1882 0.031 3.69 1883 0.031 3.76 1884 0.030 3.59 1885 0.028 3.37 0.121 3.99 1886 0.028 3.29 0.100 3.60 1887 0.029 3.30 0.091 3.71 1888 0.028 3.21 0.085 4.20 1889 0.028 3.18 0.076 4.00 1890 0.028 3.17 0.058 3.00 1891 0.029 3.22 0.062 3.13 1892 0.028 3.16 0.053 2.88 1893 0.029 3.19 0.043 2.46 1894 0.030 3.25 0.040 2.52 1895 0.030 3.25 0.036 2.51 1896 0.031 3.27 0.034 2.55 1897 0.031 3.27 0.034 2.68 1898 0.031 3.22 0.036 2.77 1899 0.031 3.14 0.038 2.82 1900 0.030 3.08 0.038 2.89 1901 0.030 3.04 0.038 3.00 1902 0.030 3.01 0.037 3.00 1903 0.030 2.99 0.043 3.35 1904 0.028 2.80 0.044 3.40 1905 0.028 2.75 0.044 3.51 1906 0.028 2.71 0.046 3.64 1907 0.027 2.63 0.051 3.82 1908 0.025 2.40 0.052 3.84 1909 0.024 2.33 0.050 3.85 1910 0.024 2.37 0.048 3.84 1911 0.025 2.38 0.046 3.83 1912 0.025 2.39 0.044 3.80 1913 0.026 2.41 0.042 3.72 1914 0.027 2.46 0.039 3.61 1915 0.028 2.51 0.039 3.69 1916 0.028 2.48 0.036 3.38 1917 0.030 2.59 0.032 2.84 1918 0.033 2.80 0.031 2.89 1919 0.033 2.89 0.030 2.89 1920 0.034 2.92 0.030 2.97 1921 0.034 2.87 0.032 3.03 1922 0.031 2.74 0.031 3.09 1923 0.033 2.73 0.031 3.12 1924 0.033 2.73 0.031 3.20 1925 0.033 2.66 0.031 3.27 1926 0.033 2.57 0.031 3.34 1927 0.032 2.51 0.032 3.39 1928 0.031 2.36 0.025 2.49 1929 0.035 2.69 0.028 2.77 1930 0.033 2.61 0.033 3.23 1931 0.030 2.29 0.042 3.90 1932 0.022 1.67 0.041 3.87 1933 0.023 1.80 0.041 3.94 1934 0.022 1.66 0.039 3.78 1935 0.024 1.82 0.041 3.80 1936 0.021 1.58 0.041 3.87 1937 0.020 1.55 0.043 3.99 1938 0.018 1.35 0.043 4.03 1939 0.017 1.33 0.042 4.02

21

1940 0.018 1.39 0.041 3.93 1941 0.020 1.52 0.039 3.74 1942 0.024 1.78 0.038 3.78 1943 0.024 1.81 0.038 3.84 1944 0.023 1.70 0.038 3.90 1945 0.023 1.63 0.038 3.95 1946 0.020 1.39 0.036 3.68 1947 0.026 1.81 0.035 3.63 1948 0.029 2.14 0.035 3.66 1949 0.029 2.15 0.035 3.71 1950 0.028 2.00 0.035 3.76

22

Forward Z-value Backward Z-value X-stat. X-Stat. 1951 0.028 1.96 0.035 3.81 1952 0.026 1.80 0.034 3.85 1953 0.025 1.72 0.034 3.85 1954 0.027 1.83 0.034 3.88 1955 0.020 1.31 0.032 3.87 1956 0.023 1.59 0.032 3.57 1957 0.023 1.61 0.033 3.56 1958 0.022 1.47 0.030 3.67 1959 0.027 1.76 0.030 3.27 1960 0.027 1.83 0.031 3.27 1961 0.027 1.72 0.030 3.31 1962 0.029 1.77 0.031 3.22 1963 0.027 1.63 0.030 3.33 1964 0.031 1.74 0.029 3.23 1965 0.033 1.84 0.029 3.17 1966 0.035 1.88 0.030 3.19 1967 0.033 1.69 0.030 3.31 1968 0.035 1.76 0.030 3.30 1969 0.037 1.80 0.032 3.33 1970 0.031 1.47 0.032 3.50 1971 0.031 1.45 0.032 3.55 1972 0.032 1.54 0.032 3.58 1973 0.038 1.92 0.033 3.56 1974 0.023 1.42 0.034 3.64 1975 0.035 3.70 1976 0.034 3.77 1977 0.033 3.77 1978 0.033 3.66 1979 0.032 3.66 1980 0.032 3.64 1981 0.031 3.66 1982 0.031 3.58 1983 0.031 3.65 1984 0.031 3.69 1985 0.031 3.69

23

Table V - Forward and Backward X-Statistics with 2% transactions costs Forward Z-value Backward Z-value X-stat. X-Stat. 1882 0.030 3.51 1883 0.030 3.57 1884 0.029 3.41 1885 0.027 3.19 0.118 3.87 1886 0.027 3.12 0.098 3.52 1887 0.027 3.13 0.089 3.57 1888 0.026 3.06 0.082 4.05 1889 0.026 3.03 0.073 3.79 1890 0.026 3.03 0.055 2.83 1891 0.027 3.08 0.059 2.98 1892 0.027 3.02 0.050 2.74 1893 0.028 3.06 0.041 2.32 1894 0.028 3.11 0.038 2.39 1895 0.029 3.11 0.035 2.37 1896 0.029 3.12 0.033 2.41 1897 0.030 3.12 0.033 2.55 1898 0.030 3.08 0.034 2.64 1899 0.029 3.00 0.036 2.67 1900 0.029 2.95 0.036 2.69 1901 0.029 2.92 0.035 2.80 1902 0.029 2.89 0.035 2.77 1903 0.029 2.89 0.041 3.15 1904 0.027 2.69 0.041 3.15 1905 0.027 2.65 0.041 3.16 1906 0.027 2.61 0.041 3.27 1907 0.026 2.55 0.042 3.39 1908 0.024 2.31 0.048 3.59 1909 0.023 2.27 0.049 3.59 1910 0.024 2.30 0.047 3.60 1911 0.024 2.31 0.045 3.59 1912 0.025 2.32 0.043 3.58 1913 0.025 2.35 0.041 3.54 1914 0.026 2.39 0.039 3.46 1915 0.027 2.44 0.036 3.35 1916 0.027 2.41 0.036 3.43 1917 0.029 2.52 0.033 3.13 1918 0.032 2.73 0.029 2.61 1919 0.032 2.82 0.029 2.66 1920 0.033 2.85 0.028 2.66 1921 0.033 2.80 0.028 2.73 1922 0.032 2.66 0.029 2.80 1923 0.032 2.66 0.029 2.87 1924 0.033 2.66 0.028 2.90 1925 0.032 2.58 0.029 3.00 1926 0.032 2.50 0.029 3.05 1927 0.031 2.44 0.029 3.12 1928 0.030 2.29 0.029 3.17 1929 0.034 2.63 0.023 2.27 1930 0.033 2.54 0.026 2.56 1931 0.029 2.22 0.031 3.02 1932 0.021 1.61 0.040 3.71 1933 0.022 1.75 0.039 3.66 1934 0.021 1.61 0.039 3.73 1935 0.023 1.77 0.037 3.58 1936 0.020 1.53 0.039 3.60 1937 0.019 1.49 0.039 3.67 1938 0.017 1.31 0.041 3.79 1939 0.017 1.31 0.041 3.81

24

1940 0.018 1.36 0.040 3.80 1941 0.019 1.50 0.038 3.71 1942 0.023 1.75 0.036 3.53 1943 0.024 1.78 0.036 3.57 1944 0.023 1.67 0.036 3.62 1945 0.022 1.61 0.036 3.68 1946 0.019 1.37 0.036 3.74 1947 0.025 1.79 0.034 3.48 1948 0.028 2.11 0.033 3.42 1949 0.029 2.12 0.033 3.45 1950 0.027 1.97 0.033 3.51

25

Forward Z-value Backward Z-value X-stat. X-Stat. 1951 0.027 1.93 0.033 3.54 1952 0.025 1.77 0.033 3.60 1953 0.024 1.69 0.033 3.64 1954 0.026 1.79 0.032 3.64 1955 0.019 1.27 0.033 3.67 1956 0.023 1.57 0.030 3.36 1957 0.023 1.60 0.030 3.35 1958 0.022 1.45 0.031 3.46 1959 0.027 1.71 0.028 3.07 1960 0.027 1.81 0.028 3.07 1961 0.027 1.71 0.029 3.12 1962 0.029 1.76 0.028 3.03 1963 0.027 1.62 0.029 3.14 1964 0.030 1.73 0.028 3.04 1965 0.033 1.84 0.027 2.99 1966 0.035 1.87 0.027 3.00 1967 0.033 1.69 0.028 3.13 1968 0.035 1.75 0.028 3.12 1969 0.037 1.80 0.028 3.14 1970 0.031 1.47 0.030 3.30 1971 0.030 1.45 0.030 3.37 1972 0.032 1.54 0.030 3.40 1973 0.038 1.92 0.030 3.39 1974 0.023 1.42 0.031 3.47 1975 0.033 3.53 1976 0.033 3.58 1977 0.033 3.58 1978 0.031 3.47 1979 0.031 3.47 1980 0.030 3.45 1981 0.030 3.48 1982 0.029 3.40 1983 0.030 3.47 1984 0.030 3.51 1985 0.030 3.51

26

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