Lead-lag relationship between markets

90 41.5

π π

as Ω which is the union of ₁

frequencies with cycles, while 2 2 2 2

( , ) ( , )

180 90 41.5 18

π π π π

∪ is Ω . Applying the Canova ₂

test, the corresponding D statistics are shown on the column 4 of Table 1.1. To be sure, 2 2

( , )

90 41.5

π π

is significant at 99% confidence, which means the four markets

follow similar cyclical mechanisms in the span of 3.5 years to 7.5 years. In fact, the short cycle peaks of 41.5 months in the S&P 500 and 10 years Treasury bond, and 67.5 months in commodity futures and industrial production are interesting, since these frequencies are within the range of the well known 3~5 year Kitchin cycle (Kitchin, 1923). Besides, the long cycle peaks of 90 months in S&P 500 and Treasury bond all are within the frequencies of 7~11 year Juglar cycle (1862). Such results also provide evidence for the existence of Kitchin cycles and Juglar cycles in those markets.

In summary, we use Canova (1996)’s test to verify the existence of the cycles in various markets. It has especially shown that the frequency peaks of the power spectrum in these markets are rather coincident. It further hints that there are common relationships behind the scenes that link the seemingly independent markets altogether.

This finding will strengthen the results of our cross-spectral analysis.

2.3.4. Lead-lag relationship between markets

Before statistically verifying the lead or lag relationships between different markets, let’s take a look at the cyclical behavior in each of the markets. Figure 1.3 shows the filtered series of these markets with frequencies 2 2

( , )

90 41.5

π π

. The arrows of Figure 1.3

show quite clearly that the momentum of the bond prices leads the stock prices, the stock prices lead the industrial production, and the commodity futures lag behind all of them for most of the time.

Figure 1.3 Filtered series of 10-years gov. bond, S&P 500, industrial productions and commodity futures

Table 1.2 and Figure 1.4 is the summary of the cross-spectrum within the frequency of 2 2

( , )

90 41.5

π π

. Instead of SP/Gov and SP/Com failing to have significant

lead or lag relationship, the other four square coherences are all above the 0.349 mark, indicating significant lead or lag relationship in these four cases. Among them, industrial production has significant lead or lag relationships with the other three

Bond

1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004

markets, in which it leads commodities for an average of 7.97 months and lags behind S&P 500 and treasury bonds for an average of 9.42 and 17.49 months, respectively.

This result indicates that economic fluctuation does influence financial markets. On the other hand, government bonds also lead commodities for 24.25 months. Noteworthy, the relationship between stock markets and economic activity and the relationship between bond markets and economic activity are similar to the results of Moore (1978), where he found that, on average, stock price peaks lead business cycle peaks for 5 months, while bond price peaks lead business cycle peaks for 14 months within the sample period 1943-73.

Table 1.2 Square coherence and average lead/lad time

IP SP GOV CRB

IP － － － －

SP 0.42(9.04) － － －

Gov 0.50(17.49) 0.17(6.15) － －

Com 0.49(-7.97) 0.21(-19.12) 0.53(-24.25) －

Note: 1.Industrial production, S&P 500, 10 years government bond and commodity are named by IP, SP, Gov and Com respectively.

2. Outside of parenthesis are the square coherence, in parenthesis are the average lead time of the row element on the column element

Figure 1.4 The significant lead/lag relations between different markets

Note: Arrows point to the lagging market

Nevertheless, although we cannot find significant lead or lag relationships between the S&P 500 index and government bond prices, and between the S&P 500 index and commodities, by indirect inferring the lag time of industrial production with S&P 500

Industrial

RJ/CRB S&P 500

10 years gov.

and with government bonds, we can find a weak support that bond prices lead the stock prices for roughly 8 months, which is similar to Moore’s (1975) results of 11 month lead. As for S&P 500 and commodities, even though the relationships are insignificant, we can also find a weak support that S&P 500 lead the commodities by indirectly inferring their individual lead or lag relationships with industrial production.

In summary, through the study of cross spectrum analysis, we verified that the commodities lag behind the other three markets, while the stock and bond markets, even tough their relationship with each other is ambiguous, both lead industrial production and commodities. Thus, our results give some evidence to Murphy (2004) and Pring’s (2002) idea that the order of lead or lag relationships among these four markets is bond market, equity market, economic activity and commodity market. Besides, the result can also reinforce the conclusion of Gorton and Rouwenhorst (2004).

2.5. Portfolio Return within Business Cycles

Are investors capable to increase their returns by implementing the aforementioned cyclical sequence among those markets? Actually, the implication of the cyclical sequence assumes investors can perfectly gauge their position in business cycles, where they should increase their stock positions before the economy reaches the trough, then switch to commodity assets before the economy reaches the peak, and then to bonds throughout most of the recession. If the investor can only choose between stocks and bonds, then the strategy is to increase stock positions before the trough and then reallocate to bonds nearing the peak. Noteworthy, the strategy with only stock and bond is similar to Siegel (1991), who has shown that portfolio returns can be enhanced significantly by switching between bonds and stocks before turning points in the

business cycle.

Table 1.3 is the summary results about whether commodity assets are a proper asset choice in the reallocation strategy based on the stages of business cycles. The first nine rows of Table 1.3 shows the summarized data of the US business cycle and the average annual return from investing in stocks, bonds and commodities over the business cycle. Over the entire period of May 1960 to December 2007, the seven recessions averaged 10.71 months in length, and expansions averaged 70.76 months in length, so that almost one-eighth of the time the economy in a recession.

Table 1.3 Average annual return of portfolio (May, 1960 － December, 2007)(bps)

(1) Average length of recession (months) 10.71

(2) Average Length of Expansion 70.67

(3) Average Length of Business cycle 81.38

(4) % of Time Economy in Recession 13.17

(5) % of Time Economy in Expansion 86.83

(6) Average Annual Return for Stock (%) 11.42 (7) Average Annual Return for Bonds (%) 7.59 (8) Benchmark Returns (6) X (5)+(7)X(4) (%) 10.92

(9) Average Annual Return for Com 12.36

(10) Average Returns of Portfolio (%)

Without Com With Com

0-month 1-month 2-month 3-month 4-month 5-month 6-month 6-month lead 14.01 15.35 15.71 16.16 15.64 15.19 15.47 15.67

From May 1960 thru December 2007, the average annual nominal return from

investing in the stock market is 11.42%, while the average return is 7.59% and 12.36%

from investing in 10-year Treasury bonds and the commodity index, respectively. The risk-adjusted return, the “benchmark” or “traditional asset class” return, is defined as the weighted average return with only stocks and bonds in the portfolio for the period and weighted according to the time the economy is in expansion (for stocks) and recession (for bonds), is 10.92%.

The column labeled “without Com” in the lower part of Table 1.3 is the return of reallocating only between stock and bond assets throughout the business cycle. The slot labeled “concurrent” reports returns from being 100% long in equities during economic expansion and 100% long in Treasury bonds during economic contractions. The returns calculated in “h-month lead” assumes an investor who leads the business cycle peaks for h-months in switching from stocks to bonds in business cycle expansions and leads the business cycle troughs also for h-months in switching from bonds to stocks in recessions. In contrast, an investor who lags the business cycle turning points to switch out of, and then into stocks an equal number of months after the peak and trough of the business cycle are labeled “h-month lag”. Actually, the results in the column labeled

“without Com” is similar to Siegel (1991), that investors can increase their returns by switching into bonds before the peak of the business cycle and into stocks before the trough of the business cycle.

The remaining part of Table 1.3 shows whether investors can increase returns by including commodity assets in their portfolio in some stages of the business cycle.

Noteworthy, as previous subsections have shown, bull commodity markets can go on even after the economy has passed its peak. Therefore, the investor can switch from stock assets to commodity assets before the peak of the business cycle and switch to

bond assets some time after the peak. The rows of lower-right part of Table 1.3 define when the investor switches its stock assets into commodity assets. The row labeled

“concurrent” means the investor becomes 100% long in commodity assets at the business cycle peak. g-month lead/lag means the investor shifts to commodity assets g months before/after the business cycle peak. The column part denotes the investor switches its commodity assets into bond assets K-months after the business cycle peak.

Note that, we still assume investors switch their bonds into stock assets h months before/after the business cycle trough.

Still, if investors can perfectly gauge the future movement of the business cycle and switch their stock assets into commodity assets before the economy reaches the peak, then switch into bonds some time after the peak, and then switch their bond assets into stock assets before the trough, they can earn more return than when their response lags the turning points of the business cycle. Besides, we can see that over most rows, investors can gain more return by switching into commodity assets before the peak of the business cycle and then switch into bond assets several months after the peak compared to the column labeled “without Com”, the case where investors only invests in stocks and bonds. The additional gains from holding commodity assets for until two months after the business cycle peak compared to the column labeled “without Com”

ranges from 23 basis points to 215 basis points per year in the span May, 1960 to December, 2007. These results reinforce that the sequential relationship among cycles in different markets do exist during the period May 1960 to December 2007. Besides, these results also echo Gorton and Rouwenhorst (2006) that the inclusion of commodities can enhance portfolio performance.

However, with the recent extraordinary spikes in commodity prices, whether this

outperformance mentioned above is due to the instable hikes or the regular intermarket sequential relationship of cycles is an issue facing scrutiny and would have to be addressed. Table 1.4 is the average return of different portfolios over each business cycle since May 1960 to December, 2007⁷. We can see that, in five out of seven business cycles since May 1960, investors would have enhanced their returns had they switched into commodity assets before the peak of the business cycle and then switch into bonds some time after the peak. The greatest addition in gains by such strategy is 1,026 basis points in the business cycle from December 1973 to January 1980. The largest additional loss of that strategy is -534 basis points at the business cycle from February 1980 to July 1981, the time just few months after the second oil crisis. During October 1978 to January 1980, cumulated rise in commodity prices reached 55.36%.

Such a extraordinary rise in commodity prices was not due to business cycles but supply shocks, thus after the crisis, even though the economic environment would favor commodity assets, the price of commodities were still falling. The other occasion that including commodity assets would result in negative additional gains was in the business cycle from August 1990 to March 2003. In fact, though the average return of including commodity assets in the portfolio cannot exceed the average returns of portfolios with only traditional asset classes, the average returns of the two are similar.

In summary, the additional gains from properly allocating commodities into the portfolio were fairly stable since May 1960.

7

We defined a complete business cycle is from peak to peak. Thus, since May 1960, there are 7 times

complete business cycle.

Table 1.4 Portfolio return over individual business cycles (bps)

1970/1~1973/11 6.30 13.35 16.69 3.34

1973/12~1980/1 8.18 11.84 22.10 10.26

1980/2~1981/7 12.93 12.25 6.91 -5.34

1981/8~1990/7 17.68 20.50 21.22 0.72

1990/8~2001/3 15.04 19.99 19.87 -0.12

2001/4~2007/12 6.19 6.24 6.88 0.64

a “

Benchmark” denotes the average annualized return defined as the weighted average of the stock and bond return for the period and weighted by the share of time the economy is in an expansion (for stocks) and a recession (for bonds).

b

“Reallocation without commodity” denotes the average annualized return in which investors switch out of stocks 6 months before the peak of the business cycle expansion and switches into stocks the same number of months before the trough of the recession.

c “

Reallocation with commodity” denotes the average return in which investors switch to commodities 6 months before the peak of business cycle, and then switch into bonds 2 month after the peak, and then switch into bonds 6 months before the trough.

2.5. Conclusions and Remarks

This section has examined the cyclical behavior of the bond market, stock market, economic activity and the commodity market. We show that: (1) The fluctuations of these four markets are governed essentially by the shorter 3~5 year Kitchin cycle and the longer 7~11 year Juglar cycle; (2) The four markets have four significant lead or lad relationships, in which economic activity leads the commodity market and lags behind both the bond market and the equity market, while the bond market leads the commodity market. The results are useful for investors to optimize their portfolio in different phases of the business cycle, and more so as we expand our discussion to include commodity markets, which is rarely discussed in previous literature. Through the empirical study by this section, readers can better understand the cyclical sequence among multiple markets. The implication of our results is straightforward and is helpful for investors to enhance their gains by incorporating such an “intermaket framework”.

Besides, the results not only can apply to asset allocation, but also on gauging business cycle turning points. For most policy makers, market participants and business managers, future economic performances are important. However, the prediction of turning points is indeed one of the most challenging aspects of economic forecasting in general, even with large-scale macroeconometric models. Zarnowitz (1992) had shown that, in history, the largest forecasting errors are all associated with business cycle turning points. Therefore, in order to overcome this challenge, forecasters should include some leading indicators in their forecasting model, as did the Wharton model (Adams and Klein, 1972; Adams and Duggal, 1974). For market participants, even though they might not be familiar with the sophisticated econometric models, they can use some kind of rule of thumb to gauge future economic movements by leading financial indicators, such as stock prices and bond prices. For example, Siegel (1991) has pointed out that, out of the forty-one recessions from 1802 through 1990, thirty-eight of them, which is 93%, have been preceded (accompanied) with declines of 8% or more (based on monthly average) in stock total return indexes. As for lagging indicators, such as commodity prices, it not only can be used to reaffirm the turning points in economic activity that precedes those of the lagging indicator’s, but its inverse can also be treated as long leaders of the next business cycle turning point.

For upcoming researchers, spectral analysis is also a possible tool for market timing decisions. Indeed, there are many markets left out of this section, such as the corporate bond market, that are grounds where later researchers can further study with spectral analysis.

III. Cycle and Performance of Mutual Funds.

3.1. Introduction

Many researchers have even applied the spectral analysis to financial economics, e.g.

Turhan-Sayan and Sayan (2001) studied the stock market and Wilson and Zurbruegg (2003) the real estate market. Nevertheless, only a few literature has applied the spectral approach to fund investment. In order to fill the gap in the literature, we will use cross-spectral analysis to find out the lead or lag relationships among various categories of funds from the data of 2,135 funds covering nine categories, namely equity funds (Eds), energy funds (ENds), currency funds (Cds), finance funds (Fds), technology funds (Tds), balanced funds (Bds), medical service funds (Mds), bond funds (BOds) and real estate funds (Rds) from 1997 to 2008, and incorporate the lead or lag relationships as a reference for investors to establish their investment portfolios.

3.2. Data and methodology

3.2.1. Data

Our data (including dividends information) comes from four websites, FundDJ, FortunEngine, E-fund and cnYES with a total of 2,135 funds. They are grouped into nine categories, namely Eds, ENds, Cds, Fds, Tds, Bds, Mds, BOds and Rds. The data covers the eight years of 2001 through 2008.

Table 1.5 Numbers of funds in each category

Note: We delete some data so that the funds of each category have to be the same starting and ending time, e.g. the source numbers of equity funds are 1,035, and, by deleting 784 numbers, 251 numbers of monthly data are left, and so forth.

For the subsequent spectral analysis, the monthly data shows the accumulated returns compiled from daily returns. The definition of returns and compound accumulated returns are defined as follows:

, , 1 ,

Prior to the cross-spectral analysis, we used uni-spectral analysis to verify if individual funds have cyclical phenomenon. Noteworthy, cross-spectral analysis is only

meaningful in finding the lead or lag relationships when individual funds have the cyclical phenomenon. In order to verify this prerequisite, we applied the test proposed by Canova (1996). We define Ω as the average frequency of the peak of power _A spectrum among funds in a fund category, plus/minus a standard deviation of their peak frequencies, and Ω is the range beyond the cycle of _R Ω . At the same time, D is _A defined as the average of

D over each fund in a given category. In conclusion, we

may determine whether if significant cyclical phenomenon exists in an interval for a specific fund category, and then we use cross-spectral analysis to verity the lead or lag relationship between specific fund categories.

3.2.3. Cross-spectral analysis

Since massive data is required in calculation when conducting cross-spectral analysis for any pair of funds, we use the following procedures in attempt to reduce computation complexity. According to uni-spectral analysis, we have a number of frequencies of power spectrum peaks for each fund category (e.g., X fund category) and average and standard deviation of frequency of power spectrum peaks calculated from X fund category is named

μ

_X and

σ

_X , respectively. The likelihood,

[

μX

−

σ μX

,

X

+

σX

]

, is regarded as a maximum interval (ignoring extreme value) of frequency of power spectrum peaks of funds in X fund category. We partition

[

μX

−

σ μX

,

X

+

σX

]

into four intervals, [a ,a ] , _{0 1} [a ,a ] , _{1 2} [a ,a ] and _{2 3} [a ,a ] , the first, second, third and fourth _{3 4} intervals, respectively:

Figure 1.5 The partition of each fund for X category funds

Assuming that X and Y are any two fund categories, we select all funds in the first interval of Y fund category and all funds in the fourth interval of X fund category; all funds in the forth interval of Y fund category and all funds in the first interval of X fund category to conduct cross-spectral analysis. We find the maximum and minimum frequency which is one to one period of cross-spectral and all lead or lag relations between X and Y fund categories will be contained in the interval in which is consists of maximum and minimum period. Without loss of generality, the procedure can help us to reduce calculation to 1/16. Notice that I define the X fund category leads Y fund category if the funds in the first interval of X lead the funds in the fourth interval of Y, and the funds in the fourth interval of X lead the funds in the first interval of Y at the same time.

3.3. Empirical results

Tables 1.6 shows the empirical results of our univariate spectral analysis. The

D

column of the Tables 1.6 present that, the

D of all fund categories are all greater than

1. In other words, it illustrates that most fund categories have the cyclical phenomenon.

In addition, the average period of power spectrum peaks almost emerges somewhere between the 17 and 22 months, which is equivalent to 1.5 – 1.8 years, a period

a 0 a ₁ a ₂ a ₃ a ₄

μ

X −

σ

1

X 2

μ

−

σ

μ

_X 1

X 2

μ

+

σ

μ

_X +

σ

obviously beyond 12 months. It implies that, other than the seasonal factors (Granger and Morgenstern, 2001) as often referred to in the literature, there is another longer regular cycle. In fact, the cycle of 17 – 22 months is very interesting. It is equivalent to half of the 3 – 5 years Kitchin Cycle. It means that on average, each short business cycle contains two bull markets and two bear markets.

In summary, we use Canova (1996)’s statistic to verify the existence of the cyclical phenomenon amid various categories of funds. It especially shows that the average frequency of power spectrum peaks amid different categories of funds is rather concentrated. It further signifies a common relationship behind the scenes to make the cycle phenomenon of the various fund categories so close. This finding will strengthen the results of my cross-spectral analysis.

Table 1.6 Univariate spectral statistic, monthly data

Ave. Freq. Std. Freq. Ave. Period Std. Period

D

Rds 0.058317 0.002594 17.174630 0.755685 1.090291 Fds 0.055397 0.003457 18.260952 1.121108 4.361164 Bds 0.053106 0.004141 18.935106 1.406434 1.868759 Cds 0.048867 0.020240 19.32595 5.30494 1.635437 Tds 0.054641 0.006403 18.490468 1.706582 5.088025 Mds 0.045363 0.003559 22.170533 1.773675 1.272006 ENds 0.054275 0.006537 18.687441 2.628904 3.634304 BOds 0.055547 0.010426 18.963197 5.563634 2.471326 Eds 0.055194 0.004070 18.223430 1.451083 2.215471

Note: 1. Average and standard deviation frequency of power spectrum is named by Ave. Freq. and Std.

Freq. respectively.

2. Average and standard deviation period of power spectrum is named by Ave. Period and Std.

Period, respectively.

Given that Bds can be regarded as the investment portfolio made of equity funds and bond funds, which cannot represent any specific industry or market, and there is no

Mds in my cross-spectral analysis.

Corresponding to maximum and minimum frequency of the cross-spectra, we have the maximum and minimum period which is the maximum and minimum of lag or lead’s time. Table 1.7 is the summary of monthly data’s cross-spectrum, in which we define that X category funds leads Y category funds if lead time of X category funds exceeds more than one month, vice versa; otherwise, they are simultaneous. Out of the summary, more than one month of the lead or lag relations neither exists between Fds

Corresponding to maximum and minimum frequency of the cross-spectra, we have the maximum and minimum period which is the maximum and minimum of lag or lead’s time. Table 1.7 is the summary of monthly data’s cross-spectrum, in which we define that X category funds leads Y category funds if lead time of X category funds exceeds more than one month, vice versa; otherwise, they are simultaneous. Out of the summary, more than one month of the lead or lag relations neither exists between Fds

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