An Independent Research Project by

YIELD CURVE INVERSION TRADING STRATEGIES

An Independent Research Project by:

Mary Rachide & Erik Schneider

Under the Direction of Faculty Advisor:

Dr. Campbell Harvey

The Fuqua School of Business

Duke University

April 28, 2003

Yield Curve Inversion Trading Strategies Mary Rachide Erik Schneider

Page 1 of 58

Hypothesis:

The ability to forecast the economic cycle will allow an informed investor to outperform the

widely discussed long-term strategies for investors – automatically rebalancing between a

diversified portfolio of equities and either a diversified portfolio of bonds or cash; or simply

buying and holding a diversified portfolio of equities.

Genesis:

Common wisdom holds that one of these scenarios must be superior over specific time

horizons. If different asset classes demonstrate mean reversion and typically outperform

during different phases of the business cycle, then an automatic rebalancing strategy will

systematically outperform a buy and hold strategy by causing an investor to sell relatively

more dear assets while acquiring ones that are relatively cheaper. Conversely, some pundits

hold that simply buying and holding equities is the superior solution over time. While

seeking to determine which of these competing camps is correct, we also sought an

opportunity to outperform either of these low-thought investment options by applying a

simple, recognizable signal that would require minimal monitoring or trading effort and

therefore be suitable for the average investor. As we believe that different asset classes do

perform in a differentiated manner across the economic cycle, we sought a signal for which

phase of the economic cycle we were experiencing (recession or expansion).

Data:

To benchmark our base case investment rules of automatic rebalancing and buy and hold we

utilized market benchmarks for the total returns of each asset class – Equity (S&P 500),

Bonds (U.S. Long-Term Government) and Cash (U.S. 30-day Treasury Bill). Additionally

we looked at an opportunity to subdivide Equity into Growth (Fama-French Large Company

Growth) and Value (Fama-French Large Company Value) components. All data for these


Page 2 of 58

total returns was provided by Ibbotson and data was utilized from January 1954 through June

2002.

To determine which phase of the economic cycle we were experiencing in the U.S. at any

given time we sought multiple signals. As a benchmark we used the National Bureau of

Economic Research official designations of economic peaks and troughs. Additionally we

utilized monthly data from the U.S. Treasury Department to identify 3-month, 5-year and 10-

year Treasury yields, from which we calculated yield spreads to determine periods of yield

curve inversion.

One Variable Data Analysis:

First we individually examined each of the asset classes to determine if the returns conform

to the normality assumption. A summary is tabulated below (full statistical analysis in

Appendix 1):

Asset Class Average Monthly

Total Return

Standard Deviation

in Monthly Total

Return

Skewness Kurtosis

Cash (T-bills) .00438379 .0028226 10.6368 8.51173

Bonds (LT Gov’t) .00509972 .0261816 4.85855 13.3507

Equity (S&P 500) .00922077 .0421526 -5.7003 12.3417

Growth .00997271 .0468033 -3.61399 9.60853

Value .0133739 .0450135 0.468686 11.6595

As presented in the summary table, none of our five asset classes pass the standard normality

tests of both skewness and kurtosis. None of the asset classes met the requirements of

kurtosis, and only one, Value, was normal by measure of skewness.


Page 3 of 58

T-bills, our proxy for holding cash, performed as expected with the lowest average total

monthly returns and a very tight standard deviation around the mean. T-bill total returns are

very positively skewed (resulting from the impossibility of negative returns) and also have

non-normal kurtosis readings.

Long-term government bonds have a slightly higher mean total return, but a much greater

standard deviation than T-bills. While this asset class is less positively skewed (some

negative returns are possible), it is still non-normal by the tests of skewness and even further

from normality as described by kurtosis. As compared to the T-bill, the long-term bond

returns a mere 7 basis point improvement in average total monthly return while the standard

deviation of returns is multiplied by about 9.3 times.

The S&P 500 is our proxy for equity, and average total returns and standard deviation are

both much higher than for either cash or long term government bonds. Interestingly, the S&P

500 is too negatively skewed to meet the requirements of a normal distribution. The negative

skewness implies that while an investor will make more on average by investing in equities

than in bonds, the investor will also be more likely to suffer from large negative returns in

equities than in bonds.

Growth equities, as defined by Fama & French, returned slightly more than all equities on an

average monthly return basis, involved a slightly higher standard deviation, and were

somewhat less negatively skewed.

Value equities, as defined by Fama & French were far and away the best performer of our

non-cash asset classes. Total average returns were significantly higher than Cash, Bonds,

S&P 500, and Growth equities. At the same time this was the equity class with the lowest


Page 4 of 58

standard deviation on monthly returns and was the only positively skewed asset class. In

terms of relating this to a trading strategy, investors always prefer positive skewness which

results in a greater likelihood of strong positive returns.

As we are attempting to determine a superior investment system over time, we also sought to

determine if it were possible to simply predict short-term returns of each of the asset classes

through time-series forecasting techniques. Unfortunately for us, but fortunately for the

proponents of efficient market theories, only Cash exhibited strong autocorrelations of

monthly returns. Assuring ourselves by this simple analysis that we would have to bring to

bear more information than a chart and a ruler, we also attempted to uncover any ability to

combine asset classes to create a portfolio with superior risk-adjusted returns.

Multiple Variable Data Analysis :

To evaluate the potential to create portfolios with superior risk-adjusted returns among our

asset classes, we first sought to uncover any clear relationships among the asset classes

selected. The degree and statistical certainty of correlation among the asset classes revealed

some interesting information. Not surprisingly, since the Growth and Value assets were

subsets of Equity, there was a very high and statistically significant correlation among these

three asset classes. This suggests that they could not be well combined to create lower risk

portfolios. Furthermore, we were surprised to discover that, while the degree of correlation

was not nearly as high as among the various subsets of equity asset classes, the correlation of

monthly bond returns with each of the other asset classes was statistically significant. Only

the T-bill, our proxy for cash, was not correlated with any of the other asset classes at the

95% confidence level. This suggests, and our later analysis confirmed, that Cash will be

superior to Bonds as an asset class to combine with any of the subsets of equity to create a

lower risk portfolio at any given level of return. A complete correlation matrix is presented in

Appendix 2.


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Correlation Matrix

LTBond

SP500

TBill

Growth

Value

ANOVA:

After creating a binomial indicator of the status of the yield curve, we applied a one-way

analysis of variance asset pairs to determine if the relative returns acted differently based on

the economic regime implied by the yield curve. Here we were testing to determine if the

yield curve determined by the 5-year bond would be more or less useful than that implied by

the 10-year bond. Using the 5-year bond determined yield curve, produced meaningfully

different relative returns for both the Bond – Cash and Cash – S&P 500 asset pairs. As we do

not believe that combining these two assets is a useful scenario, we hoped for more

interesting results from the 10-year bond determined yield curve. Asset pairs showing

statistically significant differences given both applied tests were Cash – S&P 500 and Cash –

Growth. Asset pairs showing statistically significant differences given only one of the

applied tests were Cash – Value and Bond – Growth. Details of the ANOVA analysis for

each of the 5-year and 10-year inversion indicators are included in Appendix 4 and Appendix

5, respectively.

Benchmark Returns:


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Since the ultimate measure of a superior investment system must be the level of risk-adjusted

returns, we developed benchmarks which combined each of the investment classes against

which we could compare our expected returns from implementing a predictive strategy. As

outlined above, we consider portfolios invested, 100% in Equity, 100% in Bonds or 100% in

Cash and combinations thereof. A graphical representation of the monthly returns and

standard deviation of those returns is below:

1954 - 1998

10% bonds

20% bonds

30% bonds

40% bonds

50% bonds

60% bonds

70% bonds

80% bonds

90% bonds

100% bonds

20% cash

30% cash

40% cash

50% cash

60% cash

70% cash

80% cash

90% cash

100% cash

100% equities

10% cash

0.0040

0.0050

0.0060

0.0070

0.0080

0.0090

0.0100

0.0110

0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 0.0225 0.0250 0.0275 0.0300 0.0325 0.0350 0.0375 0.0400 0.0425 0.0450

Standard Deviation

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The buy-and-hold strategy demonstrated by the extreme points on the graph is clearly

implemented. A strategy involving a combination of the asset classes assumes monthly

rebalancing to the preferred level. Therefore, for an investor holding two asset classes, he

would sell some of the asset that performed better to buy some of the asset that performed

worse. The greater the disparity in relative performance, the greater the dollar volume of

assets shifted in a given month. Note that the shift of assets to cash always decreases the

standard deviation of monthly returns while that to bonds initially decreases variability, but

eventually serves to increase it. Additionally, there is a more complex set of portfolios that

could be created by combining each of the three asset classes, whose returns and standard


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deviation of returns would be contained in the space between the two sets of portfolios

outlined above.

Additionally, we benchmarked the potential to receive superior risk-adjusted returns by

subdividing the Equity asset class into Growth and Value components:

1954 - 1998

10% growth20% growth

30% growth40% growth50% growth60% growth70% growth


100% growth

20% cash

30% cash

40% cash

50% cash

60% cash

70% cash

80% cash

90% cash

100% cash

20% bond

30% bond

40% bond

50% bond

60% bond

70% bond

80% bond

90% bond

100% bond

100% value

10% cash10% bond

0.0040

0.0060

0.0080

0.0100

0.0120

0.0140

0.0160

0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 0.0225 0.0250 0.0275 0.0300 0.0325 0.0350 0.0375 0.0400 0.0425 0.0450 0.0475 0.0500

Standard Deviation

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As was clear from the Data Analysis above, the high correlation between Growth and Value

suggests that there would be little benefit to combining them in a single portfolio. This is

born out by the tight cluster of highly variable returns shown above. Since the 100% Value

asset provided superior returns and lower standard deviation, relative to the 100% Growth

asset, we subsequently combined the Value asset with the Bond and Cash assets as described

above.

Analysis:


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In selecting the forecast of the economic cycle, the question is raised as to whether prior

knowledge of the NBER’s assignment of the peaks and troughs of each economic cycle

would enable an investor to achieve excess returns over the aforementioned trading

strategies. Because American financial markets are highly liquid and well informed, the

effects of an economic downturn are probably foreseen and discounted into the equity

markets well in advance of the actual beginning of the contraction. We tested this belief over

the period from 1954 to 2001 and found, in fact, that prior knowledge of the NBER

assignments of economic peaks and troughs would be useful in creating an effective trading

strategy.

The strategies we examined were 1) investing in bonds in NBER defined recessions and

equities in expansion periods and 2) holding cash in NBER defined recessions and equities in

expansions. While the cash/equity strategy barely beat the benchmark, the bond/equity

strategy was clearly optimal to a simple buy and hold strategy. The problem, however, is that

this trading strategy relies upon knowing ahead of time whether you are in an expansion or

contraction period. As of now, there is no reliable way to predict the NBER economic cycles.

Thus we turn next to an examination of the yield curve as a means of identifying a superior

trading strategy. Building on the research of Dr. Campbell Harvey and others suggesting that

the yield curve is a predictor of the economic cycle, we tested the five and ten-year yield

spreads to the three-month as predictive variables. We also examined the Resnick Shoemaker

Probit Model. Prior to any examination of the data, we expected the ten-year yield spread to

be a stronger indicator because a more extreme expected economic downturn is necessary to

cause an inversion here than in the five-year yield spread. A preliminary examination of the

data reveals, however, that the ten-year is actually inverted more frequently than the five-

year. Even so, the ten-year spread over the three-month is still a better predictor of regime


Page 9 of 58

changes in the market. The probit model developed by Resnick and Shoemaker was not

useful, particularly over the period from 1998 – 2002 period.

The yield curve model was first examined relative to the S&P500 portfolios created by

rebalancing with Bonds or Cash. Then it was compared to that available from the

combination of Growth and Value with Bonds or Cash. The results for each case were similar

and are demonstrated graphically below. Note that the returns for the Value and Cash

strategy are superior to the Equity and Cash strategy, the standard deviation is also higher.

1954 - 1998

5 YR Inv - Bond

10 YR Inv - Bond5 YR Inv - Cash

10 YR Inv - Cash

5 YR Inv - Short

10 YR Inv - Short

0.0040

0.0050

0.0060

0.0070

0.0080

0.0090

0.0100

0.0110

0.0120

0.0300 0.0325 0.0350 0.0375 0.0400 0.0425 0.0450

Standard Deviation

Mo

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1954 - 1998

5 YR Inv - G V Momentum10 YR Inv - G V Momentum

5 YR Inv - G V Contrarian10 YR Inv - G V Contrarian5 YR Inv - Value Cash

10 YR Inv - Value Cash

0.0040

0.0050

0.0060

0.0070

0.0080

0.0090

0.0100

0.0110

0.0120

0.0130

0.0140

0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0.0450 0.0500

Standard Deviation

Mo

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Page 10 of 58

The implication here is that the ten-year yield spread should be used to identify inversions.

Our analysis focused on only a one month lag of this variable because the information about

future economic prospects conveyed by such an inversion is immediately communicated to

the financial markets.

Mechanics:

Each trading strategy was developed in detail using Microsoft Excel. At the beginning of

each monthly period, current funds were calculated be multiplying the funds in each

investment option from the previous month by the total monthly returns for that vehicle. In

the case of a simple buy and hold strategy, this became the investment for the next month and

the process was repeated. For an automatic rebalancing strategy, the funds were reallocated

to match the desired allocation ratio after the previous monthly returns had been applied. By

this method, a small amount of whichever vehicle had returned the greatest amount was sold

and the proceeds were invested in the laggard. The return series described by these

mechanics became the benchmarks against which trading strategies were tested.

To apply each trading strategy, we inserted a check at the point of monthly reinvestment. If

the binomial indicator being examined indicated recession, all funds were switched to the

alternate investment strategy (e.g., 100% cash). Funds were maintained in the alternate

investment until the point where the binomial indicator returned to its normal state. At this

point, funds were divided again according to our benchmark strategy.

Testing a Variety of Strategies:

The buy and hold strategies we tested included: 100% growth, 100% value, 100% bonds,

100% cash. Simple, monthly rebalancing strategies included S&P to cash, bonds to S&P,

Value to Growth, Value to Bonds and cash to Value.


Page 11 of 58

We found that switching from a buy-and-hold or an auto-rebalancing strategy to allocating

monthly based on the binomial inversion factor yield excess returns. Because the cash to

Value was the most difficult of our benchmarks, we focused on this strategy in evaluating our

binomial yield curve inversion variable. Specifically, using a Cash to Value strategy based

on the binomial inversion factor outperforms holding cash, holding value, or rebalancing

among cash and Value on a monthly basis.

Out of Sample Testing:

To determine if the backward looking information provided by examining historical data

could be applied as a forward looking trading strategy, requires performing a test on data not

included in the original sample. In this case, a complicating factor is that we are seeking a

relatively rare, event-driven strategy, so there are not a large number of useful historical data

points. Additionally, each event will be driven by its own unique dynamics which may differ

radically from those of previous events. For our out of sample test, we held out data from

January 1998 through June 2002.

The result of examining Cash or Bonds versus the S&P 500 provides a much different picture

from the analysis period. This was due the collapse of long term bond rates to historic lows

during this time frame.


Page 12 of 58

1998 - 2002

10% bonds

20% bonds

30% bonds

40% bonds50% bonds

60% bonds80% bonds90% bonds

100% bonds

20% cash30% cash40% cash50% cash

60% cash

70% cash

80% cash

90% cash

100% cash

100% equities

70% bonds

10% cash

0.0035

0.0037

0.0039

0.0041

0.0043

0.0045

0.0047

0.0049

0.0051

0.0053

0.0055

0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0.0450 0.0500 0.0550 0.0600

Standard Deviation

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A similar dynamic carried over to the examination of Growth versus Value and the

combination of value with Cash or Bonds. 100% Value was not the best strategy for equity

ownership, nor was further diversifying with Cash relative to Bonds. Over our out of sample

period, there is a return penalty for Cash, but it did serve to lower monthly standard deviation

of returns.

1998 - 2002


60% growth


20% cash30% cash

40% cash

50% cash

60% cash

70% cash

80% cash

100% cash

20% bond30% bond40% bond50% bond

60% bond70% bond

80% bond

90% bond

100% bond

100% value


50% growth 70% growth80% growth

10% cash

90% cash

10% bond

0.0030

0.0035

0.0040

0.0045

0.0050

0.0055

0.0060

0.0065

0.0070

0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0.0450 0.0500 0.0550 0.0600 0.0650

Standard Deviation

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Although there was a radical change in the shape of the return profile of (and indeed, the

choice of) superior underlying strategy, the value of trading on the yield curve inversion to

lower standard deviation of returns is maintained. In this case, although standard deviation of

monthly return would be less than that of any of the all equity portfolios the penalty to

returns is severe.

1998 - 2002

5 YR Inv - G V Momentum

5 YR Inv - G V Contrarian

10 YR Inv - G V Momentum

10 YR Inv - G V Contrarian

5 YR Inv - Value Cash 10 YR Inv - Value Cash

(0.0020)

0.0000

0.0020

0.0040

0.0060

0.0080

0.0100

0.0120

0.0140

0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0.0450 0.0500 0.0550 0.0600 0.0650

Standard Deviation

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Conclusion:

Our analysis of the value of predicting the economic cycle and trading accordingly, showed a

number of interesting results. Within our benchmark period, we were surprised to discover

that Cash was superior to Bonds to combine with an equity portfolio. Further, even when we

had strong signals from our preferred yield curve inversion model, Cash was the better asset

to trade into versus Bonds or shorting the selected equity portfolio.

Across our benchmark period from 1954 to 1998, using an inverted yield curve as a trading

signal to convert from the S&P 500 to Cash served to increase monthly returns from 0.99%

to 1.09% while decreasing standard deviation of returns from 4.09% to 3.54%. Over the same

time period, we determined that a Value portfolio would have been a superior equity


Page 14 of 58

investment with a monthly return of 1.39% and a standard deviation of 4.39%. Using the

yield curve inversion as a trading signal did not return a similar increase in returns with

decreased standard deviation relative to the Value only equity portfolio. Instead, monthly

returns dropped to 1.30% while standard deviation dropped to 3.87%. It was not surprising to

discover that, compared to the particular equity class with the best ex post performance over

the time period, we would lose some return by trading into Cash. More importantly to us,

was the usefulness of the trading strategy when utilized ex ante. We therefore performed a

test on the period from 1998 to 2002 which had been held out of our benchmarking data.

Within this period, a radically inflating then deflating equity market, combined with a

simultaneous collapse in long-term interest rates, played havoc our expected buy and hold

strategies, but clearly demonstrated the value of utilizing the yield curve inversion trading

strategy. The S&P 500 returned 0.45% monthly with a standard deviation of 5.38% while

trading into Cash yield a nearly identical 0.44% monthly return, but with a standard deviation

of only 4.61%. We would however, based on our benchmark data, have been using the Value

portfolio which returned 0.63% monthly with a standard deviation of 5.57%. By applying the

yield curve inversion trading strategy, monthly returns would have dropped to 0.48% with a

commensurate decrease in standard deviation to 4.68%.

Collectively, the results point to the conclusion that compared to simply investing in the S&P

500, applying a yield curve inversion signal to trade into and out of Cash can be expected to

significantly reduce standard deviation of returns, without negatively impacting returns.

However, compared to a Value only equity investment, trading based on the yield curve

inversion will decrease standard deviation of returns, but at the cost of decreased monthly

returns.


Page 15 of 58

Bibliography Federal Reserve Statistical Release, “Selected Interest Rates.” December 2002.

http://www.federalreserve.gov/releases/h15/data.htm.

French, Kenneth R. “Description of Fama-French Benchmark Factors.”

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library

Harvey, Campbell R., “Term Structure and the Economic Growth: The Recession of 2001.”

Duke University; Fall 2002.

Ibbotson Financial Database. January 1926 – June 2002

Liu, Wei, Bruce G. Resnick and Gary L. Shoesmith, Market Timing of International Stock

Markets Using the Yield Spread. Indiana University and Wake Forest University;

2002.

Resnick, Bruce G. and Gary Shoesmith, “Using the Yield Curve to Time the Stock Market.”

Financial Analysts Journal. Charlottesville; May/Jun 2002.

Appendix 1 – One Variable Data Analysis

Page 16 of 58

Scatterplot

LTBond-0.09 -0.05 -0.01 0.03 0.07 0.11 0.15

Normal Probability Plot

LTBond

perc

enta

ge

-0.09 -0.05 -0.01 0.03 0.07 0.11 0.150.1

15

2050809599

99.9

Symmetry Plot

distance below median

dist

ance

abo

ve m

edia

n

0 0.03 0.06 0.09 0.12 0.150

0.03

0.06

0.09

0.12

0.15

Appendix 1 – One Variable Data Analysis Descriptive statistics (measures of central tendency, variability, and distribution shape) for each asset class are detailed within this appendix. None of the asset classes passes both the skewness and kurtosis tests of normality. Descriptive Statistics – Long Term Government Bonds Summary Statistics for LTBond Count = 582 Average = 0.00509972 Variance = 0.000685477 Standard deviation = 0.0261816 Minimum = -0.0878044 Maximum = 0.141801 Range = 0.229605 Stnd. skewness = 4.85855 Stnd. kurtosis = 13.3507


Page 17 of 58

Scatterplot

TBill0 3 6 9 12 15

(X 0.001)

Symmetry Plot


dist

ance

abo

ve m

edia

n

0 2 4 6 8 10(X 0.001)

0

2

4

6

8

10(X 0.001)


TBill

perc

enta

ge

0 3 6 9 12 15(X 0.001)

0.115

2050809599

99.9

Descriptive Statistics – Treasury Bills

Summary Statistics for TBill Count = 582 Average = 0.00438379 Variance = 0.00000520872 Standard deviation = 0.00228226 Minimum = 0.000296956 Maximum = 0.013385 Range = 0.0130881 Stnd. skewness = 10.6368 Stnd. kurtosis = 8.51173


Page 18 of 58

Scatterplot

SP500-0.25 -0.15 -0.05 0.05 0.15 0.25


SP500

perc

enta

ge

-0.25 -0.15 -0.05 0.05 0.15 0.250.1

15

2050809599

99.9

Symmetry Plot


dist

ance

abo

ve m

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0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

Descriptive Statistics – S&P 500

Summary Statistics for SP500 Count = 582 Average = 0.00922077 Variance = 0.00177684 Standard deviation = 0.0421526 Minimum = -0.242326 Maximum = 0.153341 Range = 0.395667 Stnd. skewness = -5.7003 Stnd. kurtosis = 12.3417


Page 19 of 58

Scatterplot

Growth-0.24 -0.14 -0.04 0.06 0.16 0.26

Symmetry Plot


dist

ance

abo

ve m

edia

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0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

Descriptive Statistics – Fama & French Growth Equities

Summary Statistics for Growth Count = 576 Average = 0.00997371 Variance = 0.00219054 Standard deviation = 0.0468033 Minimum = -0.239951 Maximum = 0.210817 Range = 0.450768 Stnd. skewness = -3.61399 Stnd. kurtosis = 9.60853


Growth

perc

enta

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-0.24 -0.14 -0.04 0.06 0.16 0.260.1

15

2050809599

99.9


Page 20 of 58

Scatterplot

Value-0.2 -0.1 0 0.1 0.2 0.3


Value

perc

enta

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-0.2 -0.1 0 0.1 0.2 0.30.1

15

2050809599

99.9

Symmetry Plot


dist

ance

abo

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0 0.04 0.08 0.12 0.16 0.2 0.240

0.04

0.08

0.12

0.16

0.2

0.24

Descriptive Statistics – Fama & French Value Equities

Summary Statistics for Value Count = 576 Average = 0.0133739 Variance = 0.00202622 Standard deviation = 0.0450135 Minimum = -0.198438 Maximum = 0.230568 Range = 0.429006 Stnd. skewness = 0.468686 Stnd. kurtosis = 11.6595

Appendix 2 – Autocorrelation Analysis

Page 21 of 58

Estimated Autocorrelations for LTBond

lag

Aut

ocor

rela

tions

0 2 4 6 8-1

-0.6

-0.2

0.2

0.6

1

Time Series Plot for LTBond

LTB

ond

1/26 1/35 1/44 1/53 1/62 1/71 1/80-0.09

-0.05

-0.01

0.03

0.07

0.11

0.15

Appendix 2 – Autocorrelation Analysis This appendix illustrates the autocorrelation analysis conducted for each asset class. Tables show the estimated autocorrelations between values of each asset at various lags. Autocorrelation Analysis – Long Term Government Bonds Data variable: LTBond Number of observations = 582 Start index = 1/26 Sampling interval = 1.0 month(s)

Estimated Autocorrelations for LTBond

Lower 95.0% Upper 95.0% Lag Autocorrelation Stnd. Error Prob. Limit Prob. Limit ---------------------------------------------------------------------------------- 1 0.0625393 0.0414513 -0.0812433 0.0812433 2 -0.00336936 0.0416131 -0.0815604 0.0815604 3 -0.0997163 0.0416136 -0.0815613 0.0815613 4 0.0541813 0.0420222 -0.0823621 0.0823621 5 0.0341596 0.042142 -0.082597 0.082597 6 0.0373293 0.0421896 -0.0826902 0.0826902 In this case, one of the 6 autocorrelation coefficients is statistically significant at the 95.0% confidence level, implying that the time series is probably completely random.


Page 22 of 58

Estimated Autocorrelations for TBill

lag

Aut

ocor

rela

tions

0 2 4 6 8-1

-0.6

-0.2

0.2

0.6

1

Autocorrelation Analysis – Treasury Bills Data variable: TBill Number of observations = 582 Start index = 1/26 Sampling interval = 1.0 month(s)

Estimated Autocorrelations for TBill Lower 95.0% Upper 95.0% Lag Autocorrelation Stnd. Error Prob. Limit Prob. Limit ---------------------------------------------------------------------------------- 1 0.95238 0.0414513 -0.0812433 0.0812433 2 0.921242 0.0695352 -0.136287 0.136287 3 0.899639 0.0880431 -0.172562 0.172562 4 0.872016 0.10263 -0.201151 0.201151 5 0.854027 0.114656 -0.224722 0.224722 6 0.837928 0.125109 -0.24521 0.24521 In this case, 6 of the 6 autocorrelation coefficients are statistically significant at the 95.0% confidence level, implying that the time series is probably completely random.

Time Series Plot for TBill

TB

ill

1/26 1/35 1/44 1/53 1/62 1/71 1/800

3

6

9

12

15(X 0.001)


Page 23 of 58

Time Series Plot for SP500

SP

500

1/26 1/35 1/44 1/53 1/62 1/71 1/80-0.25

-0.15

-0.05

0.05

0.15

0.25

Estimated Autocorrelations for SP500

lag

Aut

ocor

rela

tions

0 2 4 6 8-1

-0.6

-0.2

0.2

0.6

1

Autocorrelation Analysis – S&P 500

Data variable: SP500 Number of observations = 582 Start index = 1/26 Sampling interval = 1.0 month(s)

Estimated Autocorrelations for SP500 Lower 95.0% Upper 95.0% Lag Autocorrelation Stnd. Error Prob. Limit Prob. Limit ---------------------------------------------------------------------------------- 1 0.0206921 0.0414513 -0.0812433 0.0812433 2 -0.0319949 0.0414691 -0.0812781 0.0812781 3 0.00936037 0.0415115 -0.0813611 0.0813611 4 0.0242488 0.0415151 -0.0813683 0.0813683 5 0.0892807 0.0415394 -0.0814159 0.0814159 6 -0.0456714 0.0418678 -0.0820596 0.0820596 In this case, one of the 6 autocorrelation coefficients is statistically significant at the 95.0% confidence level, implying that the time series is probably completely random.


Page 24 of 58

Time Series Plot for Growth

Gro

wth

1/26 1/35 1/44 1/53 1/62 1/71 1/80-0.24

-0.14

-0.04

0.06

0.16

0.26

Estimated Autocorrelations for Growth

lag

Aut

ocor

rela

tions

0 2 4 6 8-1

-0.6

-0.2

0.2

0.6

1

Autocorrelation Analysis – Fama & French Growth Equities Data variable: Growth Number of observations = 576 Start index = 1/26 Sampling interval = 1.0 month(s)

Estimated Autocorrelations for Growth

Lower 95.0% Upper 95.0% Lag Autocorrelation Stnd. Error Prob. Limit Prob. Limit ---------------------------------------------------------------------------------- 1 0.0618676 0.0416667 -0.0816653 0.0816653 2 -0.0378888 0.0418258 -0.0819773 0.0819773 3 0.00111439 0.0418854 -0.082094 0.082094 4 0.0091741 0.0418854 -0.0820941 0.0820941 5 0.0363455 0.0418889 -0.082101 0.082101 6 -0.0299562 0.0419436 -0.0822082 0.0822082 In this case, none of the 6 autocorrelations coefficients are statistically significant, implying that the time series may well be completely random.


Page 25 of 58

Estimated Autocorrelations for Value

lag

Aut

ocor

rela

tions

0 2 4 6 8-1

-0.6

-0.2

0.2

0.6

1

Time Series Plot for Value

Val

ue

1/26 1/35 1/44 1/53 1/62 1/71 1/80-0.2

-0.1

0

0.1

0.2

0.3

Autocorrelation Analysis – Fama & French Value Equities Data variable: Value Number of observations = 576 Start index = 1/26 Sampling interval = 1.0 month(s)

Estimated Autocorrelations for Value Lower 95.0% Upper 95.0% Lag Autocorrelation Stnd. Error Prob. Limit Prob. Limit ---------------------------------------------------------------------------------- 1 0.0840571 0.0416667 -0.0816653 0.0816653 2 -0.0263515 0.04196 -0.0822403 0.0822403 3 -0.00753506 0.0419888 -0.0822966 0.0822966 4 -0.0106556 0.0419911 -0.0823012 0.0823012 5 0.0562525 0.0419958 -0.0823104 0.0823104 6 -0.0202184 0.0421264 -0.0825664 0.0825664 In this case, one of the 6 autocorrelation coefficients is statistically significant at the 95.0% confidence level, implying that the time series is probably completely random.

Appendix 3: Multiple Variable Analysis

Page 26 of 58

Appendix 4: Multiple Variable Analysis Data variables: LTBond, SP500, TBill, Growth, Value There are 576 complete cases for use in the correlation calculations. Correlation Matrix

LTBond SP500 TBill Growth Value ----------------------------------------------------------------------------------------------------------------------- LTBond 0.2378 0.1086 0.2223 0.2063

0.0000 0.0091 0.0000 0.0000 SP500 0.2378 -0.0783 0.9680 0.8558

0.0000 0.0604 0.0000 0.0000 TBill 0.1086 -0.0783 -0.0802 -0.0645

0.0091 0.0604 0.0545 0.1219 Growth 0.2223 0.9680 -0.0802 0.7795

0.0000 0.0000 0.0545 0.0000 Value 0.2063 0.8558 -0.0645 0.7795

0.0000 0.0000 0.1219 0.0000 ------------------------------------------------------------------------------------------------------------------------ Correlation P-Value This table shows Pearson product moment correlations between each pair of variables to measure the strength of the linear relationship between the variables. The third number in each location of the table is a P-value which tests the statistical significance of the estimated correlations. P-values below 0.05 indicate statistically significant non-zero correlations at the 95% confidence level. The following pairs of variables have P-values below 0.05, indicating that they are correlated. LTBond and SP500 LTBond and TBill LTBond and Growth LTBond and Value SP500 and Growth SP500 and Value Growth and Value SP500 and Value Growth and Value

Appendix 4: ANOVA on the 5-Year Inversion

Page 27 of 58

Box-and-Whisker Plot

BondSPDelta

lag(

Inv_

5_yr

,1)

0

1

-0.14 -0.04 0.06 0.16 0.26 0.36

Means and 95.0 Percent LSD Intervals

lag(Inv_5_yr,1)

Bon

dSP

Del

ta

0 1-9

-6

-3

0

3

6(X 0.001)

Appendix 4: ANOVA on the 5-Year Inversion Within this appendix, one-way analysis of variance tests for the deltas between monthly returns of sets of two asset classes are presented based on our binomial classification of yield curve inversions on a one month lag. This analysis allows the comparison of the relative performance of two asset classes in and out of inverted yield curves. The F-test in the ANOVA tables test whether there are any significant differences amongst the means. Because of the non-normality of the distributions of returns for our asset classes, the Kruskal-Wallis test provides a good secondary test of differences in relative asset class performance in and out of inverted yield curves. The Kruskal-Wallis test evaluates the null hypothesis that the medians of the relative performance metric within each of the lagged binomial yield curve classification are statistically the same. Use of the median instead of the mean as the test criteria eliminates the impact of outliers in the data set. To conduct the test, the data from both periods of inversion and normal yield curves are combined and ranked from smallest to largest. The average rank is then computed for the data for each of the binomially classified sets. If the P-value from this test is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.

One-Way ANOVA – Relative Performance of LT Bonds and the S&P 500 Dependent variable: BondSPDelta Factor: lag(Inv_5_yr,1) Number of observations: 581 Number of levels: 2


Page 28 of 58

One-Way ANOVA – Relative Performance of LT Bonds and the S&P 500 cont. Summary Statistics for BondSPDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 494 -0.00446731 0.0018014 0.0424429 -0.122186 1 87 -0.00170421 0.00290541 0.0539018 -0.136218 ---------------------------------------------------------------------------------------------------------------- Total 581 -0.00405356 0.00196297 0.0443054 -0.136218 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ---------------------------------------------------------------------------------------------------------------- 0 0.302746 0.424932 9.1155 27.391 1 0.148834 0.285053 0.919685 0.27451 ---------------------------------------------------------------------------------------------------------------- Total 0.302746 0.438964 8.13383 21.8447 ANOVA Table for BondSPDelta by lag(Inv_5_yr,1)

------------------------------------------------------------------------------------------------------------ Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------ Between groups 0.000564763 1 0.000564763 0.29 0.5921 Within groups 1.13796 579 0.00196538 ------------------------------------------------------------------------------------------------------------ Total (Corr.) 1.13852 580 The F-ratio, which in this case equals 0.287355, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BondSPDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondSPDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank ------------------------------------------------------------ 0 494 290.237 1 87 295.333 ------------------------------------------------------------ Test statistic = 0.0681859 P-Value = 0.793997 Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.


Page 29 of 58


lag(Inv_5_yr,1)

Bon

dBill

Del

ta

0 1-10

-7

-4

-1

2

5(X 0.001)


BondBillDelta

lag(

Inv_

5_yr

,1)

0

1

-0.1 -0.06 -0.02 0.02 0.06 0.1 0.14

One-Way ANOVA - Relative Performance of LT Bonds and T-Bills Dependent variable: BondBillDelta Factor: lag(Inv_5_yr,1) Number of observations: 581 Number of levels: 2

Summary Statistics for BondBillDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 494 0.0017383 0.000590337 0.0242969 -0.0826588 1 87 -0.00517067 0.00115448 0.0339776 -0.0964916 ---------------------------------------------------------------------------------------------------------------- Total 581 0.000703737 0.000679055 0.0260587 -0.0964916 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.121288 0.203947 3.47795 9.34901 1 0.129323 0.225814 1.46602 4.32526 ---------------------------------------------------------------------------------------------------------------- Total 0.129323 0.225814 2.98716 11.948


Page 30 of 58

One-Way ANOVA - Relative Performance of LT Bonds and T-Bills cont.

ANOVA Table for BondBillDelta by lag(Inv_5_yr,1) Analysis of Variance --------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value --------------------------------------------------------------------------------------------------------- Between groups 0.00353098 1 0.00353098 5.24 0.0225 Within groups 0.390321 579 0.00067413 --------------------------------------------------------------------------------------------------------- Total (Corr.) 0.393852 580 The F-ratio, which in this case equals 5.23784, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is less than 0.05, there is a statistically significant difference between the mean BondBillDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondBillDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank ------------------------------------------------------------ 0 494 297.067 1 87 256.552 ------------------------------------------------------------ Test statistic = 4.30909 P-Value = 0.0379059 The Kruskal-Wallis test tests the null hypothesis that the medians of BondBillDelta within each of the 2 levels of lag(Inv_5_yr,1) are the same. Since the P-value is less than 0.05, there is a statistically significant difference amongst the medians at the 95.0% confidence level.


Page 31 of 58


BillSPDelta

lag(

Inv_

5_yr

,1)

0

1

-0.15 -0.05 0.05 0.15 0.25


lag(Inv_5_yr,1)

Bill

SP

Del

ta

0 1-9

-5

-1

3

7

11(X 0.001)

One-Way ANOVA - Relative Performance of T-Bills and the S&P 500 Analysis Summary Dependent variable: BillSPDelta Factor: lag(Inv_5_yr,1) Number of observations: 581 Number of levels: 2

Summary Statistics for BillSPDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ----------------------------------------------------------------------------------------------------------- 0 494 -0.00620561 0.00162372 0.0402954 -0.121873 1 87 0.00346646 0.00271202 0.0520771 -0.1483 ------------------------------------------------------------------------------------------------------------Total 581 -0.0047573 0.00179422 0.0423583 -0.1483 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ------------------------------------------------------------------------------------------------------------0 0.248282 0.370155 6.61718 15.2549 1 0.132443 0.280743 0.225281 0.613454 ------------------------------------------------------------------------------------------------------------ Total 0.248282 0.396582 5.92577 12.3445


Page 32 of 58

One-Way ANOVA - Relative Performance of T-Bills and the S&P 500 cont. ANOVA Table for BillSPDelta by lag(Inv_5_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00692005 1 0.00692005 3.88 0.0495 Within groups 1.03373 579 0.00178537 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.04065 580 The F-ratio, which in this case equals 3.87598, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is less than 0.05, there is a statistically significant difference between the mean BillSPDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BillSPDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank ------------------------------------------------------------ 0 494 285.856 1 87 320.207 ------------------------------------------------------------ Test statistic = 3.09757 P-Value = 0.0784055 Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.


Page 33 of 58


lag(Inv_5_yr,1)

GV

Del

ta

0 1-14

-11

-8

-5

-2

1(X 0.001)


GVDelta

lag(

Inv_

5_yr

,1)

0

1

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

One-Way ANOVA - Relative Performance of Growth and Value Dependent variable: GVDelta Factor: lag(Inv_5_yr,1) Number of observations: 575 Number of levels: 2

Summary Statistics for GVDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ----------------------------------------------------------------------------------------------------------------0 488 -0.0023697 0.000801739 0.028315 -0.10753 1 87 -0.00877434 0.00164484 0.0405567 -0.14001 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00333875 0.000931937 0.0305276 -0.14001 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.12503 0.23256 -0.746931 9.11199 1 0.138661 0.278671 0.189348 4.62528 ---------------------------------------------------------------------------------------------------------------- Total 0.138661 0.278671 -1.09382 12.7949


Page 34 of 58

One-Way ANOVA - Relative Performance of Growth and Value cont. ANOVA Table for GVDelta by lag(Inv_5_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------ Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------ Between groups 0.00302873 1 0.00302873 3.26 0.0714 Within groups 0.531903 573 0.000928278 ------------------------------------------------------------------------------------------------------------ Total (Corr.) 0.534932 574 The F-ratio, which in this case equals 3.26274, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean GVDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for GVDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 488 293.214 1 87 258.753 -------------------------------------------------------------- Test statistic = 3.17705 P-Value = 0.0746756 The Kruskal-Wallis test tests the null hypothesis that the medians of GVDelta within each of the 2 levels of lag(Inv_5_yr,1) are the same. Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.


Page 35 of 58


lag(Inv_5_yr,1)

Bon

dGD

elta

0 1-9

-6

-3

0

3

6(X 0.001)


BondGDelta

lag(

Inv_

5_yr

,1)

0

1

-0.17 -0.07 0.03 0.13 0.23 0.33

One-Way ANOVA - Relative Performance of LT Bonds and Growth

Dependent variable: BondGDelta Factor: lag(Inv_5_yr,1) Number of observations: 575 Number of levels: 2

Summary Statistics for BondGDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ----------------------------------------------------------------------------------------------------------------0 488 -0.00541221 0.00209873 0.0458119 -0.136733 1 87 -0.00164752 0.00368114 0.0606724 -0.163053 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.0048426 0.00233398 0.0483113 -0.163053 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.30037 0.437104 6.88741 19.2342 1 0.162041 0.325094 0.423122 0.510276 ---------------------------------------------------------------------------------------------------------------- Total 0.30037 0.463424 5.91763 15.4029


Page 36 of 58

One-Way ANOVA - Relative Performance of LT Bonds and Growth cont.

ANOVA Table for BondGDelta by lag(Inv_5_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00104648 1 0.00104648 0.45 0.5036 Within groups 1.33866 573 0.00233623 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.3397 574 The F-ratio, which in this case equals 0.447935, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BondGDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondGDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank ------------------------------------------------------------ 0 488 286.299 1 87 297.54 ------------------------------------------------------------ Test statistic = 0.338046 P-Value = 0.560959 The Kruskal-Wallis test tests the null hypothesis that the medians of BondGDelta within each of the 2 levels of lag(Inv_5_yr,1) are the same. Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.


Page 37 of 58


lag(Inv_5_yr,1)

Bon

dVD

elta

0 1-18

-15

-12

-9

-6

-3(X 0.001)


BondVDelta

lag(

Inv_

5_yr

,1)

0

1

-0.21 -0.11 -0.01 0.09 0.19 0.29

One-Way ANOVA - Relative Performance of LT Bonds and Value Dependent variable: BondVDelta Factor: lag(Inv_5_yr,1) Number of observations: 575 Number of levels: 2

Summary Statistics for BondVDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ----------------------------------------------------------------------------------------------------------------0 488 -0.00778192 0.00203464 0.045107 -0.202883 1 87 -0.0104219 0.00330588 0.0574968 -0.208345 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00818135 0.00222246 0.0471429 -0.208345 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.258857 0.461741 2.62287 14.5755 1 0.105941 0.314285 -2.18773 2.57142 ---------------------------------------------------------------------------------------------------------------- Total 0.258857 0.467202 0.497432 13.9685


Page 38 of 58

One-Way ANOVA - Relative Performance of LT Bonds and Value cont.

ANOVA Table for BondVDelta by lag(Inv_5_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.000514591 1 0.000514591 0.23 0.6308 Within groups 1.27518 573 0.00222544 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.27569 574 The F-ratio, which in this case equals 0.231231, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a\ statistically significant difference between the mean BondVDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondVDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank ------------------------------------------------------------ 0 488 288.631 1 87 284.46 ------------------------------------------------------------ Test statistic = 0.0465502 P-Value = 0.829179 The Kruskal-Wallis test tests the null hypothesis that the medians of BondVDelta within each of the 2 levels of lag(Inv_5_yr,1) are the same. Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.


Page 39 of 58


lag(Inv_5_yr,1)

BillG

Del

ta

0 1-11

-7

-3

1

5

9

13(X 0.001)


BillGDelta

lag(

Inv_

5_yr

,1)

0

1

-0.21 -0.11 -0.01 0.09 0.19 0.29

One-Way ANOVA - Relative Performance of T-Bills and Growth

Dependent variable: BillGDelta Factor: lag(Inv_5_yr,1) Number of observations: 575 Number of levels: 2

Summary Statistics for BillGDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ----------------------------------------------------------------------------------------------------------------0 488 -0.0071054 0.00196346 0.0443109 -0.137027 1 87 0.00352315 0.00356451 0.0597036 -0.205777 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00549725 0.00221445 0.0470579 -0.205777 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.245906 0.382933 4.99172 10.1325 1 0.14565 0.351426 -0.697078 1.85611 ---------------------------------------------------------------------------------------------------------------- Total 0.245906 0.451683 3.91312 9.52679


Page 40 of 58

One-Way ANOVA - Relative Performance of T-Bills and Growth cont.

ANOVA Table for BillGDelta by lag(Inv_5_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00834101 1 0.00834101 3.78 0.0522 Within groups 1.26275 573 0.00220376 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.27109 574 The F-ratio, which in this case equals 3.7849, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BillGDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BillGDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank --------------------------------------------------------------- 0 488 282.561 1 87 318.506 --------------------------------------------------------------- Test statistic = 3.45638 P-Value = 0.0630045 The Kruskal-Wallis test tests the null hypothesis that the medians of BillGDelta within each of the 2 levels of lag(Inv_5_yr,1) are the same. Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.


Page 41 of 58


lag(Inv_5_yr,1)

BillV

Del

ta

0 1-13

-10

-7

-4

-1

2(X 0.001)


BillVDelta

lag(

Inv_

5_yr

,1)

0

1

-0.23 -0.13 -0.03 0.07 0.17 0.27

One-Way ANOVA - Relative Performance of T-Bills and Value

Dependent variable: BillVDelta Factor: lag(Inv_5_yr,1) Number of observations: 575 Number of levels: 2

Summary Statistics for BillVDelta lag(Inv_5_yr,1) Count Average Variance Standard dev. Minimum ----------------------------------------------------------------------------------------------------------------0 488 -0.0094751 0.00182446 0.0427137 -0.207198 1 87 -0.0052512 0.00326529 0.0571427 -0.224761 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00883601 0.00203945 0.0451602 -0.224761 lag(Inv_5_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.204393 0.411591 1.27693 9.91818 1 0.149888 0.374649 -2.05064 4.2917 ---------------------------------------------------------------------------------------------------------------- Total 0.204393 0.429154 -0.329068 11.8175


Page 42 of 58

One-Way ANOVA - Relative Performance of T-Bills and Value cont.

ANOVA Table for BillVDelta by lag(Inv_5_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00131734 1 0.00131734 0.65 0.4220 Within groups 1.16933 573 0.00204071 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.17064 574 The F-ratio, which in this case equals 0.645533, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BillVDelta from one level of lag(Inv_5_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BillVDelta by lag(Inv_5_yr,1) lag(Inv_5_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 488 284.9 1 87 305.391 -------------------------------------------------------------- Test statistic = 1.1233 P-Value = 0.289206 The Kruskal-Wallis test tests the null hypothesis that the medians of BillVDelta within each of the 2 levels of lag(Inv_5_yr,1) are the same. Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.

Appendix 5: ANOVA on 10-Year Inversion

Page 43 of 58


lag(Inv_10_yr,1)

Bon

dSP

Del

ta

0 1-9

-5

-1

3

7

11(X 0.001)


BondSPDelta

lag(

Inv_

10_y

r,1)

0

1

-0.14 -0.04 0.06 0.16 0.26 0.36

Appendix 5: ANOVA on 10-Year Inversion One-Way ANOVA – Relative Performance of LT Bonds and the S&P 500 Analysis Summary Dependent variable: BondSPDelta Factor: lag(Inv_10_yr,1) Number of observations: 581 Number of levels: 2

Summary Statistics for BondSPDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 479 -0.00564214 0.00180792 0.0425196 -0.122186 1 102 0.0034065 0.00264803 0.051459 -0.136218 ---------------------------------------------------------------------------------------------------------------- Total 581 -0.00405356 0.00196297 0.0443054 -0.136218 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.302746 0.424932 9.4434 28.1298 1 0.148834 0.285053 0.296377 0.521339 ---------------------------------------------------------------------------------------------------------------- Total 0.302746 0.438964 8.13383 21.8447


Page 44 of 58

One-Way ANOVA – Relative Performance of LT Bonds and the S&P 500 cont. ANOVA Table for BondSPDelta by lag(Inv_10_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00688535 1 0.00688535 3.52 0.0610 Within groups 1.13164 579 0.00195447 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.13852 580 The F-ratio, which in this case equals 3.52288, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BondSPDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondSPDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 479 285.015 1 102 319.108 -------------------------------------------------------------- Test statistic = 3.46879 P-Value = 0.0625334 Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.


Page 45 of 58


lag(Inv_10_yr,1)

Bon

dBill

Del

ta

0 1-68

-48

-28

-8

12

32(X 0.0001)


BondBillDelta

lag(

Inv_

10_y

r,1)

0

1

-0.1 -0.06 -0.02 0.02 0.06 0.1 0.14

One-Way ANOVA - Relative Performance of LT Bonds and T-Bills Analysis Summary Dependent variable: BondBillDelta Factor: lag(Inv_10_yr,1) Number of observations: 581 Number of levels: 2

Summary Statistics for BondBillDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 479 0.00152147 0.000590955 0.0243096 -0.0826588 1 102 -0.0031364 0.00108467 0.0329343 -0.0964916 ---------------------------------------------------------------------------------------------------------------- Total 581 0.000703737 0.000679055 0.0260587 -0.0964916 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.121288 0.203947 3.69659 9.75244 1 0.129323 0.225814 0.991146 4.23979 ---------------------------------------------------------------------------------------------------------------- Total 0.129323 0.225814 2.98716 11.948


Page 46 of 58

One-Way ANOVA - Relative Performance of LT Bonds and T-Bills cont. ANOVA Table for BondBillDelta by lag(Inv_10_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00182446 1 0.00182446 2.69 0.1012 Within groups 0.392028 579 0.000677077 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 0.393852 580 The F-ratio, which in this case equals 2.6946, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BondBillDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondBillDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 479 295.136 1 102 271.578 ------------------------------------------------------------ Test statistic = 1.65612 P-Value = 0.198125 Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.


Page 47 of 58


lag(Inv_10_yr,1)

BillS

PD

elta

0 1-10

-6

-2

2

6

10

14(X 0.001)


BillSPDelta

lag(

Inv_

10_y

r,1)

0

1

-0.15 -0.05 0.05 0.15 0.25

One-Way ANOVA - Relative Performance of T-Bills and the S&P 500 Analysis Summary Dependent variable: BillSPDelta Factor: lag(Inv_10_yr,1) Number of observations: 581 Number of levels: 2

Summary Statistics for BillSPDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 479 -0.0071636 0.00161614 0.0402013 -0.121873 1 102 0.0065429 0.00249836 0.0499836 -0.1483 ---------------------------------------------------------------------------------------------------------------- Total 581 -0.0047573 0.00179422 0.0423583 -0.1483 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.248282 0.370155 6.83975 15.9861 1 0.132443 0.280743 -0.113135 0.902562 ---------------------------------------------------------------------------------------------------------------- Total 0.248282 0.396582 5.92577 12.3445


Page 48 of 58

One-Way ANOVA - Relative Performance of T-Bills and the S&P 500 cont. ANOVA Table for BillSPDelta by lag(Inv_10_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.0157984 1 0.0157984 8.93 0.0029 Within groups 1.02485 579 0.00177004 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.04065 580 The F-ratio, which in this case equals 8.92547, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is less than 0.05, there is a statistically significant difference between the mean BillSPDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BillSPDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 479 281.66 1 102 334.863 ------------------------------------------------------------ Test statistic = 8.44723 P-Value = 0.00365483 Since the P-value is less than 0.05, there is a statistically significant difference amongst the medians at the 95.0% confidence level.


Page 49 of 58


GVDelta

lag(

Inv_

10_y

r,1)

0

1

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

One-Way ANOVA - Relative Performance of Growth and Value Analysis Summary Dependent variable: GVDelta Factor: lag(Inv_10_yr,1) Number of observations: 575 Number of levels: 2

Summary Statistics for GVDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 473 -0.00235475 0.000840287 0.0289877 -0.10753 1 102 -0.00790182 0.00134391 0.0366594 -0.14001 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00333875 0.000931937 0.0305276 -0.14001 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.12503 0.23256 -0.591573 8.99359 1 0.138661 0.278671 -0.303424 6.75281 ---------------------------------------------------------------------------------------------------------------- Total 0.138661 0.278671 -1.09382 12.7949


lag(Inv_10_yr,1)

GV

Del

ta

0 1-13

-10

-7

-4

-1

2(X 0.001)


Page 50 of 58

One-Way ANOVA - Relative Performance of Growth and Value cont. ANOVA Table for GVDelta by lag(Inv_10_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00258179 1 0.00258179 2.78 0.0961 Within groups 0.53235 573 0.000929058 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 0.534932 574 The F-ratio, which in this case equals 2.77894, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean GVDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for GVDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 473 293.043 1 102 264.613 --------------------------------------------------------------- Test statistic = 2.45729 P-Value = 0.116977 Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.


Page 51 of 58


BondGDelta

lag(

Inv_

10_y

r,1)

0

1

-0.17 -0.07 0.03 0.13 0.23 0.33

One-Way ANOVA - Relative Performance of LT Bonds and Growth Analysis Summary Dependent variable: BondGDelta Factor: lag(Inv_10_yr,1) Number of observations: 575 Number of levels: 2

Summary Statistics for BondGDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 3 -0.00665392 0.00211249 0.0459618 -0.136733 1 2 0.00355696 0.00330558 0.0574941 -0.163053 ---------------------------------------------------------------------------------------------------------------- Total 5 -0.0048426 0.00233398 0.0483113 -0.163053 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.30037 0.437104 7.09525 19.6985 1 0.162041 0.325094 -0.116461 0.927028 ---------------------------------------------------------------------------------------------------------------- Total 0.30037 0.463424 5.91763 15.4029


lag(Inv_10_yr,1)

Bon

dGD

elta

0 1-10

-6

-2

2

6

10

14(X 0.001)


Page 52 of 58

One-Way ANOVA - Relative Performance of LT Bonds and Growth cont. ANOVA Table for BondGDelta by lag(Inv_10_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00874821 1 0.00874821 3.77 0.0528 Within groups 1.33096 573 0.00232279 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.3397 574 The F-ratio, which in this case equals 3.76626, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BondGDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondGDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 473 281.617 1 102 317.598 ------------------------------------------------------------ Test statistic = 3.93572 P-Value = 0.0472676 Since the P-value is less than 0.05, there is a statistically significant difference amongst the medians at the 95.0% confidence level.


Page 53 of 58


lag(Inv_10_yr,1)

Bon

dVD

elta

0 1-13

-9

-5

-1

3(X 0.001)


BondVDelta

lag(

Inv_

10_y

r,1)

0

1

-0.21 -0.11 -0.01 0.09 0.19 0.29

One-Way ANOVA - Relative Performance of LT Bonds and Value Analysis Summary Dependent variable: BondVDelta Factor: lag(Inv_10_yr,1) Number of observations: 575 Number of levels: 2

Summary Statistics for BondVDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 473 -0.00900867 0.00204726 0.0452466 -0.202883 1 102 -0.00434487 0.00304515 0.0551829 -0.208345 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00818135 0.00222246 0.0471429 -0.208345 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.258857 0.461741 2.44631 14.9675 1 0.105941 0.314285 -2.50202 3.31854 ---------------------------------------------------------------------------------------------------------------- Total 0.258857 0.467202 0.497432 13.9685


Page 54 of 58

One-Way ANOVA - Relative Performance of LT Bonds and Value cont. Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00182505 1 0.00182505 0.82 0.3653 Within groups 1.27387 573 0.00222315 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.27569 574 The F-ratio, which in this case equals 0.820927, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BondVDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BondVDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 473 284.482 1 102 304.314 ------------------------------------------------------------ Test statistic = 1.19565 P-Value = 0.274191 Since the P-value is greater than or equal to 0.05, there is not a statistically significant difference amongst the medians at the 95.0% confidence level.


Page 55 of 58


lag(Inv_10_yr,1)

Bill

GD

elta

0 1-12

-7

-2

3

8

13

18(X 0.001)


BillGDelta

lag(

Inv_

10_y

r,1)

0

1

-0.21 -0.11 -0.01 0.09 0.19 0.29

One-Way ANOVA - Relative Performance of T-Bills and Growth Analysis Summary Dependent variable: BillGDelta Factor: lag(Inv_10_yr,1) Number of observations: 575 Number of levels: 2

Summary Statistics for BillGDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 473 -0.00812609 0.00195698 0.0442378 -0.137027 1 102 0.00669335 0.00325716 0.0570715 -0.205777 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00549725 0.00221445 0.0470579 -0.205777 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.245906 0.382933 5.13232 10.634 1 0.14565 0.351426 -1.034 2.39953 ---------------------------------------------------------------------------------------------------------------- Total 0.245906 0.451683 3.91312 9.52679


Page 56 of 58

One-Way ANOVA - Relative Performance of T-Bills and Growth cont. ANOVA Table for BillGDelta by lag(Inv_10_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.0184271 1 0.0184271 8.43 0.0038 Within groups 1.25267 573 0.00218616 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.27109 574 The F-ratio, which in this case equals 8.429, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is less than 0.05, there is a statistically significant difference between the mean BillGDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BillGDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 473 278.638 1 102 331.412 ------------------------------------------------------------ Test statistic = 8.46667 P-Value = 0.00361597 Since the P-value is less than 0.05, there is a statistically significant difference amongst the medians at the 95.0% confidence level.


Page 57 of 58


BillVDelta

lag(

Inv_

10_y

r,1)

0

1

-0.23 -0.13 -0.03 0.07 0.17 0.27


lag(Inv_10_yr,1)

BillV

Del

ta

0 1-14

-10

-6

-2

2

6(X 0.001)

One-Way ANOVA - Relative Performance of T-Bills and Value Analysis Summary Dependent variable: BillVDelta Factor: lag(Inv_10_yr,1) Number of observations: 575 Number of levels: 2

Summary Statistics for BillVDelta lag(Inv_10_yr,1) Count Average Variance Standard dev. Minimum ---------------------------------------------------------------------------------------------------------------- 0 473 -0.0104808 0.00182653 0.042738 -0.207198 1 102 -0.00120847 0.00298321 0.0546188 -0.224761 ---------------------------------------------------------------------------------------------------------------- Total 575 -0.00883601 0.00203945 0.0451602 -0.224761 lag(Inv_10_yr,1) Maximum Range Stnd. Skewness Stnd. kurtosis ----------------------------------------------------------------------------------------------------------------0 0.204393 0.411591 0.988825 10.2473 1 0.149888 0.374649 -2.18256 5.10111 ---------------------------------------------------------------------------------------------------------------- Total 0.204393 0.429154 -0.329068 11.8175


Page 58 of 58

One-Way ANOVA - Relative Performance of T-Bills and Value cont. ANOVA Table for BillVDelta by lag(Inv_10_yr,1) Analysis of Variance ------------------------------------------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ------------------------------------------------------------------------------------------------------------- Between groups 0.00721398 1 0.00721398 3.55 0.0599 Within groups 1.16343 573 0.00203042 ------------------------------------------------------------------------------------------------------------- Total (Corr.) 1.17064 574 The F-ratio, which in this case equals 3.55296, is a ratio of the between-group estimate to the within-group estimate. Since the P-value of the F-test is greater than or equal to 0.05, there is not a statistically significant difference between the mean BillVDelta from one level of lag(Inv_10_yr,1) to another at the 95.0% confidence level. Kruskal-Wallis Test for BillVDelta by lag(Inv_10_yr,1) lag(Inv_10_yr,1) Sample Size Average Rank -------------------------------------------------------------- 0 473 281.693 1 102 317.245 -------------------------------------------------------------- Test statistic = 3.84241 P-Value = 0.0499681 The StatAdvisor --------------- Since the P-value is less than 0.05, there is a statistically significant difference amongst the medians at the 95.0% confidence level.

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