Introduction:

Building upon the previous three studies on gold, we conclude with a technical analysis of gold’s secular bull market trend since 2001. We will use time series analysis to determine gold’s secular trend growth rate, and historical volatility to set the trend boundaries. The trend and its historical boundaries will offer a reference point for identifying where we are today – a first step in any decision making process.

Technical analysis, with its proclivity to market timing, is a critical part of the investment decision process regardless of methodology or time horizon. I hope that this article will make it increasingly obvious how closely related technical analysis is to fundamental analysis and how the two disciplines are indeed inseparable.

Scope:

As always, I want to first establish the scope of our discussion. In this article we will:

1. Identify the trend growth rate of gold’s secular bull market since 2001.
2. Set the historical boundaries of gold’s price swing around the trend growth rate.
3. Observe where we are now in the context of the secular trend.

We begin by marking the start of the most recent secular bull market in gold using the monthly closing prices from our first article. Gold bottomed in July 1999 at $253.20. That bottom was retested 21 months later in April 2001 at $255.80. Our study will begin with the closing price as of December 2000.

The price action from the bottom in 1999 to the subsequent retest in 2001 comprises a bottoming pattern separating two trends, the new secular bull trend from the preceding secular bear trend. It is the nature of asset prices to take time to change their secular trend. Topping and bottoming patterns act to separate the old trend from the new.

Assembling the Data:

Having decided upon our data set, we then feed it into a time series analysis producing the following table. This is similar to the cross-sectional regression analysis we used in our first gold study, except that the independent variable is time.

where…

• t : monthly time period 1-127, from December 2000 through June 2011
• GLDt : observed monthly closing price of gold at the close of each period-t
• Yt : time series regression, ln(Yt) = ln(b) + t * ln(m), where ln(m) = rate/mo
• Resid : regression error term or residual, actual – regression,  Resid = GLDt – Yt
• Ytlow: lower channel historical support line, Ytlow = Yt * (1+ %ResidMin)
• Ytup: upper channel historical resistance line, Ytup = Yt * (1+ %ResidMax)

The error term is normally distributed about an expected value of zero, but it is serially correlated as is often the case with a financial series. I am primarily concerned with extracting the trend growth rate from this time series, and no further transformation is necessary. Note that this regression explains 98% gold’s price movement.

Identifying the Overall Trend Growth Rate:

A financial time series usually will exhibit an exponential path governed by its growth rate. Technical analysis is used to visually identify the trend direction and rate, and to observe when changes occur. This requires that asset prices be plotted on a log-linear scale with time. I cannot stress this point enough!

Our eyes, our minds, and our time can be put to better use by first translating the above data into pictures. In the following chart, we plot the price of gold on a log-linear scale overlaid with its time series regression line. This line describes the mean growth trend about which the price swings as it “reverts to the mean”. ¹ The average trend growth rate for the overall secular bull market has been 1.42% per month or an annualized 18.4%. Pause and take that in for a moment.

“A financial time series usually will exhibit an exponential path governed by its growth rate. Technical analysis is used to visually identify the trend direction and rate, and to observe when changes occur. This requires that asset prices be plotted on a log-linear scale with time.”

Setting the Historical Boundaries:

The red and green dashed lines enveloping the price in the chart above depict the historical boundaries of the price swing around its mean trend. In technical analysis parlance, these boundaries define the price channel, the upper line being resistance, and the lower support.

There are many ways to describe price channels and more will be discovered I am sure. ² Our channel lines are defined simply by the historically highest and lowest price swing around the mean trend, as given by the residual. Drawing the lines parallel to the mean on a log-linear chart creates a constant percentage rather than a constant price channel. The upper red channel line is a constant 22.2% above the mean, and the lower green channel line is a constant 18.0% below, as measured vertically from the mean.

Introducing High-Low-Close Charts:

Our time series so far has been plotted a line chart using only closing prices, a good starting point in helping to define the trend growth rate. But prices move every minute of every day, not just once per month. The preceding line chart has been reproduced below using a high-low-close chart (HLC) adding the intraday price range to the monthly close.

The blue dashed line is the same 18.4% secular time series regression line as before. But the channel lines on either side include the intraday price movement, and are incidentally an equidistant +/-26% about the mean, totaling 52% wide measured vertically.

We are now ready to consider “where we are today.”

Where We Are Today:

The preceding charts describe at a glance how that the price of gold has moved within a price channel +/-26% about its mean secular trend growth rate in excess of 18% annually. Three years ago, an eight month correction took gold down 34% across its entire channel. Since then, gold has been in a cyclical uptrend at a hot 26% annual rate and penetrated the upper channel resistance level this August. It was rudely slapped back to find support again at its cyclical trend line, where we are today.

So, let’s take a closer look at this cyclical uptrend that started in 2009. ³

This chart is identical to the one preceding it except that I have added a trend support line connecting the intra month lows of the cyclical trend. This support line functions like the lower support channel of our time series, but can be drawn visually in a matter of seconds without all the math.

A break of this trend line would be an early indication that the 2009 cyclical uptrend may be changing rate or direction. The probability of a trollop down to the lower channel support line increases significantly, especially if the regression line (blue) is also broken. ⁴ This early warning is something every investor should want to see, wouldn’t you?

Let’s take a closer look at our nearly three year uptrend.

This weekly HLC chart of the cyclical uptrend since 2009 shows our support line in more detail. The squiggly red line is an arithmetic 40-week moving average which can be automatically calculated and plotted by any charting application. It closely approximating our trend line, a reason why it is so popular in technical analysis. Notice the intra-month challenged of our support line and moving average in September as well as multiple other hits since 2009. That’s what makes it a support line.

Let’s add one more indicator to our preceding chart…

The three parallel blue lines on this chart represent the time series regression and channel lines of the three year cyclical uptrend. The trend’s 26% annual rate closely parallels the 40-week moving average and our green trend line, a break of which would be an early warning of a probable test of the lower (blue) support channel. The green trend line is the practical support level for a reversion to the cyclical mean.

Notice also that the upper channel resistance line has been tested three times since 2009, each time followed by a reversion to the mean. The price movement and support and resistance action which I have described are not just coincidence, but rather characteristic of price movement in a time series.

Decisions, Decisions:

So, should you add gold to your portfolio today? If you want to construct the best answer for your specific situation, then use technical analysis to know where we are today within the context of our eleven year secular bull market trend in gold. Here’s how.

The closer you buy to an upper channel or resistance of some kind, the greater the risk of a quick loss from a reversion to the mean or worse yet, a correction to the lower channel support line. You would be vindicated if the secular trend was to resume, but boy would it sting in the short term. And what if unable to tolerate the pain, you sell near the bottom of the of the 30-40% correction? ⁵

It stands to reason therefore that the closer you by to some support level, the lower the risk of loss because the quicker you can tell if you are wrong and modify your decision when support is broken. The less money you lose, the less you have to gain to break even. Profit and loss is an asymmetrical exercise. ⁶ The reverse is also true of selling near a resistance level. ⁷

This is why technical analysis!

Conclusion:

In the four articles on gold written this month of October 2011, we have learned that the data does not support a significant correlation between monthly returns in gold and inflation or equities. But there is a strong inverse relationship between the secular price trends of gold and equities. Therefore, adding gold to an equity portfolio significantly alters relative risk adjusted return. Meanwhile, technical analysis helps reduce the probability of sudden losses in absolute terms when we know where we are in relation to the overall secular trend.

Combining both fundamental and technical analysis techniques is critical to understanding financial markets. My primary application of “the technicals” has been to forth-tell where we are in the trend using sound techniques supported by reasonable basis, rather than fore-tell where we will be.

Thank you for reading this article. Your comments are welcomed!

Related articles suggested for your reading:

Adding Gold To An Equity Portfolio

Relationship Between Gold, Inflation, and Equities II

Relationship Between Gold, Inflation, and Equities

Merging Fundamental and Technical Analysis

Footnotes:

1. I am not implying that prices must revert to the mean. I am simply acknowledging that “reversion to the mean” is a popular belief with reference a financial time series.
2. Other methods of describing price boundaries include: Bollinger Bands, Keltner Channels, Moving Average Envelopes, Parabolic SAR, Pivot Points, Price Channels (Turtles), and the well known statistical standard error of the estimate and predictive interval. Some methods are more grounded in a reasonable basis than others.
3. Within the 11 year secular trend, there are three distinct cyclical trend growth rates: 13.5% between 2001-2005, 18.0% between 2006-2008, and 25.8% between 2009-2011-1/2.
4. Using the 2008 correction as a guide, an eight month correction from August 2011 to April 2012 would put the lower channel at $1307.4, which is a 24% drop from the end of October, and a 32% drop from the high of August, an annualized rate of -44% .
5. Buying at the top and selling at the bottom is a reason often quoted by opponents of market timing and by implication of technical analysis. And I would agree if the so called market timing were attempted with complete ignorance of technical analysis. The alternative to this haphazard emotional response is to use sound technical analysis technique to mitigate risk of loss. Another oft quoted myth is that a market timer must be right twice. Serious practitioners know that the more right you are the first time, the more room there is for error the second time (see note-6).
6. A 33% loss requires a subsequent 50% gain to break even, this is elementary. A market decline can take out years of gains in a few months, which only exacerbates the first problem. Thus profit and loss are an asymmetrical exercise in the sense of both price movement and time.
7. When support areas are broken, they often switch roles and serve as psychological resistance or areas. The same is true of resistance areas. A successful break above our upper channel resistance line can be an indication that the uptrend has become steeper. This happened to the equity market in the late 1990′s just before the secular bull market ended. This price behaviour pattern is often referred to as a “blow-off” implying that the trend growth rate is unsustainable in the long term.

Introduction:

Using correlation and regression analysis, we learned in a previous article that monthly returns of gold since 1979 have not been statistically correlated with U.S. inflation or equities as represented by the CPI-U and the S&P 500 index, respectively. And using long term price charts, we learned in another article that the long term trend in gold prices has been strongly related inversely to the secular trend in equities.

In this article, we will examine the effect of adding gold to a diversified equity portfolio using another statistical tool, mean-variance analysis.

 Scope:

As always, let’s first establish the scope of our discussion. We will address two questions:

1. How are an equity portfolio’s expected return and risk characteristics affected by adding gold to the mix?
2. With respect to question-1, are the effects significantly different when equities are in a secular bear market as compared with a secular bull market?

Many asset management firms assign gold to an asset class of “other” investments that tend toward a more concentrated portfolio compared with the total stock and bond markets. This “other” asset class is then limited to some percentage of the equity allocation of the total portfolio.

For example, a client whose portfolio is allocated to 60% equities and 40% bonds wants to invest separately also in REITs, commodity and precious metal securities. The firm’s policy would limit these assets in the aggregate to no more than 10% of the equity portfolio, so that the final allocation becomes 54% equity, 40% bonds, and at most 6% other.

Following this policy example, we will analyze the effects of adding gold on the equity side of a portfolio only. However, the principals presented here can be applied to the analysis of more than two asset classes.

Definition of Terms:

An asset’s expected return is the sample average return calculated over a given period. Expected return comes at the cost of risk, which for this study we will define as the variance or standard deviation of returns over the same period.

In a two-asset portfolio, such as stocks and bonds, or gold and equities, the portfolio’s expected return is the dollar-weighted average of the expected return of each asset. Represented mathematically:

 The variance of a two-asset portfolio is defined by a quadratic which includes each asset’s weight and variance, as well as the correlation between the two assets. Represented mathematically:

We have already defined correlation (shown above as rho) in some detail in the afore-mentioned articles, as well as price trends, and several other terms.

Assembling the Data:

We begin by recycling the original data gathered in the first article in this series, the monthly closing prices of gold and the S&P 500 index (SPX). We convert the monthly prices to quarterly and then calculate the percent change in prices of gold and equities as shown below.

We then split the data where the secular bull yielded to secular bear market in equities. While the actual day that divided these two secular trends in the SPX was 3/24/2000, the topping pattern lasted more than 6 months. I will grossly approximate the dividing line to be between 2000 and 2001. The bull market data is represented in our table above by the background color green and the bear market by red.

Since there are 42 secular bear market quarters in the given data set, I will balance this using the final 42 secular bull market quarters preceding 2001. Albeit this appears somewhat arbitrary, the result is that I have 20 years of continuous data to work with.

After performing some basic and necessary statistical calculations on our data set, we can assemble the following table of long-only portfolios each having a different mix of gold and equities. For each portfolio, we then calculate the expected return and variance. From this table we can now construct an important tool in our analysis called the minimum variance frontier.

where…

• Wgld: is the portion of the equity allocation weighted in gold
• Wspx: is the portion of the equity allocation weighted in SPX
• E(Rp): is the combined portfolio’s expected return
• VARp: is the combined portfolio’s variance
• Sharpe: is simply the ratio of expected return per unit risk, E(Rp)/STDEVp ¹

We repeat this table for three data sets,

1. the SPX bull market data from quarter ending 9/28/1990 through 12/29/2000
2. the SPX bear market data from quarter ending 3/30/2001 through 6/30/2011
3. and finally the combined data from quarter ending 9/28/1990 through 6/30/2011

The Effect of Adding Gold in an Equity Secular Bull Market:

Now, let’s see what the data is telling us beginning with a secular bull market in equities. And what better way to “see” data than to chart it?

In the following chart, the green line represents the combined gold and equity portfolio’s expected or average return, and the red line (curve) represents its risk in terms of return variance. The horizontal scale marks the portfolio’s weighting in gold from 0% (all equities) to 100% (no equities). Reading from left to right, the chart depicts the effect of adding increasingly more gold on portfolio expected return and variance.

As we increasingly displace equities with gold, portfolio expected return decreases linearly from approximately +3.4% per quarter to -0.44%. This is because during the last ten years of the equity bull market, the average or expected return for equities was 3.4% while that for gold was -0.44%. This should come as no surprise considering what we learned in the previous article, that there is an apparent inverse relationship between the secular trends in gold and equities.

What is more interesting is that the portfolio’s variance decreased rapidly at first from 0.48% at 100% equity, slowed to 0.11% at 57% gold, and then climbed rapidly to 0.32% at 100% gold. At 57% gold and 43% equity, variance had dropped to just 23% of that in an all equity portfolio.

Therefore, when adding gold to an equity portfolio during a secular bull market in equities, there is a non-linear trade-off between risk and expected return whose benefit diminishes rapidly as the gold allocation approaches 57%.

Incidentally, the Sharpe ratio is higher than an all-equity portfolio until gold’s allocation reaches 39%, peaking at approximately 25% gold.

Why was the change in portfolio variance non-linear? The answer lies in that the correlation between gold and equities was less than +1.0, and was in fact -0.4131. ² The closer correlation approaches -1.0, the steeper the parabolic shape of the variance curve becomes. And the more asymmetrical the trade-off becomes between expected return and risk, the more the benefit of diversification increases. But as correlation approaches +1.0, the trade-off becomes so linear that the benefits of diversification with regard to volatility cease.

The Effect of Adding Gold in an Equity Secular Bear Market:

Let’s apply this analysis when equities were in a secular bear market. Compare the following chart with the preceding one for a bull market. The two charts are very similar except for one significant difference.

The expected return increases linearly from 0.42% per quarter for an all-equity portfolio to 4.28% for an all gold portfolio. This is again attributable to the inverse relationship between the secular trends in gold and equities.

Variance is again non-linear, dropping to 28% of that of an all equity portfolio at a 75% gold allocation. Variance did not drop here as much as it did in the case of the bull market in equities because bear market correlation of 0.0797 was closer to +1.0  compared with the bull market correlation of -0.4131. The Sharpe ratio rose as the allocation to gold rose.

The Effect of Adding Gold in an Equity Secular Bull and Bear Market:

So what would have happened if we had held some gold the whole time from quarter ending 9/28/1990 through 6/30/2011? Look closely at the following chart:

The inverse relationship between the secular trends in gold and equity prices was such that no matter how much gold we add, the expected return is virtually unchanged at approximately 1.91% per quarter! The effect was isolated to variance alone.

Variance is again non-linear dropping to 28% of that of an all equity portfolio at a 63% gold allocation. The correlation between gold and equity returns in the combined bull-bear market measured -0.1939, and was again statistically insignificant. Though the Sharpe ratio was higher than an all equity portfolio when any gold was added, the ratio peaked at around 63% allocation to gold.

Mean-Variance Analysis and the Minimum Variance Frontier:

The preceding three charts depict expected portfolio return and variance at different allocations from 100% equities to 100% gold. If we plot each portfolio allocation as a point where its variance is read on the horizontal axis and its expected return on the vertical axis, the result would be a single line called the minimum variance frontier. We do just that in the following chart, plotting all three scenarios at once.

Starting at the upper right point on the green line and working down and to the left, portfolio variance drops at the expense of expected return as we increasingly add gold. Variance finally reaches the minimum 0.11% at 57% gold and 43% equity known as the minimum variance portfolio (MVP). This part of the curve is called the efficient frontier. All portfolios below the MVP are inefficient because variance increases while expected return declines.

Starting at the lower right point on the red line and working up and to the left, portfolio variance drops while expected return increases as we increasingly add gold. What a sweet deal! This continues until we reach the MVP and the efficient frontier at 75% gold and 25% equity. Note that at 0.30% variance your expected return can be either 2.35% or 4.28% depending on whether your allocation is 50% gold or 100%, respectively.

Now consider the blue line. No matter what gold allocation you use, expected return remains practically constant. But variance is reduced by 72% when allocating 63% of the equity portfolio to gold.

Conclusion:

In this study, we used our conclusions about correlation to advantage by combining gold with equities thereby reducing portfolio variance. And we used our observations about secular price trends to advantage by at least acknowledging that portfolio expected return and variance trade-offs differ significantly between a secular bull and a secular bear market.

Adding some gold to an equity portfolio can reduce portfolio variance. This was a no-brainer in the case of the secular bear market, but came with a trade-off of expected return in the secular bull market scenario.

As with most things in the natural world, timing is everything. But if your investment philosophy disregards consideration for the secular trend, the decision regarding some allocation to gold is a not a question of if but of how much, a problem best determined using optimization methodologies.

So, should you add gold to your portfolio? As one who merges technical and fundamental analysis, I would have to say that the more correct question is whether to add gold today. The answer to that question will have to wait to be addressed in the next article.

Thank you for reading this article. Your comments are welcomed!

Related articles suggested for your reading:

A Technical Analysis of Gold’s Secular Uptrend

Relationship Between Gold, Inflation, and Equities II

Relationship Between Gold, Inflation, and Equities

Footnotes:

1. The proper Sharpe ratio compares excess return per unit of excess risk. I hope that having used Mr. Sharpe’s name in haste, I have not also used it in vain.
2. The correlation of negative 0.4131 is still statistically insignificant at the 99% confidence level, i.e. it is statistically unlikely to be different from zero. The calculated student’s t-statistic of 2.87 is less than the critical student’s t-statistic of 2.97.

Introduction:

In part-1 of this study, we considered the question of whether monthly returns of gold were correlated with those of inflation or equities in the United States of America. We analyzed three decades of monthly returns using two statistical techniques, correlation and regression analysis. Were you surprised by the results?

Here in part-2 we will investigate whether there has been a relationship between the long term price trends of gold, inflation, and equities. But before we do, let us first clearly distinguish between “correlation of returns” and “price trends“.

Definition of Terms:

Sometimes, the words correlation and trend are loosely defined or used interchangeably. But each has a distinct meaning. Two assets may have a strong relationship between their long term price trends, while at the same time having little correlation between their monthly or weekly price returns.

As we discussed in detail in part-1, the statistical definition of the word correlation describes the degree to which the average variation of returns (deviation from the sample average return) between two assets are related. The mathematical formula once again is:

Trend analysis on the other hand examines the movement of an asset’s price, not returns, over a period time, typically months or years. This trend is often obvious by visible inspection on a chart whose x-axis represents time, and y-axis represents price.

Trend analysis seeks to uncover the historical direction and degree (slope) of price movement, and to detect if and when changes occur. Trend is related to the discount rate applied to an asset by market participants in the aggregate. For this reason, the vertical scale must be exponential and not linear so that rates of change become evident.

Prices seldom move in a straight line, preferring instead to ungulate in waves. An uptrend is defined by a series of waves having of higher highs and higher lows, while lower highs and lower lows define a downtrend. Trends can also exhibit serial correlation.

As explained in part-1, the S&P 500 index (SPX) monthly closing prices will act as the proxy for the equity market price. ¹ And gold prices will be represented by the monthly closing US dollar price of one troy ounce. ² Inflation will be defined as a general rise the price level of goods and services due to monetary causes, and will be represented by the U.S. Bureau of Labor Statistics’ Consumer Price Index for All Urban Consumers: All Items (CPI-U). ³

Comparing the Long Term Trend of Gold Prices and Inflation:

The following historical price chart will help determine whether the long term trends of gold, inflation and equity prices are related. The red line represents the inflation index, the green line represents gold prices, and the blue line represents equities, all end-of-month values from January 1979 to June 2011. Note that the vertical axis is plotted logarithmically, representing a proportional (percentage) rate of change in monthly values.

Note: horizontal axis markings represent the end and not the beginning of a year.

During the three decades after the U.S. became the last nation to abandon the gold standard, gold prices rose sharply at first and volatility was high. But by late 1999, gold had declined to a low of $254.80 as the wild price volatility also subsided.

Meanwhile, inflation rose sharply in the 1970’s before setting a slower pace in 1982, and slower still in 1990. The average annual rate of inflation from 1974 through 1979 was approximately 8.8% per year. From 1981 through 1989, the average annual rate dropped to 4.3%. And from 1991 through 1999, the average annual rate dropped to 2.6%.

Since the general price level rose in US dollar terms albeit at a declining rate, gold priced in dollars should have risen also. Therefore this chart does not indicate that the long term trend of gold since 1979 has been related to that of inflation in the United States as represented by the CPI-U.

Consider what happened next. Gold rose steadily at an average annual rate in excess of 16% per year from 2001 through 2009 while the rate of inflation declined to less than 2.5% per year during the same period.

If the uptrend in gold prices since 2001 has been indicative of monetary inflation, and the value of gold is assumed to be constant, then that which cost one dollar in 2000 should cost $5.22 today. This level of inflation should therefore be evidenced in the general price level of goods and services. Though some commodities like oil or cigarettes have risen sharply, the general price level of goods and services as defined by the CPI-U has risen at a much lower rate in U.S. dollars compared with gold.

Comparing the Long Term Trend of Gold and Equity Prices:

Let’s look at the first half of the price chart once again. While gold prices gold were declining to the low of $254.80 1999, equity prices as represented by the S&P 500 index were in a major secular bull market. From 1980 through 1999, the S&P 500 index rose at an average annual rate of 14%. In fact it is clearly evident that gold’s rate of decline steepened within a year after the S&P 500′s climb accelerated in 1995.

The new millennium ushered in a change in the secular trends ⁴ of both gold and equities. When gold began its clearly defined upward trend in 2001, the nearly two decade bull market in equities had already come to an end a year earlier, and a new secular bear market in equities had begun. While equities gyrated through several cyclical bear and bull markets, gold was climbing at an average rate of approximately 16% per year!

Conclusions:

The evidence in the U.S. for the past three decades indicates that the long term trend in gold prices has not been significantly related to inflation, but rather to the long term trend in equities. Gold prices may be affected at least to some degree by the same domestic and international factors which manifest themselves in equity prices, factors which are largely secular in nature.

Factors which encourage investor confidence in paper assets perhaps also make them indifferent to gold. Conversely, in times when acute levels of uncertainty and bear market volatility challenge investors’ confidence, they sell equities and seek to replace their fiat currencies for hard assets, such as gold.

Thank you for reading this article. Your comments are welcomed!

Read part-1 of this study

Related articles suggested for your reading:

A Technical Analysis of Gold’s Secular Uptrend

Adding Gold To An Equity Portfolio

Relationship Between Gold, Inflation, and Equities

Merging Fundamental and Technical Analysis

Footnotes:

1. (Sep. 2011). In Yahoo! Finance. Retrieved from http://finance.yahoo.com/
2. (Sep. 2011). In World Gold Council. Retrieved from http://www.gold.org/
3. (Sep. 2011). In Federal Reserve Bank of St. Louis. Retrieved from http://stlouisfed.org/
4. Secular trends last approximately 9-25 years. Several cyclical trends each approximately 1-3 years in duration can exist within one secular bull or bear trend. The cyclical bear from 2000 until early 2003, followed by the cyclical bull ending late 2007, and the second cyclical bear ending in early 2009 all comprise the same secular bear market which began in 2000 as measured by the S&P 500 index.

Introduction:

No doubt you have often heard it stated as a matter of fact that gold is a hedge against inflation or, what may at first glance seem synonymous, that gold is correlated with inflation. Have you wondered to what degree this may be true, if it is true at all? Perhaps curiosity about gold has been piqued in light of recent record gold prices coupled with economic conditions in general.

In this first of two articles we will assemble and analyze data describing the statistical correlation between the monthly returns of gold, inflation, and equities. And in part-2 of this study we will compare the long term price trends of gold, inflation, and equities.

Scope:

We will consider two specific questions within the context of the U.S. markets. The answers will impact the asset allocation decisions we make when managing an investment portfolio.

1. Have gold returns after the end of the Bretton Woods agreement been correlated with actual inflation or with equity market returns? ¹
2. Has there been a relationship between the long term price trends of gold, inflation, and equities? We will address this question in part-2 of this study.

Definition of Terms:

Part of the problem is the ambiguity that arises due to the different usage of words. So let’s begin by defining some important terms. The S&P 500 index (SPX) monthly closing prices will act as the proxy for the equity market price. ² And gold prices will be represented by the monthly closing US dollar price of one troy ounce. ³

In this study inflation will be defined as a general rise the price level of goods and services due to monetary causes, i.e. due to the supply and demand dynamics of money and not of goods and services. Inflation will be represented by the U.S. Bureau of Labor Statistics’ Consumer Price Index for All Urban Consumers: All Items (CPI-U). ⁴ Though I acknowledge that its adequacy as said measure is often debated, it is none the less what many investors reference when quoting inflation.

By “correlation” I mean the statistical definition of the word, and not its colloquial meanings. Correlation analysis verifies if returns are related, while regression analysis attempts to define the precise mathematical form of this relationship, such as y=mx+b.

Of specific interest is the sample linear correlation coefficient of monthly returns between two assets. This coefficient describes the degree to which the average variation of returns (deviation from the sample average return) between two assets is linearly related. Values range from -1 to +1, where zero means no linear relationship exists, and +/-1 means a perfect linear relationship exists. Mathematically, the correlation coefficient is:

For example, if gold’s monthly return rises above (or declines below) its average value coincident with inflation rising above its average, then the correlation coefficient will be positive (negative). The more closely the variation of returns of one asset is synchronized with the other regardless of scale, ⁶ the closer the correlation coefficient approaches to +/-1. The more random the relationship, then the closer correlation approaches zero.

Scatter plots help visualize the relationship and help determine if it is linear or non-linear. A strong non-linear correlation could result in a low linear correlation coefficient with misleading results. Always keep in mind that correlation does not imply causation, and it should be suspect in the absence of a reasonable basis.
Have Gold Returns Been Correlated With Inflation or Equity Returns?

We will attempt to answer this question by performing two exercises using monthly return data gathered from January 1979 through June 2011:

1. Calculate and analyze the correlation coefficient between gold, the CPI-U, and the SP500.
2. Perform a linear cross-sectional regression analysis, regressing gold against the CPI-U and then against the SP500.

We can do this quite adequately using Microsoft Excel. Note in the following table that we are calculating correlation and regression using monthly price change in percent, as shown in the columns labeled “Gold%”, “CPIU%”, and “SPX%”, rather than using the raw prices.

Over the period from January 1979 through June 2011, correlation (in green) between monthly gold returns and inflation was a paltry 0.1252. Gold and equity returns correlated even less, a miniscule 0.0270. The fact that the “tcalc” values (in red) are all less than the critical value 2.8232, confirms that correlations are indeed all statistically insignificant or likely equal to zero at the 99% confidence level. Incidentally, the correlation between equities and inflation was only 0.0706.

The lack of significant linear correlations between gold, inflation and equity returns does not necessarily mean that there is no correlation at all. After all a strong non-linear correlation could exist, producing low linear correlation values.

The following scatter plots will visually confirm the absence of strong linear and non-linear correlations. Notice the randomness of these patterns – much like a shotgun blast. The first chart regresses gold against inflation and the second regresses gold against equities.

These regression charts can each be summarized as a table sometimes referred to as an ANOVA, or analysis of variance table. Notice in the table below that both coefficients of determination, R², are virtually zero. ⁵ On average, the variation in inflation returns explains 1.57% of the variation in gold returns, leaving more than 98% unexplained by inflation. The explanatory factor between gold and equities is even less, a mere 0.07%.

“The absence of significant correlation in monthly returns between gold and equities is a delectable fact of prime significance when analyzing the diversification role of gold in an investment portfolio’s asset allocation.”

But Have Gold Returns Been Correlated Over Shorter Periods?

What we have examined thus far is the correlation of returns over a secular ⁷ time frame, namely over three decades. But what about cyclical periods of say 1-3 years? A simple way to begin to answer this question is by plotting the data visually as in the following chart.

The red line represents 3-year moving correlation between gold and inflation measured using the scale on the left. Each point on this line is correlation calculated for the preceding three years between monthly gold returns and the percentage change in inflation. The green and dashed blue lines represent gold and equity prices measured using the price scale on the right and are offered as a historical backdrop for the correlation line.

As the high rate of inflation of the 1970’s declined significantly in the early 1980’s, the 3-year  moving correlation between gold and inflation dropped suddenly from a positive .2505 to a negative .3963 in less than one year. This correlation then trended gradually to its highest level at .5222 in 1993 only to again decline sharply to a negative .3164 by late 1996.

Correlation treaded above zero during the cyclical bear then bull markets in equities between 2000 and late 2007. Then by late 2008, correlation rose swiftly to 0.4511 as the second cyclical bear market of the 2000 secular bear market was well underway. I will leave it to the reader to extrapolate a meaningful conclusion from this correlation data.

Conclusions:

The data presented here does not support a significant correlation between monthly gold returns and inflation or equities in the U.S.A. during the three decades since 1979. But the absence of significant correlation in monthly returns between gold and equities is a delectable fact of prime significance when analyzing the diversification role of gold in an investment portfolio’s asset allocation.

In part-2 of this study, we will address the second question namely whether there has been a relationship between the long term price trends of gold, inflation, and equities.

 Thank you for reading this article. Your comments are welcomed!

Read part-2 of this study

Related articles suggested for your reading:

A Technical Analysis of Gold’s Secular Uptrend

Adding Gold To An Equity Portfolio

Relationship Between Gold, Inflation, and Equities II

Merging Fundamental and Technical Analysis

Footnotes:

1. Equity valuation theory and practice includes expected inflation in the required return estimate or discount rate.
2. (Sep. 2011). In Yahoo! Finance. Retrieved from http://finance.yahoo.com/
3. (Sep. 2011). In World Gold Council. Retrieved from http://www.gold.org/
4. (Sep. 2011). In Federal Reserve Bank of St. Louis. Retrieved from http://stlouisfed.org/
5. The coefficient of determination, R-squared, is the percentage of the total variation explained by the regression equation, Y=mX + b. It is equal to SSreg/SStot, the variation explained by the regression as a proportion of the total variation, where SStot = SSreg + SSresid. In a simple (not multiple) linear regression, R-squared is also equal to the square of the correlation coefficient.
6. A hypothetical correlation coefficient of +1 does not mean that the dependent variable moves 1% for a 1% move in the independent variable. The dependent variable may for example be related by a factor or 1.2 and thus moves 1.2% for a 1% move in the independent variable.
7. Secular trends last approximately 9-25 years. Several cyclical trends each approximately 1-3 years in duration can exist within one secular bull or bear trend. The cyclical bear from 2000 until early 2003, followed by the cyclical bull ending late 2007, and the second cyclical bear ending in early 2009 all comprise the same secular bear market which began in 2000 as measured by the S&P 500 index.

The principles discussed in last month’s articles on chart patterns, the overview and the head and shoulders, underlie reversal chart patterns in general. They apply to this article’s topic as well: the double-top reversal pattern. 

Perhaps the most common mistake when identifying patterns is neglecting to consider the preceding trend. This is akin to neglecting the context. To emphasize an important point made in the overview article: 

“Chart patterns are divided into two main groups: reversal patterns which occur at the end of a trend, or continuation patterns which reside within the trend. It logically follows then that a prerequisite to any chart pattern is the existence of a prior trend.” 

A price movement is not a certain chart pattern just because it looks like it, as if we are playing find the horsey in cloud. Analyzing chart patterns involves a critical process of applying a set of criteria drawn from a rationale based on investor psychology. Invariably, the first step is to identify the prior major trend. Let’s look at a real example.


The Prior Trend: 

Figure-1 is a weekly chart of the 2003-2009 S&P500 (SPX). The uptrend preceding the 2007 double-top began in 2003. After leveling out to a more sustainable rate in 2004, the uptrend accelerated again in 2006, forming a minor trend within the four year major trend ¹. The 6-month double-top defined a transitional sideways move, reversing the 2003 cyclical uptrend. 

Figure 1: Major uptrend in S&P500 from 2003 - 2008

After the 2000-2002 bear market, volatility (VIX) settled down below 20%. Then it rose suddenly through resistance in 2007 as the first top was forming. Volatility remained range bound between 20%-30% for the duration of the double-top. In late 2008 volatility spiked to nearly 80%, after nearly a year and a 50% drop since the double-top formation.
 

The Double-Top: 

The double-top is a reversal consolidation pattern very similar to a head and shoulders. As the name suggests, the double-top is comprised of two tops that share the same resistance level. Like most reversal patterns, the double-top occurs both at the top and bottom of a major trend in somewhat mirror images ². It forms almost identically in both individual stocks and indices. 

Figure-2 is weekly chart of SPX in 2007 including volume overlaid with a simple moving average for reference. The wave from March 2007 to point-A set a new bull market high. It was partially retraced by wave A-B on heavier and above average volume. Point-B completes the first top after breaking the minor uptrend. Volatility rose noticeably as described earlier by figure-1. 

Figure 2: Double-Top in S&P500, 2007.

Point-B forms the neckline, a critical support level which, if broken, can become major resistance. This is later evidenced by point-H. 

Wave B-C begins the second top, testing the first ³. Interest waned as evidenced by the lower volume as the 2003 cyclical bull market came to an end. The sell-off from C-D is again accompanied by heavier and above average volume, this time retracing the entire preceding rally. 

Our example finds support near the neckline followed by a brief rally from D-E on lower volume. This rally is not a requirement for the pattern; the breakdown could have happened at point-D. The market falls from point-E towards the neckline as volume increases sharply ⁴.
 

Completing the Pattern: 

The double-top reversal pattern is complete only after the neckline is decidedly breached on a closing basis (point-F). Characteristics of this breach include: 

• the day’s close is decidedly below the neckline ⁵
• heavier and preferably above average volume
• a wide intraday price range (volatility)

The higher highs and higher lows that defined the prior major uptrend have been replaced with a lower high and a lower low from C-D-E through F. An occasional minor bounce on low volume back to neckline-resistance is characteristic, but not a requirement for the pattern.
 

Price Objective: 

The height of the pattern gives us a glimpse into the degree of decline. Measure the price differential between the apex and the neckline, then subtract it from the neckline for the minimum price target. Using volatility to determine the minimum price target is a general principal in technical analysis. 

In figure-2, the pattern is about 129 points tall. Subtracting 129 from neckline gives us a minimum price target of 1304. Price can decline well below the minimum target as our example illustrates in figure-1. The price objective is met by point-G, and then significantly exceeded below point-J.
 

A Closer Look: 

Figure-3 is a daily bar chart of the second top in the SPX as 2007 comes to a close. The second top began on August 16 with a reversal day ⁶. The rally occurred on mostly below average volume. The last two weeks of the rally came on steadily declining and below average volume including October 5 and 9 which set new closing highs in the major uptrend. 

Figure 3: Daily bar chart of the S&P500, 2007

The 2003 bull market ended on an intraday high on October 11, also a reversal day. Volume was the highest in three weeks and above average. The single day’s swing from high to low measured 1.88% of the previous close. The right side of top #2 was marked by several distribution days ⁷ in close proximity.


Conclusion: 

The aforementioned harbingers appeared before the pattern itself was complete. The preponderance of the evidence behooves the trader or investor to take shelter if not take advantage:

• Volatility breaking out during the first top
• Pattern of rising volume in down legs, waning in up legs
• Concentration of distribution days
• Leading stocks breaking out of poorly formed late stage bases and failing

The double-top reversal pattern is very similar to the head and shoulders. It is identified by distinct criteria and characteristics. The consolidation forms over a period of weeks to months, not days, and certainly not intraday. It is a rare but reliable pattern that when spotted, take note, and take action!


Thank you for reading this article. Your comments are welcomed! 

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1. A major uptrend often ends by accelerating away from its mean as if escaping its gravitational hold, only to revert back to the mean. This is sometimes called a blow-off, and is especially evident at the end of a secular trend like 1982-2000.
2. The double bottom is similar to the double-top with a few important differences. Lack of buying pressure is enough to allow prices to move lower. The double bottom usually takes longer to form. A significant increase in volume is needed for a market to break out of the neckline.
3. The action of the second top is often described as “testing” the highs. When the second top slightly exceeds the first, it is often called a “bull trap” (bear trap in a double-bottom), or a false breakout. Sometimes the second top falls just short of the first. The two tops are generally within 3% of each other.
4. In our example, the decline from point-E towards the neckline occurred during the Christmas holiday, skewing the volume lower. Volume rose back above average after the holidays.
5. About three closes or a 3% decline below the neckline are added confirmation.
6. Reversal day: a sharp single day price reversal. In an uptrend, new highs are set intraday, and then close near the bottom of the day’s range. Wider price range and heavier volume increase the probability of a short term trend reversal.
7. Distribution day: a significantly lower close on higher volume. The larger the percentage-decline relative to average volatility and the higher the volume, the more significant the distribution day. Churning is also a type of distribution day marked by higher volume with almost no change in the day’s close.

• Murphy, John J. Technical Analysis of the Financial Markets. Paramus: NYIF, 1999
• Pring, Martin J. Technical Analysis Explained, 3rd ed. McGraw Hill, 1991
• O’Neil, William J. How to Make Money Selling Stocks Short. Hoboken: Wiley, 2005
• Kaufman, Perry J. New Trading Systems and Methods, 4th ed. Hoboken: Wiley, 2005
• Edwards, Robert D. & Magee, John. Technical Analysis of Stock Trends, 7th ed. Boca Raton: St. Lucie, 1997

Myron J. Gordon 1920 - 2010

Introduction:

In part-2 of this series we applied the concept of the present value of all future cash flows in the form of the dividend discount model. A variation of this is called the Gordon Growth Model ¹ which deals with the infinite summation problem more directly.  

The Gordon Growth Model (GGM):  

The summation in the present value model is an infinite geometric series. It can be mathematically transformed ² into what is known as the Gordon Growth Model, or GGM for short. Although cash flow can be represented by several measures, let’s use dividends for illustration purposes.  

The GGM’s inputs are the next period’s cash flow (Div1), an appropriate required rate of return (r), and a dividend growth rate (g). By varying these inputs, we can calculate a minimum/maximum range for V0. 

Variations of the GGM can be constructed by simple substitution. Below are two examples. Et represents earnings, and k is the payout ratio k=Div/E. 

Assumptions and Economic Rationale: 

When using any valuation model, its underlying assumptions, economic rationale, and intended application should be carefully understood. The assumptions behind the GGM include:  

• The dividend is consistent with the firm’s profitability
• The growth rate is constant and sustainable indefinitely ³
• The required rate of return is greater than the growth rate (r > g)
• Price tracks value over the long term Pt = Vt
• Returns are approximately equal to the cost of capital
• The payout ratio is constant, thus the dividend and earnings growth rates are equal
• The growth rate of the firm is in line with or slower than that of GDP ⁴
• The required return and growth rate reflect long term assumptions

The GGM is useful for valuing a mature firm expected to experience stable growth over the long term. The model is also useful in valuing broadly based equity indices. It is simple to use and easy to understand. The GGM is best applied as the mature stage of a multi-stage valuation model.  

A major drawback of the GGM is its extreme sensitivity to input data. Some underlying assumptions may unrealistic, such as a stable growth rate and constant payout ratio. The spread (r – g) is not necessarily constant. The required rate of return can vary over time independently from the growth rate.  

Application Notes:  

Rearranging the GGM explicitly shows that (r-g) represents the dividend yield. Solving for total return (r) reveals its two components: dividend yield  and capital appreciation.  

Total Return = Dividend Yield + Capital Appreciation

The GGM can be used to explain or “justify” the popular Price-Earnings or PE ratio. Note that trailing PE is higher than leading PE by the growth rate, (1+g). The PE ratio is directly proportional to the payout ratio ⁵, and inversely proportional to (r-g). 

  
Let’s Ty An Example: 

Your manager has asked you to determine, based on the GGM, the implied market discount rate for the S&P500 index (SPX) as of yearend 2009. You research data on the SPX and find that the long term SPX dividend growth rate has averaged 5.24% annually. The cash dividend in 2009 was $22.41, and the SPX closed at 1115.10. Rearranging the GGM and solving for required return ⁶: 

Conclusion: 

The Gordon Growth Model is a variation of the basic dividend discount model. It deals directly with the infinite summation problem. The GGM is easy to understand and very versatile in application. But it requires calculating long term parameters to which it is extremely sensitive. 

Thank you for reading this article. Your comments are welcomed!

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1. Also known as the constant growth model. Myron J. Gordon, and Eli Shapiro, 1956. Mr. Gordon, born October 15, 1920, passed away on July 5, 2010.
2. Consult a math book for details. As for me, I will take it on faith.
3. Sustainable growth rate can be approximated by g = ROE x (1-k)
4. Since we are calculating the present value of all future dividends, the growth horizon is infinite. If a growth rate greater than that of the long term GDP were sustainable, the firm would eventually overtake the economy.
5. The sustainable growth rate is a function of the payout ratio. A change in the payout ratio is partially offset by a change in the growth rate. Increasing k in the numerator decreases g, increasing the denominator.
6. Your manager is busy working on your review, so you decide to go a step further and calculate the average implied required return for the past 30 years. I get 8.36% with a standard deviation of 1.41%.

I enjoy learning, and I don’t mind taking the time I need to understand the material. A bowl of gourmet ice cream is to be enjoyed before it melts, but slowly enough to avoid brain freeze.

When I study for the CFA exam, my objectives are to:

• Achieve a high level of confidence of scoring 70% on the exam ¹
• Deepen and broaden my knowledge of the field
• Enjoy the learning process

Strategy:

The learning outcome statements (LOS) are an important tool to scoring well on the exam. Consider the following excerpts from the CFA Institute article, The CFA Program Our Fifth Decade ² .

“The purpose of the LOS is to enhance candidate learning while guiding examination writers…which helps candidates prepare for the exacting standards of the investment profession.” p 5

“The [exam teams] write examination questions and guideline answers within the bounds of the LOS. The COE views the LOS as a contract with the candidates: If candidates can do what LOS indicate, they should be well prepared for the examinations.” p 5

“To be included on the CFA examination, a question must relate directly to one or more curriculum LOS.” p 7


Taking Notes:

Given these insights, I begin by downloading a list of all learning outcome statements for my current level. At the beginning of a reading, I cut and paste each individual LOS onto one sheet of loose leaf paper. As I work through the reading, I note the answers I find to each LOS on its own sheet until I have finished the reading.

These initial loose-leaf notes are somewhat sloppy, disorganized, and hard to read especially later at final review. So, I reorganize and consolidate them into a permanently bound notebook, no more than one notebook for each curriculum volume. This process of transcribe the original notes doubles as my first review of the reading.

I then begin to work the reading’s practice problems with my notes open, supplementing them where necessary. The practice serves as my second review, and tests whether my notes are adequate for the final review later. You can’t review what you don’t understand, so I want to be sure that I understood the reading before moving on to the next one.

After completing all the readings and practice problems, I take a few days to review all my notes again. Then I rework all the practice problems, but this time with notes closed. I grade my answers and review my notes where needed. I am shooting for 80% or better on each reading. It is time for the final review.

The Final Review:

With less than two months remaining to exam day, I turn my attention to building speed, endurance, and familiarity while testing my retention. I simulate test conditions as much as possible, even to the extent of using a similar table and chair as in the actual exam: narrow, rickety, and uncomfortable.

I pull out the mock exams which I have been accumulating from various sources. At first, I will complete just one 3-hour exam at a sitting. Over the next day or two I grade the exam, review my notes, and rework practice problems where I am still having trouble. I keep track of time and score, but I allow myself more than three hours as needed.

Then with only a few weeks left, I begin to take two full 3-hour exams at a sitting, simulating test day. I then grade, review, and rework practice problems as before. My focus changes to scoring well within the time allotted, and refining my test-taking strategy and technique.

My ultimate goal is to achieve a 90% confidence level of scoring at least 70% overall with 15 minutes to spare in each 3-hour session. Believe me, it takes practice. While the process accelerates as test day approaches, the ideal aim is to peak on test day.

T-Minus-One:

I do not study the day before the exam as this only increases anxiety at a time when confidence is crucial. I drive the route to my exam location to make sure there are no surprises, and I eat lunch where I will eat on test day. I gather everything I will need into a brief case and put it in my car. All systems are “Go“. I relax and enjoy the day; the long months of study are over!

Exam Day:

It is time to put my game face on. I have done everything possible to learn the material. I have a simple strategy to attack the test — it will never know what hit it. I am well prepared for a day of fun competition!

If you have tips which you would like to share, your comments are welcomed.

1. Why not put the CFA level-1 quant to use from the start!
2. Robert R. Johnson, PhD, CFA; Jan R. Squires, DBA, CFA; Peter B. Mackey, CFA; Bobby Lamy, PhD, CFA; The CFA Program Our Fifth Decade. http://www.cfainstitute.org/cfaprogram/Documents/the_cfa_program_our_fifth_decade.pdf

In part-1 of this series, we discussed the concept that a stock’s intrinsic value is the present value of its cash flows. Here in part-2 we will introduce the general form of the present value model, and discuss the dividend discount model in more detail. 

The present value model discounts all future cash flows to determine a stock’s present value, V₀, which is its intrinsic value at t=0. Discounting all future cash flows necessitates using the infinity sign (∞) in the summation. The foundational formula for the present value model is: 

 The Dividend Discount Model (DDM): 

One specific measure of cash flow (CF_t) which we can discount is a stock’s dividend stream, hence the name “Dividend Discount Model” or DDM for short. The inputs to the DDM are the time horizon (t), the dividend stream (DIV_t), and an appropriate discount rate ¹ or required rate of return (r). After substituting dividends for cash flow in the above equation, the formula now reads: 

Discounting dividends into infinity presents an obvious problem. One approach to dealing with this problem is to use a two-stage version of the model which changes our time horizon to one that is finite. We will discuss another approach that addresses the infinity problem directly in a subsequent article. 

The Two-Stage DDM: 

A simplified two-stage version of the DDM assumes a finite distribution period of dividends from t=1 to t=n, followed by a resale price, P_n. Essentially this is the present value of an income stream plus a terminal value. You can think of this terminal value as the discounted present value at t=n of all future cash flows into infinity, that is to say from t= n → ∞. 

The dividend stream can be represented as a series of individual dividends [Div₁, Div₂,…,Div_n], or as an initial value (Div₀) and a growth rate of dividends (g). Our basic present value formula can then be expanded into these two forms as follows: 

By varying the inputs to the equation [Div, r, P_n], we can calculate a minimum/maximum range of values. Note that the infinite t=∞ has now become a finite t=n. 

An Example: 

Raytheon Company (RTN:NYSE) paid a dividend of $1.24 in the last fiscal year ². Over the past 5-years, dividends grew at an annualized rate of 9.2%. Based on further analysis, let us say that you have estimated the following variables: 

• The average dividend growth rate of 9.2% is sustainable for another 5 years.
• The fair sale price of RTN will be approximately $65/share at the end of the 5-years.
• Your required rate of return is 15%

Plugging these variables into the two stage formula above, you calculate the intrinsic value as follows: 

The present value of the dividend stream is $5.33 and of the terminal value is $32.32, or $37.65 total. Paying less than this for RTN will add a margin of safety. Note how that the terminal value is a large portion of the present value in this example.

 

“Rational men, when they buy stocks…would never pay more than the present worth of the expected future dividends.” — John Burr Williams ³ 


Economic Rationale: 

When using any valuation model, care must be taken in understanding the assumptions and economic rationale behind it and the application for which it is intended. Investors who use the DDM take a long term, non-controlling ownership perspective at the stockholder level. 

Common stockholders have an ownership claim on future cash flows. Dividends are actual distributions made to shareholders, a direct access to a company’s value. Dividends are more stable than earnings over the long term, and are less sensitive to short term fluctuations. 

The DDM may be more appropriate for a mature, historically dividend-paying company with few investment opportunities. The company’s board of directors has established a consistent dividend policy that is related to profitability and that has a constant payout ratio. 

Conclusion: 

The DDM is one example of the present value model where dividends are the cash flow being discounted. In application, this is easier for bonds than it is for stocks where the magnitude and timing of cash flows and the proper discount rate are more difficult to determine. 

In following articles we will present: 

• The Gordon Growth Model which addresses the t=∞ issue
• Free cash flow and residual income applications
• Methods of calculating the terminal value, Pn

Thank you for reading this article. Your comments are welcomed. 

Previous Article — Next Article 

1. The discount rate can be a market rate determined by an analyst for the public, or a personal required rate of return determined by each individual investor. A different rate could be applied for each cash flow period being discounted. Lacking a defensible rationale, our formulas use the same rate in all periods. Determining the discount rate is beyond the scope of this series.
2. Data for RTN compliments of the Charles Schwab equity research website.
3. Williams, John Burr. The Theory of Investment Value. Fraser, 1997: p. 6.
   Originally published in 1938, the full text of the quote is worth presenting.
   “The definition for investment value which we have chosen is in harmony with the time honored method of economic theory, which always begins its investigations by asking, ‘What would men do if they were perfectly rational and self-seeking?’ The answer is that rational men, when they buy stocks and bonds, would never pay more than the present worth of the expected future dividends, or of the expected future coupons and principal; nor could the pay less, assuming perfect competition, with all traders equally well informed.”

We discussed important concepts common to chart patterns in a previous article. Having laid the foundation, we now focus on a very popular reversal pattern called the “Head and Shoulders”. Remember that reversal patterns are found at the end of a major trend, while continuation patterns form within a trend. 

The head and shoulders consolidation is composed of three humps: the apex of the prior rally known as the head, and two lower peaks called shoulders, one on either side of the head. I like to use the abbreviation “SHS” for shoulder-head-shoulder.    

Like most reversal patterns, the SHS occurs both at the top and bottom ¹ of a trend in somewhat mirror images, but with some important differences. It is also worth noting that SHS tend to form differently in individual stocks compared with indices. Our discussion will focus on the SHS topping pattern as it forms primarily in major indices.    

Chart Example:    

Figures 1 and 3 describe a SHS found in the Dow Jones Industrial Average during 2007. The charts plot the weekly closing price on a logarithmic scale below which the weekly volume is overlaid with a simple moving average. Please refer to figure-1 below.   

Figure-1: Dow Jones Industrial Average, 2007

The Left Shoulder:    

The wave which began in March 2007 is partially retraced by A-B on heavier and above average volume. And thus the left shoulder is formed.  Volatility rises noticeably. And many any leading stocks are forming late stage bases which are prone to failure.   

The Head:    

The market begins a new wave B-C and sets a new high for the major uptrend, but interest is waning as evidenced by the lower volume. Point-C sets the apex of the pattern and marks the end of the 2003 cyclical bull market.    

The sell-off from C-D is again accompanied by heavier and above average volume. Unlike A-B, approximately the entire preceding rally is retraced until support is found near the level of the previous low. Volatility remains higher.   

Figure 2: Price and volume action in the head of the pattern.

The daily bar chart in figure-2 allows a closer examination of the price and volume action during the formation of the head. Shown are three significant up days on steadily declining volume. In fact, the highest close on October 9 came on below average volume.    

The intraday high, the apex of the 2003 bull market, occurred on October 11. Volume was the highest in three weeks and well above average. The day’s swing from high to low measured 1.76% of the previous close. However, the day closed well into the lower part of its trading range.    

October 11 was by definition a key reversal day ². The right side of the head was marked by many distribution days ³ in close proximity.    

The Neckline:    

Returning to figure-1, the two lows at points B and D form the neckline, an area of support which will become resistance when broken. Although it is common to draw support/resistance as a line through B-D, I prefer defining support/resistance levels by a horizontal area as shown in figure-1, rather than by a sloping line.    

The Right Shoulder:    

After finding support at point-D, the market rallies from D-E only to falter at resistance usually found near the peak of the left shoulder, well below the head. The market then plummets towards the neckline as volume increases sharply. Sometimes there is a small bounce at neckline support, but this is not a requirement for the pattern.    

In our example, the decline from point-E towards the neckline occurred during the Christmas holiday, skewing the volume lower. Volume was back above average after the holidays.    

Completing the Pattern:    

The SHS reversal pattern is not complete until after the neckline is decidedly breached below point-F on a closing basis. About three closes or a 3% decline below the neckline are added confirmation. On the day the neckline is unmistakably breached,  

• the volume is preferably heavier and above average
• the intraday price range is wide (volatile)
• the day’s close is decidedly below the neckline

The previous rally has been squelched. Three peaks are evident defining the head and shoulders pattern. The higher highs and higher lows characteristic of the prior uptrend have been replaced with a lower high and a lower low from C-D-E through F. Occasionally, there is a minor bounce on low volume back to neckline-resistance, but this is not a requirement for the pattern.    

Price Objective:    

The height of the pattern gives us a glimpse into the degree of the impending decline. Measuring the price differential between the apex and the neckline, and subtracting it from the neckline at point-F yields the minimum price target. ⁴ 

It is important to stress that the price target is a minimum. Prices can and often do move well beyond this target as our example later illustrates. If point-G met the price objective, point-J definitely exceeded it.   

Figure 3: Dow Jones Industrial Average, 2003-2009

The Larger Context:    

Figure-3 shows that our SHS was preceded by a four year uptrend which defined the major uptrend line. Volatility (VXD) settled down mostly below 17%. After mid-2006, a steeper minor uptrend emerged, which was later broken by the decline from the head. The major trend line was not broken until well after the neckline was breached.    

During the left shoulder, volatility rose suddenly through resistance, ranging approximately 15-30% for the duration of the SHS formation. Once it became obvious that this may be more than a 20% correction, volatility spiked to around 70%. But by then we were in the fourth quarter of 2008, nearly a year and some 3000 points after the head and shoulders formation. 

The Warning Signs:    

Harbingers of the bear appeared before the SHS pattern became evident. The preponderance of the evidence should move the trader or investor to take shelter if not take advantage. Warning signs included… 

• The rise in volatility during the left shoulder
• A pattern of rising volume in down legs, declining in up legs
• Increasing distribution days in close proximity
• Leading stocks breaking out of sloppy late stage bases and failing

Conclusion:    

It is worth repeating that the SHS forms over a period of weeks to months. Though many practitioners will point out a SHS on an intraday chart, I do not believe the rationale behind the pattern supports this practice.    

As popular as the SHS pattern is among traders, it occurs less frequently than one might think. During my experience as a trader and an active trader broker, I have heard of more SHS sightings than of Elvis. But it is reliable enough that when you spot it, take notice, and take action!

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1. The bottoming or reverse SHS is similar to the SHS top with a few important differences. The bottom SHS usually takes longer to form. A significant increase in volume is needed for a market to break out of the neckline, while a lack of buying pressure is enough to allow prices to move lower. 
2. Reversal day: A sharp single day price reversal. In an uptrend, a new high is made only to close near the bottom of the day’s range. A wider price range and heavier volume increase the probability of a short term trend reversal. See also 03-12-03, 02-19-04, 03-07-05, 11-29-05, 07-18-06, 03-14-07, 08-16-07, 11-21-08, 02-05-2010, 06-21-2010.
3. Distribution day: A lower close on higher volume. The larger the decline relative to average volatility and the higher the volume, then the more significant the distribution day. (Churning is a type of distribution day marked by almost no change in the day’s close.) See also in 2010 04-06, 04-27, 04-30, 05-04, 05-06, 05-07, 05-14, 05-18, 05-20.
4. Using volatility to determine the minimum price target is a general principal in technical analysis.  

Introduction:

Consider any field of study, and it soon becomes apparent that topics can be categorized in a nested fashion, much like a set of wooden Russian dolls. In the field of technical analysis, chart patterns are one of the bigger dolls inside of the big doll.

Having a detailed understanding of chart patterns will dramatically improve your technical analysis skill regardless of your trading time frame. The interpretation of shorter term price movement will become clearer when taken in light of the larger chart pattern.

The focus of this article is on important chart patterns that occur in major indices. Important chart patterns include but are not limited to:

• Reversal Patterns: Head & Shoulders, Double Top, Triple Top, Saucer, and Island Reversal.
• Continuation Patterns: Flag, Pennant, Wedge, Rectangle, and Triangle.

Up, Down, and Sideways:

Markets trend, either in an uptrend or a downtrend. Within these trends, the market can consolidate, or move sideways because major trends tend not to change abruptly. These sideways transitional periods signal by their very shape either a continuation of the previous trend albeit at a different rate, or a reversal of the previous trend.

Therefore, chart patterns are divided into two main groups: reversal patterns which occur at the end of a trend, or continuation patterns which reside within the trend. Most patterns occur in both uptrends and downtrends as almost mirror images. It logically follows then that a prerequisite to any chart pattern is the existence of a prior trend, and that the prior trend line is broken by the consolidation pattern.

The larger the consolidation pattern in terms of its price action, volume, and time in development, the larger its importance in relation to what is to follow.

Chart Patterns and Sheet Music:

For me, recognizing chart patterns is much like learning sheet music. Both sheet music and chart patterns mark time. The treble-clef is like the price action and bass-clef is like the volume action. Chart patterns are as important to investing as the sound track is to a movie. The music during the final victory scene in the movie Invictus is quite distinguishable from say when someone is about to get wacked in a scene from Jason.

The Rationale: During a consolidation, the forces of supply and demand are relatively in balance. The rationale behind chart formations is tied to actions of institutional-size investors. As they reallocate their portfolios in light of their latest analysis, the size and scope of the resulting distribution and accumulation are such that the prior trend is interrupted, maybe even reversed.

Conclusion:

It is important to understand that these formations occur over a period of many weeks to months. Therefore, chart patterns should be plotted on daily and weekly charts using a logarithmic price scale.

Though many technical analysis practitioners will point out one of these chart patterns on an intraday chart, I question whether the rationale behind some of these patterns supports this practice.

Important chart formations help put shorter term movements into context. The knowledge derived from understanding one formation can be readily applied to others. It behooves us therefore to learn as much detail as possible about each chart pattern, its price and volume action over time.

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