Adding Gold To An Equity Portfolio
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 CPIU 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, meanvariance analysis.
Scope:
As always, let’s first establish the scope of our discussion. We will address two questions:
 How are an equity portfolio’s expected return and risk characteristics affected by adding gold to the mix?
 With respect to question1, 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 twoasset portfolio, such as stocks and bonds, or gold and equities, the portfolio’s expected return is the dollarweighted average of the expected return of each asset. Represented mathematically:
The variance of a twoasset 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 aforementioned 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 longonly 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,
 the SPX bull market data from quarter ending 9/28/1990 through 12/29/2000
 the SPX bear market data from quarter ending 3/30/2001 through 6/30/2011
 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 nonlinear tradeoff between risk and expected return whose benefit diminishes rapidly as the gold allocation approaches 57%.
Incidentally, the Sharpe ratio is higher than an allequity portfolio until gold’s allocation reaches 39%, peaking at approximately 25% gold.
Why was the change in portfolio variance nonlinear? 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 tradeoff becomes between expected return and risk, the more the benefit of diversification increases. But as correlation approaches +1.0, the tradeoff 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 allequity 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 nonlinear, 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 nonlinear 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 bullbear 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.
MeanVariance 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 tradeoffs 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 nobrainer in the case of the secular bear market, but came with a tradeoff 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:
 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.
 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 tstatistic of 2.87 is less than the critical student’s tstatistic of 2.97.
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Other Links to this Post

A Technical Analysis of Gold’s Secular Uptrend  Sargon Y. Zia, merging fundamental analysis with technical analysis — November 2, 2011 @ 10:03 am

Relationship Between Gold, Inflation, and Equities II  Sargon Y. Zia, merging fundamental analysis with technical analysis — November 2, 2011 @ 5:54 pm

Relationship Between Gold, Inflation, and Equities  Sargon Y. Zia, merging fundamental analysis with technical analysis — November 4, 2011 @ 11:24 am
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By RakeMeBack, July 6, 2012 @ 10:45 am
First of all I would like to say fantastic blog!
I had a quick question which I’d like to ask if you do not mind. I was interested to know how you center yourself and clear your mind before writing. I have had a tough time clearing my thoughts in getting my ideas out there. I do enjoy writing however it just seems like the first 10 to 15 minutes are wasted just trying to figure out how to begin. Any recommendations or tips? Thank you!
By Sargon Zia, July 16, 2012 @ 8:44 pm
Thank you for the kind words, and I apologize for taking so long to respond. Work and house guests both required some overtime for a while there. I usually begin an article with a question that I would like to explore the answers to personally. This question then often becomes the beginning of the article and that makes the process of writing flow more for me. The first of the four articles on gold, “Relationship Between Gold, Inflation, and Equities” is a good example of this. Searching for the answers becomes the motivation behind the work of doing the research which later comprises the body of the artcle. Before I begin the actual writing of the article I prepare an outline to gather my thoughts. The writing process can take several major rewrites (no less than three) before I am satsified enough to post. Thank you again for writing me and for your interest in the articles.