All that Glitters is Gold?
University of Groningen Faculty of Economics and Business
MScBA: Finance Master thesis Arend Horst 1479822 October 2010 Instructor: Dr. P.P.M. Smid Abstract
In this master thesis I examine the potential added value of gold in a diversified portfolio. I find that gold does not serve as a hedge for stocks, bonds and real estate. Gold can, however, be a safe haven for stocks and real estate, meaning that the returns on gold are uncorrelated with the returns of those assets in times of a market crash. Through historical simulation of multiple portfolios, I find that allocating a part of the portfolio to gold can increase the performance of the portfolio, but this increase in performance is often not significant.
TABLE OF CONTENTS
Chapter 1 – Introduction ... - 3 -
Chapter 2 - Literature review ... - 3 -
2.1 Returns: gold vs. stocks, bonds and real estate ... - 5 -
2.2 Added value of gold in a portfolio ... - 7 -
Chapter 3 - Data and Methodology ... - 9 -
3.1 Asset selection ... - 10 - 3.2 Portfolio management ... - 11 - 3.3 Performance measurement ... - 11 - 3.4 Data analysis ... - 15 - 3.5 Data description ... - 15 - Chapter 4 – Analysis ... - 16 -
4.1 Gold as a hedge for stocks, bonds and real estate ... - 17 -
4. 2 Gold as a safe haven for stocks, bonds and real estate ... - 18 -
4.3 Performance of portfolios containing gold ... - 19 -
CHAPTER 1 – INTRODUCTION
The following anecdote is a true story. Nearly three years ago, a man I knew decided to sell a part of his investment portfolio, which consisted mostly of stocks. The reason for selling shares was that he wanted to invest in gold bars. The credit crisis had already initiated the plummeting of stock prices, and he wanted to secure his wealth by buying something he believed would stay valuable no matter what. But not only did gold keep its value in those turbulent times, it generated some interesting profits. After a few months, the man could no longer resist taking his profit, and he sold his gold bars. Thereby, without realizing it, he stopped to see gold as a safe haven, which was the initial reason for investing in it, and started to treat it as an investment asset that is expected to generate positive returns. But what is gold actually?
Throughout history, gold has always been seen as a robust store of value. For the last 3.000 years, one ounce of gold has at least been worth 400 leaves of bread. For centuries, the currencies of most countries were attached to the price of gold (Diebold, Husted and Rush [1991]). Gold had and still has an important role in our financial system. Nowadays, gold is also applied in industrial processes, for its resistance to oxidizing corrosion and its ability to conduct electricity. Since 1973, when the U.S.A. untied the link between the U.S. dollar and the gold price, gold became a tradable asset. As an investment class, gold also has some interesting characteristics. Although it does not pay dividends or interest, and it is not involved in any kind of economic activity, it has returns comparable to common stocks, both in value and in volatility (see for example Jaffe [1989] and Davidson et al. [2003]). Also, the price of gold is insulated from the business cycle (Lawrence [2003]), which suggests that the returns on gold might exhibit a very low correlation with the returns of other assets which are more connected to the business cycle. Put differently, gold might be a hedge for other assets. Following Baur and Lucey [2009], I define a hedge as an asset that is on average uncorrelated or negatively correlated with other assets such as stocks or bonds. However, gold may be more than a hedge; it could also be a safe haven. I define a „safe haven‟ as an asset that is uncorrelated or negatively correlated with other assets such as stocks and bonds in extreme market situations. During a market crash such as the recent credit crunch, investors would benefit much more from a safe haven than from a hedge, as a hedge can show positive correlation in extreme market situations as it is only on
average uncorrelated to stocks and bonds.
gold coins or gold bars (World Gold Council [2010]), therefore the price of gold bullion is used as a proxy for gold investment.
The main research question that will be answered in this thesis is whether gold, as a potential safe haven, adds value to a diversified portfolio. To do so, the literature about gold returns is examined to develop expectations about the characteristics of the relation between the returns on gold, stocks, bonds and real estate. This literature is also used to derive appropriate portfolio performance measures. The performance measures that are used are the Sharpe ratio, Jensen‟s alpha, value-at-risk and skewness. These performance measures are used to quantify the potential added value of gold to a portfolio. Portfolio performance is replicated by an historical simulation. To be able to this, I have gathered monthly data on stocks, bonds, real estate and gold prices for the period 1973 – 2010. MSCI and FTSE trackers are used as proxies for stocks, bonds and real estate. I have constructed five different portfolios. These portfolios vary in their allocation to risky assets to represent investors‟ various tolerance for risk, and serve as the benchmark portfolios. For every portfolio and performance measure, the optimal allocation to gold is calculated. The performance of the portfolios including the optimized allocation to gold is compared to their respective benchmark portfolios with respect of the different performance measures. From the dataset, the logarithmic returns, standard deviations and correlations of the five portfolios are calculated, using 3-month treasury bills as a proxy for the risk-free rate. Besides historically simulating the five portfolios, the dataset is also used to calculate the correlations between the returns on gold and the returns on stocks, bonds and real estate. This is done for both the entire sample period as well as crisis periods, to test whether gold is a hedge and a safe haven, respectively.
I find that gold is not a hedge but is a safe haven for stocks, bonds and real estate. Consequently, I find that adding gold to a portfolio can improve the performance of that portfolio, regardless of performance measure or risk attitude. Except for the Sharpe ratio and skewness, the increase in performance is never significant though.
CHAPTER 2 - LITERATURE REVIEW
In this chapter the foundations of this thesis are laid. First I will deal with the characteristics of the relationship between gold and stocks, bonds and real estate and whether gold can be described as a hedge for those assets. Then I will examine the performance of portfolios consisting gold. Each section ends with the formulation of a hypothesis.
2.1 RETURNS: GOLD VS. STOCKS, BONDS AND REAL ESTATE
“According to the Capital Asset Pricing Model (CAPM), two different risk measures are relevant to a portfolio manager – the portfolio‟s total risk and its systematic risk. Total risk is measured by the variance of the portfolio. Systematic risk, also known as market-related risk, is measured by the portfolio‟s beta. As the performance of an institutional portfolio manager is typically evaluated in comparison with that of a market index, a portfolio‟s systematic risk is more pertinent to the manager. The total risk of a portfolio is a function of the correlation coefficients and variances of the different investments that form the portfolio. The key of reducing the total risk of a portfolio by adding gold is the correlation coefficient between the returns on gold and the returns on the other assets present in the portfolio. If the correlation coefficient is low enough; the presence of gold will lower the total risk of the portfolio without decreasing its expected return.” (Chua et al. [1990, p.76])
An asset with a very low correlation with another asset can be seen as a hedge for that asset. Following the definition of a hedge by Baur and Lucey [2009]: an asset that is uncorrelated or negatively correlated with another asset or portfolio on average. The question remains whether gold can be such a hedge for stocks, bonds and real estate. The academic coverage of the relation between the returns on gold and the returns on real estate and bonds is very limited. However, many authors have investigated the relation between the returns on gold and the returns on stocks, although most of this research is U.S. based. Table 1 presents the findings of many authors on either the correlation between the returns on stocks and gold, the beta of gold versus the stock market, or both.
Table 1: Gold as a zero-beta asset, overview of the literature
Author Year Subject Data Correlation* Beta
Herbst 1983
Long-term relationship between gold and US common stocks
1934 - 1976 -0.115
Jaffe 1989
Gold and gold stocks as investments for institutional portfolios
1971 - 1987 0.054 0.09
Chua 1990 Diversifying a portfolio with
Blose 1996
Gold stocks as an alternative to gold bullion as investment instrument
1973 - 1982 -0.127
Smith 2001 Relationship between gold
price and U.S. stock indices 1991 - 2001 -0.052
Lawrence 2003 Difference between gold and
other assets 1975 - 2001 -0.07
Davidson, Faff, Hillier 2003 Gold factor exposure in
international asset pricing 1975 - 1994 0.11
Baryshevsky 2004 Interrelation of gold yield with
other assets yields 1984 - 2004 -0.23
Daly 2005 Added value of gold in a
portfolio 1968 - 2004 -0.01 -0.01
McCown, Zimmerman 2006 Gold as a zero-beta asset 1970 - 2003 -0.166 -0.036
Hillier, Draper, Faff 2006 The investment role of precious
metals in financial markets 1976 - 2004 -0.03 -0.067
Ratner, Klein 2008 Portfolio implications of gold
investment 1975 - 2005 -0.01
Conover, Jensen,
Johnson, Mercer 2009
Benefits of investing in precious metals, direct versus indirect
1973 - 2006 -0.03
Baur, Lucey 2009 Gold as a hedge or safe haven
for stocks and bonds 1995 - 2005 -0.071 -0.11
Baur, McDermott 2010
Gold as a hedge or safe haven for stocks and bonds,
international evidence
1979 - 2009 -0.02
Note: the correlation and beta are between the returns on gold and stocks *) significance is not reported as the authors did not provide this information
crash. They find that gold is indeed a safe haven for stocks. Baur and McDermott [2010] expand the research by Baur and Lucey [2009] to 53 countries, both developed and emerging. They confirm the finding that gold is a safe haven for stocks. However, the evidence is much stronger for developed countries than for emerging countries. In particular, they find that gold acted as a safe haven during the stock market crash of 1987 (Black Monday) and the current financial crisis. Their results for the Asian crisis in 1998 do not indicate a safe haven effect of gold, for any of the markets tested. In general, they conclude that gold has the potential to act as a stabilizing force for the global financial system by reducing losses when it is most needed.
Herbst [1983], Blose [1996], Davidson et al. [2003] and McCown and Zimmerman [2006] have investigated the hedging capabilities of gold against inflation and the stock market, and have found that gold is indeed a hedge for stocks and inflation. Lawrence [2003] explains that gold has hedging capabilities because the price of gold is insulated from the business cycle. This insulation is caused by the large above-ground inventory of gold relative to the supply flow. A sudden surge in gold demand can be quickly and easily met through sales of existing holdings of gold jewelry or other products. The academic coverage of the relationship between gold and stocks is vast. The relationship between bonds and gold is limited, and the relationship between real estate and gold has received little attention. Georgiev et al. [2003] find that direct investment in real estate offers some diversification benefits to an established stock and bond portfolio and Hoesli et al. [2003] investigate the added value of real estate in an institutional portfolio and find that a 15 to 20% allocation to real estate is optimal. The investable universe of real estate has been growing, and in a well diversified portfolio, real estate should be present (Hudson-Wilson et al. [2003]).
Expectations
In general, most authors find that the returns on gold exhibit a very low or negative correlation to the returns on stocks. Baur and Lucey [2009] and Baur and McDermott [2010] found that this characteristic of gold returns makes gold a hedge and a safe haven for stocks. Therefore I expect that gold returns will have a low or negative correlation to stocks, and that gold might increase portfolio performance through diversification benefits.
2.2 ADDED VALUE OF GOLD IN A PORTFOLIO
most risky as portfolio one and the least risky as portfolio four. Portfolio one contains 85% in stocks, 10% in real estate and 5% in treasury bills while portfolio four contains 45% in stocks, 10% in real estate, 35% in government bonds and 10% in treasury bills. To each portfolio, 5 and 10% gold is added and the change in performance is measured by the Sharpe ratio based on geometric buy and hold returns. The Sharpe ratio, originally named the reward-to-risk ratio [Sharpe, 1966, 1994], is a measure of the excess return per unit of risk, where the risk free rate is the benchmark. See the methodology chapter for a more advanced description of the Sharpe ratio. Jaffe [1989] finds that the Sharpe ratio rises for each portfolio when 5% gold is added, and rises even more when 10% gold is added. He concludes that this evidence suggests that adding gold to a portfolio would have boosted portfolio performance in the period 1971-1987.
A closely related study by Chua et al. [1990] extends the data of Jaffe [1989] by 18 months to include the market crash of October 1987 (Black Monday), and divided the test period into two sub periods. The first is from Sep. 1971 until Dec. 1979, the second is from Jan. 1980 until Dec. 1988. They find that although gold outperforms the S&P 500 over the entire period, it has a negative return in the second sub period and is outperformed by the S&P 500, despite Black Monday. Nonetheless, they find that in both sub periods, adding 25% gold bullion to a common stock portfolio lowers the portfolio‟s total risk while it increases its average return. They do not explicitly use the Sharpe ratio; however, a reduced total risk combined with an increased return would increase the Sharpe ratio. Hillier et al. [2006] investigate the role of precious metals in financial markets for the period 1976 until 2004. They use gold, silver and platinum to assess the hedging capabilities of precious metals. All three have a very low correlation with stock returns, and they find that gold is the optimal metal to supplement a stock portfolio. They construct buy and hold portfolio‟s consisting of the stock market index and some percentage of gold, ranging between 1% and 40%. As a performance measure they use the relative change in Sharpe ratio caused by increasing the allocation to gold. They find that the portfolio‟s efficiency measured by the Sharpe ratio initially increases with the amount of gold added, reaching an optimum when approximately 30% is allocated to gold, after which adding more gold results in a gradually declining Sharpe ratio.
Sharpe ratio, they find that except for the last period (2001-2005), portfolios without gold outperformed portfolios with gold. Using Jensen‟s alpha, they find that the benefit of a 5% allocation to gold is both positive and negative depending on the time period. Over longer time periods the benefit of a 5% allocation to gold becomes negligible. So Ratner and Klein [2008] do not only find that gold as a stand-alone investment cannot outperform a diversified stock portfolio, but also that gold does not add value to a portfolio in terms of increased Sharpe ratio or Jensen‟s alpha.
Daly [2005] describes alpha as the holy grail of portfolio managers, and wonders whether gold can add alpha to a portfolio. Alpha is a measure of excess return relative to the expected equilibrium return based on the security market line. Daly [2005] uses the S&P 500 as a benchmark and adds 10% and 20% gold to the benchmark portfolio. For his portfolio analysis he uses 10, 20 and 30 year periods of buy and hold returns. He finds that although gold lowers the volatility, it also lowers the returns, leading to a reduced Sharpe ratio and no added value measured by alpha.
Conover et al. [2009] find that adding gold to a portfolio increases return and decreases standard deviation. These results are stronger when there is a restrictive monetary policy rather than an expansive monetary policy.
Expectations
CHAPTER 3 - DATA AND METHODOLOGY
In order to achieve meaningful results, it does not only matter what is analyzed, but also how it is analyzed. In the literature chapter the groundwork was laid concerning what I am going to analyze, and I have briefly shown how other authors have conducted similar research. In this chapter I will elaborate on how the analysis will be done. The chapter is divided in five sections. The first three sections discuss the main choices I had to make regarding the simulation. These choices concerned asset selection, portfolio management strategy and portfolio performance measurement. The fourth section summarizes how the hypotheses will be tested, and the final section covers the data used.
3.1 ASSET SELECTION
Following Jaffe [1989], I have chosen the following asset classes to represent my benchmark portfolio: stocks, bonds and real estate. Since stocks and real estate historically have high returns and high volatility, they are labeled as risky assets. Bonds historically have relatively low returns and low volatility, and are therefore considered a safe asset. To prevent the outcome of the analysis to be determined too much by the specific assets I have chosen, I use indices as proxies for these assets. As the risk free rate, I use the 3 month rate for US Treasury Bills. For gold, I use gold bullion, as gold stocks or gold mutual funds can behave quite differently (Chua, Sick and Woodward [1990]). The choice of using indices has multiple consequences. First, the perspective is more likely to represent a large (institutional) investor than a small private investor, since an internationally diversified portfolio requires a certain amount of capital. Second, since I expect gold to lower volatility and increase returns in part because of the low correlation, the results I find will only be stronger for less diversified portfolios.
Table 2: weights of benchmark portfolios Benchmark Portfolio Allocation to 1 2 3 4 5 Stocks 0.60 0.50 0.40 0.30 0.20 Real Estate 0.30 0.25 0.20 0.15 0.10 Bonds 0.10 0.25 0.40 0.55 0.70
Note: the weights are value-weighted
To every portfolio, varying degrees of gold will be added while the other weights in the portfolio are reduced proportionally. This method allows me to measure the effect gold has on each portfolio. As each portfolio represents a different tolerance for risk, the results are robust for different risk attitudes. More details about the allocation to gold will follow in section 3.4.
3.2 PORTFOLIO MANAGEMENT
After deciding on the asset classes and the subsequent weights in the portfolio, I need to choose how I will maintain the portfolio. Basically I will have to choose to either actively or passively manage the portfolio. Following the authors who previously investigated the performance of portfolios containing gold (Jaffe [1989], Daly [2005], Hillier et al. [2006] and Ratner and Klein [2008]), I choose a passively managed portfolio. A passively managed portfolio is here defined as a portfolio where long-term goals are the main focus, whereas active management seeks to capture short-long-term profits by actively buying and selling assets. The most passive example of a passive strategy is a buy and hold strategy, as the above mentioned authors have used. One important implication of a buy and hold strategy is that the weights of the most risky assets, those with the highest returns, will increase over time. This means that for longer time periods, the portfolio weights may significantly deviate from the original weights. As I have constructed different portfolios to capture the effect of adding gold for different risk attitudes, it is important to ensure that the portfolios‟ risk and return characteristics remain consistent over time (Tokat [2006]). To achieve this, I use a slightly more active strategy. I rebalance the portfolio weights when they deviate more than 5%, relative to their weight, from their intended weights. Note that transaction costs are ignored.
3.3 PERFORMANCE MEASUREMENT
Sharpe ratio
ratio, equals the risk premium per unit of risk, which is quantified by the standard deviation of the portfolio. The Sharpe ratio is defined as (1):
(1)
Where Ri is the return of the asset or portfolio, Rf is the risk free rate and σi is the standard deviation
of the return of the asset or portfolio. All returns are continuously compounded, so the return in period
t, Rt , is calculated as in (2):
(2)
Where ln stands for the natural logarithm, Pt is the price or value at time t and Pt-1 is the price or value one period before t. In the original Sharpe ratio, the returns are calculated arithmetically. However, since arithmetic returns are biased upwards (Ziemba [2005]), I use logarithmic returns.
The standard deviation is calculated as in (3):
(3) Where xi is the observation, x is the average of the observations and n is the amount of observations.
The Sharpe ratio is widely used as a simple and efficient way to rank assets and portfolios, despite the fact that it uses total volatility and thus its use is justified only for diversified portfolios without idiosyncratic risk. The intuition of the Sharpe ratio is simple. When the return of the asset increases, or the volatility decreases, both labeled as “good” events, the Sharpe ratio rises. Therefore, a higher Sharpe ratio is better. Initially this intuition was used to determine whether an asset could add value to a portfolio. When the asset‟s Sharpe ratio was higher than the portfolio‟s Sharpe ratio, the asset should be included. However, this assumes that the correlation of the asset with the portfolio is zero. As this is not a very realistic assumption, the authors mentioned in the literature chapter actually use a Generalized Sharpe Ratio, a definition used by Hodges [1998] and Dowd [2000]. This Generalized Sharpe Ratio is still quite straightforward. To determine whether an asset adds value to a portfolio, the Sharpe ratio of the portfolio before adding the asset is compared to the Sharpe ratio of the portfolio
after the asset has been added. A higher Sharpe ratio indicates that the asset does add value and vice versa. Whenever I calculate the Sharpe ratio, I do this in the generalized form.
Jensen’s alpha
Some of the authors (Daly [2005] and Ratner and Klein [2008]) use a measure of excess return such as Jensen‟s alpha, besides the Sharpe ratio. Jensen‟s alpha, from now on „alpha‟, measures the value added by an asset or a portfolio relative to a benchmark, typically the stock market index. Hence, alpha is the excess return over the expected return based on the benchmark. It is a powerful tool because unlike the Sharpe ratio, it does not require a portfolio to be fully diversified. Therefore, I use this performance measure as a complement to the Sharpe ratio. Alpha,
i, is defined as (4):(4)
Where Ri is the return of the portfolio, Rf is the risk free rate, βi is the beta of the portfolio and Rm is the return of the market. Beta, βi, is calculated as the covariance between the portfolio‟s return and the market return divided by the variance of the market return (5):
(5)
Value-at-Risk
Value-at-risk (VaR) has become a popular risk measure in portfolio performance measurement (Alexander and Baptista [2002], Dowd [1999]). It quantifies the downside risk of a security or portfolio, given a certain confidence interval and time period. I use VaR as a proxy for the portfolio‟s exposure to losses, which is very relevant from an institutional investment perspective.
I use VaR (95%) to define which losses will not be exceeded with 95% certainty. The formula to calculate VaR is based on the fact that the 5%-quartile of a normal distribution always occurs 1.645 standard deviations below its mean. The formula for VaR is in (6):
(6) Where σi is the portfolio‟s standard deviation.
Skewness
When returns are not normally distributed, higher moments of the distribution such as skewness become relevant. The skewness of a distribution refers to its asymmetry. A positive (negative) skew
means that the bulk of the values, including the median, lies to the left (right) of the mean. A normal distribution has zero skewness.
Investors are assumed to have a preference for positive over negative skewness (Kraus, Litzenberger, [1976]). This is based on the three desirable properties for an investor‟s utility function that is (a) positive marginal utility for wealth, i.e., nonsatiety with respect to wealth, (b) decreasing marginal utility for wealth, i.e., risk aversion, and (c) non-increasing absolute risk aversion, i.e., risk assets are not inferior goods. A formal derivation of the utility function can be found in Kraus and Litzenberger [2006, p. 1086].
Thus simultaneously the portfolio is to maximize return and skewness for a given level of risk. Therefore I will also calculate the effect gold has on the skewness of the portfolios‟ returns distribution. The formula for skewness, S, is given in (7):
(7)
Where xi is the observation, x is the average of the observations.
Expectations
3.4 DATA ANALYSIS
So far I have discussed how I have constructed my portfolios and how I will measure their performance. In the next chapter I will test the expectations I formulated in the chapters two and three. These expectations are tested in the following way. I will test whether gold can serve as a hedge for risky assets, by testing the correlations between the returns on gold and the returns on the risky assets. I will test whether gold is a safe haven for stocks, bonds and real estate, by measuring the correlation between the returns on those assets in extreme market situations. To test the influence the addition of gold has on portfolio performance, I will use historical simulation of several portfolios. For each performance measure and portfolio, the percentage of gold added is chosen to optimize that performance measure. By using this procedure, I can distinguish the effect gold can have on each risk attitude and chosen performance measure. This allows me to choose a perspective, and find the optimal allocation to gold for that perspective. It also sheds light on how gold can add value, as I will show the relation between the portfolio performance and the percentage of gold added to a portfolio. In order to maintain broad applicability, I will not synthesize the impact gold has as indicated by each performance measure, as there is no objective way to weigh the specific performance measures. The specific hypotheses that I will test, will be discussed in more detail in chapter 4, analysis and results.
3.5 DATA DESCRIPTION
1 ) 1 ( 12 r 12 * i
Table 3: summary statistics for March 1973 - May 2010
Stocks Bonds Real Estate Gold
Mean return (%) 6.12 0.12 5.64 7.90
Standard deviation (%) 16.28 5.41 23.08 22.08
Highest monthly return (%) 17.28 9.04 25.61 37.67
Lowest monthly return (%) -21.89 -6.11 -26.99 -28.90
Skewness 9.58** 6.86** -0.59 0.91
Note: returns are continuously compounded as in (2) and annualized by standard deviation is calculated as in (3) and annualized by
** indicates significance at the 1% level
CHAPTER 4 – ANALYSIS
In this chapter I will use the methodology developed in chapter 3 to test the expectations I formulated in chapters 2 and 3, by transforming these expectations into testable hypotheses. First, I will investigate whether gold can serve as a hedge or a safe haven for stocks, bonds and real estate. Second, I will assess portfolio performance using various performance measures while adding various percentages of gold.
4.1 GOLD AS A HEDGE FOR STOCKS, BONDS AND REAL ESTATE
I will now test whether gold can serve as a hedge for stocks, bonds and real estate. To test the hedging capabilities of gold, I will use the full sample period, as a hedge is defined as an asset that is on average uncorrelated to, in this case, stocks, bonds and real estate.
In this subsection I will test the following hypotheses:
H. 1: On average, the correlation between the returns on gold and stocks is equal to zero. H. 2: On average, the correlation between the returns on gold and bonds is equal to zero. H. 3: On average, the correlation between the returns on gold and real estate is equal to zero.
The alternative hypotheses are:
H. 1a: On average, the correlation between the returns on gold and stocks is not equal to zero. H. 2a: On average, the correlation between the returns on gold and bonds is not equal to zero. H. 3a: On average, the correlation between the returns on gold and real estate is not equal to zero.
I will test these hypotheses one-sided, because in my definition of a hedge, a negative correlation still qualifies.
Table 4 reports on the correlations between the returns on these assets.
Table 4: correlations between the returns on assets, 1973 - 2010
Stocks Real Estate Bonds Gold
Stocks 1.00
Real Estate 0.78** 1.00
Bonds 0.40 0.05 1.00
Gold 0.10* 0.11* 0.06 1.00
* and ** indicate significance at the 5 and 1% level (one-sided)
correlation would approach +1 and 0 is interpreted as no correlation. The second aspect is the significance of the correlation, which indicates whether a correlation coefficient is significantly different from zero. For example, the correlation coefficient between the returns on stocks and the returns on bonds is 0.40, which indicates a positive correlation between the returns on those assets. However, as this correlation is not significantly different from zero, there is no evidence that this correlation is caused by anything other than pure chance.
In the sample, gold returns have a correlation of 0.10, 0.06 and 0.11 with the returns on stocks, bonds and real estate, respectively. This means that there is a very weak but significant relationship between the returns on gold and the returns on stocks and real estate. Therefore I reject hypothesis 1 and 3, and accept hypothesis 1a and 3a. The returns of gold have low correlation with the returns on stocks and real estate, but this correlation is not zero. The relation between the returns on gold and the returns on bonds is weak, but also insignificant. This means that the correlation between gold and bonds is insignificantly different from zero, I therefore cannot reject hypothesis 2. According to my definition of a hedge, an asset uncorrelated to another asset on average, gold is indeed a hedge for bonds. Gold is not a hedge for stocks and real estate. However, given that the correlation coefficients are so low, gold might be able to decrease a portfolio‟s total risk through diversification.
4. 2 GOLD AS A SAFE HAVEN FOR STOCKS, BONDS AND REAL ESTATE
So far I have showed that the returns on gold on average have a low correlation to the returns on stocks and real estate and that gold is a hedge for bonds. However, to be a safe haven, the returns on gold have to be uncorrelated with the returns of stocks, bonds and real estate in times of a market crash. To test whether gold can serve as a safe haven for stocks, bonds and real estate I will define crisis periods and test for the correlations of gold with the other assets in these periods. Crisis periods are defined endogenously by looking at the data‟s worst 5 percentile returns.
In this subsection I will test the following hypotheses:
H. 4: In extreme market situations, the correlation between the returns on gold and the returns on
stocks is equal to zero.
H. 5: In extreme market situations, the correlation between the returns on gold and the returns on
bonds is equal to zero.
H. 6: In extreme market situations, the correlation between the returns on gold and the returns on
H. 4a: In extreme market situations, the correlation between the returns on gold and the returns on
stocks is not equal to zero.
H. 5a: In extreme market situations, the correlation between the returns on gold and the returns on
bonds is not equal to zero.
H. 6a: In extreme market situations, the correlation between the returns on gold and the returns on
real estate is not equal to zero.
Again, I will test these hypotheses one-sided, because in my definition a safe haven, a negative correlation still qualifies.
Following Baur and Lucey [2009], I define returns at the lowest 5 percentile as extreme market situations. I have sorted the returns on stocks ascending, and I used the worst 5 percent of stocks returns as a proxy for extreme market situations. With this data, I calculated the correlations between the returns on all assets on those days. Note that specific periods cannot be determined with this method, as the worst returns are found all over the thirty year period. The resulting correlations are in table 5.
Table 5: Correlations between assets in case of extreme market situations
Stocks Real Estate Bonds Gold
Stocks 1.00
Real Estate 0.72** 1.00
Bonds -0.46** -0.29 1.00
Gold -0.17 -0.09 0.37* 1.00
* and ** indicate significance at the 5 and 1% level (one-sided)
Note: extreme market situations are defined as losses that are in the lowest 5-percentile
Gold is indeed a safe haven for risky assets such as stocks and real estate, as the returns on gold are negatively correlated to both stocks (-0.17) and real estate (-0.09) in extreme market situations, although these correlations are not significant. Put differently, the correlations are insignificantly different from zero. The intuition is that investors turn to a safe haven when all else fails (Baryshevsky [2004] and Belsky and Alderman [1991]), but they trade in this safe haven in favor of stocks and real estate when those two are bullish. So even though gold by itself is a risky asset with returns and volatility comparable with stocks and real estate, it serves, like bonds, as a safe place to go when the market suffers from large adverse shocks.
4.3 PERFORMANCE OF PORTFOLIOS CONTAINING GOLD
table 2 from chapter 3 for the characteristics of the five portfolios. As described in section 3.4, the allocation to gold is chosen to optimize the performance measure that is used.
I start with the performance of the benchmark portfolios. Each of the performance measures is calculated on a yearly basis. This gives me a total of 37 observations for every performance measure and every portfolio. From these observations, I calculated the average value of the performance measure and its standard deviation. Then, for every performance measure, I calculated what allocation to gold would optimize that performance measure. I did this separately for each portfolio. For these new portfolios I also calculated the average value of the performance measure and its standard deviation. So now I have the performance of portfolios containing optimized allocations to gold, which I can compare to the performance of the benchmark portfolios. The results of this comparison are in table 6.
Table 6: portfolio performance with and without optimized allocation to gold
Portfolio 1 2 3 4 5
Gold (%) P.M. Gold (%) P.M. Gold (%) P.M. Gold (%) P.M. Gold (%) P.M.
Sharpe ratio 0.0 0.19 0.0 0.23 0.0 0.19 0.0 -0.02 0.0 -0.03 5.3 0.21 4.9 0.23 8.6 0.21 21.8 0.12 42.2 0.09 [-0.17] [-0.02] [-0.13] [-2.04]* [-1.87]* Alpha (%) 0.0 -0.60 0.0 1.09 0.0 -1.07 0.0 -4.47 0.0 -3.66 32.5 0.91 0.4 1.66 81.9 1.02 81.8 1. 17 81.8 1.33 [-0.14] [-0.05] [-0.25] [-0.88] [-0.79] VaR (%) 0.0 22.69 0.0 19.23 0.0 15.83 0.0 13.82 0.0 11.78 39.0 19.60 38.0 17.63 11.3 15.26 13.8 12.82 4.6 11.35 [1.37] [0.86] [0.38] [0.88] [0.43] Skewness 0.0 -0.21 0.0 -0.30 0.0 -0.11 0.0 -0.12 0.0 0.05 84.9 0.08 82.2 0.09 81.5 0.12 78.7 0.14 78.5 0.15 [-1.89]* [-2.42]* [-1.38] [-1.43] [-0.53]
Note: P.M. stands for performance measure, t-values are in parentheses * indicates significance at the 5% level
Sharpe ratio
I start with the Sharpe ratio by testing the following hypotheses:
H. 7.1: an optimized allocation to gold does not increase the average Sharpe ratio of portfolio 1 H. 7.2: an optimized allocation to gold does not increase the average Sharpe ratio of portfolio 2. H. 7.3: an optimized allocation to gold does not increase the average Sharpe ratio of portfolio 3. H. 7.4: an optimized allocation to gold does not increase the average Sharpe ratio of portfolio 4. H. 7.5: an optimized allocation to gold does not increase the average Sharpe ratio of portfolio 5.
The alternative hypotheses are:
H. 7.1a: an optimized allocation to gold increases the Sharpe ratio of portfolio 1. H. 7.2a: an optimized allocation to gold increases the Sharpe ratio of portfolio 2. H. 7.3a: an optimized allocation to gold increases the Sharpe ratio of portfolio 3. H. 7.4a: an optimized allocation to gold increases the Sharpe ratio of portfolio 4. H. 7.5a: an optimized allocation to gold increases the Sharpe ratio of portfolio 5.
For every portfolio, the Sharpe ratio increases when an optimized amount of gold is added to the portfolio, although for portfolio 2 the increase is not visible in table 6 due to the rounding. In general, the increase for portfolios 1-3 is rather small. Portfolios 4 and 5, however, exhibit a larger increase in Sharpe ratio. These portfolios have a negative Sharpe ratio without gold, and a positive Sharpe ratio when an optimal percentage of the portfolio is allocated to gold. However, the optimal allocation to gold is quite large for these portfolios, considering that these are supposed to be the safer portfolios. So it seems that safer portfolios, or investors with a low tolerance for risk, benefit the most from adding gold to their portfolio, in terms of an increased Sharpe ratio.
Graph 1 confirms that only a minor allocation to gold could improve the Sharpe ratios of portfolios 1-3, and that allocations of more than 10% decrease the Sharpe ratios for those portfolios. The Sharpe ratios of portfolios 4 and 5 increase with a 25-30% allocation to gold.
Because the increase in Sharpe ratio is insignificant for portfolios 1-3, I cannot reject hypotheses 7.1-7.3. Gold does not significantly increase the Sharpe ratio for those portfolios. The increase in Sharpe ratio is significant for portfolios 4 and 5; therefore I reject hypotheses 7.4 and 7.5. Gold does increase the Sharpe ratio for safer portfolios. This result should be interpreted with caution, however, since the optimal allocation to gold for portfolio 5 is over 40%, which might significantly alter the portfolio‟s risk profile.
Alpha
The effect gold has on a portfolio‟s alpha, is tested through the following hypotheses:
H. 8.1: an optimized allocation to gold does not increase the average alpha of portfolio 1. H. 8.2: an optimized allocation to gold does not increase the average alpha of portfolio 2. H. 8.3: an optimized allocation to gold does not increase the average alpha of portfolio 3. H. 8.4: an optimized allocation to gold does not increase the average alpha of portfolio 4. H. 8.5: an optimized allocation to gold does not increase the average alpha of portfolio 5.
H. 8.1a: an optimized allocation to gold increases the average alpha of portfolio 1. H. 8.2a: an optimized allocation to gold increases the average alpha of portfolio 2. H. 8.3a: an optimized allocation to gold increases the average alpha of portfolio 3. H. 8.4a: an optimized allocation to gold increases the average alpha of portfolio 4. H. 8.5a: an optimized allocation to gold increases the average alpha of portfolio 5.
Except for portfolio 2, all portfolios have a negative alpha when there is no gold in the portfolio, and a positive alpha when an optimal percentage of gold is added. The optimal percentage of gold varies for every portfolio. Portfolio 2 benefits from a very small allocation to gold, while portfolio 1 has the highest alpha when 32.5% gold is added. The portfolios 3-5 have the highest alpha when approximately 82% gold is added. The relation between the allocation to gold and the alpha of the portfolios is visualized is graph 2.
Graph 2 shows that on average, the alpha both increases and decreases with the amount of gold added for portfolios 1-3. For portfolios 4 and 5, the adding of gold generally has a positive effect on the alpha of the portfolios. But as indicated by table 6, for none of the portfolios the increase in alpha is significant. I therefore cannot reject hypotheses 8.1-8.5. Gold does not significantly increase the average alpha for any of the portfolios.
VaR
To examine the effect gold has on a portfolio‟s Value-at-Risk, I test the following hypothesis:
H. 9.3: an optimized allocation to gold does not decrease the average VaR of portfolio 3. H. 9.4: an optimized allocation to gold does not decrease the average VaR of portfolio 4. H. 9.5: an optimized allocation to gold does not decrease the average VaR of portfolio 5.
The alternative hypotheses are:
H. 9.1a: an optimized allocation to gold decreases the average VaR of portfolio 1. H. 9.2a: an optimized allocation to gold decreases the average VaR of portfolio 2. H. 9.3a: an optimized allocation to gold decreases the average VaR of portfolio 3. H. 9.4a: an optimized allocation to gold decreases the average VaR of portfolio 4. H. 9.5a: an optimized allocation to gold decreases the average VaR of portfolio 5.
In general, VaR is lower for less risky portfolios. For every portfolio, the VaR is lower when an optimal percentage of gold is added. This optimal allocation to gold is the largest for portfolio 1, the most risky portfolio, and the smallest for portfolio 5, the safest portfolio. The relation between the allocation to gold and the VaR of the portfolios is visualized is graph 3.
Skewness
The presence of gold also influences the distribution of the returns of the portfolio. I use the following hypothesis to test this effect:
H. 10.1: an optimized allocation to gold does not increase the average skewness of portfolio 1. H. 10.2: an optimized allocation to gold does not increase the average skewness of portfolio 2. H. 10.3: an optimized allocation to gold does not increase the average skewness of portfolio 3. H. 10.4: an optimized allocation to gold does not increase the average skewness of portfolio 4. H. 10.5: an optimized allocation to gold does not increase the average skewness of portfolio 5.
The alternative hypotheses are:
H. 10.1a: an optimized allocation to gold increases the average skewness of portfolio 1. H. 10.2a: an optimized allocation to gold increases the average skewness of portfolio 2. H. 10.3a: an optimized allocation to gold increases the average skewness of portfolio 3. H. 10.4a: an optimized allocation to gold increases the average skewness of portfolio 4. H. 10.5a: an optimized allocation to gold increases the average skewness of portfolio 5.
The allocation to gold that optimizes the portfolio‟s skewness is very high for every portfolio, about 80% on average. Without gold, the portfolios have a negative skewness. Remember that investors‟ are assumed to prefer a positive over a negative skewness. The relation between the allocation to gold and the skewness of the portfolios‟ returns is visualized in graph 4.
CHAPTER 5 – CONCLUSION
In this thesis I have investigated the potential added value of gold in an internationally diversified portfolio consisting of stocks, bonds and real estate. First, it is tested whether gold is a hedge and a safe haven for stocks, bonds and real estate. Second, it is tested how these hedging capabilities of gold could influence the performance of different investment portfolios. Five portfolios were created to represent five different risk attitudes.
I have found that gold does not serve as a hedge for stocks, bonds and real estate. The correlation coefficient is significantly larger than zero. However, the correlation coefficients are low, so although strictly taken gold is not a hedge, it can be useful in a portfolio as a diversification tool. I did find that gold is a safe haven for stocks and real estate. In extreme market situations, the returns on gold are uncorrelated to the returns on stocks and real estate. This might stabilize the value of an investment portfolio in times when this is needed the most.
For the performance measures, I have found that adding gold can increase the Sharpe ratio, although this increase was only significant for the safer portfolios. For none of the portfolios is gold able to significantly add alpha, although there seems to be a positive relation between the allocation to gold and alpha for the safer portfolios. Value at risk is reduced the most for the riskier portfolios, although none of the reductions is significant. The skewness of the returns of the portfolios can significantly be increased by adding gold for the riskier portfolios. However, since the optimal allocation to gold is on average 80% for this performance measure, the potential benefit an institutional investor would have from the increased skewness might not be very realistic, as an 80% allocation to one asset is hardly prudent portfolio diversification, and has severe negative consequences for any other performance measure.
In my research design I have made several choices which affect the applicability of the results. First, by using indices, the portfolios used might represent those owned by large institutional investors rather than small investors. Second, since I used historical simulation to evaluate portfolio performance, the results are based on historic prices and returns. There is no guarantee that the relations that I have found will remain constant over time. Third, I chose four performance measures to evaluate portfolio performance. Each of the performance measures had a different perspective, i.e. added value, risk reduction, risk-reward tradeoff, but there are many more performance measures which may have relations to gold different than the ones I used.
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