Download Do Precious Metals Shine in the Portfolio of a Nordic

Do Precious Metals Shine in the Portfolio of a Nordic Investor?
Milla Ilona Pohjasvaara
Department of Finance and Statistics
Hanken School of Economics
Helsinki
2016
HANKEN SCHOOL OF ECONOMICS
Department of:
Type of work:
Finance and Statistics
Thesis
Author:
Date:
Milla Ilona Pohjasvaara
30.12.2016
Title of thesis:
Do Precious Metals Shine in the Portfolio of a Nordic Investor?
Abstract:
The purpose of this thesis is to investigate if precious metals can act as diversifiers in a
portfolio of Finnish, Swedish, Norwegian, Danish or USA stocks, and if it is better to
follow a buy-and-hold strategy or a more tactical switching strategy to complement a
portfolio of stocks with precious materials. The study covers 28 years of daily data in
the sample period extending from March 1988 to August 2016.
A GARCH type model is used to investigate the diversifying qualities of precious metal
in a portfolio of a country’s stocks. The model is also extended with two dummy
variables to investigate if the diversifying qualities of precious metals are greater in
periods of low market returns and/or in periods of high market volatility. The portfolio
efficiencies of the two different investment strategies are compared to find out which
way of including precious metals in an equity portfolio brings more improvement in
efficiency. In the dynamic switching strategy GARCH(1,1) model is fitted each
consecutive day of the simulation in a rolling window analysis to indicate when to
switch, since in that strategy precious metals are only held in periods of abnormally
high market volatility (when the diversification benefits of precious metals are
proposed to be most pronounced).
The empirical results show that precious metals have potential to act as diversifiers in
a portfolio of Finnish, Swedish, Norwegian, Danish or USA stocks. Most interestingly,
evidence is found of diversification benefits especially in times of extremely high
market volatility for Finland with silver, for Sweden with gold, for both countries with
platinum, and for Norway with gold and platinum. A buy-and-hold strategy is found to
be superior to switching the strategy, which is in line with earlier research.
Keywords:
Precious metals, Gold, Silver, Platinum, Finland, Sweden, Norway, Denmark, USA,
Portfolios simulations, Portfolio theory, Volatility modeling, GARCH model, R,
Rugarch package
CONTENTS
1 INTRODUCTION.......................................................................................1
1.1
Contribution of the Study ..................................................................................... 1
1.2
Aim of the Study ...................................................................................................2
1.3
Restrictions ..........................................................................................................2
1.4
Structure of the Study ..........................................................................................2
2 THEORY.................................................................................................... 3
2.1
Portfolio Theory ...................................................................................................3
2.2
Stressed Markets and Precious Metals in a Portfolio .......................................... 5
3 MODELING STOCK MARKET RETURNS AND VOLATILITY.............. 7
3.1
Efficient Market Hypothesis ................................................................................ 7
3.2
Characterization of Financial Time Series.......................................................... 8
3.3
ARCH models ......................................................................................................9
4 PREVIOUS RESEARCH ......................................................................... 12
4.1
Do precious Metals Shine? An Investment Perspective – Hillier, Draper and
Faff (2006) ........................................................................................................ 12
4.2
Is gold a Hedge or a Safe Haven? An Analysis of Stocks, Bonds and Gold –
Baur and Lucey (2010) ...................................................................................... 13
4.3
Is gold a safe haven? International evidence – Baur and McDermott (2010) .. 13
4.4
Gold and the US dollar: Hedge or haven? – Joy (2011) .................................... 14
4.5
Summary............................................................................................................ 15
5 DATA ....................................................................................................... 16
5.1
Descriptive Statistics .......................................................................................... 16
5.2
Correlation Tables ............................................................................................. 19
6 METHODOLOGY ................................................................................... 25
6.1
Econometric models...........................................................................................25
6.1.1
GARCH (1, 1) with Student-t Innovations.............................................25
6.1.2
Regression Models for Examining Diversification Properties of
Precious Metals ..................................................................................... 27
6.2
Portfolio Simulations with Precious Metals ..................................................... 30
6.3
Implementation of the Switching Strategy ........................................................ 31
7 RESULTS ................................................................................................ 35
7.1
Investment Properties of Precious Metals ......................................................... 35
7.2
Portfolio Simulations with Precious Metals ...................................................... 41
8 DISCUSSION AND CONCLUSION ....................................................... 47
8.1
Discussion of the Results ................................................................................... 47
8.2
Critical Overview of the Results ....................................................................... 49
8.3
Concluding Remarks and Suggestions for Further Research........................... 50
SVENSK SAMMANFATTNING ..................................................................51
REFERENCES ............................................................................................ 58
APPENDICES
Appendix 1
Correlation Structure of Gold, Silver, Platinum, Nordic Indexes and
USA (in $) ...................................................................................................... 60
Appendix 2
Autocorrelation and Heteroscedasticity Tests for Winsorized Return
Series ..............................................................................................................62
Appendix 3
Plots of Country Indexes and Precious Metals (in $) ...........................63
Appendix 4
Plots of Nordic Indexes and Precious Metals (in local currency) ........ 64
Appendix 5
Student-t GARCH (1,1) and Market Model Estimation Results .......... 66
Appendix 6
Autocorrelation and Heteroscedasticity Tests for Regression
Residuals ....................................................................................................... 69
Appendix 7
R Code For Selected Parts of the Analysis ............................................ 71
TABLES
Table 1
Summary of Previous Research ....................................................................... 15
Table 2
Descriptive Statistics – Precious Metals and Country Indexes........................ 18
Table 3
Correlation Table of Precious Metals and Country Indexes – Time periods:
Full Period, Before, During and After Dotcom Crash .................................... 21
Table 4
Correlation Table of Precious Metals and Country Indexes – Time Periods:
During and After Credit Crash .......................................................................22
Table 5
Diversification Properties of Precious Metals for Finland and Sweden, Data
for March 1988–August 2016........................................................................ 38
Table 6
Diversification Properties of Precious Metals for Norway, Denmark and
USA, Data for March 1988–August 2016 ...................................................... 40
Table 7
Finland and Sweden – Relative Efficiency of Financial Portfolios
Incorporating Various Weights of Precious Metals, Data for March 1988–
August 2016 ....................................................................................................43
Table 8
Norway, Denmark and USA – Relative Efficiency of Financial Portfolios
Incorporating Various Weights of Precious Metals, Data for March 1988–
August 2016 ................................................................................................... 46
Table 9
Summary of Simulation Results for Finland, Sweden, Norway, Denmark
and USA ......................................................................................................... 48
A1.1
Correlation Table of Precious Metals and Country Indexes – Time periods:
Full Period, Before, During and After Dotcom Crash ................................... 60
A1.2
Correlation Table of Precious Metals and Country Indexes – Time
periods:During and After Credit Crash .......................................................... 61
A2.1
Results from Ljung-Box and ARCH LM tests on the Winsorized (on 1 %
level) Return Series ........................................................................................62
A5.1
GARCH (1,1) – Finland, Sweden, Norway and Precious Metals ..................... 66
A5.2
GARCH (1,1) – Denmark, USA and Precious Metals ....................................... 67
A5.3
Basic Diversification Properties of Precious Metals, Data for March 1988–
August 2016 ................................................................................................... 68
A6.1
Results from Ljung-Box and ARCH LM tests for Market Model Residuals ... 69
A6.2
Results from Ljung-Box and ARCH LM tests for Modified Market Model
Residuals ........................................................................................................70
FIGURES
Figure 1
A3.1
Implementation of the Switching Strategy – A “for loop” in R ......................32
Plot of Nordic Indexes (in$) ............................................................................63
A3.2
Plot of USA Index and Precious Metals (in $) .................................................63
A4.1
Plot of Finland Index and Precious Metals (in €) ........................................... 64
A4.2
Plot of Sweden Index and Precious Metals (in SK) ........................................ 64
A4.3
Plot of Norway Index and Precious Metals (in NK).........................................65
A4.4
Plot of Denmark Index and Precious Metals (in DK) ......................................65
1
1
INTRODUCTION
Gold has facilitated exchange throughout history. It has a unique role culturally and
socially embedded in our society. It is viewed as a store of value, and has been desired
by humans since the earliest civilizations. Even our “paper money” has a linkage to gold
through the gold standard system that bases the value of currency to physical gold. In
more recent times, the price of gold has seen a sharp appreciation since the beginning
of the current millennium. It has risen to never-before-seen heights, which in turn has
inspired investors all around the world to take an interest in gold, and even other
precious metals.
Another reason for gold’s newfound popularity is its presumed quality as a “safe haven”
asset. Investors believe that gold is an asset that holds, or even increases, its value in
times of market turbulence. Such a quality has become even more coveted following the
most recent severe global financial crisis of 2008 that brought the world’s financial
system to its knees, nearly bringing it down altogether. Baur and Lucey (2010), and
Baur and McDermott (2010) research gold’s statistical qualities, and find evidence of its
hedge and safe haven aspects with respect to the stock markets in various countries. 1 In
a portfolio setting that would imply advantageous diversification qualities for gold, and
possibly for other precious metals as well.
1.1
Contribution of the Study
Earlier studies have found evidence of diversification benefits from allocating a part of
holdings to gold, or even other precious metals (silver and platinum), in a portfolio of
stocks. Regardless, the research on the investment aspects of precious metals in
general, and more specifically when concerning the Nordic markets, is still relatively
scarce. There is no extensive research done in the Finnish, Swedish, Norwegian or
Danish markets, which is a gap this study intends to fill.
Furthermore, gold price has, after rallying since the end of the 20th century, and
reaching its historical peak in 2011, been more or less declining since 2013. Most of the
earlier studies on the diversification aspects of gold have been done before the turning
point of 2011 (before which gold price had been booming for ten years). These relatively
They define a hedge as an asset that is, on average, negatively correlated with the stock market. A safe
haven is defined as an asset that is negatively correlated with the stock market particularly in stressed
market conditions.
1
2
recent changes in gold price are quite different from the vigorous soar upward that has
been seen in history, and that already by its own merit gives cause for an up-to-date
research. Interestingly, the price also seems to have started to sharply climb starting
from the beginning of 2016. The data sample of this study also covers the period of the
financial crisis in 2008 and changes in the stock market since then, which is of interest
due to the proposed safe haven aspects of gold.
This study closely follows the earlier study of Hillier, Draper and Faff (2006). However,
in addition to fresh data and new markets, the most significant contribution of this
study is the implementation of the switching strategy. More specifically, the way to
code the switching strategy is new and unique to this thesis. A closer introduction to the
R code for the switching strategy is presented in Appendix 7.
1.2
Aim of the Study
The aim of this study is to investigate if precious metals can act as diversifiers in a
portfolio of Finnish, Swedish, Norwegian, Danish or USA stocks, and if it is better to
follow a buy-and-hold strategy or a switching strategy.
1.3
Restrictions
This study is restricted to investments in physical gold, silver and platinum. An
alternative way of investing in precious metals would be indirectly via exchange-traded
funds (ETFs).
1.4
Structure of the Study
The remainder of this paper is organized as follows: Section 2 presents the theoretical
reference frame of the study. Section 3 discusses the general concept of modeling stock
market returns and volatility. Section 4 briefly reviews the previous research in the area
of the thesis. Section 5 describes the data and reports the summary statistics. An
overview of the methodology is presented in Section 6. The estimation results of this
study are found in Section 7, and finally Section 8 concludes.
3
2
THEORY
This section provides a view of the investment aspects of precious metals in the light of
basic financial theory. First, the modern portfolio theory is introduced. The purpose is
to provide a theoretical background for assessing the qualities of precious metals in an
investment portfolio. Second, the concept of stressed markets and its connection to the
unique investment aspects of precious metals are briefly discussed.
2.1
Portfolio Theory
The reason why we are interested in the low correlations that precious metals have with
the equity market is the Modern portfolio theory by Harry Markowitz (1952). The
theory states that investors can reduce the risk in their investment portfolios by
choosing assets that are not perfectly correlated with each other. In fact, the less
correlated the individual assets are, the better. The modern portfolio theory has
revolutionized the way in which investors around the world construct their portfolios.
Markowitz assumes that all investors are rational and risk averse, which would mean
that they only accept a higher level of risk if it’s compensated by a higher level of
expected return. Investors aim to maximize a portfolio’s expected return given the
specific level of risk, or instead minimize a portfolio’s risk given a specific level of
expected return. (Markowitz 1952.) The risk in a portfolio is uncertainty over how the
individual assets’ prices will develop in the future. The fact that there exists uncertainty
is of material consequence in the analysis of investors’ rationality. (Markowitz 1991.)
The theory combines logical assumptions and statistical measures and models the
expected return of an asset as a stochastic variable. Therefore, also the return of a
portfolio of assets, that is a weighted sum of the individual assets’ returns, becomes a
stochastic variable. The advantage is, that stochastic variables have an expected value
and a variance. The expected return of a portfolio is given by the following formula
(1)
,
where E(Rp) is the expected return of the portfolio, wi is asset i’s weight in the portfolio
and E(Ri) is asset i’s expected return.
4
Risk is defined as the standard deviation (or variance) of the return. The standard
deviation is a measure of how big the spread around the mean is in a series of
observations. Variance is the standard deviation squared. The variance and standard
deviation of a portfolio are given by following formulas
(2)
,
(3)
,
where σp is the volatility of the portfolio, σp2 is the variance of the portfolio, ρij is the
correlation coefficient between the two variables (asset i and j), σi and σj are the
standards deviations of the respective assets, wi and wj are the weights of the
respective assets in the portfolio.
Diversification
Diversification is a method of minimizing the risk (given the level of expected return)
by investing in several different assets. The development of the prices of the individual
assets in relation to each other is fundamental to the risk of the portfolio. An investor
can, for example, spread her capital to different companies in different industries. A
mixture of stocks, that react differently to outside factors and that move relatively
independent of each other, makes a portfolio’s risk lower and increases the potential of
higher returns. Thus, a diversified portfolio generates a lower risk than its individual
component assets, which in turn leads to a higher risk-adjusted return (Markowitz
1952).
Correlation
Correlation is a measure of strength and direction of a linear relationship between two
variables. The strength of the correlation is measured by the correlation coefficient (ρ).
The coefficient can take on values between 1 and -1, where 0 means that there is no
relationship between the variables. Plus one means a perfect positive relationship and
minus one a perfect negative relationship. Zero value signifies that the variables are not
linearly correlated. What makes, for example, gold an attractive investment alternative
is precisely the low, often negative, correlation with other asset classes, which implies it
could function as a natural hedge.
5
Covariance is another measure of (linear) relationship between two variables. It is
obtained by the following formula
,
(4)
where the covariance (σij) between two variables is calculated as the correlation (ρij)
between the two variables multiplied by the respective variables’ standard deviations
(σiσi).
Diversifiable and Non-Diversifiable Risk
The risk that is left after considerable diversification is called the market risk, or even
systemic risk or non-diversifiable risk. Market risk is dependent on fluctuations of the
market and is thus affected by all the companies in a market. In comparison, the part of
risk that can be diversified away is called unique risk or even company specific risk,
non-systemic risk or diversifiable risk. (Bodie, Kane and Marcus 2008.)
Asset Allocation
Asset allocation decision deals with the issue of how the capital is divided (allocated)
between different asset classes, e.g., stocks, bonds, commodities or real estate.
Diversification, on the other hand, is concerned with how capital is divided within a
particular asset class (for example stocks). Careful allocation can further reduce the
effect of non-diversifiable risk in a portfolio. Although, it is not possible to diversify
away market risk, a portfolio that contains many different markets will be less affected
by the risk of the individual markets. (Bodie et al. 2008.)
2.2
Stressed Markets and Precious Metals in a Portfolio
As the correlation tables in the data section of this study demonstrate, precious metals
have low, occasionally even negative, correlations with countries’ stock market indexes.
That is what implies that they could possibly hedge portfolio risk. A hedge is an asset
that is, on average, negatively correlated with the stock market. A safe haven, on the
other hand, is an asset that is negatively correlated with the stock market especially
during stressed market conditions (economic crisis). (Baur and Lucey 2010.)
It should be noted that a hedge asset is not automatically also a safe haven asset, and
vice versa. It has, however, been proposed in earlier research that gold could, in fact, be
both a hedge and a safe haven against the stock market. That possible double benefit
has attracted attention to precious metals as potential portfolio diversifiers.
6
A lot of research has been done on the evolution of correlations in stressed markets.
Stressed market conditions are defined as times of abnormally high volatility. Longin
and Solnik (1995), Ramchad and Sysmel (1998), Ang and Chen (2002), Campbell,
Forbes, Koedijk and Kofman (2008) and also Büyükşahin and Robe (2014) among
others, find that correlations between financial assets tend to generally increase in
stressed market conditions. That general increase in correlations in turn would imply,
that an asset that is a hedge under normal market conditions, is not necessarily a hedge
under stressed market conditions.
Consequently, a safe haven asset, which is particularly negatively correlated with the
market under turbulent market conditions, would be an especially valuable diversifier
(given that it is also a hedge). This is because portfolio variance is dominated by the
covariance between the financial assets. Therefore, if correlations between the financial
assets in the market in general rise, then also the portfolio variance (risk) increases. It
follows, that any asset that can help correct the increasing risk in stressed market
conditions, by being less correlated with the others, is a valuable one. The hypothesized
safe haven qualities of precious metals, combined with their hedge qualities, make
them particularly attractive in this respect. Some of the earlier research results on the
hedge and/or safe haven qualities of precious metals are more closely introduced in
Section 4 of this thesis.
7
3
MODELING STOCK MARKET RETURNS AND VOLATILITY
This section first presents the efficient market hypothesis, and the concept of a random
walk, since they form the theoretical basis for modeling stock market returns. Then,
some well-known features common to many financial time series are briefly reviewed.
These features are presented, since their occurrence creates the need for volatility
models like the GARCH model. Finally, the ARCH model and the GARCH model are
introduced.
3.1
Efficient Market Hypothesis
The general notion of efficient markets seems to emphasize a lack of return
predictability as the criterion for efficiency. Markets are efficient when all available
information is reflected in the stock prices at that point in time. It follows, that
according to the theory future movements in stock prices cannot be predicted based on
historical information. In efficient markets, the prices fully and instantly reflect all
available relevant information for the pricing of stocks at all times. Thus, according to
the efficient market hypothesis, an investor should not be able to make abnormal
profits using past price data alone. Jensen (1978) formulates this in a more exact way:
“A market is efficient with respect to information set θt if it is impossible to make
economic profits by trading on the basis of information set θt.”
Fama (1970) specifies three forms of market efficiency. The weak conditions for market
efficiency are satisfied when prices reflect all relevant historical information. The semistrong conditions are satisfied when prices reflect all publicly available information.
The strong conditions are satisfied when prices reflect all publicly and privately
available information. If the markets satisfy the weak form market efficiency, stock
prices are said to follow a random walk. In an efficient market, according to Fama
(1965), “stock prices follow random walks and at every point in time actual prices
represent good estimates of intrinsic values”.
According to that, future price changes are only dependent on surprising news (new
information) that cannot be predicted. The best predictor for tomorrow’s price is
therefore today’s price, since we cannot know in advance what news we’ll get in the
future. In the model, today’s price is equal to yesterday’s price plus an added factor
(white noise series) that reflects new information. In a random walk consecutive
8
changes in a stock’s price are independent, identically distributed random variables
(Fama 1965).
The random walk model was for long the standard in modeling stock market prices. It
has later been empirically shown that stock prices do not always follow a random walk.
The existence of market anomalies questions the empirical validity of the efficient
market hypothesis, since they allow for the possibility of arbitrage. Such anomalies are,
for example, calendar anomalies (prices may vary depending on the day, month or
season), momentum effect (stocks that have done well recently will continue to perform
well), and the existence of price bubbles, to name a few.
3.2
Characterization of Financial Time Series
This section reviews some of the qualities related to the volatility of financial time
series. In literature, financial time series are often assumed to be normally distributed
and stationary. Such assumptions are, however, unrealistic.
Heteroscedasticity
The variance of financial time series is not constant throughout the sample. This quality
is called heteroscedasticity. Conditional heteroscedasticity is when volatility of a series
cannot be predicted over any time period (stock prices are often like this). Changes in
volatility are unpredictable and cannot be related to particular events or to for example
cyclical trends, like toy stores every year selling more toys in December. Unconditional
heteroscedasticity means that periods of higher or lower future volatility can be
predicted (cyclical variables for example).
Leptokurtosis
Financial time series are often not normally distributed, and tend to exhibit excess
kurtosis. Such a leptokurtic distribution has a “higher peak” and “fatter tails” than a
normal distribution with the same mean and variance.
Volatility clustering
In the case of financial data, large and small errors tend to occur in clusters, i.e., large
returns are followed by more large returns, and small returns by more small returns.
This is a sign of persistence of volatility shocks, which is a tendency the GARCH model
aims to capture.
9
Autocorrelation
Autocorrelated residuals are common in financial time series estimation. Engle (1982)
developed the ARCH model in an effort to model such qualities. Autocorrelation, or
serial correlation, occurs for example when you have volatility clustering. That is to say,
financial time series often exhibit positive autocorrelation. An increase in one series
leads to an increase in the other. Negative autocorrelation would mean that an increase
in one series would lead to a decrease in the other. Autocorrelation is a measure
between minus one and one. It expresses how much a current value of a time series is
related to the values that came before it.
Asymmetric Volatility Phenomenon
Volatility reacts more to decreasing stock prices than to increasing stock prices. In
other words, stock market volatility tends to raise more in market downturns than in
market upturns. This is a tendency that, for example, the EGARCH model in which the
conditional variance reacts asymmetrically to positive and negative residuals aims to
capture.
Co-movements in Volatility and Spillovers
Markets today seem to be more interdependent due to increased globalization. In other
words, different financial time series have a tendency to co-move. An intense volatility
somewhere in the world, say in a dominant financial market, can also spillover through
to other markets. A volatility spillover by Engle’s (1990) definition means that the
variance of market A at time t-1 affects the variance of market B at time t. The BEKKGARCH model, for example, is defined in a way that allows it to detect volatility
spillovers. In other words, it allows the conditional variances and covariance of two
different stock markets to influence each other.
3.3
ARCH models
ARCH models were developed to model qualities of financial return series discussed in
Section 3.2. They are nonlinear and stationary time series models. In ARCH models,
the parameters of a model describing the conditional variance are estimated from
historical observations of the time series. In these models, historical observations give
information about the variance of the subsequent period.
Robert Engle (1982) first introduced the ARCH (Autoregressive Conditional
Heteroscedasticity) model, and then his student Tim Bollerslew (1986) came up with its
generalized
form
the
GARCH
(Generalized
Autoregressive
Conditional
10
Heteroscedasticity) model. Since then also further variations have been developed on
the basis of these models to try and better capture the different qualities of financial
time series. For example, the previously mentioned EGARCH model that aims to
capture asymmetry in volatility.
ARCH(p)
The ARCH model accounts for the time dependence of variance and volatility
clustering. If the variance of the error terms is in reality not constant, but is assumed
constant in the model, it can cause the standard error estimates to be false. Therefore,
since the variance of the errors in financial data rarely is constant, it is better to use a
model that does not assume constant variance. Also the volatility clustering tendency of
financial data motivates the use of ARCH models. The ARCH model inherently assumes
that the variance of today’s error term is linked to how large previous day’s error terms
were.
Mean model (with ARCH(p) errors) for the returns Rt is defined as follows
(5)
(6)
.
(7)
If the returns are centered, then the parameter μ need not be estimated. In an ARCH(p)
model for {ut} the variance σt2 depends on p lags of ut2. {εt} are i.i.d disturbances with
zero mean and unit variance, and σt is the conditional volatility of the (centered) return
at time t.
GARCH(p,q)
In practise, some problems have emerged with ARCH models. It turns out, that p
(number of lagged squared residuals) often has to be very large for the model to really
be able to efficiently capture the time dependence of volatility. That in turn, makes the
computation more difficult and time-consuming. Furthermore, the positivity constraint
on the coefficients ai becomes harder to maintain when p grows larger.
11
Mean model (with GARCH(p,q) errors for the returns Rt is defined as follows
(8)
(9)
(10)
(11)
.
In a GARCH(p,q) model for {ut} the variance σt2 depends on p lags of ut2 and q lags of
σt2. {εt} are i.i.d disturbances with zero mean and unit variance, and σt is the
conditional volatility of the (centered) return at time t.
The specific model from the GARCH family chosen for this study is the GARCH(1,1)
with Student-t innovations. In the model, an additional shape parameter is added to
the equation for the residual to describe the shape of the distribution (since we use
Student-t innovations). The model is described in greater detail in the methodology
section of this study.
12
4
PREVIOUS RESEARCH
This section presents some of the previous research results on the investment aspects of
precious metals, gold in particular. This thesis, for the most part, closely follows the
methodology used in the Hillier, Draper and Faff study that is introduced first.
4.1
Do precious Metals Shine? An Investment Perspective – Hillier,
Draper and Faff (2006)
Hillier, Draper and Faff investigate the US market (S&P 500) and the EAFE index
(MSCI Europe/Australasia/Far East) in an analysis of daily data in the period of 1976–
2004. They focus on the investment role of gold, silver and platinum and test them in a
portfolio setting. The authors find that all the precious metals have low correlations
with the market, which would suggest potential diversification benefits when held in a
portfolio of US or EAFE index stocks.
They also investigate diversifying capabilities of precious metals specifically in volatile
market conditions and during poor market return periods. Most interestingly, they find
that the data reveals particular hedging capability of precious metals during periods of
abnormally high stock market volatility. That, in turn, implies potential safe haven
qualities. However, during poor market return periods the results show no indication of
added hedging capability of precious metals.
The authors also test two different investment strategies: a buy-and-hold strategy and a
switching strategy. They measure portfolio efficiency as the relative risk-to-reward
ratio, that is introduced in Section 6 of this thesis. In the switching strategy precious
metals are held only in periods of abnormally high stock market volatility, otherwise
the market index is held. In the passive buy-and-hold strategy, a portfolio is
constructed in the beginning of the period with a specified weight in the precious metal,
and the portfolio is not rebalanced during the entire investment period.
The results show that, during the period from 1976 to 2004, financial portfolios with
varying weights of precious metals perform substantially better in comparison to only
holding a standard equity portfolio (the S&P 500 or the EAFE). Gold is found to be the
most useful diversifier and platinum the second best. Also silver provides clear
efficiency gains for the portfolios, although not as large as the other two metals.
13
The passive investment strategy, where the portfolios are not rebalanced, is in most
cases clearly superior to the active one with regular portfolio rebalancing. The authors
suggest, that this could be because the added return performance from precious metals
during high volatility periods is simply not enough to offset the long-term strong
performance of the constant buy-and-hold portfolio.
4.2
Is gold a Hedge or a Safe Haven? An Analysis of Stocks, Bonds and
Gold – Baur and Lucey (2010)
Baur and Lucey search for a constant time-varying relationship between the stock and
bond markets and the gold price in USA, Great Britain and Germany in the period of
1995–2005. They use daily data and test gold as a safe haven and/or a hedge asset in
those markets.
They find that gold, on average, in all the researched markets functions as a hedge
against stocks but not for bonds. Furthermore, gold is also as a safe haven under
extremely negative market conditions for stocks. They show, however, that the safe
haven quality is very brief and only lasts approximately fifteen trading days. According
to them, that could imply that gold is not a safe haven in the long term.
In other words, those investors who hold gold over fifteen trading days after an
extremely negative shock are going to lose money. The authors reason that this could
mean, that investors hold gold under extremely negative market conditions, but then
sell when the market participants regain their trust in the market and the economy
stabilises.
4.3
Is gold a safe haven? International evidence – Baur and McDermott
(2010)
Bauer and McDermott further investigate gold’s safe haven and/or hedge qualities by
extending the research to the developed markets and the largest growing markets. 2
They compare daily, weekly and monthly observations during the period of 1979–2009.
Note, that the period is longer than in Baur and Lucey (2010).
They show that gold is both a hedge and a safe haven during the sample period for the
largest European stock markets and the USA, but not for Australia, Canada, Japan or
Australia, Brazil, Canada, China, France, Germany, India, Italy, Japan, Russia, Switzerland, Great Britain
and Germany.
2
14
the larger emerging markets (Brazil, China, India, Japan and Russia). Particularly
noteworthy is, that they find that gold is a strong safe haven for the most of the
developed markets during the most severe financial crisis period in 2008. Similar
observations cannot, however, be made for the Asian crisis in 1997.
Concerning the optimal choice for data frequency, it should be noted that they get the
most significant research results by using daily data. This is especially true for periods
of extreme negative shocks that occur with the probability of one per cent. They reason,
that this means that investors react to short term and extreme sudden shocks by
seeking safe haven in gold. That could be interpreted as panicked investors hastily
investing their capital after sudden extreme losses to perceived safe haven assets. They
conclude, that a more gradual trend (weekly or daily losses) does not seem to cause
such impulsive behaviour in the investors.
4.4
Gold and the US dollar: Hedge or haven? – Joy (2011)
The hedge qualities of gold have also been investigated in the context of currency
portfolios. Joy investigates if gold is a hedge and/or a safe haven against the US dollar
in the period of 1986–2008. He examines weekly data of dollar pairs to 16 currencies
over a 23-year period with the help of a DCC-GARCH model.
He finds that gold is, on average, negatively correlated with dollar returns, and is thus
indeed a hedge with respect to the US dollar during the period from 1986 to 2008. He
also investigates how this correlation changes through time. He finds that towards the
end of the research period the correlation grows increasingly negative. That means that
towards the end of the period, gold is an even better hedge against the dollar.
He also investigates gold’s safe haven qualities, but does not find evidence in his data
that gold is a safe haven with respect to the dollar. That means that his results show
that gold does not act as a safe haven for extreme losses in the value of the dollar.
O’Connor, Lucey, Batten and Baur (2015) figure one reason for this could be that gold
and the dollar are both considered to act as safe havens, which could, in turn, mean
that they might co-move in times of market distress.
15
4.5
Summary
The table below summarizes the results from the four studies presented in the previous
research section.
Table 1
Summary of Previous Research
Authors
Data
Hillier, Draper
Equity
and Faff
markets:
(2006)
Time period:
Conclusion
USA and
Gold, silver and platinum exhibited
Europe/Australasia/Far
hedging capability, particularly in
East
high volatility periods. Although,
1976–2004
both the buy-and-hold and the
switching
strategies
provided
efficiency gains, the former was
Frequency:
Baur and
Equity
and
Lucey (2010)
bond markets:
Daily
USA, UK and Germany
superior.
Gold is a hedge and a safe haven for
stocks, but not for bonds. The safe
haven quality is found only after
Time period:
Frequency:
Baur and
Equity
McDermott
markets:
(2010)
Time period:
Frequency:
1995–2005
extreme negative shocks and lasts
only approximately 15 days.
Daily
Australia, Brazil,
Gold is both a hedge and a safe
Canada, China, France,
haven for the largest European
Germany, India, Italy,
stock markets and the USA, but not
Japan, Russia,
for Australia, Canada, Japan or the
Switzerland, Great
larger emerging markets. Gold is
Britain and Germany
also a strong safe haven for the
1979–2009
Daily, weekly and
most of the developed markets
during the most severe financial
crisis period in 2008
monthly
Joy (2011)
Exchange
rates:
16 major dollar-paired
Gold is a hedge against the dollar
exchange rates
but not a safe haven. Furthermore,
Time period:
1986–2008
Frequency:
Weekly
gold’s hedge qualities strengthen
towards the end of the research
period.
16
5
DATA
This section introduces the data of the study: five country indexes and three time series
for precious metals. Descriptive statistics and correlation tables are presented for all of
the daily return series in varying currencies in the period 25.3.1988–10.8.2016. The
correlation tables are further divided into three subperiods in order to examine the
time-varying nature of the correlations between the precious metals and the indexes.
5.1
Descriptive Statistics
Nordic Countries, USA and Precious Metals
The data for this study is gathered from Datastream. The raw data set consists of 7410
daily observations in a period extending from 25 March 1988 to 10 August 2016. The
equity markets of the five countries are proxied by Datastream’s own total market
country equity indexes: “TOTMKFN” for Finland, “TOTMKSD” for Sweden,
“TOTMKNW” for Norway, “TOTMKDK” for Denmark and “TOTMKUS” for USA. The
Datastream datatype for the country indexes is RI (return index), and thus accounts for
the re-investment of dividends.
The precious metals, in turn, are represented by the London Gold Bullion Market price
in US$/Troy ounce “GOLDBLN”, the Zurich Silver price in US$/kilogram “SFSILVR”
and London Free Market platinum price in US$/Troy ounce “PLATFRE”. The
Datastream datatype for the precious metal price series is P#S (unpadded price), which
means that Datastream displays “NA” (missing value) on non-trading days rather than
pads with the latest known value. Rows with missing values are omitted which leaves us
with 7016 observations. Then, the raw data is transformed into continuously
compounded returns according to the formula below, where Rt is the return at time t,
Pt is the price at time t, and Pt-1 is the price at time t-1. Thus, in the end the final
dataset consists of 7015 observations for each series.
(12)
Furthermore, the data contains some outliers and is therefore winsorized at the one
percent level. Extreme values are thus replaced with the chosen percentile value from
17
each end. The data is winsorized only at the one percent level in order to avoid altering
the sample too much. Winsorising is chosen rather than trimming (completely
removing the extreme values) to avoid censoring the data.
Table 2 below displays the descriptive statistics for the different series. Finland has the
most volatile return series of the country indexes, with an annualized standard
deviation of 25.46 percent in euros and 26.81 percent in dollars. Furthermore, Finland
has the lowest annualized mean return of 10.45 percent in euros and 9.48 in dollars.
Denmark in Danish krone experienced the highest annual mean return over the sample
period of all the series (13.23 percent). USA has the least volatile country index return
series, with an annualized standard deviation of 16.06 percent.
Out of the precious metals, silver has the most volatile returns, with an annualized
standard deviation of more than 25 percent in all currencies. Silver also has the largest
spread between the highest maximum daily returns and lowest minimum returns in all
currencies, which can be seen as a reflection of the great volatility of the silver return
series. Silver also has the highest annualized mean returns in all currencies
(approximately 8 percent). Gold is the least volatile of the precious metal series, with an
annualized standard deviation of approximately 15 percent in all currencies.
Excess kurtosis is greater than zero for all the series, which is an indication of a
leptokurtic distribution. Skewness is slightly negative for all the series. Negative skew
implies an asymmetrical distribution with a longer tail to the left. In most (but not all)
cases the median is somewhat larger than the mean. The Jarque-Bera test rejects the
null hypothesis of normal distribution (“skewness is zero and excess kurtosis is zero”)
for all the return series.
The results for the autocorrelation and heteroscedasticity tests for the series can be
found in Appendix 2. All the series, in all different currencies, exhibit
heteroscedasticity. Also autocorrelation is found in most of the cases. However, the null
hypothesis of “no autocorrelation” is not rejected for the gold series, when expressed in
euros or Danish krone. The null hypothesis of “no autocorrelation” is also not rejected
for the platinum series, when expressed in euros, Swedish krona, Danish krone or
dollars.
18
Table 2
Descriptive Statistics – Precious Metals and Country Indexes
Return Series
and Currency
Gold
Silver
Platinum
Finland
Sweden
Norway
Denmark
USA
$
€
SK
NK
DK
$
€
SK
NK
DK
$
€
SK
NK
DK
€
$
SK
$
NK
$
DK
$
$
N
Mean
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
7015
0.017
0.020
0.022
0.020
0.019
0.030
0.031
0.035
0.033
0.029
0.015
0.015
0.018
0.018
0.014
0.041
0.038
0.044
0.040
0.048
0.046
0.052
0.052
0.044
Ann.
Mean
4.18
5.06
5.65
5.08
4.71
7.59
7.71
8.70
8.20
7.24
3.83
3.84
4.43
4.48
3.50
10.45
9.48
11.19
10.02
12.11
11.7
13.23
13.02
11.09
Std.
0.915
0.926
0.968
0.947
0.929
1.699
1.627
1.650
1.635
1.634
1.278
1.303
1.319
1.299
1.302
1.604
1.689
1.348
1.571
1.294
1.543
1.012
1.176
1.012
Ann.
Std.
14.52
14.70
15.36
15.04
14.75
26.97
25.82
26.20
25.96
25.93
20.28
20.69
20.95
20.62
20.68
25.46
26.81
21.4
24.95
20.54
24.5
16.07
18.66
16.06
Min
Median
Max
Skew
-2.93
-2.71
-2.77
-2.70
-2.70
-5.39
-5.15
-5.18
-5.14
-5.20
-4.11
-4.07
-3.97
-3.91
-4.07
-4.97
-5.13
-4.11
-4.81
-4.20
-4.97
-3.13
-3.47
-3.09
0.013
0.019
0.032
0.021
0.016
0.047
0.000
0.036
0.031
0.022
0.000
0.028
0.032
0.031
0.026
0.050
0.059
0.065
0.068
0.054
0.068
0.065
0.077
0.057
2.68
2.63
2.81
2.76
2.64
4.99
4.69
4.91
4.82
4.69
3.66
3.62
3.70
3.73
3.62
4.99
5.08
4.05
4.61
3.71
4.54
3.02
3.25
3.24
-0.18
-0.06
0.00
0.02
-0.05
-0.14
-0.13
-0.07
-0.08
-0.13
-0.18
-0.15
-0.08
-0.08
-0.14
-0.09
-0.11
-0.11
-0.16
-0.23
-0.20
-0.17
-0.15
-0.13
Excess
Kurtosis
Statistic
Jarque-Bera
P-value
1.51
0.87
0.81
0.82
0.83
1.37
1.39
1.33
1.33
1.36
1.20
1.00
0.86
0.93
0.99
1.54
1.20
1.26
1.24
1.30
1.43
1.31
0.94
1.60
707
226
193
198
205
576
589
524
523
566
462
317
227
260
309
704
432
482
480
559
646
537
285
768
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
Note: The time series have been transformed into daily continuously compounded returns in percentage form (%) and then winsorized on 1 % level. The annualized
mean is calculated as 252*daily mean. The annualized standard deviation is calculated as √252*daily standard deviation.
19
5.2
Correlation Tables
This section shows the correlations between the five country indexes with the three
precious metals. All the time series for the individual countries are expressed in local
currency. That is to say, instead of expressing everything in dollars. That way, the
possibility of dollar exchange rate affecting the correlations is eliminated (gold is
believed to be a dollar-hedge). The correlation of a specific country index with the
precious metals are presented for both the full sample period (A) and for five shorter
subperiods: “Before Dotcom Crash” (B), “During Dotcom Crash” (C), “After Dotcom
Crash and Before Credit Crash” (D), “During Credit Crash” (E) and “After Credit Crash”
(F).
The division to subperiods is done to examine how the correlation varies through time.
The financial crisis periods are of particular interest, since earlier research has provided
some support to gold’s (and possibly other precious metal’s) hypothesized safe haven
qualities. A safe haven asset is particularly negatively correlated during periods of
financial distress. Such qualities would make an asset an especially useful diversifier.
Table 3 displays periods from A to D, and Table 4 continues with periods from E to F.
The first column in the two tables gathers correlations of the metals with the Finnish
index. The gold return for the full sample period has a zero correlation with the Finnish
index return (0.02). Silver and platinum have slightly higher correlations with the
index but still relatively low (0.11 and 0.10). These correlations remain low throughout
the subperiods from B to F. The correlation between gold and the Finnish index even
turns negative during the Credit crash period (-0.14). In conclusion, the responsiveness
of the precious metal returns to Finnish equity market returns is low. That in turn,
might imply diversification benefits.
The second column in the tables displays correlations of the metals with the Swedish
index. Gold returns are negatively correlated with the Swedish index for the full period
(-0.11). Silver and platinum have zero correlations with the Swedish index (0.03 and
0.00). That would, once again, indicate possible diversification benefits. The
correlations of the metals with the Swedish index seem to be particularly low during the
Dotcom crash (-0.16, -0.04 and -0.09), as well as during the Credit crash (-0.33, 0.00
and -0.05). That might indicate particular hedging capability of precious metals during
periods of abnormally high stock market volatility.
20
The third column in the tables presents the correlations of the metals with the
Norwegian index. First, just as the Finnish and Swedish indexes, also the Norwegian
index has a low correlation with gold (-0.03), silver (0,09) and platinum (0.06) for the
full period. The correlations of the metals with the Norwegian index remain in general
low throughout all the subperiods, but are clearly at their highest during the period
after the Dotcom crash but before the Credit crash (0.21, 0.17, 0.19). The correlation of
gold with the index is at its lowest during the Credit crash period (-0.19).
The fourth column in the tables presents the correlations of the metals with the Danish
index. For the full period, the precious metals correlations with the index are once
again low (0.0, 0.11 and 0.02), and they remain low throughout all the subperiods. As
for the previous countries, the correlation of gold with the index is at its lowest during
the Credit crash period (-0.09). That could, yet again, be hypothesized to be related to
the turmoil of the financial crisis and gold’s proposed safe haven qualities.
Finally, the fifth column in the tables presents the correlations of the metals with the
USA index. Once more, for the full period, the precious metals correlations with the
index are low (-0.05, 0.07 and 0.04), and they remain low throughout all the
subperiods. The correlation of gold with the index is particularly low. It is either zero or
negative throughout all the periods, which might indicate particular hedging capability
of gold with the USA index. The correlation of gold with the index is at its lowest during
the Dotcom crash period (-0.15).
In panel A of Table 3 for the full sample period, it can be observed, that the correlations
between the precious metal returns are high for all the countries (0.27 to 0.53). The
precious metal correlations remain relatively high throughout the subperiods, but seem
to generally clearly drop during the Dotcom crash for all the countries (0.02 to 0.35).
They then start rising again during the period after the Dotcom crash (0.10 to 0.53).
21
Table 3
G
Correlation Table of Precious Metals and Country Indexes – Time periods: Full Period, Before, During and After Dotcom Crash
S
P
Fin
G
S
P
Swe
G
S
P
Nor
1
0.47
0.48
-0.03
G
1
0.31
0.09
S
1
0.06
P
G
S
P
Den
1
0.48
0.43
0.00
G
1
0.27
0.11
S
1
0.02
P
G
S
P
USA
1
0.53
0.45
-0.05
1
0.34
0.07
1
0.04
A. Full sample period, March 1988 – August 2016
G
1
S
0.47
0.47
0.02
G
1
0.31
0.11
S
1
0.10
P
P
Fin
1
1
0.49
0.50
-0.11
G
1
0.32
0.03
S
1
0.00
P
Swe
1
Nor
1
Den
1
USA
1
B. Before Dotcom Crash, March 1988 – February 2000
G
1
S
0.51
0.58
0.11
G
1
0.46
0.09
S
1
0.09
P
P
Fin
1
1
0.53
0.59
0.00
G
1
0.47
0.02
S
1
0.04
P
Swe
1
1
0.52
0.57
0.02
G
1
0.46
0.06
S
1
0.04
P
Nor
1
1
0.52
0.53
0.03
G
1
0.47
0.06
S
1
0.02
P
Den
1
1
0.48
0.47
-0.08
1
0.41
-0.06
1
-0.04
USA
1
C. During Dotcom Crash, March 2000 – October 2002
G
1
S
0.27
0.33
0.04
G
1
0.09
0.09
S
1
0.05
P
P
Fin
1
1
0.32
0.35
-0.16
G
1
0.11
-0.04
S
1
-0.09
P
Swe
1
1
0.25
0.30
0.07
G
1
0.07
0.01
S
1
-0.04
P
Nor
1
1
0.27
0.26
0.04
G
1
0.05
0.04
S
1
-0.02
P
Den
1
1
0.29
0.14
-0.15
1
0.02
-0.03
1
0.00
USA
1
D. After Dotcom Crash and before Credit Crash, November 2002 – September 2007
G
S
P
Fin
1
0.39
0.39
0.12
G
1
0.13
0.12
S
1
0.14
P
1
1
0.40
0.41
0.05
G
1
0.15
0.13
S
1
0.08
P
Swe
Note: “G” = Gold, “S” = Silver and “P” = Platinum.
1
Nor
1
0.41
0.41
0.21
G
1
0.15
0.17
S
1
0.19
P
1
Den
1
0.39
0.33
0.13
G
1
0.10
0.17
S
1
0.05
P
1
USA
1
0.53
0.45
-0.04
1
0.25
0.09
1
0.04
1
22
Table 4
G
Correlation Table of Precious Metals and Country Indexes – Time Periods: During and After Credit Crash
S
P
Fin
G
S
P
Swe
G
S
P
Nor
1
0.39
0.39
-0.19
G
1
0.21
0.12
S
1
0.07
P
G
S
P
Den
1
0.39
0.34
-0.09
G
1
0.16
0.20
S
1
0.04
P
G
S
P
USA
1
0.50
0.50
-0.06
1
0.35
0.18
1
0.10
E. During Credit Crash, October 2007 – October 2009
G
1
S
0.39
0.37
-0.14
G
1
0.22
0.15
S
1
0.11
P
P
Fin
1
1
0.41
0.41
-0.33
G
1
0.23
0.00
S
1
-0.05
P
Swe
1
Nor
1
Den
1
USA
1
F. After Credit Crash, November 2009 – August 2016
G
S
P
Fin
1
0.57
0.50
-0.07
G
1
0.32
0.14
S
1
0.15
P
1
1
0.57
0.53
-0.20
G
1
0.33
0.05
S
1
0.01
P
Swe
Note: “G” = Gold, “S” = Silver and “P” = Platinum.
1
Nor
1
0.56
0.52
-0.18
G
1
0.31
0.08
S
1
0.03
P
1
Den
1
0.57
0.44
-0.05
G
1
0.25
0.10
S
1
0.01
P
1
USA
1
0.64
0.54
0.03
1
0.41
0.20
1
0.15
1
23
Correlations Between the Country Indexes
Appendix 1 presents the correlation structure of gold, silver, platinum and the country
indexes, when all the series are nominated in dollars. The notion behind nominating
everything in a common currency was initially to be able to compare the correlation
structure of the entire dataset in a single table. However, due to the unique (inverse)
relationship of gold and the dollar, all the Nordic country indexes are more correlated
with gold when expressed in dollars. 3 Baur and McDermott (2010) analyse the role of
exchange-rate effects and find that when an index and gold returns are both commonly
nominated in dollars, their co-movement in all market conditions is greater, which
according to them cancels or considerably reduces gold’s safe haven property.
Sweden for example in panel A of Table A1.1 has a positive correlation of 0.08 with gold
when expressed in dollars. The corresponding correlation in panel A of Table 3, where
both are expressed in the Swedish krona is negative (-0.11). Similarly, in panels E and F
in Table A2.2 Sweden has positive correlations with gold (0.11 and 0.15) when
expressed in dollars. That also is in sharp contrast to panel A of Table 4 in Swedish
krona, where the corresponding correlations are very negative (-0.33 and -0.20).
Similar observations can be made for the other Nordic countries with gold. In
summary, the observed increased correlations in the dollar-expressed-correlations are
likely to be driven by changes in the dollar.
All the Nordic indexes in Appendix 1 appear to be more correlated with each other than
with USA, which is to be expected. During the full sample period in panel A, the returns
of the Finnish index are most correlated with the returns of the Swedish index. The
correlation coefficient between the two is 0.72. That is the highest coefficient in the
correlation matrix for the full time-period. This high correlation might also be observed
from Figure A3.1 (in Appendix 3). The Finnish index seems to move very similarly with
the Swedish index. Both indexes also have peaks around the year 2000 (IT-boom).
Interestingly, the country correlations can be generally characterized to be rising over
time (from subperiod B to F). Thus, the country correlations are for the most cases at
their highest during the latest subperiod in panel F. One could speculate, that this could
be due to the most recent severe worldwide economic crisis. All correlations usually
increase during periods of financial distress. In times of such extreme worldwide
Gold has been found to be a dollar-hedge in several previous studies. It is often argued to be one of the
main determinants on the price of gold.
3
24
economic turbulence all different markets around the world often co-move more
strongly. Also correlations between different asset classes increase.
25
6
METHODOLOGY
This section is divided into three subsections: (1) econometric models, (2) portfolio
simulations with precious metals, and (3) implementation of the switching strategy.
The first subsection presents the econometric models used for examining the
diversification properties of precious metals. The next subsection presents the two
different strategies for portfolio simulations: a passive and an active investment
strategy. Also, the relative risk-to-reward ratio is introduced. It is used for measuring
the relative efficiency of financial portfolios in the investment strategies. The final
subsection describes the execution of the switching strategy in greater detail.
6.1
Econometric models
This section describes the GARCH(1,1) model with Student-t innovations for returns,
and also two regression models that are fitted to examine the investment properties of
precious metals. The two last regression models are adapted from Hillier, Draper and
Faff (2006). The estimations are performed with the R rugarch package.
6.1.1
GARCH (1, 1) with Student-t Innovations
Mean model (with GARCH(1,1) errors, whose innovations are t-distributed) for the
returns Rt is defined as follows
(13)
(14)
(15)
where Stv stands for standardized t-distribution with v degrees of freedom (v is the
shape parameter.
26
The conditional variance σt2 of the (index or metal) return at time t, is given by
Equation 14, where a0 > 0, a1 ≥ 0 and b1 ≥ 0. The variance is calculated from the lagterm ut-12 (lagged squared return) and σt-12 (lagged variance). The variance equation
also assigns a weight γ to the long-run average variance rate VL. It follows, that a0 is
defined as the weighted long-run average variance rate. It is the constant term in the
variance equation. The ARCH-coefficient a1 indicates the weight that is given to the
squared return from the previous time-period (day in this case). The GARCHcoefficient b1 indicates the weight of the previous day’s variance.
The persistence of volatility is given by the sum of a1 and b1, where a1 + b1 < 1
(stationarity condition). The model is not stationary if persistence is not less than unity.
If the stationarity condition is met the conditional variance will converge towards the
unconditional variance VL. Persistence measures how quickly (or slowly) the variance
reverts or “decays” toward its long-run mean VL. If persistence is high, it means that
the regression of the variance toward its long-run mean is slow (slow decay). If
persistence is low, it implies rapid decay and a faster reversion of the variance toward
its long-run mean.
In conclusion, since we know financial time series tend to exhibit volatility clustering,
this type of persistence of volatility is modeled with a GARCH(1,1) model in this thesis. 4
By volatility clustering we mean, that the variance of return can sometimes be high or
low for extended periods. The markets respond to new information with price
movements (volatility). Thus, if the markets respond to new information with a large
price movement, the volatility clustering tendency means that such high volatility
events tend to persist a while after the first volatility shock. The first volatility shock is
the sudden increase in volatility, which was brought on by new information.
In a GARCH(1,1) model, a1 measures how much a volatility shock (sudden change) this
period feeds through into next period’s volatility and persistence measures the rate at
which this effect decays over time. In other words, the parameters a1 and b1 measure
how quickly the variance changes with respect to new information and how quickly the
variance estimate reverts to its long-run mean. The restrictions on the model are all
gathered below.
The preliminary tests in Appendix 2 showed significant autocorrelation in most series and
heteroscedasticity in all series and lags that were tested.
4
27
(16)
The model is restricted, since the conditional variance must always be positive and
finite. If it was the case that v ≤ 2, the conditional variance would not be finite. The
restrictions on a0, a1 and b1 are imposed in order to ensure that the conditional
variance is positive.
6.1.2
Regression Models for Examining Diversification Properties of
Precious Metals
The following section presents the two regression models that are fitted to investigate
the diversification properties of gold, silver, and platinum. The first model is a market
model-type regression, and we will from now on refer to it as the “market model” for
simplicity. The second regression, we refer to as the “modified market model”, since
two dummy variables are added into the mean equation.
The two market models are also adjusted for GARCH (1,1) effects in the precious metals
series as was also done in Hillier, Draper and Faff (2006). Also, Corhay and Tourani
(1996) note, that even if there is no intrinsic interest in estimating the conditional
variance, the market model should be estimated by maximum likelihood to produce a
more efficient estimator of the regression parameters. The lack of efficiency in the least
square estimator could lead to spurious conclusions from the empirical study, they
underline.
Market Model – Basic Diversification Properties of Precious Metal Returns
Basic diversification properties of the precious metals are examined by estimating
market model-type regressions of the following form
(17)
(18)
(Adapted from Hillier, Draper and Faff 2006)
where RPM,t is the daily return on the precious metal (PM) and Ri,t is the daily return on
the market index (i) of the country for the corresponding period. The error ut is
modeled by the GARCH(1,1) model with Student-t innovations, that was presented in
28
the previous section. Both αPM and βPM are parameters to be estimated in the
regression. αPM is the constant. βPM is called the precious metal beta. The aim of fitting
the regression is to investigate whether precious metals returns account for variability
in the index returns. A significant negative estimate of βPM would signify clear
diversification properties. That is to say, that it would imply that adding the precious
metal in a portfolio with the index would result in diversification benefits. The more
negative the estimated coefficient, the more impressive diversification benefits can be
achieved from including that metal in a portfolio. Presuming of course that the
estimated βPM is statistically significant. If the estimated precious metal beta was
significantly positive and also very large, that would in turn mean, that the precious
metal in question had absolutely no diversifying properties.
Modified Market Model – Diversification Properties of Precious Metals in Periods of
High Volatility and Poor Stock Returns
In order to capture more subtle diversification properties of precious metals, a
modified version of the Market Model is also fitted. It is otherwise exactly the same as
the market model presented above, but two multiplicative dummy variables are added
into the mean equation. The model is as follows
(19)
(20)
(Adapted from Hillier, Draper and Faff 2006)
where VolDumt is what we will call the “volatility dummy”. It is a multiplicative
dummy variable, that indicates when the GARCH(1, 1) volatility of the country index in
question is more than two standard deviations (2σt) greater than mean market
volatility. In such high volatility periods, it assumes the value of the country index.
Therefore, it is equal to either zero (in low volatility periods), or the index return (in
high volatility periods). The volatility dummy is added in order to be able to
differentiate between periods of high and low stock market volatility. The parameter γ
is to be estimated in the regression. If the estimated parameter is negative (γ < 0) and
statistically significant, it means that we might speculate a precious metal to be more
negatively correlated with the market during periods of abnormally high volatility.
29
In addition to the volatility dummy, the simple market model also grows by an added
“return dummy” (RetDumt). The return dummy is also a multiplicative dummy. It
indicates periods of “low market returns”. That is to say, it assumes the value of the
country index in periods when the market return was more than two standard
deviations (2σt) lower than the mean market return. In other periods (when returns
were “not low”) the return dummy is equal to zero. The parameter η is to be estimated
in the regression. If the estimated parameter is negative (η < 0) and statistically
significant, it means that we might speculate a precious metal to be more negatively
correlated with the market during periods of lower than average returns.
A more exact description of the construction of the volatility and return dummies are
presented in Appendix (A7.1). Furthermore, the R code used for implementing the
modified market model regressions is also provided in the appendix.
Statistical hypotheses
We set the following hypotheses for signs of the estimated parameters for βPM, γ and η
in the market model and the modified market model regressions:
H01: βPM < 0
H02: γ < 0
H03: η < 0.
We hypothesize βPM to be negative which would mean that precious metals give clear
diversifying benefits when held with the index. Furthermore, a significant estimate of
βPM near zero would for example also imply that the metal does not seem to be
correlated with the index, and thus might provide diversifying benefits. A significant
large positive estimated βPM, on the contrary, would be a sign of no diversifying benefits
at all from adding the precious metal in the market portfolio. We also hypothesize γ
and η to be <0. That means that we speculate precious metals to be particularly
negatively correlated with the market during periods of abnormally high volatility
(γ<0) and during “poor” market periods of lower than average returns (η<0).
30
6.2
Portfolio Simulations with Precious Metals
Portfolios containing either gold, silver or platinum are compared with each other to
investigate which metal provides the most diversification benefits. Various weightings
are tested for each metal. The purpose is to investigate which (if any) weighting of each
metal increases portfolio efficiency when compared to the market portfolio.
Relative Reward-to-Risk Ratio
The measure that is used to determine and compare the efficiency of the different
portfolios is the relative risk-to-reward ratio (λ). It is written as follows
(21)
(Adapted from Hillier, Draper and Faff 2006)
where
is the mean return of the hybrid metals-and-market portfolio during the full
sample period and
is the mean return of the market portfolio of the country during
that same period. The variables
and
are, respectively, the mean GARCH(1,1)
variance estimates of the hybrid metals-and-market portfolio and market index return.
The variable λ represents the relative increase or decrease in the reward-to-risk ratio
that a hybrid metals-and-market portfolio can give the investors. A reward-to-risk ratio
greater than one signifies an improvement in the efficiency of a portfolio from adding
precious metals. Similarly, a value less than one means a decline in efficiency. That, in
other words, means that the standard market portfolio (country index) is better than
the hybrid metals-and-market portfolio in such cases.
It is assumed that the inclusion of precious metals in a market index portfolio could
reduce systematic risk, particularly during times of high market volatility. Therefore,
two different investment strategies are compared: a passive “buy-and-hold strategy”
and an active “switching strategy”. In the switching strategy the precious metal is only
included in the portfolio in times of high market volatility.
Buy-and-Hold-Strategy
In the passive strategy the investor constructs a portfolio in the beginning of the period
with a specific weight in precious metal. The weight of the metal stays constant
31
throughout the entire period. The rest of the wealth is in the market portfolio. The
portfolio is held from the beginning until the end of the period without rebalancing the
weight of the metal versus the market portfolio. The relative performance of the
strategy is assessed by calculating λ that measures portfolio efficiency.
For this strategy, the return of the hybrid metals-and-market portfolio in the λ equation
is simply equal to the expected return of the buy-and-hold portfolio. The variance of the
hybrid metals-and-market portfolio, in turn, is the variance of the buy-and-hold
portfolio. In order to be able to calculate the variance of the buy-and-hold portfolio,
GARCH(1,1) is fitted to both the country index (market portfolio) and the metal return
series. The exact implementation of the buy-and-hold strategy is further described in
the Appendix (A7.2). Also R code, that was used to generate the lambdas for the buyand-hold strategy, is presented in the Appendix.
Switching Strategy
The switching strategy is active and thus involves rebalancing the weights of the
investment portfolio. The investors switch to a hybrid metals-and-market portfolio in
periods of high market volatility. Consequently, precious metal is held in the portfolio
only when market volatility is higher than two standard deviations from the mean
standard deviation. Otherwise, the market portfolio is held.
For this strategy, the calculation of the return and variance of the hybrid metals-andmarket portfolio in the λ equation is more complex. The implementation of the
switching strategy is explained more thoroughly in the following section and also in the
Appendix (A7.3).
6.3
Implementation of the Switching Strategy
The switching strategy is implemented on
5 countries * 3 precious metals * 12 portfolio weights
= 180 different permutations.
The run time for the code for one of these permutations (1 weight, 1 country, 1 metal)
was approximately 25 minutes 26 seconds. For each of the five countries, 3 precious
metals * 12 portfolio weights = 36 permutations were tested. The run time for the code
per country was approximately 15 hours 27 minutes. Consequently, it took
32
approximately 77 hours 17 minutes for the code to run for all the 180 different
permutations in the switching strategy. R code for the switching strategy and more
detailed explanations of the calculations are presented in Appendix (A7.3).
In the switching strategy, investors fit the GARCH(1,1) each day to the latest 500
observations of the historical market return series available thus far to obtain a series of
return volatility estimates for the index return. 5 Then, they compare the latest volatility
estimate to the mean of these volatility estimates in order to determine if today is
considered a period of high market volatility. If today’s volatility is more than two
standard deviations above the mean of the GARCH volatility series, investor holds a
weight w % of precious metal in her portfolio and a weight of (1-w) % of the index. If
today is not considered a period of high volatility, then the investor just holds the index.
Thus, the market portfolio is held unless a “switch signal” is obtained from the GARCH
analysis
Figure 1 Implementation of the Switching Strategy – A “for loop” in R
Start
Volatility is “very high”
if the latest value in the
GARCH market
volatility series is more
than 2σ above the
mean market volatility
Rhp and σhp2 for today are
equal to those of a Buyand-Hold portfolio with a
w % holding in precious
metal
Fit GARCH to
[today + past 499]
observations of index and
metal to get today’s σhp
Fit GARCH to [past 500]
observations of the index
return to get a vector of
volatility estimates
Condition true?
Yes
if volatility was
“very high”
No
otherwise
Calculate the Risk-to-Reward ratio
λ=
Rhp/σhp2
for
Ri/σi2
today
Rhp and σhp for today are
equal to Ri and σi2 since we
just held the market
porfolio yesterday
Fit GARCH to
[today + past 499]
observations of
index to get
today’s σhp
Repeat
LOOP
Save today’s λ in the same
place with the earlier ones
5 Only historical observations that would have been available up until that point in time are looked at (to
avoid Look-ahead-Bias).
33
The procedure is performed inside a “for loop” in R for a rolling window of 500
observations. In one iteration inside the loop we first (1) check if we got a “switch
signal” yesterday: “Yes” or “No” in Figure 1. That means that we fit GARCH(1,1) to
index observations [i:(499+i)], where i=(1,2,…,6516). Then we (2) calculate λ to
estimate how well the switching strategy did today.
To clarify, the first day in the simulation that λ is calculated for is observation number
501 that we refer to as “today”, and the preceding 500 observations are considered to be
“in the past” at that point. However, during the first step inside the loop observation
500 is actually the present time for the imagined investor. So, she makes her switch
decision based on all information available up until that day (also the quote for that day
is included). Thus, she does not look at “future” data, but only data that would have
been available to her on that day. She then proceeds to calculate the λ in the following
time step based on information that would have been available to her at that point
(observation 501 has also become known by that point).
The return and variance of the hybrid metals-and-market portfolio in the λ equation
are equal to those of a buy-and-hold portfolio, in case we got a “switch signal” yesterday
and ended up holding w % of metal in our portfolio and the rest in the index. The
return and the variance of the hybrid metals-and-market portfolio in the λ equation
are, in turn, simply equal to those of the index in case we did not get a “switch signal”
yesterday and just decided to hold the market portfolio. This is illustrated by the orange
boxes in the above figure. To obtain coefficients for the λ equation for today
GARCH(1,1) needs to be fitted to both the index and the metal return series. That
means
that
GARCH(1,1)
is
fitted
to
observations
[(i+1):(499+i+1)],
where
i=(1,2,…,6516), of the index and the metal return series.
After obtaining the lambda for the current day, the loop then starts from the beginning.
It runs over and over again (6515 iterations), each time moving one time step ahead. It
stops when the final historical observation in the index return series is reached. Thus,
the loop repeats the analysis 6515 times per 1 permutation (7015 historical observations
- first 500 observations = 6515 iterations). The first 500 observations are considered
“past data” needed to begin the rolling window, and therefore there are only 6515
iterations per loop and not 7015, which is the actual number of observations in the data
set. As Figure 1 illustrates, in a single iteration in the loop GARCH model needs to be
34
fitted three times. Therefore, it is fitted 6515 iterations/loop * 3 GARCH fits/iteration =
19545 times per 1 permutation. And as it follows, GARCH(1,1) is fitted
180 permutations * 19545 fits/permutation
= 3 518 100 times in the entire switching strategy.
Because a rolling window approach is used, the switching strategy lambda can be
tracked through time. In the final stage of the analysis, after omitting outliers and cases
when the lambda is not finite, we take the mean of the lambda vector to get a single
lambda value (1 weight for 1 country with 1 metal). This stage of the analysis is
described in more detail in the Appendix (A7.3 “Calculating the lambda”).
35
7
RESULTS
This section presents the research results. It is divided into two subsections: (1)
investment properties of precious metals, and (2) portfolio simulations with precious
metals. The first subsection discusses the estimation results from the three different
regression models that were fitted in the thesis. The second subsection exhibits the
results from the portfolio simulations with the two different investment strategies.
7.1
Investment Properties of Precious Metals
This section first discusses the results from the GARCH(1,1) regressions. Next, the
results from the market model regression are introduced. Finally, the results from the
modified market model are presented. The two market model-type regressions are
fitted in order to investigate the diversification properties of the precious metals.
Datastream Finland, Sweden, Norway, Denmark, and USA total market return indexes
are the market proxies, and the research period extends from March 1988 to August
2016.
Conditional Variance Properties of the Series – GARCH(1,1) with Student-t Innovations
The student-t distribution is selected after trying the model with different distribution
specifications and comparing the Log-Likelihood values. The results from fitting the
GARCH(1,1) on the return series are presented in the Appendix (A5). The estimated
parameters in the table are as they were presented in the methodology section. The
“shape” parameter denotes the estimated degrees of freedom in the t-distribution.
Robust standard errors are used in reporting the p-values for the estimated parameters.
The regression parameters in the first column of tables A5.1 and A5.2 are nearly all
positive and statistically significant for all the five country indexes, which is a sign that
the model fits the country index return series well. The only exception is USA for which
the a0 coefficient is not significant. Furthermore, the ARCH LM tests on the residuals
of the model indicate that the model fits the country indexes well. That is, the ARCH
LM tests for lags 5 and 7 are insignificant for all the country indexes, which means that
the model has successfully captured the heteroscedasticity in all the five country index
return series up until those lags.
As was earlier noted, in Appendix 2 we can see that all the index return series indeed
exhibit significant heteroscedasticity in the preliminary tests. The five index return
36
series also exhibit significant autocorrelation in the preliminary tests. However, the
results of the Ljung-Box test on the residuals of the model hint, that all the
autocorrelation in the series is perhaps not fully captured by the model. Regardless,
GARCH(1,1) model specification is the most convenient and often still a “good enough”
choice.
The results in the second column of the tables show that the model fits the gold return
series the best. The estimated regression parameters are significant for the gold series
in all five currencies. Also, the Ljung-Box test and the ARCH LM test on the residuals
imply that the model fits the data well. The estimated regression parameters in the
third column are significant also for the silver return series, except when expressed in
euros or Danish krone. In those cases, the a0 and a1 coefficients are statistically
insignificant. In other words, the ARCH-term in the variance equation is insignificant
for silver when expressed in those two currencies.
A different model specification might potentially be more suitable for silver in these
cases (or not), but testing for multiple alternative models is not feasible in the scope of
this thesis, and thus GARCH(1,1) is deemed adequate. Ljung-Box test on the residuals
of the model is significant, which hints that the significant autocorrelation that is found
in the silver series might not be fully captured by the model. The ARCH LM test on the
residuals on the other hand is generally insignificant, which implies that the model fits
well for that part, and has successfully captured the heteroscedasticity in the silver
series.
The final column gathers the results for the platinum series. The estimated parameters
are mostly very significant, which is a good sign. However, a0 is not significant for the
platinum return series when expressed in euros, Swedish krona, Danish krone or
dollars. The Ljung-Box test on the residuals is generally significant for the platinum
series, except when expressed in dollars. A significant Ljung-Box test generally means
that there is leftover autocorrelation (that the model has not succeeded in capturing).
It should be noted, however, that the preliminary tests on the platinum series most
often failed to detect autocorrelation in the series in the first place. The ARCH LM test
is insignificant for platinum expressed in the Swedish krona and dollars, which is good
and implies no leftover heteroscedasticity in those cases. The test is significant for
platinum in euros, Norwegian krone and Danish krone, which implies the opposite.
37
Thus, according to the ARCH LM test, in those cases the model has not succeeded in
capturing all the heteroscedasticity in the platinum series.
Basic Diversification properties of Precious Metals – Market Model Regressions
Table A5.3 summarizes the estimation results for the market model-type regressions.
The estimated parameters in the table are, once again, as they were presented in the
methodology section. The regressions for each country are estimated from data
nominated in local currency.
The most noteworthy result in the market model regression results is that both the
Swedish and the US market indexes have a significant negative elasticity with gold.
That is, the estimated precious metal beta
is negative and statistically significant
at a 1 % significance level. That would imply a negative responsiveness of gold return to
those indexes. Negative responsiveness means potential for gold to act as a particularly
good diversifier, when held in a portfolio of Swedish or US stocks. The magnitude of
these two negative estimated betas is, however, less than 0.1.
Most of the regressions, on the other hand, show significant positive estimated betas
for the market index return. It should be noted though, that these estimated betas are
very low and most certainly substantially less than unity. Most are, in fact, near zero. In
terms of portfolio diversification, a low positive correlation or a correlation near zero is
also desirable, even though a negative correlation would be most optimal. A statistically
significant near zero beta estimate for a metal means that its price does not move with
the index.
Hillier, Draper and Faff (2006) found significant negative estimated precious metal
betas for all three precious metals with the USA index. The results in this thesis, on the
other hand, show a significant negative estimated precious metal beta for USA only
with
gold.
However,
when
Hillier,
Draper
and
Faff
used
the
MSCI
Europe/Australasia/Far East (EAFE) index as an alternative market proxy, all the
estimated precious metal betas were low but significantly positive. That result is more
similar to the results from the market model regressions presented in this thesis. That
is, in the results for the market model regressions in this thesis, nine out of the fifteen
estimated precious metal betas were low but significantly positive, four were
insignificant and two were significantly negative.
38
Diversification properties in High-Volatility/Poor-Performance Periods – Modified
Market Model Regressions
Table 5 below presents the estimation results from the modified market model
regressions for Finland and Sweden. The signs and magnitudes of the estimated
precious metal betas in the table below are similar to what they were in the market
model regressions presented in the previous section. However, without a doubt, the
most fascinating and important finding in the below regressions is that the estimated
coefficient for the volatility dummy is both negative and statistically significant for
Finland with Silver, for Sweden with gold, and for both the countries with platinum.
Table 5
Diversification Properties of Precious Metals for Finland and Sweden, Data for
March 1988–August 2016
Mean Equation:
Variance Equation:
Gold
Silver
Platinum
A: Datastream Finland Total Market Return as market proxy
0.0105
(0.23)
0.0182
(0.21)
0.0134
αPM
0.0258 ***
(0.00)
0.0757 ***
(0.00)
0.0889 ***
βPM
-0.0166
(0.27)
-0.0542
**
(0.01)
-0.1035
***
γ
0.0088
(0.56)
0.0308
(0.17)
0.0182
η
0.0056
**
(0.04)
0.0234
(0.46)
0.0105
a0
0.0513 ***
(0.00)
0.0506
(0.14)
0.0537 ***
a1
0.9440 ***
(0.00)
0.9443 ***
(0.00)
0.9416 ***
b1
7.9882 ***
(0.00)
4.8697 ***
(0.00)
9.6401 ***
Shape
1.22
4.75
2.31
VL
B: Datastream Sweden Total Market Return as market proxy
0.0074
(0.44)
0.0200
(0.19)
0.0123
αPM
-0.0399
***
(0.00)
0.0070
(0.67)
0.0211
βPM
-0.0571 **
(0.02)
-0.0260
(0.50)
-0.1430 ***
γ
-0.0349
(0.16)
0.0389
(0.28)
-0.0075
η
0.0062 **
(0.04)
0.0226 ***
(0.00)
0.0128
a0
0.0479
***
(0.00)
0.0444
***
(0.00)
0.0553 **
a1
0.9466 ***
(0.00)
0.9497 ***
(0.00)
0.9389 ***
b1
9.5909 ***
(0.00)
5.3809 ***
(0.00)
10.3682 ***
Shape
1.15
3.92
2.26
VL
(0.26)
(0.00)
(0.00)
(0.46)
(0.19)
(0.00)
(0.00)
(0.00)
(0.33)
(0.15)
(0.00)
(0.83)
(0.36)
(0.03)
(0.00)
(0.00)
Note: ***, **, * indicate significance levels of 1%, 5%, and 10% respectively. Robust standard errors are
used and the rounded P-values are in the brackets.
The negative volatility coefficient indicates diversifying benefits particularly in volatile
markets (when diversification is most important) from including precious metals in a
stock portfolio. That is an important finding. The detection of a significant volatility
dummy also lends support to the intuition on which the switching strategy is based on.
39
That is to say, that we switch to a hybrid market-to-metals portfolio only in times of
high stock market volatility, since we expect metals to be particularly negatively
correlated in such periods. In fact, the volatility dummy (calculated again and again
each new day in the simulation) functions as the “switch signal” in the switching
strategy in this thesis, as was discussed in Section 6.
The estimated coefficient for the return dummy is, on the other hand, statistically
insignificant for both countries with all the metals. Thus, no evidence of diversifying
benefits in times of poor market performance is found in the above regressions. An
insignificant estimate for the return dummy is not an entirely unexpected result, since
also Hillier, Draper and Faff (2006) failed to find evidence of diversifying benefits in
times of poor market performance for all three precious metals with both the USA and
EAFE indexes in their research period extending from January 1976 to April 2004.
Table 6 below summarizes the results from the modified market model regressions for
Norway, Denmark and USA. The estimated precious metal betas, once more, mirror the
results from the simple market model regressions. The estimated coefficient for the
volatility dummy is negative and significant for gold and platinum when regressed
against the Norway index. This, once again, would indicate significant diversification
benefits from holding gold or platinum in a portfolio of Norwegian stocks in times of
high stock market volatility. In other words, it means that during the sample period
extending from March 1988 to August 2016, gold and platinum are more negatively
correlated with Norway stock market returns in periods of high market risk. The
estimated volatility coefficient for Norway with silver is statistically insignificant.
The volatility coefficients for the three metals with Denmark and USA indexes do not
reach the traditional level of significance of 10 percent. It is worth noting, however, that
the estimated negative coefficient for the volatility dummy for gold when regressed
against the USA index, although not significant, approaches statistical significance with
an exact p-value of 0.103. The negative volatility coefficient for gold with the Denmark
index is also not too distant from statistical significance at the 10 percent level with an
exact p-value of 0.130. It could therefore be speculated, that there might still be
diversifying benefits in volatile markets from holding gold in a portfolio with the
Denmark or the USA market index.
The return dummies are once more statistically insignificant, except for silver with the
Norway index for which the return coefficient is significantly positive. Thus, no
40
evidence of diversifying benefits in times of poor market performance is found in these
regressions either.
Table 6
Diversification Properties of Precious Metals for Norway, Denmark and USA,
Data for March 1988–August 2016
Mean Equation:
Variance Equation:
Gold
Silver
Platinum
C: Datastream Norway Total Market Return as market proxy
0.0020
(0.82)
0.0133
(0.36)
0.0067
αPM
***
0.0203
(0.12)
0.0635
(0.00)
0.0647 ***
βPM
-0.0890 **
(0.02)
-0.0734
(0.21)
-0.0839 *
γ
**
-0.0307
(0.25)
0.0682
(0.07)
0.0006
η
0.0056 **
(0.02)
0.0253
(0.10)
0.0092 **
a0
***
***
0.0487
(0.00)
0.0466
(0.00)
0.0486 ***
a1
0.9465 ***
(0.00)
0.9463 ***
(0.00)
0.9472 ***
b1
8.8914 ***
(0.00)
5.4325 ***
(0.00)
10.3680 ***
Shape
1.20
3.65
2.24
VL
D: Datastream Denmark Total Market Return as market proxy
0.0070
(0.44)
0.0074
(0.62)
0.0121
αPM
0.0337 **
(0.02)
0.1131 ***
(0.00)
0.1041 ***
βPM
-0.0690
(0.13)
0.0539
(0.46)
-0.0883
γ
-0.0207
(0.49)
-0.0080
(0.89)
0.0496
η
**
0.0057
(0.02)
0.0226
(0.16)
0.0099 *
a0
0.0489 ***
(0.00)
0.0487 ***
(0.00)
0.0510 ***
a1
0.9460 ***
(0.00)
0.9463 ***
(0.00)
0.9445 ***
b1
8.1895 ***
(0.00)
4.9718 ***
(0.00)
9.5556 ***
Shape
1.16
4.56
2.29
VL
E: Datastream USA Total Market Return as market proxy
0.0101
(0.16)
0.0217
(0.16)
0.0122
αPM
***
-0.0474
(0.00)
0.0314
(0.29)
0.0414 **
βPM
-0.0506
(0.10)
-0.0856
(0.19)
0.0354
γ
0.0143
(0.60)
0.0503
(0.31)
-0.0277
η
0.0015 *
(0.07)
0.0144 ***
(0.00)
0.0084
a0
***
***
0.0457
(0.00)
0.0402
(0.00)
0.0595 ***
a1
0.9532 ***
(0.00)
0.9570 ***
(0.00)
0.9379 ***
b1
5.5324 ***
(0.00)
5.3317 ***
(0.00)
7.9931 ***
Shape
1.54
5.41
3.36
VL
(0.57)
(0.00)
(0.08)
(0.98)
(0.02)
(0.00)
(0.00)
(0.00)
(0.32)
(0.00)
(0.28)
(0.17)
(0.08)
(0.00)
(0.00)
(0.00)
(0.32)
(0.02)
(0.45)
(0.45)
(0.23)
(0.00)
(0.00)
(0.00)
Note: ***, **, * indicate significance levels of 1%, 5%, and 10% respectively. Robust standard errors are
used and the rounded p-values are in the brackets.
Autocorrelation and Heteroscedasticity Tests for the Market Models
The results from the autocorrelation and heteroscedasticity tests on the residuals of the
market model and the modified market model are presented in the Appendix (A6). The
results are very similar to what they were for the single series GARCH(1,1) models for
41
the precious metals. That is to say, that when the country indexes are regressed against
the gold series, no leftover autocorrelation or heteroscedasticity is detected in either the
market model or the modified market model residuals.
In the case of the silver series, once more, the Ljung-Box test rejects the null hypothesis
of “no autocorrelation” for all countries (for both models). The ARCH LM test, on the
other hand, rejects the null hypothesis of “no heteroscedasticity” only for Norway with
silver for the market model. For the platinum series, the Ljung-Box test rejects the null
hypothesis of “no autocorrelation” for all countries but the USA.
However, just as previously, it should be noted yet again that the preliminary tests on
the platinum series most often failed to detect autocorrelation in the series in the first
place. The ARCH LM test again rejects the null hypothesis of “no heteroscedasticity” for
Finland, Norway and Denmark with platinum. Thus, in those cases, according to the
test, the models have not succeeded in capturing all the heteroscedasticity in the
platinum series.
7.2
Portfolio Simulations with Precious Metals
Table 7 reports the λ estimates for different portfolios with varying weighs of precious
metals held with the Finnish or the Swedish stock market index. Lambda values are
calculated for two different investment strategies. The risk-to-reward ratio reported in
the table, is the relative efficiency of an investment portfolio in the respective
investment strategy when compared with just holding either the Finnish or the Swedish
stock market portfolio.
The efficiency gains from holding gold in a hybrid portfolio with the Finnish index in
the buy-and-hold strategy would, judging by these figures, be immense. The numbers
seem to suggest, that the optimal weighting of gold could be 70 percent, which is a
remarkably high weight. For comparison, Hillier, Draper and Faff (2006) found an
optimal portfolio weight of gold with the USA index of 30 percent during their sample
period extending from March 1988 to August 2016.
However, when looking at Figure A4.1 (in Appendix 4) of the Finnish index and gold, it
is noteworthy, that they do seem to be moving in quite opposite directions. One might
almost say, that they oscillate around each other. The behaviour of the lines around
year 2008 is of particular interest. The lines intersect sharply and the gold price seems
42
to develop entirely in the opposite direction than the Finnish index. That might suggest
an especially negative correlation during the crisis period, and in turn lend support to
the hypothesis of gold as a safe haven asset. Also, Table 4 reported a relatively high
negative correlation between the Finnish index and gold during the period of the credit
crisis. It should be noted here, that the previous similar study by Hillier, Draper and
Faff did not, for instance, include the period of the most recent major financial crisis in
2008.
Platinum buy-and-hold portfolios also performed remarkably well, with an optimal
weighting in platinum of 50 percent with the Finnish market portfolio. Silver too,
provided notable efficiency gains, and its optimal weight in a portfolio was 40 percent
for Finland in the buy-and-hold strategy. That is to say, that in a buy-and-hold strategy
incorporating 40 percent weight of silver in a portfolio with the Finnish index provided
an efficiency gain of approximately 52 percent.
Similarly to Hillier, Draper and Faff, the efficiency gains provided by the switching
strategy were significantly more modest than with the buy-and-hold strategy for all the
metals. For example, an optimal 50 percent weighting in gold in the switching strategy
provided only an efficiency gain of approximately 4 percent. With greater weights in
silver and platinum, it seems, that the Finnish market portfolio, in fact, dominated the
hybrid market-to-metal portfolios in the switching strategy.
The results for Sweden were similar. The optimal portfolio weight for gold was 60
percent in the buy-and-hold strategy and 40 percent in the switching strategy. The
efficiency gains provided by the buy-and-hold strategy were, once more, substantially
greater. The optimal weight for silver in both strategies was 30 percent. In the
switching strategy, the Swedish market index outperformed the hybrid metals-andmarket portfolio for the portfolio weights above 40 percent.
Similarly to Finland, the results from the switching strategy for Sweden were only
marginally better than just holding the market index for all the metals. A switching
strategy in platinum with the optimal portfolio weight of 50 percent, for instance,
provided only an efficiency gain of approximately 2 percent. The efficiency gains from
holding lower weights of precious metals in the switching strategy were almost nonexistent for all three metals.
43
Table 7
Finland and Sweden – Relative Efficiency of Financial Portfolios Incorporating Various Weights of Precious Metals, Data for March
1988–August 2016
Market Proxy
Wmetal =
0.01
0.02
0.05
and Strategy
A: Datastream Finland Total Market Return as market proxy
Gold portfolios
Buy and hold
Switching
Silver portfolios
Buy and hold
Switching
Platinum portfolios
Buy and hold
Switching
0.15
0.20
0.30
0.40
0.50
0.60
0.70
0.80
1.0147
1.0005
1.0297
1.0010
1.0768
1.0034
1.1623
1.0070
1.2574
1.0126
1.3629
1.0180
1.6063
1.0275
1.8872
1.0329
2.1759
1.0364
2.4003
1.0313
2.4531
1.0213
2.2643
1.0055
1.0151
1.0010
1.0304
1.0023
1.0772
1.0044
1.1573
1.0070
1.2380
1.0091
1.3165
1.0118
1.4509
1.0157
1.5244
1.0094
1.5078
1.0062
1.3992
0.9973
1.2274
0.9865
1.0330
0.9737
1.0121
1.0003
1.0243
1.0007
1.0619
1.0018
1.1269
1.0085
1.1939
1.0109
1.2614
1.0132
1.3884
1.0206
1.4803
1.0193
1.5008
1.0183
1.4216
1.0080
1.2459
0.9947
1.0123
0.9800
1.0168
1.0003
1.0341
1.0007
1.0882
1.0034
1.1871
1.0080
1.2972
1.0102
1.4190
1.0143
1.6934
1.0214
1.9839
1.0255
2.2179
1.0235
2.2833
1.0155
2.1074
1.0015
1.7462
0.9847
1.0170
1.0011
1.0342
1.0020
1.0864
1.0043
1.1741
1.0069
1.2592
1.0079
1.3367
1.0101
1.4445
1.0106
1.4538
1.0052
1.3538
0.9956
1.1780
0.9857
0.9772
0.9762
0.7891
0.9635
1.0139
1.0001
1.0281
1.0002
1.0713
1.0017
1.1455
1.0062
1.2207
1.0092
1.2938
1.0113
1.4165
1.0167
1.4708
1.0182
1.4175
1.0205
1.2543
1.0101
1.0246
1.0011
0.7860
0.9874
B: Datastream Sweden Total Market Return as market proxy
Gold portfolios
Buy and hold
Switching
Silver portfolios
Buy and hold
Switching
Platinum portfolios
Buy and hold
Switching
0.10
Note: Relative efficiency is given by the reward-to-risk ratio calculated according to Equation 21. Darker shade indicates optimal portfolio weight.
44
Table 8 below shows the results for the portfolio simulations with Norway, Denmark
and USA indexes as the market proxy. Once more, the efficiency gains from the buyand-hold strategy are substantially greater than from the switching strategy. For
Norway, the optimal weight of gold is 50 percent in both strategies. The optimal weight
for silver was 30 percent in both strategies.
The Norway market portfolio seems to have outperformed both the buy-and-hold and
the switching strategies for silver with portfolio weights higher than 50 percent. The
optimal weight for platinum was 40 percent in the buy-and-hold strategy and 30
percent in the switching strategy for Norway. The Norway market portfolio
outperformed the switching strategy with weights higher than 50 percent in platinum.
The λ estimates for Denmark in Table 8 seem somewhat more underwhelming than for
the other Nordic countries. They are, however, quite similar than those for Finland,
Sweden and Norway but slightly lower for all the metals in both strategies. The optimal
weight of gold was 40 percent in the buy-and-hold strategy and 30 percent in the
switching strategy. The optimal weight of silver and platinum was 20 percent in both
strategies for Denmark.
The Denmark market portfolio outperformed both strategies in higher portfolio weights
for both silver and platinum, and also for the switching strategy for gold. The lower
weights for all the metals give relatively modest efficiency gains in both strategies.
Regardless, overall, the efficiency gains from the buy-and-hold strategy are still quite
substantial for Denmark with all the metals, just as they were for the other Nordic
countries.
The results for USA suggest that the optimal weight of gold is 40 percent in the buyand-hold strategy and 30 percent in the switching strategy. For comparison, Hillier,
Draper and Faff found the optimal portfolio weight of gold with the USA index to be 30
percent in both strategies. Thus, the optimal portfolio weight in the buy-and-hold
strategy was greater in this study, while the optimal weight in the switching strategy
was the same for both studies.
However, the magnitudes of efficiency gains implied by the lambdas were larger in
these results. For example, the efficiency gains from holding gold in a hybrid portfolio
with the USA index in the buy-and-hold strategy were approximately 60 percent with
the optimal portfolio weight of 40 percent. In Hillier, Draper and Faff the equivalent
45
efficiency gains were approximately 34 percent with their optimal portfolio weight of
30 percent.
However, it should be noted, yet again, that their sample period extending from
January 1976 to April 2004 did not include the most recent major financial crisis of
2008. Therefore, since precious metals might be particularly good diversifiers in times
of high market volatility, the difference in time periods could potentially be a
contributor in the greater efficiency gains for USA with gold obtained in this study
when compared with Hillier, Draper and Faff.
The optimal weight for silver was 20 percent in both strategies for USA in this study.
The optimal weight for platinum was 30 percent in the buy-and-hold strategy and 15
percent in the switching strategy. Similarly to the other countries, the buy-and-hold
strategy clearly dominated the switching strategy. The USA market portfolio
outperformed both strategies in higher portfolio weights for both silver and platinum,
and also for the switching strategy for gold.
46
Table 8
Norway, Denmark and USA – Relative Efficiency of Financial Portfolios Incorporating Various Weights of Precious Metals, Data for
March 1988–August 2016
Market Proxy
Wmetal =
0.01
0.02
0.05
and Strategy
C: Datastream Norway Total Market Return as market proxy
Gold portfolios
Buy and hold
Switching
Silver portfolios
Buy and hold
Switching
Platinum portfolios
Buy and hold
Switching
0.20
0.30
0.40
0.50
0.60
0.70
0.80
1.0298
1.0024
1.0767
1.0031
1.1608
1.0045
1.2522
1.0075
1.3503
1.0131
1.5595
1.0185
1.7598
1.0248
1.8936
1.0262
1.8883
1.0181
1.7097
1.0052
1.4052
0.9909
1.0145
1.0001
1.0291
1.0007
1.0727
1.0019
1.1431
1.0034
1.2076
1.0044
1.2613
1.0078
1.3177
1.0121
1.2849
1.0092
1.1667
1.0016
0.9974
0.9902
0.8176
0.9751
0.6543
0.9661
1.0125
1.0015
1.0252
1.0010
1.0636
1.0033
1.1280
1.0054
1.1910
1.0065
1.2498
1.0096
1.3383
1.0145
1.3583
1.0116
1.2843
1.0089
1.1225
0.9997
0.9121
0.9876
0.6989
0.9741
1.0136
1.0002
1.0274
1.0019
1.0697
1.0026
1.1429
1.0080
1.2180
1.0112
1.2927
1.0144
1.4257
1.0157
1.5010
1.0152
1.4738
1.0097
1.3278
0.9982
1.0963
0.9846
0.8410
0.9687
1.0118
1.0003
1.0233
1.0014
1.0557
1.0030
1.0995
1.0064
1.1264
1.0075
1.1320
1.0082
1.0730
1.0068
0.9367
0.9976
0.7660
0.9920
0.6007
0.9786
0.4609
0.9679
0.3507
0.9597
1.0102
1.0001
1.0204
1.0006
1.0498
1.0006
1.0938
1.0046
1.1283
1.0057
1.1493
1.0087
1.1372
1.0076
1.0429
1.0030
0.8826
0.9954
0.6948
0.9856
0.5161
0.9728
0.3662
0.9633
1.0147
1.0016
1.0296
1.0034
1.0757
1.0040
1.1567
1.0097
1.2417
1.0150
1.3284
1.0156
1.4916
1.0231
1.6010
1.0224
1.5991
1.0178
1.4565
1.0072
1.2072
0.9896
0.9253
0.9690
1.0143
1.0010
1.0283
1.0004
1.0684
1.0030
1.1254
1.0075
1.1645
1.0092
1.1802
1.0095
1.1319
1.0084
0.9945
0.9981
0.8176
0.9900
0.6463
0.9760
0.5022
0.9644
0.3891
0.9520
1.0122
1.0018
1.0245
1.0030
1.0607
1.0020
1.1174
1.0072
1.1664
1.0139
1.2030
1.0128
1.2205
1.0120
1.1458
1.0116
0.9894
1.0032
0.7933
0.9922
0.6014
0.9758
0.4385
0.9659
E: Datastream USA Total Market Return as market proxy
Gold portfolios
Buy and hold
Switching
Silver portfolios
Buy and hold
Switching
Platinum portfolios
Buy and hold
Switching
0.15
1.0147
1.0007
D: Datastream Denmark Total Market Return as market proxy
Gold portfolios
Buy and hold
Switching
Silver portfolios
Buy and hold
Switching
Platinum portfolios
Buy and hold
Switching
0.10
Note: Relative efficiency is given by the reward-to-risk ratio calculated according to Equation 21. Darker shade indicates optimal portfolio weight.
47
8
DISCUSSION AND CONCLUSION
This section first briefly summarizes the results from the analysis, and compares them
to the earlier study of Hillier, Draper and Faff. Next, follows a critical overview of the
results. Finally, concluding remarks and suggestions for further research are offered.
8.1
Discussion of the Results
The results from the market model and modified market model regressions suggest
that there is indication of possible diversification benefits from adding precious metals
(low or even negative estimated precious metal betas). The estimated market model
precious metal betas are significantly negative for gold when held in a portfolio with the
Swedish or the USA market index, which indicates hedging qualities. Also the
remaining estimated betas are far less than unity, and most of them are statistically
significant, as is illustrated by Table 9.
Most interestingly, evidence is found of diversification benefits especially in times of
extremely high market volatility, as can also be seen from the table below. The
estimated coefficient for the volatility dummy is both negative and statistically
significant for Finland with silver, for Sweden with gold, and for both countries with
platinum. The estimated volatility coefficient is also significantly negative for Norway
with gold and platinum. Furthermore, the estimated negative volatility coefficients for
USA with gold and Denmark with gold, are marginally non-significant with p-values of
0.103 and 0.130 respectively.
In the portfolio simulations, the buy-and-hold strategy outperforms the switching
strategy for all the countries, and produces considerably more substantial efficiency
gains. Thus, evidence is found, that a long-term commitment to precious metals is
superior to tactical allocation. The efficiency gains from holding lower weights of
precious metals in the switching strategy are, in fact, almost non-existent for all three
metals for all the countries, which was also the case in Hillier, Draper and Faff.
They suggest that the clear dominance of the buy-and-hold strategy could be because
the added return performance from precious metals during high volatility periods is
simply not enough to offset the long-term strong performance of the constant buy-andhold portfolio. One could also argue, that a possible reason for the superiority of the
buy-and-hold strategy could be that commodities have experienced a bull-market. Gold
48
returns, for example, seem to have been steadily rising, particularly with respect to the
Finnish, Swedish and Norwegian market indexes (see graphs in Appendix 3). It could
thus be argued that the difference in the performance of the two strategies could simply
be due to less exposure to the booming precious metals. It should be noted, however,
that not all the metals outperform their stock market indexes. The inferior performance
of the switching strategy could also be due to the way it’s constructed. The investor first
observes the high volatility, and only afterwards makes the switch for the next period,
and is therefore systematically late to the party. Both strategies are, non the less, found
to be superior to just holding the market portfolio, except for the very highest portfolio
weights in some instances.
Table 9
Summary of Simulation Results for Finland, Sweden, Norway, Denmark and USA
Sample period extends from March 1988 to August 2016.
Country
Finland
Sweden
Norway
Denmark
USA
Precious
Metal
Gold
Silver
Platinum
Gold
Silver
Platinum
Gold
Silver
Platinum
Gold
Silver
Platinum
Gold
Silver
Platinum
Precious Metal Beta
Volatility
Coefficient
Optimal Portfolio
Weight
Significant and
negative
Significant and
negative
Buy-andhold
X
X
Significant
and near zero
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
*
*
70 %
40 %
50 %
60 %
30 %
40 %
50 %
30 %
40 %
40 %
20 %
20 %
40 %
20 %
30 %
Switching
50 %
30 %
30 %
40 %
30 %
50 %
50 %
30 %
30 %
30 %
20 %
20 %
30 %
20 %
15 %
Note: * indicates that the result is not significant, but relatively close.
The optimal portfolio holdings for both strategies are gathered in the above table, and
are clearly highest for Finland, Sweden and Norway. The optimal portfolio weights for
USA are, although slightly higher, remarkably similar to what they are in the Hillier,
Draper and Faff study. That could be viewed as a type of robustness check, whereby
their findings could be replicated by the methods used in the current paper. The
difference in time periods, particularly the inclusion of the latest crisis period in this
49
study, could potentially be a contributor to the generally greater weights and efficiency
reported here.
Similarly to Hillier, Draper and Faff, gold provides, by far, the most substantial
efficiency gains, and silver the least. Allocating, for example, five percent of holdings to
gold in the buy-and-hold strategy, leads to efficiency gains varying from six to eight
percent for all the countries. When comparing the countries, the efficiency gains from
adding any of the precious metals are clearly highest for Finland, second highest for
Sweden and third highest for Norway. The efficiency gains are fourth highest for USA
and, judging by the results, generally the lowest for Denmark. The efficiency gains from
the buy-and-hold strategy are still quite substantial even for Denmark with all of the
three metals, and gold in particular.
8.2
Critical Overview of the Results
It should be noted that the estimated volatility dummy is not statistically significant for
all the cases. That, in turn, could be seen as problematic for the switching strategy,
since the volatility dummy indicates when the switch is made. It could be argued of
course, that an estimated volatility dummy might be significant during some shorter
subperiods in the sample, even if it’s not significant during the full period, and vice
versa. Such considerations could be of consequence due to the fact that the switching
strategy is a rolling window analysis.
It could also be questioned how tangible the switching strategy is for actual investors.
The real world applicability of the results is affected by the fact that transaction costs,
for practical reasons, are entirely neglected in the analysis. In reality, they could be a
major issue in the implementation of the switching strategy as the portfolio is
constantly rebalanced. Furthermore, in many cases, the results in the portfolio
simulations suggest portfolio weights higher than 30 percent. It should be noted, that
realistically a portfolio manager would probably not very often choose an allocation
higher than 25 percent. However, even the lower weights seem to provide good
efficiency gains in the buy-and-hold strategy.
This thesis also fits a GARCH model with dummy variables in the mean equation,
which could, according to Doornik and Ooms (2003), in some cases lead to
multimodality in the GARCH likelihood. According to them, that could cause standard
inference on the estimated coefficients not to be accurate. Another criticism to the
50
model choice could be that it neglects feedback. Joy (2011), for example, chooses a
DCC-GARCH model to allow for such effects.
Furthermore, the Ljung-Box and ARCH LM tests, in some cases, showed signs of
possible need for further model specification. An approach specifying different models
for different countries with different metals could potentially bring even better results
from the regressions.
8.3
Concluding Remarks and Suggestions for Further Research
The empirical results in this thesis show that, in the sample period extending from
March 1988 to August 2016, precious metals have potential to act as diversifiers in a
portfolio of Finnish, Swedish, Norwegian, Danish or USA stocks. A buy-and-hold
strategy is found to be superior to the switching strategy, which is in line with earlier
research. The results offer new insight into the Nordic markets, but a lot more future
research remains to be done in the field of precious metals investments.
Possible areas for further research include an alternative way to implement the
switching strategy. Some other indicator could be used for the switch, for example a
volatility index like VOX or CBOE, which could potentially make the strategy more
realistic and accessible to actual investors. When it comes to real world applicability of
these results, a suggestion for further research could also be to investigate having
holdings in precious metals indirectly via exchange-traded funds (ETF), rather than
investing in them directly. This variation has been researched before but not in the
context of Finnish, Swedish, Norwegian or Danish markets.
Furthermore, the finding of significant estimated volatility dummies, and also earlier
research results on the safe haven aspects of gold raise thoughts on investor psychology
related to investing in precious metals. The psychology behind investor’s decisions
related to gold investing could be an interesting research topic in the field of
behavioural finance.
51
SVENSK SAMMANFATTNING
En sammanfattning av avhandlingen presenteras på svenska i det här kapitlet.
BAKGRUND
Guldpriset har ökat markant under tjugohundratalet: år 2005 var guldpriset 444,75
amerikanska dollar, men år 2011 hade guldpriset nått den svindlande siffran på
1 384,38 amerikanska dollar, vilket är den högsta nivån som någonsin har rapporterats
för guld. Guldpriset har således ökat väldigt kraftigt sedan 2005, vilket i sin tur har lett
till att guld samt andra ädelmetaller har fått ett uppsving i sin popularitet bland
investerare efter denna tidpunkt. Dessutom efterfrågas ädelmetaller i högre
utsträckning i tider av ekonomisk turbulens eftersom en del investerare tror att
ädelmetaller är speciellt lågt korrelerade med avkastningen på aktier när
finansmarknaderna stormar. Detta skulle betyda att ädelmetaller har förmågan att
minska risken i en aktieportfölj. Guld, silver och platina har också historiskt setts som
värdebevarare, vilket har bidragit till deras popularitet som diversifieringstillgångar i
aktieportföljer, speciellt efter finanskrisen år 2008.
KONTRIBUTION
Tidigare forskningar har visat att det finns betydande diversifieringsfördelar med att
inkludera ädelmetaller i en portfölj av aktier. Det finns emellertid inga studier som
fokuserar på den finska, svenska, norska eller danska marknaden. Till skillnad från
tidigare studier undersöker denna magistersavhandling ädelmetallers eventuella
egenskaper som diversifieringskomponent i samband med speciellt finska, svenska,
norska och danska börsen.
Guldpriset har också gradvis börjat minska sedan 2013 efter att ha stigit exponentiellt
från och med början av tvåtusentalet. De flesta av de tidigare studierna har gjorts före
denna vändpunkt år 2013, vilket leder till ett behov av en studie som görs med färska
prisdata. Förutom den nordiska marknaden görs analysen också på den amerikanska
marknaden. Detta görs både för att underlätta jämförandet av resultaten med tidigare
studier (som ofta fokuserar på den amerikanska marknaden) men också för att det inte
finns nyligen genomförda undersökningar på den amerikanska marknaden. Fastän den
största kontributionen av avhandlingen inte kommer från nya data är de dock ett
viktigt tillägg till befintliga forskningsresultat.
52
Undersökningen avser också att jämföra två olika investeringsstrategier: ”köp och
behåll”-strategin och en mer dynamisk investeringsstrategi där metallens vikt i
portföljen återbalanseras beroende på marknadssituationen. Genomförandet av den
dynamiska strategin är den största kontributionen av avhandlingen. Det specifika sättet
att koda återbalanseringen av portföljen är ny och unik för just denna studie.
SYFTE
Syftet med den här avhandlingen är att studera om det är fördelaktigt att diversifiera en
finsk, svensk, norsk, dansk eller amerikansk aktieportfölj med ädelmetaller samt om
det är bättre att följa en aktiv eller en passiv investeringsstrategi.
TIDIGARE FORSKNING
”Do Precious Metals Shine? An Investment Perspective” − Hillier, Draper och Faff
(2006)
Hillier, Draper och Faff (2006) undersöker om det är fördelaktigt att komplettera en
aktieportfölj med guld, silver eller platina i sin studie ”Do Precious Metals Shine? An
Investment Perspective”. Undersökningen fokuserar på den amerikanska marknaden
och undersökningsperioden sträcker sig från 1976 till 2004. De använder sig av dagliga
tidsseriedata under en period om 28 år. De finner att portföljer med en viss viktning i
guld, silver eller platina presterade bättre än portföljer med enbart aktier.
Det intressantaste resultatet i studien är emellertid att ädelmetaller är speciellt lågt
korrelerade med börsindexen i tider av ekonomisk turbulens (då volatiliteten på
aktiemarknaden är hög). Detta skulle innebära att det skulle löna sig att ha en viktning i
ädelmetaller speciellt under finansiell marknadsoro eller stress. Som följd av detta
resultat bestämmer Hillier, Draper och Faff sig att jämföra en passiv ”köp och behåll”strategi och en mer dynamisk investeringsstrategi där metallens vikt i portföljen
återbalanseras beroende på marknadssituation. Med andra ord, den aktiva strategin
innebär att portföljen har en större viktning i guld, silver eller platina då volatiliteten på
aktiemarknaden är hög. I en passiv strategi hålls andelen guld, silver eller platina i
portföljen konstant under hela investeringsperioden. Slutsatsen i analysen blir att den
passiva ”köp och behåll”-strategin presterar bättre för både guld, silver och platina
under perioden 1976−2004.
53
BESKRIVNING AV DATA OCH METOD
Datamaterialet för hela undersökningsperioden består av 6515 dagliga observationer
för både guld-, silver- och platinapriset samt observationer av de fem olika
marknadsindexen. Det finska, svenska, norska, danska och amerikanska börsindexet i
studien är av typen ”Total Return (TR)” vilket innebär att utdelningar återinvesteras i
indexvärdet. Alla aktieavkastningar är beräknade i lokal valuta. Den undersökta
perioden sträcker sig från mars 1988 till augusti 2016. Datamaterialet består således av
dagliga tidsseriedata under en period om 28 år. Rådata har hämtats ur Thomson
Reuters Datastream-databasen och har därefter behandlats i Excel och R. Före
analysen omvandlades rådata till avkastningar i kontinuerligt beräknad form (Rt) enligt
nedanstående formel. Pt är priset vid tidpunkt t och Pt-1 är priset vid tidpunkt t-1.
De metoder som används i avhandlingen kan delas i två grupper: (1) metoder som
syftar till att studera ädelmetallers egenskaper som diversifieringskomponent och (2)
portföljsimuleringar för att undersöka ifall det är bättre att välja en aktiv eller en passiv
investeringsstrategi.
Första steget i metoden är GARCH(1,1)-modellen som finns nedan.
,
,
(1)
(3)
(4)
Där Rt är avkastningen för antingen ädelmetall eller börsindex på tidpunkt t, ut är
feltermen i regressionen och μ är en parameter som ska estimeras. Ekvation 2
estimerar den uppskattade betingade variansen. Där betecknar a0 en konstant som
skattats under den föregående tidsperioden, a1 är ARCH-koefficienten, b1 är GARCH-
54
koefficienten, ut-12 är de kvadrerade residualerna av medelvärdesekvationen från den
föregående tidpunkten och σt-12 är föregående tidpunkts prognostiserade varians. VL i
ekvation 3 är den långtgående variansen.
Det andra steget i metoden presenteras i marknadsmodellregressionen nedan.
(4)
Där är RPM,t avkastningen för ädelmetallen och Ri,t är avkastningen för börsindexet, ut
är feltermen i regressionen, αPM och βPM är parametrar som ska estimeras.
Det tredje steget i metoden är den modifierade marknadsmodellen nedan.
(5)
Där är VolDumt en dummyvariabel för hög marknadsvolatilitet, RetDumt är en
dummyvariabel för låga marknadsavkastningar, γ och η är parametrar som ska
estimeras. Både marknadsmodellen och modifierade marknadsmodellen justeras för
GARCH(1,1)-effekter i ädelmetalpriser.
Det sista steget i metoden är portföljsimuleringar. λ i ekvation 6 är ett mått som mäter
relationen mellan risk och avkastning i en viss portfölj.
(6)
Där är
avkastningen för en portfölj med en viss viktning i guld, silver eller platina.
är risken för en portfölj med en viss viktning i guld, silver eller platina.
avkastningen för en portfölj med enbart aktier och
enbart aktier.
är
är risken för en portfölj med
55
GENOMFÖRANDET AV UNDERSÖKNINGEN
Ädelmetallers diversifieringsegenskaper analyseras med en GARCH(1,1)-modell, en
marknadsmodell och en modifierad marknadsmodell. Båda slags marknadsmodeller
justeras för GARCH(1,1)-effekter i ädelmetalpriser. Först analyseras ädelmetallers
egenskaper som diversifieringskomponent i en finsk, svensk, norsk, dansk och
amerikansk aktieportfölj genom att utföra marknadsmodellregressioner med prisdata
för guld, silver och platina.
Sedan används en modifierad marknadsmodell för att analysera om ädelmetallers
diversifieringsegenskaper
påverkas
av
marknadsläget.
Den
modifierade
marknadsmodellen analyserar det eventuella sambandet mellan tider av ekonomisk
turbulens eller låga avkastningar och ädelmetallers diversifieringsegenskaper under
sådana marknadsförhållanden. Det vill säga att den modifierade marknadsmodellen
används för att granska ifall ädelmetaller är speciellt lågt korrelerade med börsindexen
i tider av ekonomisk turbulens.
Ekonomisk turbulens definieras i avhandlingen som tider av hög marknadsvolatilitet
eller som tider av låga marknadsavkastningar. Den modifierade marknadsmodellen har
en dummyvariabel för båda av dessa två extrema marknadsförhållanden: en
dummyvariabel för hög marknadsvolatilitet och en dummyvariabel för speciellt låga
marknadsavkastningar. En signifikant negativ betakoefficient för dummyvariabeln för
hög marknadsvolatilitet skulle indikera att ädelmetallen ifråga är speciellt negativt
korrelerad med börsindexen under volatila marknadsförhållanden. En signifikant
negativ betakoefficient för dummyvariabeln för speciellt låga marknadsavkastningar
skulle indikera att ädelmetallen ifråga är speciellt negativt korrelerad med börsindexen
då aktieavkastningarna är låga.
Till sist utförs portföljsimuleringar för att undersöka ifall det är bättre att välja en aktiv
eller en passiv investeringsstrategi. En aktiv investeringsstrategi innebär att andelen
guld,
silver
eller
platina
i
aktieportföljen
hålls
konstant
genom
hela
investeringsperioden. En passiv investeringsstrategi däremot innebär att metallens vikt
i portföljen återbalanseras beroende på den rådande situationen på marknaden.
Ekvation 6 används för att jämföra olika investeringsportföljer.
56
RESULTAT OCH KONKLUSIONER
Resultaten för marknadsmodellregressionerna antyder att det finns potential för guld
att fungera som en diversifieringskomponent på finska, svenska, danska och
amerikanska börsen. Betakoefficienterna i ekvation 4 var således både låga och
statistiskt signifikanta för dessa länder. Betakoefficienten för Norge var inte statistiskt
signifikant.
Marknadsmodellregressionerna för silver visade att det finns potential för silver att
fungera som en diversifieringskomponent på finska, norska och danska och börsen. Det
vill säga att betakoefficienten i ekvation 4 var både låg och statistiskt signifikant för
Finland, Norge och Danmark.
Marknadsmodellregressioner för platina visade att det finns potential för platina att
fungera som en diversifieringskomponent i Finland, Norge, Danmark och Amerika.
Betakoefficienterna i ekvation 4 var således både låga och statistiskt signifikanta för
alla länder förutom Sverige.
Det viktigaste resultatet från den modifierade marknadsmodellen för guld var att
dummyvariabeln för hög marknadsvolatilitet var negativ för alla länder. Den negativa
koefficienten var emellertid statistiskt signifikant endast för Sverige och Norge. Detta
skulle indikera att guld har förmågan att minska risken i en svensk eller en norsk
aktieportfölj speciellt under volatila marknadsförhållanden.
Modifierade marknadsmodellregressioner för silver visade att dummyvariabeln för hög
marknadsvolatilitet var negativ för alla länder förutom Danmark. Den negativa
koefficienten var emellertid statistiskt signifikant endast för Finland. Således har silver
förmågan att minska risken i en finsk aktieportfölj speciellt under volatila
marknadsförhållanden.
Modifierade marknadsmodellregressioner för platina visade att dummyvariabeln för
hög marknadsvolatilitet var negativ för alla länder förutom Amerika. Den negativa
koefficienten var emellertid statistiskt signifikant endast för Finland, Sverige och
Norge. Således har platina förmågan att minska risken i en finsk eller en svensk eller en
norsk aktieportfölj speciellt under volatila marknadsförhållanden.
Portföljsimuleringar visade att det är bättre att följa en passiv investeringsstrategi.
Resultaten var likadana för alla länder. I en passiv strategi hålls andelen guld, silver
57
eller platina i portföljen konstant under hela investeringsperioden. Slutsatsen i
analysen blev således att det är fördelaktigt att diversifiera en finsk, svensk, norsk,
dansk och amerikansk aktieportfölj med ädelmetaller, och ytterligare att den passiva
”köp och behåll”-strategin presterade bättre för både guld, silver och platina under
perioden 1988–2016 för dessa länder.
58
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60
APPENDIX 1 CORRELATION STRUCTURE OF GOLD, SILVER,
PLATINUM, NORDIC INDEXES AND USA (IN $)
A1.1 Correlation Table of Precious Metals and Country Indexes – Time periods: Full Period,
Before, During and After Dotcom Crash
Gold Silver Platinum Finland Sweden Norway Denmark USA
A. Full sample period, March 1988 – August 2016
Gold
1
0.53
0.45
0.08
0.08
0.18
0.15
-0.05
Silver
1
0.34
0.18
0.19
0.25
0.23
0.07
Platinum
1
0.14
0.14
0.21
0.18
0.04
Finland
1
0.72
0.57
0.57
0.37
Sweden
1
0.65
0.63
0.42
Norway
1
0.63
0.34
Denmark
1
0.27
USA
1
B. Before Dotcom Crash, March 1988 – February 2000
Gold
1
0.48
0.47
0.06
-0.02
0.00
0.06
-0.08
Silver
1
0.41
0.10
0.04
0.07
0.11
-0.06
Platinum
1
0.05
0.02
0.02
0.06
-0.04
Finland
1
0.52
0.40
0.42
0.17
Sweden
1
0.51
0.43
0.24
Norway
1
0.44
0.20
Denmark
1
0.07
USA
1
C. During Dotcom Crash, March 2000 – October 2002
Gold
1
0.29
0.14
-0.13
-0.07
0.12
0.11
-0.14
Silver
1
0.02
0.02
0.06
0.10
0.15
-0.03
Platinum
1
-0.07
-0.07
-0.08
-0.03
0.00
Finland
1
0.76
0.47
0.38
0.37
Sweden
1
0.54
0.47
0.42
Norway
1
0.57
0.28
Denmark
1
0.14
USA
1
D. After Dotcom Crash and before Credit Crash, November 2002 – September 2007
Gold
1
0.53
0.45
0.29
0.23
0.39
0.34
-0.04
Silver
1
0.25
0.23
0.29
0.32
0.35
0.09
Platinum
1
0.14
0.15
0.30
0.17
0.04
Finland
1
0.73
0.53
0.55
0.39
Sweden
1
0.58
0.63
0.42
Norway
1
0.58
0.23
Denmark
1
0.22
USA
1
61
A1.2 Correlation Table of Precious Metals and Country Indexes – Time periods: During and
After Credit Crash
Gold Silver Platinum Finland Sweden
E. During Credit Crash, October 2007 – October 2009
Gold
1
0.50
0.47
0.11
0.11
Silver
1
0.35
0.33
0.32
Platinum
1
0.32
0.32
Finland
1
0.86
Sweden
1
Norway
Denmark
USA
F. After Credit Crash, November 2009 – August 2016
Gold
1
0.64
0.54
0.16
0.15
Silver
1
0.41
0.30
0.29
Platinum
1
0.33
0.30
Finland
1
0.88
Sweden
1
Norway
Denmark
USA
Norway
Denmark
USA
0.23
0.40
0.40
0.78
0.79
1
0.17
0.37
0.37
0.80
0.82
0.82
1
-0.06
0.18
0.10
0.46
0.48
0.42
0.40
1
0.22
0.34
0.37
0.79
0.82
1
0.14
0.26
0.32
0.80
0.80
0.76
1
0.03
0.20
0.15
0.59
0.59
0.55
0.50
1
62
APPENDIX 2
AUTOCORRELATION AND HETEROSCEDASTICITY TESTS FOR WINSORIZED RETURN SERIES
A2.1 Results from Ljung-Box and ARCH LM tests on the Winsorized (on 1 % level) Return Series
Test and Lag
Length
Currency
Fin
Swe
Nor
Den
Usa
Gold
€
SK
NK
DK
$
€
Silver
SK
NK
DK
$
€
Platinum
SK
NK
DK
$
€
SK
NK
DK
$
Ljung-Box Test
Lag 10
Test Statistic
P-value
Lag 15
Test Statistic
Lag 20
Test Statistic
P-value
ARCH LM Test
Lag 10
Test Statistic
P-value
Lag 15
Test Statistic
P-value
Lag 20
Test Statistic
P-value
45
0.00
30
0.00
32
0.00
81
0.00
29
0.00
14
0.19
19
0.03
17
0.06
13
0.25
12
0.25
45
0.00
52
0.00
72
0.00
49
0.00
48
0.00
15
0.11
19
0.04
23
0.01
13
0.21
11
0.35
58
0.00
46
0.00
33
0.00
86
0.00
38
0.00
15
0.44
23
0.07
28
0.06
14
0.50
18
0.24
47
0.00
54
0.00
74
0.00
50
0.00
52
0.00
16
0.37
19
0.21
24
0.06
14
0.53
16
0.37
59
0.00
58
0.00
36
0.00
97
0.00
52
0.00
26
0.15
35
0.02
49
0.06
25
0.19
33
0.03
72
0.00
77
0.00
101
0.00
78
0.00
78
0.00
22
0.32
23
0.30
29
0.08
20
0.43
22
0.34
1386
0.00
1159
0.00
1258
0.00
1175
0.00
1406
0.00
5519
0.00
604
0.00
631
0.00
537
0.00
690
0.00
458
0.00
423
0.00
448
0.00
453
0.00
489
0.00
581
0.00
550
0.00
565
0.00
574
0.00
728
0.00
1477
0.00
1212
0.00
1315
0.00
1216
0.00
1445
0.00
594
0.00
662
0.00
677
0.00
584
0.00
786
0.00
496
0.00
461
0.00
485
0.00
491
0.00
554
0.00
626
0.00
593
0.00
605
0.00
620
0.00
788
0.00
1517
0.00
1246
0.00
1332
0.00
1241
0.00
1472
0.00
634
0.00
700
0.00
713
0.00
626
0.00
850
0.00
535
0.00
496
0.00
525
0.00
530
0.00
585
0.00
642
0.00
606
0.00
621
0.00
638
0.00
814
0.00
Note: The null hypotheses for the tests are “no autocorrelation” (Ljung-Box) and “no heteroscedasticity” (ARCH LM). The values are rounded up to the nearest integer for the
Ljung-Box and ARCH LM test statistics and to two decimal places for the P-values. Darker shade indicates cases when the null hypothesis cannot be rejected at any traditional
level of significance.
63
APPENDIX 3
(IN $)
PLOTS OF COUNTRY INDEXES AND PRECIOUS METALS
A3.1 Plot of Nordic Indexes (in$)
A3.2 Plot of USA Index and Precious Metals (in $)
64
APPENDIX 4 PLOTS OF NORDIC INDEXES AND PRECIOUS METALS (IN
LOCAL CURRENCY)
A4.1 Plot of Finland Index and Precious Metals (in €)
A4.2 Plot of Sweden Index and Precious Metals (in SK)
65
A4.3 Plot of Norway Index and Precious Metals (in NK)
A4.4 Plot of Denmark Index and Precious Metals (in DK)
66
APPENDIX 5 STUDENT-T GARCH (1,1) AND MARKET MODEL
ESTIMATION RESULTS
A5.1 GARCH (1,1) – Finland, Sweden, Norway and Precious Metals
Mean Equation:
Variance Equation:
Country
Finland (€)
a0
a1
b1
shape
Ljung-Box
ARCH LM
Sweden (SK)
VL
Lag 10
Lag 45
Lag 5
Lag 7
a0
a1
b1
shape
Ljung-Box
ARCH LM
Norway
(NK)
VL
Lag 10
Lag 45
Lag 5
Lag 7
a0
a1
b1
shape
Ljung-Box
ARCH LM
VL
Lag 10
Lag 45
Lag 5
Lag 7
0.0114**
(0.03)
0.0716***
(0.00)
0.9258***
(0.00)
10.6701***
(0.00)
4.64
69.22
(0.00)
71.76
(0.00)
0.01
(0.99)
0.23
(0.99)
0.0157***
(0.00)
0.0772***
(0.00)
0.9154***
(0.00)
11.7263***
(0.00)
2.15
20.68
(0.00)
21.38
(0.00)
5.17
(0.09)
5.62
(0.17)
0.0286***
(0.00)
0.0942***
(0.00)
0.8900***
(0.00)
11.5641***
(0.00)
1.83
24.83
(0.00)
25.88
(0.00)
2.31
(0.41)
2.53
(0.60)
Gold
0.0055**
(0.04)
0.0510***
(0.00)
0.9444***
(0.00)
7.8784***
(0.00)
1.24
0.70 (0.61)
1.49 (0.74)
0.57 (0.86)
1.11 (0.90)
0.0059**
(0.02)
0.0471***
(0.00)
0.9478***
(0.00)
9.4032***
(0.00)
1.18
0.90 (0.53)
1.88 (0.65)
0.68 (0.83)
0.88 (0.93)
0.0057**
(0.02)
0.0489***
(0.00)
0.9462***
(0.00)
8.8790***
(0.00)
1.20
0.46 (0.71)
1.52 (0.74)
2.70 (0.34)
3.25 (0.47)
Silver
Platinum
0.0235
(0.61)
0.0518
(0.30)
0.9435***
(0.00)
4.7918***
(0.00)
5.16
20.56 (0.00)
21.81 (0.00)
4.19 (0.16)
5.45 (0.18)
0.0226***
(0.00)
0.0443***
(0.00)
0.9499***
(0.00)
5.3438***
(0.00)
3.95
24.89 (0.00)
26.72 (0.00)
2.49 (0.37)
3.78 (0.38)
0.0243**
(0.03)
0.0463***
(0.00)
0.9472***
(0.00)
5.3734***
(0.00)
3.78
40.85 (0.00)
42.30 (0.00)
5.40 (0.08)
6.70 (0.10)
0.0103
(0.19)
0.0535***
(0.00)
0.9421***
(0.00)
9.4217***
(0.00)
2.34
8.22 (0.01)
9.54 (0.01)
6.52 (0.05)
9.07 (0.03)
0.0128
(0.35)
0.0553
(0.03)**
0.9390***
(0.00)
10.4000***
(0.00)
2.26
9.03 (0.00)
10.80 (0.01)
4.55 (0.13)
6.42 (0.12)
0.0091**
(0.01)
0.0479***
(0.00)
0.9479***
(0.00)
10.1666***
(0.00)
2.25
9.95 (0.00)
12.48 (0.00)
5.56 (0.08)
7.04 (0.08)
Note: ***, **, * indicate significance levels of 1%, 5%, and 10% respectively. Rounded P-values in parenthesis.
67
A5.2 GARCH (1,1) – Denmark, USA and Precious Metals
Mean Equation:
Variance Equation:
Country
Denmark
(DK)
a0
a1
b1
shape
Ljung-Box
ARCH LM
Usa ($)
VL
Lag 10
Lag 45
Lag 5
Lag 7
a0
a1
b1
shape
Ljung-Box
ARCH LM
VL
Lag 10
Lag 45
Lag 5
Lag 7
0.0195***
(0.00)
0.1032***
(0.00)
0.8820***
(0.00)
7.6998***
(0.00)
1.33
114.40
(0.00)
118.60
(0.00)
3.53
(0.22)
5.06
(0.22)
0.0061
(0.17)
0.0671***
(0.00)
0.9290***
(0.00)
7.6689***
(0.00)
1.58
2.56
(0.18)
8.15
(0.02)
0.14
(0.97)
0.47
(0.98)
Gold
0.0059**
(0.03)
0.0493***
(0.00)
0.9454***
(0.00)
8.1604***
(0.00)
1.15
0.37 (0.76)
0.94 (0.87)
0.61 (0.85)
1.15 (0.89)
0.0015*
(0.06)
0.0453***
(0.00)
0.9536***
(0.00)
5.5676***
(0.00)
1.53
0.64 (0.63)
1.28 (0.79)
2.83 (0.32)
3.48 (0.43)
Silver
0.0235
(0.49)
0.0511
(0.16)
0.9440***
(0.00)
4.9024***
(0.00)
4.92
23.36 (0.00)
24.81 (0.00)
3.78 (0.19)
4.88 (0.24)
0.0141***
(0.00)
0.0401***
(0.00)
0.9572***
(0.00)
5.3018***
(0.00)
5.60
33.83 (0.00)
34.93 (0.00)
2.81 (0.31)
3.88 (0.37)
Platinum
0.0109
(0.16)
0.0526***
(0.00)
0.9425***
(0.00)
9.5798***
(0.00)
2.26
8.56 (0.00)
9.63 (0.01)
5.63 (0.07)
7.70 (0.06)
0.0083
(0.22)
0.0595***
(0.00)
0.9380***
(0.00)
7.9965***
(0.00)
3.46
1.23 (0.43)
2.13 (0.59)
3.59 (0.22)
4.59 (0.27)
Note: ***, **, * indicate significance levels of 1%, 5%, and 10% respectively. P-values are in parenthesis and
are rounded up to two decimal places.
68
A5.3 Basic Diversification Properties of Precious Metals, Data for March 1988–August 2016
Mean Equation:
Variance Equation:
Gold
Silver
A: Datastream Finland Total Market Return as market proxy
0.0093
(0.28)
0.0130
(0.36)
αPM
0.0243 ***
(0.00)
0.0712 ***
(0.00)
βPM
0.0056 **
(0.04)
0.0233
(0.46)
a0
0.0511 ***
(0.00)
0.0506
(0.13)
a1
***
***
0.9442
(0.00)
0.9444
(0.00)
b1
7.9738 ***
(0.00)
4.8705 ***
(0.00)
Shape
1.22
4.54
VL
B: Datastream Sweden Total Market Return as market proxy
0.0109
(0.24)
0.0162
(0.28)
αPM
-0.0531 ***
(0.00)
0.0110
(0.47)
βPM
0.0062 **
(0.03)
0.0226 ***
(0.00)
a0
0.0476 ***
(0.00)
0.0443 ***
(0.00)
a1
0.9469 ***
(0.00)
0.9498 ***
(0.00)
b1
9.5907 ***
(0.00)
5.3704 ***
(0.00)
Shape
1.15
3.92
VL
C: Datastream Norway Total Market Return as market proxy
0.0044
(0.61)
0.0072
(0.61)
αPM
***
0.0066
(0.60)
0.0693
(0.00)
βPM
0.0057 **
(0.02)
0.0247 *
(0.05)
a0
***
***
0.0493
(0.00)
0.0464
(0.00)
a1
0.9459 ***
(0.00)
0.9469 ***
(0.00)
b1
8.8956 ***
(0.00)
5.4179 ***
(0.00)
Shape
1.21
3.70
VL
D: Datastream Denmark Total Market Return as market proxy
0.0086
(0.33)
0.0081
(0.57)
αPM
0.0220 *
(0.09)
0.1171 ***
(0.00)
βPM
**
0.0059
(0.03)
0.0231
(0.23)
a0
0.0495 ***
(0.00)
0.0490 **
(0.01)
a1
***
***
0.9453
(0.00)
0.9457
(0.00)
b1
8.1881 ***
(0.00)
4.9771 ***
(0.00)
Shape
1.15
VL
E: Datastream USA Total Market Return as market proxy
0.0092
(0.20)
0.0172
(0.24)
αPM
-0.0494 ***
(0.00)
0.0309
(0.25)
βPM
0.0015 *
(0.06)
0.0143 ***
(0.00)
a0
***
***
0.0456
(0.00)
0.0400
(0.00)
a1
0.9533 ***
(0.00)
0.9573 ***
(0.00)
b1
***
***
5.5356
(0.00)
5.3323
(0.00)
Shape
1.54
5.41
VL
Platinum
0.0116
0.0793
0.0102
0.0531
0.9424
9.5612
2.22
0.0136
0.0052
0.0127
0.0553
0.9390
10.4051
2.27
0.0064
0.0572
0.0091
0.0485
0.9473
10.3998
2.24
0.0081
0.1062
0.0100
0.0511
0.9444
9.6345
2.28
0.0142
0.0395
0.0084
0.0594
0.9379
8.0017
3.33
***
***
***
(0.32)
(0.00)
(0.16)
(0.00)
(0.00)
(0.00)
**
***
***
(0.27)
(0.71)
(0.35)
(0.02)
(0.00)
(0.00)
***
**
***
***
***
(0.58)
(0.00)
(0.02)
(0.00)
(0.00)
(0.00)
***
*
***
***
***
(0.49)
(0.00)
(0.09)
(0.00)
(0.00)
(0.00)
***
**
***
***
***
(0.23)
(0.01)
(0.23)
(0.00)
(0.00)
(0.00)
Note: ***, **, * indicate significance levels of 1%, 5%, and 10% respectively. Robust standard errors are
used and the rounded P-values are in the brackets.
69
APPENDIX 6 AUTOCORRELATION AND HETEROSCEDASTICITY TESTS
FOR REGRESSION RESIDUALS
A6.1 Results from Ljung-Box and ARCH LM tests for Market Model Residuals
Mean Equation:
Variance Equation:
Gold
Finland (€)
Ljung-Box
ARCH LM
Sweden (SK)
Ljung-Box
ARCH LM
Norway
(NK)
Ljung-Box
ARCH LM
Denmark
(DK)
Ljung-Box
ARCH LM
Usa ($)
Ljung-Box
ARCH LM
Lag 10
Lag 45
Lag 5
Lag 7
Lag 10
Lag 45
Lag 5
Lag 7
Lag 10
Lag 45
Lag 5
Lag 7
Lag 10
Lag 45
Lag 5
Lag 7
Lag 10
Lag 45
Lag 5
Lag 7
0.84
1.66
0.51
1.00
0.69
1.48
0.84
1.10
0.47
1.53
2.74
3.28
0.44
0.97
0.56
1.10
0.65
1.20
2.89
3.66
Silver
(0.55)
(0.70)
(0.88)
(0.91)
(0.61)
(0.74)
(0.78)
(0.90)
(0.71)
(0.73)
(0.33)
(0.46)
(0.72)
(0.87)
(0.87)
(0.90)
(0.63)
(0.81)
(0.31)
(0.40)
23.57
24.65
4.36
5.26
25.15
26.95
2.58
3.87
43.73
45.11
5.64
7.00
26.04
27.31
4.51
5.42
34.20
35.37
2.90
3.91
Platinum
(0.00)
(0.00)
(0.14)
(0.20)
(0.00)
(0.00)
(0.36)
(0.37)
(0.00)
(0.00)
(0.07)
(0.09)
(0.00)
(0.00)
(0.13)
(0.19)
(0.00)
(0.00)
(0.30)
(0.36)
10.87
11.90
6.22
8.56
9.09
10.85
4.51
6.38
11.46
13.85
5.35
6.82
11.64
12.55
5.64
7.74
1.37
2.29
3.73
4.72
(0.00)
(0.00)
(0.05)
(0.04)
(0.00)
(0.01)
(0.13)
(0.12)
(0.00)
(0.00)
(0.08)
(0.10)
(0.00)
(0.00)
(0.07)
(0.06)
(0.39)
(0.55)
(0.20)
(0.25)
Note: The null hypotheses for the tests are “no autocorrelation” (Ljung-Box) and “no heteroscedasticity” (ARCH
LM). All the values in the table are rounded up to two decimal places.
Darker shade indicates rejection of the null hypothesis at a 10 % significance level.
70
A6.2 Results from Ljung-Box and ARCH LM tests for Modified Market Model Residuals
Mean Equation:
Variance Equation:
Gold
Finland (€)
Ljung-Box
ARCH LM
Sweden (SK)
Ljung-Box
ARCH LM
Norway
(NK)
Ljung-Box
ARCH LM
Denmark
(DK)
Ljung-Box
ARCH LM
Usa ($)
Ljung-Box
ARCH LM
Lag 10
Lag 45
Lag 5
Lag 7
Lag 10
Lag 45
Lag 5
Lag 7
Lag 10
Lag 45
Lag 5
Lag 7
Lag 10
Lag 45
Lag 5
Lag 7
Lag 10
Lag 45
Lag 5
Lag 7
0.86
1.69
0.50
0.99
0.65
1.42
0.75
0.95
0.64
1.74
2.92
3.55
0.47
1.04
0.54
1.12
0.71
1.29
2.80
3.55
Silver
(0.54)
(0.69)
(0.88)
(0.91)
(0.63)
(0.76)
(0.81)
(0.92)
(0.63)
(0.68)
(0.30)
(0.42)
(0.71)
(0.85)
(0.87)
(0.89)
(0.60)
(0.79)
(0.32)
(0.42)
23.40
24.45
4.24
5.15
25.34
27.16
2.68
3.88
43.74
45.16
6.03
7.31
26.08
27.35
4.58
5.51
33.99
35.05
3.14
4.13
Platinum
(0.00)
(0.00)
(0.15)
(0.21)
(0.00)
(0.00)
(0.34)
(0.36)
(0.00)
(0.00)
(0.06)
(0.07)
(0.00)
(0.00)
(0.13)
(0.18)
(0.00)
(0.00)
(0.27)
(0.33)
11.13
12.17
6.08
8.51
8.72
10.56
4.19
6.06
11.58
13.85
5.03
6.52
11.64
12.56
5.30
7.50
1.36
2.26
3.76
4.74
(0.00)
(0.00)
(0.06)
(0.04)
(0.00)
(0.01)
(0.16)
(0.14)
(0.00)
(0.00)
(0.10)
(0.11)
(0.00)
(0.00)
(0.09)
(0.07)
(0.40)
(0.56)
(0.20)
(0.25)
Note: The null hypotheses for the tests are “no autocorrelation” (Ljung-Box) and “no heteroscedasticity” (ARCH
LM). All the values in the table are rounded up to two decimal places.
Darker shade indicates rejection of the null hypothesis at a 10 % significance level.
71
APPENDIX 7
R CODE FOR SELECTED PARTS OF THE ANALYSIS
The following section presents the R code that was used to generate results for selected
parts of the analysis in this thesis.
A7.1 R code for the Modified Market Model – Construction of the Volatility
Dummy
We start by loading packages that will be used. Then we choose which country index
and which metal will be analysed. “GoldD” is a vector of continuously compounded gold
returns with length = 7015. “UsaD” is a vector of continuously compounded USA index
returns with length = 7015. Thus the following example is performed on USA index and
gold (in $).
In the actual analysis the following procedure is performed for 5 countries * 3 metals =
15 different permutations.
library(psych)
library(rugarch)
a<-GoldD
b<-UsaD
m<-winsor(a, trim = 0.01, na.rm = TRUE)
i<-winsor(b, trim = 0.01, na.rm = TRUE)
# choose metal
# choose index
# winsorize on 1 % level
Then GARCH(1,1) model is fitted on the USA index in order to obtain a series of
GARCH(1,1) volatility estimates. We save them in a vector “si”.
spec<-ugarchspec(mean.model=list(armaOrder=c(0,0), include.mean = TRUE),
distribution="std")
garchresults<-ugarchfit(spec, i)
si<-garchresults@fit$sigma
# GARCH volatility estimate for the index
Then a dummy “vD” is created: It is equal to one when index GARCH(1,1) volatility
series is greater than two standard deviations above the mean index volatility,
otherwise zero. Then we multiply “vD” with the index return series to get the volatility
dummy “VD”. Volatility dummy is the index return in “risky times”, otherwise zero.
vD<-as.numeric(si > (mean(si)+2*sd(si)))
VD<-vD*i
rD<-as.numeric(i < (mean(i)-2*sd(i)))
RD<-rD*I
# volatility dummy
# return dummy
72
Then a dummy “rD” is created: It is one when index return is more than two standard
deviations lower than the mean index return, otherwise zero. We then multiply “rD”
with the index return to get “RD”. Return dummy “RD” is index return in times of “very
low historical returns”, otherwise zero.
Finally, we perform the analysis according to the Modified Market model equation.
data <- cbind(m,i,VD,RD)
# data frame with variables for the regression
spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder =
c(1, 1), submodel = NULL, external.regressors = NULL, variance.targeting =
FALSE), mean.model = list(armaOrder = c(0, 0), include.mean = TRUE,
external.regressors = matrix(data[,2:4], ncol=3)),distribution.model =
"std")
garch <- ugarchfit(spec=spec,data=data[,1],solver.control=list(trace=0))
garch
# print out estimation results
We also calculate the GARCH long-run average variance.
omega<-garch@fit$matcoef[5,1]
alpha1<-garch@fit$matcoef[6,1]
beta1<-garch@fit$matcoef[7,1]
alpha1+beta1
df<-garch@fit$matcoef[8,1]
VL<-omega/(1-alpha1-beta1)
VL
#
#
#
#
save omega estimate from the regression
save alpha1 estimate from the regression
save beta1 estimate from the regression
check (alpha1+beta1) is below one
# calculate long-run variance
# print out long-run variance
A7.2 R code for the Buy-and-Hold Strategy
Again, we start by loading packages. Then we choose vectors for index and metal
returns and winsorize. The following example is performed on USA index and gold (in
$). “GoldD” is a vector of continuously compounded gold returns with length = 7015.
“UsaD” is a vector of continuously compounded USA index returns with length = 7015.
In the actual analysis the following procedure is performed for 5 countries * 3 metals =
15 different permutations. So all in all, 5 * countries * 3 metals * 12 weights = 180
lambda values are calculated in the Buy-and-Hold strategy.
library(psych)
library(rugarch)
a<-GoldD
b<-UsaD
m<-winsor(a, trim = 0.01, na.rm = TRUE)
i<-winsor(b, trim = 0.01, na.rm = TRUE)
# choose metal
# choose index
# winsorize on 1 % level
73
We fit GARCH(1,1) to both index and metal returns in order to obtain a series of
GARCH(1,1) variance estimates for both return series.
spec<-ugarchspec(mean.model=list(armaOrder=c(0,0), include.mean = TRUE),
distribution="std")
garchresults<-ugarchfit(spec, m)
hm<-garchresults@fit$sigma^2
# GARCH variance estimate for the metal
spec<-ugarchspec(mean.model=list(armaOrder=c(0,0), include.mean = TRUE),
distribution="std")
garchresults<-ugarchfit(spec, i)
hi<-garchresults@fit$sigma^2
# GARCH variance estimate for the index
The following values need to be calculated in order to be able to calculate the Risk-toReward ratio (λ) of the portfolio.
Erm<-mean(m)
Eri<-mean(i)
mhm<-mean(hm)
mhi<-mean(hi)
cor(m,i)
#
#
#
#
#
daily mean of the metal
daily mean of the index
mean of the daily GARCH variance estimates for the metal
mean of the daily GARCH variance estimates for the index
correlation between the metal and the index
Then the Risk-to-Reward ratio is calculated for all 12 different portfolio weights
(“0.01,0.02,0.05,0.1,0.15,0.20,0.3,0.4,0.5,0.6,0.7,0.8”) with a loop.
temp1<-NULL
# initialize vectors
all.l<-NULL
for (a in c(0.01,0.02,0.05,0.1,0.15,0.20,0.3,0.4,0.5,0.6,0.7,0.8))
{wm<-a
# weight of the metal
wi<-1-wm
# weight of the index
# Expected return of the hybrid-metal-market portfolio:
ErHybrid<-wm*Erm+wi*Eri
# Variance the hybrid-metal-market portfolio:
hHybrid<-(wm)^2*mhm+(wi)^2*mhi+2*wm*wi*sqrt(mhm)*sqrt(mhi)*cor(m,i)
l<-(ErHybrid/hHybrid)/(Eri/mhi) # calculate lambda for a specific weight
temp1<-l
# save it to temp1
all.l<-c(all.l, temp1)
# add it to a data frame with the others
}
all.l
# print out lambdas for 12 different portfolio weights
74
A7.3 R code for the Switching Strategy
The following example is only performed for one country, one metal and one weight. In
the actual analysis the procedure is repeated for 5 countries * 3 metals * 12 portfolio
weights = 180 different permutations. The following example is executed on Swedish
index and gold returns (both nominated in SK) for the portfolio weight of 1 %.
We start by loading packages that will be used. Then we choose which country index
and which metal will be analysed. “GoldSK” is a vector of continuously compounded
gold returns with length = 7015. “SweSK” is a vector of continuously compounded
Swedish index returns with length = 7015.
library(tseries)
library(evir)
library(fGarch)
library(psych)
a<-GoldSK
b<-SweSK
m<-winsor(a, trim = 0.01, na.rm = TRUE)
i<-winsor(b, trim = 0.01, na.rm = TRUE)
wm<-0.01
wi<-1-wm
# choose metal
# choose index
# winsorize
# select weight for the metal
# weight for the index
The next step of the procedure is preformed inside a loop. In the loop the two steps
described below are repeated each day in the simulation. The loop repeats the analysis
7015 historical observations - 500 observations (formation period) = 6515 times.
1) We first run the GARCH on past 500 index observations to see if we get a signal
that “times are abnormally risky” (compared to the past 500 observations). If
we get a “switch signal” we decide to have a weighting “wm” in metal on that
day. If we get a signal that “times are not abnormally risky” (compared to the
past 500 observations) we just hold the index on that day.
2) We then calculate the Risk-to-Reward ratio (λ) for today (to investigate how
good our “switching strategy” did today). To obtain lambda we first need to
calculate the “ingredients” in the Risk-to-Reward equation. If we got a signal in
step one (based on past 500 index observations) that “times are abnormally
risky”, then the hybrid portfolio return for today is the realized historical return
for today on a Buy-and-Hold portfolio with a “wm” weighting in metal and “wi”
weighting in index. If we got a signal that “times are not abnormally risky”
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(compared to the past 500 observations), then the realized hybrid portfolio
return for today is just the index return (since we didn’t get a “switch signal”
yesterday and ended up just holding the index for today). The hybrid portfolio
variance for today is constructed similarly. If we got a “switch signal” yesterday,
it is equal to the GARCH variance estimate on a Buy-and-Hold portfolio (with a
“w” weighting in metal and “wi” weighting in index) for today. If we did not get
a switch signal yesterday, the hybrid portfolio variance for today is equal to the
GARCH volatility estimate for the index today (since we decided to just hold the
index yesterday).
We start by initializing empty vectors that will be needed inside the loop to collect
results as it runs over and over again each “new” day and produces more values:
all.c <- NULL
all.hh <- NULL
all.index <- NULL
all.hi <- NULL
temp1 <- NULL
temp2 <- NULL
temp3 <- NULL
temp4 <- NULL
Step 1
We fit GARCH(1,1) to index returns in order to obtain a series of GARCH(1,1) variance
estimates. “si” is index volatility estimates for past 500 observations of index. (To
clarify, the first day in the simulation that λ is calculated for is actually observation
number 501 that we refer to as “today”, and the preceding 500 observations are
considered to be “in the past” at that point.)
# beginning of the loop
for(i in 1:6516){ # run GARCH on past 500 observations of the index:
garchresultsi<-garchFit(formula = ~ garch(1, 1), data = index[i:(499+i)],
cond.dist = c("std"), include.mean=TRUE,trace=FALSE)
si<[email protected]
# length 500
Then, based on index GARCH(1,1) volatility series we just obtained for the past 500
observations, we create a dummy “vD” with length = 500. It is equal to one when index
GARCH(1,1) volatility series is greater than two standard deviations above the mean
index volatility, otherwise zero.
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vD<-as.numeric(si > (mean(si)+2*sd(si)))
# also create an exact opposite of the dummy “vD”:
reversevD<-as.numeric(si <= (mean(si)+2*sd(si)))
# length 500
# length 500
We also create a dummy “reversevD” that is exactly the opposite of “vD”. It is one when
“vD” is zero and zero when “vD” is one. So it is one when “volatility is low” and zero in
times of “high volatility”. “vD” indicates times when we want to hold precious metals
(“switch signal”). “reversevD” indicates times when we do not want to hold metals in
our portfolio.
Step 2
Then we calculate the “ingredients” for the lambda equation. We first calculate “erp”
which is realized return on a Buy-and-Hold portfolio (with a “w” weighting in metal
and “wi” weighting in index) for today. Then we calculate “c”, which is realized hybridportfolio return for day today. The realized hybrid-portfolio return for today depends
on if we got a “switch signal” in step one or not.
1) “c” is equal to “erp” (the realized return on a Buy-and-Hold portfolio today) if
“vD[500]” is 1 (latest past volatility observation is “abnormally high”).
2) “c” is equal to “(index[499+i+1])” (the realized return on the index today) if
“reversevD[500]“ is 1. And then of course “vD[500]” is 0, which means latest
past volatility observation is “not abnormally high”.
erp<-wm*metal[499+i+1]+wi*index[499+i+1]
# length 1
# “observation number [499+i+1] of erp”
c <- vD[500]*erp+reversevD[500]*index[499+i+1]
# length 1
Then we fit GARCH(1,1) to the index and metal again but this time we move one time
step forward. In step one we fit GARCH on past observations [i:(499+i)] to see if we get
a “switch signal”. Now we fit GARCH on observations [(i+1):(499+i+1)] so that we can
calculate the lambda for today (to investigate how good our “switching strategy” did
today). So we fit GARCH on past 499 observations plus today.
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garchresultsi2<-garchFit(formula = ~ garch(1,
index[(i+1):(499+i+1)], cond.dist = c("std"),
include.mean=TRUE,trace=FALSE)
garchresultsm2<-garchFit(formula = ~ garch(1,
metal[(i+1):(499+i+1)], cond.dist = c("std"),
include.mean=TRUE,trace=FALSE)
hi<[email protected]^2
hm<[email protected]^2
1), data =
# GARCH to index
1), data =
# GARCH to metal
# length 500
# length 500
Index and metal variance estimates are stored in vectors “hi” and “hm” respectively.
“hi” is the index variance estimates for past 499 observations plus today. “hm” is the
metal variance estimates for past 499 observations plus today. The variance of a Buyand-Hold portfolio “hp” for today is then calculated with the latest values of “hi” and
“hm” as inputs.
# “hp” is GARCH variance estimate on a Buy-and-Hold portfolio (with a “w”
# weighting in metal and “wi” weighting in index) for today:
hp<
(wm)^2*hm[500]+(wi)^2*hi[500]+2*wm*wi*sqrt(hm[500])*sqrt(hi[500])*cor(meta
l[(i+1):(499+i+1)],index[(i+1):(499+i+1)])
#length 1
With the help of “hp” we can then calculate the hybrid portfolio variance for today “hh”.
“hh” is equal to “hp”, if “vD[500]” is 1 (latest past volatility observation was
“abnormally high”). “hh” is equal to “hi[500]”, if “reversevD[500]“ is 1 (latest past
volatility observation is “not abnormally high”).
hh<-vD[500]*hp+reversevD[500]*hi[500]
#length 1
Now we have produced all the “ingredients” for calculating the Risk-to-Reward ratio.
temp1
temp2
temp3
temp4
<<<<-
c
# save “realized hybrid-portfolio return for day today”
hh
# save hybrid portfolio variance for today
index[499+i+1]
# save index return for today
hi[500]
# save index GARCH variance estimate for today
all.c <- c(all.c, temp1)
all.hh <- c(all.hh, temp2)
all.index <- c(all.index, temp3)
all.hi <- c(all.hi, temp4)
} #end "for" loop
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The results from this one iteration are saved and the loop then starts again from the
beginning. It runs over and over again (6515 iterations), each time moving one time
step ahead. It stops when the final historical observation in the index return series is
reached.
After the loop has finished, we then save the results from all the 6515 iterations in a
data frame. “GoldSK.SweSK.01” is a data frame with 4 variables that are required for
calculating the Risk-to-Reward ratio. There are 7015-500 = 6515 observations of each
variable, since 500 is the length of the rolling window. Therefore, the first 500 index
and metal observations are needed as inputs for the first iteration of the process, which
is why there will be only 6515 lambdas produced.
GoldSK.SweSK.01 <- data.frame(all.c, all.hh, all.index, all.hi)
# set working directory to save results in the correct place
setwd("File Location Omitted")
write.csv(GoldSK.SweSK.01, file="GoldSK.SweSK.01.csv") # write a CSV file
The results are saved in a cvs-file (file location is omitted in the code) and then
calculated according to the Risk-to-Reward ratio equation in to a vector of lambda
values. Then cases when lambda is not finite are removed. Lambda is for example not
finite in such days of the switching strategy when index return was zero, since then we
would divide by zero in the Risk-to-Reward equation for that day. Also outliers are
removed with the help of plots and boxplots. Finally, we take the average of the vector
of the remaining lambda values to get one single lambda value for the switching
strategy for the portfolio weight of 1 % gold with the Swedish index.
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Calculating the lambda
Below is an example on how the “lambda” vector was checked for outliers and cases
when it is not finite. We first retrieve the csv-file containing the results from the loop.
# set working directory to retrieve the data frames from the correct place
setwd("File Location Omitted")
# get the 12 data frames produced in the “switching strategy”
# 6515 rows and 4 columns in each of the 12 data frames
# one data frame per each weight (12 weights in total):’
GoldSK.SweSK.80 = read.csv("GoldSK.SweSK.80.csv", header = TRUE)
GoldSK.SweSK.70 = read.csv("GoldSK.SweSK.70.csv", header = TRUE)
GoldSK.SweSK.60 = read.csv("GoldSK.SweSK.60.csv", header = TRUE)
GoldSK.SweSK.50 = read.csv("GoldSK.SweSK.50.csv", header = TRUE)
GoldSK.SweSK.40 = read.csv("GoldSK.SweSK.40.csv", header = TRUE)
GoldSK.SweSK.30 = read.csv("GoldSK.SweSK.30.csv", header = TRUE)
GoldSK.SweSK.20 = read.csv("GoldSK.SweSK.20.csv", header = TRUE)
GoldSK.SweSK.15 = read.csv("GoldSK.SweSK.15.csv", header = TRUE)
GoldSK.SweSK.10 = read.csv("GoldSK.SweSK.10.csv", header = TRUE)
GoldSK.SweSK.05 = read.csv("GoldSK.SweSK.05.csv", header = TRUE)
GoldSK.SweSK.02 = read.csv("GoldSK.SweSK.02.csv", header = TRUE)
GoldSK.SweSK.01 = read.csv("GoldSK.SweSK.01.csv", header = TRUE)
Then we calculate a vector of lambda values.
lambda01<((GoldSK.SweSK.80$all.c)/(GoldSK.SweSK.80$all.hh))/((GoldSK.SweSK.80$all.i
ndex)/(GoldSK.SweSK.80$all.hi))
# length 6515
max(lambda80)
# see if the max and min values look very low/high
min(lambda80)
Next we inspect the series with different tools. We set it to be finite and remove
extreme outliers. For example:
plot(lambda80)
# plot lambda vector and for example
plot(lambda80, ylim=c(-3,5))
# try different limits
plot.ts(lambda80, main=plot.ts(lambda80))
# plot
lambda80NoJunk=lambda80[is.finite(lambda80)] # set lambda to be finite
out=boxplot(lambda80NoJunk)
# see boxplot
boxplot(lambda80, main=plot(lambda80), ylim=c(-3,5))
# try limits
We choose suitable limits for lambda and thus remove outliers that are outside those
limits. Finally, we take the mean of the lambda vector. “laJ80” is then the lambda value
for Gold and Swedish index for this portfolio weight.
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lambda80NoJunk=lambda80NoJunk[(lambda80NoJunk< 5) & (lambda80NoJunk > -3)]
summary(lambda80NoJunk)
# see summary of the “cleaned” lambda vector
summary(lambda80)
# compare to lambda vector before “cleaning”
length(lambda80NoJunk)
# see how many observations were removed
length(lambda80)
length(lambda80)-length(lambda80NoJunk)
plot.ts(lambda80NoJunk,ylab="Lambda for Finland...");grid()
laJ80<-mean(lambda80NoJunk) # mean of the vector of lambdas
laJ80
# print lambda (length 1)