post bretton woods thesis
TRANSCRIPT
Department of Economics School of Business, Economics and Law at University of Gothenburg Vasagatan 1, PO Box 640, SE 405 30, Göteborg, Sweden +46 37 786 0000, +46 31 786 1326 (fax) www.handels.gu.se infohandels.gu.se
Bachelor Thesis 2011 (15 ECTS)
Department of Economics
Authors:
Roderick Nilsson 881221 [email protected] Victor Rohlin 861110 [email protected]
Tutor: Lars-Göran Larsson ([email protected])
Economics Spring 2011
Post Bretton Woods study of an alternative efficient portfolio using
CAPM.
2
Post Bretton Woods study of an alternative efficient
portfolio using CAPM.
By
Roderick Nilsson Victor Rohlin
June 2011
Abstract
As global markets are flooded by fiat currencies since the discontinuation of Bretton Woods by Nixon
in 1971, we seek to identify the optimal alternative portfolio by applying financial theory, primarily
CAPM and statistical measurements. The alternative portfolio constructed consists of precious
metals, foreign exchange rates, commodities and stock indices measured with prices on a monthly
basis. During the time period 1971-2011 precious metals and oil displayed high returns and risk levels
while foreign exchange rates provided very little return but low risk. The designed optimal risky
portfolio outperformed all other assets on a risk-return basis with a return of almost two percent per
month on average with a volatility level of ten percent. In addition this portfolio provided the
investor with a significant alpha return on average 1,2% every month. Thus any arbitrary risk-willing
rational U.S. investor would choose to passively invest in the optimal risky portfolio between 1971
and 2011. Furthermore we identified strong linear relationships in the foreign exchange rates.
Keywords: CAPM, Quantitative Analysis, Market Portfolio, Bretton Woods, Gold, Silver, Bear Market,
Bull Market, Fiat Money, Foreign Exchange Rates, Correlation and Alternative Historical Investments.
JEL Classification: G10 & G11.
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Acknowledgments Lars-Göran Larsson: For steering us in the right direction and for his excellent support.
Joakim Lennartsson, Librarian at the Economics Library, Gothenburg: For pinpointing
essential databases, saving a lot of time.
Dr. Jianhua Zhang, University of Gothenburg: For help during labs in Portfolio Investments.
Federal Reserve Bank of St. Louis: For providing excellent, easy to use data.
Federal Reserve Board of Governors: For providing excellent, easy to use data.
Dow Jones: For providing excellent, easy to use data.
Thomson Reuters: For providing us with easy to access data.
Table of Contents
Post Bretton Woods study of an alternative efficient portfolio using CAPM. ........................................ 2
Abstract ................................................................................................................................................... 2
Acknowledgments ................................................................................................................................... 3
Introduction ............................................................................................................................................. 5
Background .......................................................................................................................................... 5
Aim of thesis ........................................................................................................................................ 6
Limitations and demarcations of study ............................................................................................... 6
Methodology ........................................................................................................................................... 6
Literature Review .................................................................................................................................... 7
Capital Asset Pricing Model (CAPM) .................................................................................................... 7
Assumptions of CAPM ..................................................................................................................... 7
Criticisms and limitations to CAPM ................................................................................................. 7
Expected return ................................................................................................................................... 8
Risk level & Beta-value ........................................................................................................................ 8
Financial ratios .................................................................................................................................... 9
Jensen’s alpha .................................................................................................................................. 9
Variance & Covariance Matrix (VCM) & Correlation Matrix ............................................................... 9
Minimum Variance Portfolio (MVP) .................................................................................................... 9
Optimal Risky Portfolio (ORP).............................................................................................................. 9
Capital market line (CML) .................................................................................................................. 10
Data ....................................................................................................................................................... 11
Stock indices ...................................................................................................................................... 11
Dow Jones Industrial Average ....................................................................................................... 11
S&P 500 COMPOSITE - PRICE INDEX (U$) ...................................................................................... 11
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Commodities ..................................................................................................................................... 12
Gold Bullion LBM (U$ per Troy ounce) .......................................................................................... 12
Silver Fix LBM (U$ per Troy ounce) ............................................................................................... 12
Oil Spot Price: West Texas Intermediate (U$ per barrel) .............................................................. 12
Wheat U.S. Cash price at principal markets (U$ per bushel) ........................................................ 12
Foreign exchange rates ..................................................................................................................... 13
Foreign exchange rate JPY/USD .................................................................................................... 13
Foreign exchange rate CHF/USD ................................................................................................... 13
Foreign exchange rate GBP/USD ................................................................................................... 13
Interest rates ..................................................................................................................................... 14
3 Month U.S. Treasury (risk free rate) ........................................................................................... 14
Results ................................................................................................................................................... 14
Return and risk .................................................................................................................................. 14
Historical returns ........................................................................................................................... 15
VCM & Correlation Matrix ................................................................................................................. 17
Portfolios ........................................................................................................................................... 17
Financial ratios .................................................................................................................................. 18
Summary and conclusions ..................................................................................................................... 20
Summary ........................................................................................................................................... 20
Conclusions ........................................................................................................................................ 20
Suggestions for further studies ......................................................................................................... 21
Bibliography ........................................................................................................................................... 22
Literature: .......................................................................................................................................... 22
Web & Data Sources:......................................................................................................................... 22
Tables .................................................................................................................................................... 23
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Introduction As global markets are flooding in paper currencies without real backing, so called fiat currencies,
since the end of Bretton Woods by Nixon in 1971 we seek the optimal historical alternative portfolio
by applying financial theory and statistical modeling. The alternative portfolio constructed consists of
precious metals, foreign exchange rates, commodities and stock indices.
Background From the beginning of time the main fuel for the global economy has been trade. In order for trade
to occur exchanges of goods and/or services must take place. Historically an advanced trading
environment without bartering requires a monetary medium to enable price discovery and liquefy
the market. Thus facilitating trade of goods and services requires a monetary medium that grants its
holder future value, i.e. purchasing power when traded for e.g. a good. Evidently the currency must
be limited if not scarce and universally tradable with perpetual demand if it should be able to store
purchasing power over time.
Presently the financial world is in disarray with governments and large institutions receiving
tremendous amounts of credit and bailouts with huge deficits as a result. The current monetary
regime began after the Nixon shock in 1971 when the U.S. discontinued the redeemability of an
American dollar into gold, thus formally ending the dollar’s scarcity as the Bretton Woods treaty
ended. As money supply increased over time through quantitative easing, credit growth and deficit
spending, financial markets developed and prices adjusted thus potentially weakening the currency’s
confidence - the ability to store value. Historically, gold and other precious metals have been safe
storages of value. Though they offer no interest in return their inherent scarcity and high demand in
combination with global confidence provides stable purchasing power over time.
Since 1971 the money supply, measured by M2 Money stock, as depicted above increased over
1800%, thus indicating an enormous increase in the supply of U.S. Dollars in circulation. This measure
Figure 1: U.S. Money supply (M2 Measure)
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includes a broader set of financial assets held principally by households. (M2 | Federal Reserve Bank
of St. Louis, 2011)
In modern times an Investor has vast amounts of investment opportunities available and it has never
been easier to invest in different securities, e.g. stocks, indices, bonds, commodities and foreign
exchange rates. Therefore the purpose of this thesis is to evaluate portfolio allocations from the
discontinuation of Bretton Woods in order to maximize return and optimize risk exposure for an
arbitrary rational risk-willing U.S. investor.
Aim of thesis The objective is to evaluate portfolio allocations between 1971 and 2011 in order to maximize return
and optimize risk exposure for an arbitrary rational risk-willing U.S. investor with hindsight, given the
plethora of investment opportunities available to investors today.
Limitations and demarcations of study For the purpose of this study it is assumed that all assumptions inherent to the CAPM model are fully
satisfied. Inflation will not be taken into account. The portfolio created and all assets evaluated will
be priced in dollars. Neither counterparty nor interest risk will be accounted for when setting a risk
free benchmark rate. Selected time period stretches from 1971 to 2011 and all investment
opportunities available in 2011 are assumed to be available throughout the whole period.
Regarding portfolio constructions the total portfolio units must equal (+1) and each asset is limited to
its weight being at least (-1) unit, therefore the modeled portfolios will account for the ability of
short selling assets. As markets are assumed to have a mean efficient variance, the efficient market
theorem as a whole is presumed to be fulfilled. Furthermore the portfolios are passive in their nature
and will therefore not be reallocated during the selected period.
Methodology Being financial students at the Economics department at the School of Business, Economics and law
grants us theoretical insights and understanding of portfolio theory and its practical applications.
Courses, such as econometrics and portfolio investments have been undertaken adding further
knowledge and a deeper interest of finance. Skills acquired include financial analysis using e.g. CAPM,
calculating and analyzing various financial/statistical key performance indicators.
Thru financial quantitative analysis on the empirical data acquired for all of the selected investments,
ranging from indices to precious metals and global currencies. The objective is to reveal the
historically optimal risky portfolio. Applying financial theory, primarily the CAPM model, and
statistical methods will enable the calculation of the optimal risky portfolio. Accordingly performed
by solving for the maximal return in relation to risk exposure, as found in the modern portfolio
theory. In addition the minimum variance portfolio, i.e. the portfolio with maximized diversification
effect, will also be solved for in order to find the efficient frontier of the portfolio and thus enabling
valuations with the benchmark market via the capital market line. All calculations will be performed
by using spreadsheet programming in Microsoft Excel with the analysis toolpak kit and the solver
add-in for statistical calculation respectively for finding the optimal portfolio weightings.
Ultimately this analysis will result in an optimal risky portfolio, revealing how an U.S. investor should
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have passively invested since the U.S. discontinued the monetary system Bretton Woods in order for
the rational risk-willing investor to maximize return in terms of risk.
Literature Review The analysis will be based upon the CAPM model. This model was developed by Sharpe, Lintner and
Mossin in several articles published in 1964, building on Markowitz foundation of modern portfolio
theory from 1952. (Bodie, 2008)
Capital Asset Pricing Model (CAPM) CAPM is commonly used to obtain the expected return on an asset when it is incorporated in a well-
diversified portfolio. CAPM suggests that the relationship between an asset’s expected return and its
beta is positively related. Also CAPM is implying that market return beats the risk-free return because
investors doubtlessly want reward for taking risk. (Bodie, 2008)
Assumptions of CAPM Perfect competition exists, i.e. investors are price-takers, none of the individual investments affect security prices
All investments are held for identical time periods
There are no transaction costs or taxes
Investors are able to borrow and lend at the risk-free rate
Identical information and expectations for all investors, e.g. same probability and variance
Investors are risk-averse
Assets are infinitely divisible
Investors are mean variance optimizers
Criticisms and limitations to CAPM
Due to the several assumptions behind CAPM it has naturally been challenged and criticized for not
being a model holding up to the scrutiny of the real world, as the assumptions are almost impossible
for the real market to fulfill. E.g. it does not account for interest rate risk, counter-party risk,
transactional costs or inflationary effects.
Consequently CAPM has been vigorously tested by researches since its inception and inconsistent
results have been acquired. Using data from 1930 - 1960 evidence was found that suggested that the
average return of a portfolio was positively linked to its beta. On the contrary, Farma and French
found no relationship between average return and beta while analyzing a dataset over the period
1963-1990. (Hillier, Ross, Westerfield, Jaffe, & Jordan, 2009)
Ultimately the CAPM model can provide a good benchmark if investors hold highly diversified
portfolios with mostly passive investments assuming that correlations between assets remain fixed.
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Therefore the selected time period is of utmost importance in order to create a reliable model.
Expected return In order to properly evaluate an asset the actual return has to be calculated. It is calculated by taking
the difference in price 2 consecutive periods and put it in relation to the former period, thus resulting
in the nominal return. Forecasting the expected return for an arbitrary period is calculated by the
average return of all return datasets for each asset. (Bodie, 2008)
Risk level & Beta-value In financial theory, the risk is the probability that the actual return will differ from the expected
return. It can be measured in various ways but the most common is by the standard deviation of an
asset, which is retrieved by taking the square root of the variance. The standard deviation shows not
only downside risk but also returns that exceed the expected return. (Bodie, 2008)
Beta depicts the relationship of the asset or portfolio with the market as a whole. It measures the
variance that cannot be removed by diversification, i.e. the systemic risk. (Hillier, Ross, Westerfield,
Jaffe, & Jordan, 2009)
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Financial ratios A key feature in measuring risk-return ratios is by measuring how much an extra unit of excess return
will cost in terms of added risk for that very asset. Two major ratios are used; The Sharpe
ratio measures risk by the standard deviation of the asset, the Treynor ratio uses the systematic risk,
the asset’s beta as the risk factor. (Bodie, 2008)
A larger ratio is preferable in both cases.
Jensen’s alpha
Alpha is a measured value of the excess return for a portfolio given risk exposure, asset correlation
and the risk free rate. Created by M. Jensen (1968) it is the main indicator of investor performance
compared to market.
Variance & Covariance Matrix (VCM) & Correlation Matrix Calculations of the VCM and the correlation matrix are done with Excel analysis toolkit and are used
to construct the portfolios. Variances, i.e. the inherent covariance of a single asset, appear along the
diagonal and covariances between different assets appear in the off-diagonal sections. (Bodie, 2008)
Minimum Variance Portfolio (MVP) The MVP achieves the greatest power of diversification. It is constructed by weighing all the
individual assets in order to attain the portfolio with the greatest return to lowest risk. The MVP will
according to theory and empirical evidence display a lower standard deviation - risk - than any of the
individual assets. Finding the individual weights is done by equaling all the assets in the minimum
variance portfolio to have the same covariance - thus, the same contribution to variance. Therefore
the asset with the smallest weight has the largest contribution per unit weight and should be
dropped first. (Bodie, 2008)
Optimal Risky Portfolio (ORP) ORP achieves the greatest return in relation to amount of risk taken. Thus a rational and risk-willing
investor would combine a set off assets in order to acquire higher return and benefit from the
diversification effect that lowers the portfolio risk level. Calculating the weighting of assets in a
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composite portfolio enables the construction of a portfolio efficient frontier, i.e. a linear relationship
between risk and return (see figure 1). (Bodie, 2008)
Capital market line (CML) CML is the line of an investment opportunity set that illustrates the risk-return combinations
available to investors when combining a market portfolio and a risk free asset. The slope of the CML
has the coefficient of the Sharpe ratio starting off at the risk free rate. All portfolios on CML
represent the highest possible Sharpe ratio thus maximizing return in terms of risk. (Bodie, 2008)
Figure 2: The Capital Market Line (CAPM)
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Data The data used in this study is collected from various sources depending on the nature of the
commodity. Included are four major commodities, gold, silver, oil and wheat, all whose price is
denominated in U.S. dollars. As well as three of the world’s most widely traded exchange rates, the
Japanese Yen, the Swiss Franc and the British Pound, all directly paired with the U.S. dollar. Also
included are the assumed risk free asset, the 3-month U.S. Treasury bill and the Dow Jones Industrial
Average. The selected benchmark is the S&P 500 due to its specification and linkage to high market
cap equities, the largest 500 companies - thus creating a suitable proxy for the overall market.
Selected data consists of monthly observations stretching 40 years from January 1971 to January
2011, with oil as an exception as oil data ranges from the end of December 1970 to the end of
December 2010, which is assumed to be equal the following period (end of Dec 70 ~ beginning Jan
71). 1971 being a suitable start off point due to the U.S. suspension and later discontinuation of the
Bretton Woods system where they no longer could guarantee the redeemability of a dollar into gold,
i.e. the dollar became a fiat currency. (IMF, 2011)
Stock indices
Dow Jones Industrial Average
The Dow Jones I. A. is one of the oldest market indices in U.S. starting as a 12-stock industrial average
and currently grown to a 30-stock average consisting mostly of manufacturers of industrial and
consumer goods but also companies in the entertainment and financial industry as well as companies
in the information technology business. At the present time it is not calculated as a standard average
but as a price-weighted average that also takes the effects of stock splits into consideration. It is the
highest quoted index in all media. (CME Group Index Services, LLC, 2011)
Data, monthly end prices due to limitations assumed to be same as following months start price, for
the period 1970-12-31 - 2010-12-31 was acquired from CME Group. (*CME Group Index Services,
LLC, 2011)
S&P 500 COMPOSITE - PRICE INDEX (U$)
The S&P 500 has its focus on large cap companies traded on NYSE or NASDAQ. Since 1957 it has
provided a free-float capitalization-weighted index containing 500 leading U.S. companies in principal
industries. It reflects approximately 75% of the U.S. equities market and is a leading barometer in the
market due to its wide range of companies and sectors, not only including value stocks but also
growth stocks. Due to the characteristics of the index it is the benchmark. (Standard & Poor's, 2011)
Data, monthly start prices for the period 1971-01-01 - 2011-01-01 was acquired from Reuters
through DataStream. (Thomson Reuters , 2011)
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Commodities
Gold Bullion LBM (U$ per Troy ounce)
Gold has historically been regarded as a monetary medium and also as a safe haven for investors
when times have been unstable due to economic or political uncertainty. It is a frequently traded
precious metal and is sometimes almost thought of as a currency because of how it behaves in the
market, i.e. as a safe storage of purchasing power. Since 1919 markets have used The London Gold
Fixing as a pricing medium to price gold products and derivatives. The price is set two times a day by
the five members of the London Gold Market Fixing LTD and is determined by speculation thru
supply and demand in the market. (The London Gold Market Fixing Ltd, 2011)
Data, monthly start prices for the period 1971-01-01 - 2011-01-01 was acquired from Reuters
through DataStream. (Thomson Reuters , 2011)
Silver Fix LBM (U$ per Troy ounce)
As gold, silver is and has been a major tradable precious metal for investors for a very long time. It
usually tracks the changes in gold price with high correlation. However, inherently it is more volatile
than gold, mostly because of lower liquidity in the market. Due to lack of physical demand for the
commodity, large investors and speculators are able to influence the price. The price of silver is
determined almost in the same way as gold. The difference is that only three members of The
London Silver Market Fixing LTD set the price, and only once every day. (The London Silver Market
Fixing Ltd, 2005)
Data, monthly start prices for the period 1971-01-01 - 2011-01-01 was acquired from Reuters
through DataStream. (Thomson Reuters , 2011)
Oil Spot Price: West Texas Intermediate (U$ per barrel)
Oil has been drilled and used for centuries but it was not until the oil boom in the late 19th century
that it earned the status it has today. Because of its easy transportability and high energy density it
quickly became one of the most important energy sources man has come across. It is often referred
to as the “black gold”. The three main crude oil benchmarks are West Texas Intermediate (WTI),
refined in the Midwest and Gulf Coast regions, it is predominantly used in the U.S. and is the
commodity underlying the oil’s futures price on the New York Mercantile Exchange. Brent blend
found in the North Sea and mainly used in Europe and the Dubai crude extracted from Dubai. (BBC,
2007)
Data, monthly start prices for the period 1971-01-01 - 2011-01-01 was acquired from the Federal
Reserve Bank of St. Louis. (WTI | Federal Reserve Bank of St. Louis, 2011)
Wheat U.S. Cash price at principal markets (U$ per bushel)
Wheat has become one of the most important crops largely because it is easy to grow on a large
scale and it is easy to store for long periods of time. The large demand for wheat has mainly come
from demand of flour globally, due to increased population, but lately the main drivers of wheat
prices are the increase in use of bio fuels and the increase of meat consumption in the developing
world, as it requires a lot of grain to produce 1 kilogram of meat. Wheat in the commodities market
is classified in many different classes, based on the quality of the grain.
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Data, monthly start prices for the period 1971-01-01 - 2011-01-01 was acquired from U.S.
Department of Agriculture. In order to create a fair price, a wheat basket was constructed to provide
an average price independent from quality and location in the U.S. This basket was created with
equal weighting from these types of wheat: (USDA, 2011)
No. 1 hard red
winter (ordinary
protein), Kansas
City, MO
No. 1 hard red
winter (13%
protein), Kansas
City, MO
No. 2 soft red
winter, Chicago,
IL
No. 2 soft red
winter, St. Louis,
MO
No. 2 soft red
winter, Toledo,
OH
No. 1 soft white,
Portland, OR
Foreign exchange rates As for the foreign exchange rates included in this thesis, the Japanese Yen, the Swiss Franc and the
British Pound Sterling, are denominated in U.S. dollars. The currencies included all have strong ties to
the current reserve currency U.S. dollar and are all considered global currencies. They also place
among the top traded currencies in the foreign exchange market.
Because neither the European Euro nor the German D-mark fulfilled the 40-year time period
requirement both currencies were omitted for the purpose of this study.
Foreign exchange rate JPY/USD
Japan has a major capital market and is a big trading partner to the U.S. Their currency, the Japanese
Yen, has for long functioned as the borrowing currency in carry trades between exchanges due to
Japan’s almost non-existing interest rates since the 90’s. The Japanese Yen is a global currency and a
highly traded one in the foreign exchange market. Data, monthly start prices for the period 1971-01-
01 - 2011-01-01 was acquired from the Federal Reserve Bank of St. Louis, priced as JPY/USD and was
hence inverted in order to provide the dollar value for a Japanese Yen. (JPY | Federal Reserve Bank
of St. Louis, 2011)
Foreign exchange rate CHF/USD
The Swiss Franc has similarly to gold played the role as a so-called safe haven currency for investors
when global financial unrest has been present. It is a small economy relative to the U.S. The Swiss
Franc is also considered a global currency and places highly among the top traded currencies in the
foreign exchange market. Data, monthly start prices for the period 1971-01-01 - 2011-01-01 was
acquired from the Federal Reserve Bank of St. Louis, priced as CHF/USD and was hence inverted in
order to provide the dollar value for a Swiss Franc. (CHF | Federal Reserve Bank of St. Louis, 2011)
Foreign exchange rate GBP/USD
The British Pound Sterling serves as a global currency as well and is also considered one of the top-
traded currencies on the foreign exchange market. It is a major currency with a large capital market
and Britain has always been a large trading partner to the U.S. Historically it has been and still is a
very important market for trading commodities, such as gold and silver. Data, monthly start prices
for the period 1971-01-01 - 2011-01-01 was acquired from the Federal Reserve Bank of St. Louis,
priced as GBP/USD and was hence inverted in order to provide the dollar value for a British Pound.
(GBP | Federal Reserve Bank of St. Louis, 2011)
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Interest rates
3 Month U.S. Treasury (risk free rate)
U.S. three-month Treasury bill priced at secondary market rate on yearly discount basis in percent. In
order to calculate monthly returns rates have been divided by twelve, providing a monthly return.
Further divided by a 100 in order to obtain nominal return in its decimal form. (UST 3M | Board of
Governors of the Federal Reserve System, 2011)
Results Below is a presentation of relevant information and statistics extrapolated from the selected data
from 1971 to 2011.
Return and risk Gold Silver Dow Oil JPY CHF GBP Wheat S&P UST
Avg. return 0,9 % 1,1 % 0,6 % 1,1 % 0,3 % 0,4 % 0,1 % 0,5 % 0,7 % 0,5 % Table 1: Average monthly returns for each asset
Highest returning assets are oil and silver while a dollar investment in Pound Sterling yields very little.
Regarding returns, as depicted in Figure 2 above it is clear that all selected assets display a positive
average return on a monthly basis. Precious metals and oil are top performers with an approximate
average return of one percent every month. Worst performer is a dollar based investment in Pound
Sterling, which barely is positive and all dollar investments in foreign exchange return less than the
benchmark risk free rate.
Concerning risk, as measured and displayed by standard deviation in Figure 3, it can be seen that
both oil and silver exhibits relatively high levels of risk. Lowest risk is achieved through dollar
investments in foreign exchange rates. The risk free benchmark is close to zero, thus making that
assumption plausible. Visually combining risk and return it seems that gold offers highest return per
unit of risk.
Figure 4: Average return 1971-2011 Figure 3: Risk, standard deviation 1971-2011
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Historical returns
Further analysis was performed in order to graph and outline returns over a historical perspective
which has been divided into decennial data groups. As this is a mere overview we will not put any
changes into a historical event perspective, as we are only interest in prices and returns.
Figure 5: Nominal returns 1971-1981, observed returns range (-50% → +150%) Nota Bene: Return axis is elongated by one unit.
Figure 4 reveals an extraordinary volatility in oil and silver returns during the period 1971-1981.
In addition wheat returns fluctuate heavily between 1972 and 1975. Few but very large chocks in
returns over the period 1971-81.
Figure 6: Nominal returns 1981-1991 observed returns range (-50% → +50%)
During 1981-1991 silver returns display large bidirectional movements. Additional and similar
movements can be seen in oil returns, which also are very volatile. Around 1987 a large drop (~20%)
occurs on the Dow asset. During 1981-91 more frequent chocks in returns but intensities are lower
than previous period.
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Figure 7: Nominal returns 1991-2001 observed returns range (-50% → +50%)
Frequent moves in oil and silver similar to previous periods. Gold displays a steady trend until 1999
when returns soared to 20%. During 1991-01 more frequent chocks in return but intensities are
lower than previous periods.
Figure 8: Nominal returns 2001-2011 observed returns range (-50% → +50%)
As previously silver and oil returns are very volatile but in addition several chocks hit wheat returns
between 2002 and 2011. Also noted are two chocks to the Dow and the S&P 500 in 2001 and 2008.
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VCM & Correlation Matrix VCM Gold Silver Dow Oil JPY CHF GBP Wheat S&P UST
Gold 0,004 0,004 0,000 0,000 0,000 0,001 0,000 0,000 0,000 0,000
Silver 0,004 0,010 0,000 0,001 0,000 0,000 0,000 0,001 0,000 0,000
Dow 0,000 0,000 0,002 0,000 0,000 0,000 0,000 0,000 0,002 0,000
Oil 0,000 0,001 0,000 0,009 0,000 0,000 0,000 0,001 0,000 0,000
JPY 0,000 0,000 0,000 0,000 0,001 0,000 0,000 0,000 0,000 0,000
CHF 0,001 0,000 0,000 0,000 0,000 0,001 0,000 0,000 0,000 0,000
GBP 0,000 0,000 0,000 0,000 0,000 0,000 0,001 0,000 0,000 0,000
Wheat 0,000 0,001 0,000 0,001 0,000 0,000 0,000 0,004 0,000 0,000
S&P 0,000 0,000 0,002 0,000 0,000 0,000 0,000 0,000 0,002 0,000
UST 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 Table 2: Variance and Covariance Matrix
Largest variance along the diagonal above is observed for Silver and Oil, matching the visual
representation in figure 3. Small covariances overall, highest are gold and silver (0,004).
CORREL. Gold Silver Dow Oil JPY CHF GBP Wheat S&P UST
Gold 1,00
Silver 0,69 1,00
Dow -0,05 0,08 1,00
Oil 0,07 0,07 0,00 1,00
JPY 0,17 0,06 -0,05 -0,06 1,00
CHF 0,28 0,10 -0,14 0,00 0,58 1,00
GBP -0,21 -0,10 -0,02 -0,09 -0,37 -0,64 1,00
Wheat 0,04 0,13 0,00 0,10 -0,01 -0,02 0,01 1,00
S&P -0,01 0,11 0,90 0,03 -0,05 -0,12 -0,05 0,01 1,00
UST -0,07 -0,09 -0,03 0,01 -0,10 -0,07 0,06 -0,03 -0,02 1,00 Table 3: Correlation Matrix
Confirming findings in table 2 the correlation between gold and silver reveals a strong positive linear
relationship (+0,69). The Swiss Franc displays highly negative correlation with the British Pound
Sterling (-0,64) but a positive correlation to the Japanese Yen (+0,58). Further findings include very
strong correlation between S&P 500 and the Dow Jones (+0,90).
Portfolios Further statistical analysis in combination with the results, included above, provides the foundation
on which an efficient portfolio frontier can be calculated, by finding the optimal risky portfolio (ORP)
and the minimum variance portfolio (MVP).
ORP E ( r ) 1,9% Std. dev. 10,0% Sharpe R. 14,4%
Asset Gold Silver Dow Oil JPY CHF GBP Wheat
Weights 1,10 0,01 1,07 0,45 -0,03 -0,80 -1,00 0,20 Table 4: Optimal Risky Portfolio, on a monthly basis
Through solving for the maximal Sharpe ratio the optimal risky portfolio is constructed given the
portfolio constraints. The resulting portfolio shorts all foreign currencies and invests heavily into gold
and the Dow Jones index. Smaller positions in oil and wheat are taken as well as a very tiny position
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in silver. The ORP displays a very high return (1,9%), high risk (10,0%) with a positive Sharpe ratio
(+14,4%) thus rewarding an investor for the amount of risk taken w.r.t. risk free return.
MVP E ( r ) 0,3% Std. dev. 1,0% Sharpe R. -16,7%
Asset Gold Silver Dow Oil JPY CHF GBP Wheat
Weights 0,01 0,00 0,08 0,02 0,09 0,32 0,45 0,02 Table 5: Minimum Variance Portfolio, on a monthly basis
By solving for the minimal standard deviation, i.e. risk, the minimum variance portfolio is constructed
given the portfolio constraints. The resulting portfolio goes long in all foreign currencies, primarily in
British Pound Sterling and Swiss Franc. Also the portfolio takes very small long positions in precious
metals, oil, and wheat. The long position in the Dow Jones is approximately as large as the one in the
Japanese Yen. The MVP displays a very small return (0,3%)and low risk (1,0%) with a negative Sharpe
ratio (-16,7%). Thus not rewarding an investor for the amount of risk taken w.r.t. risk free return.
Plotting the efficient portfolio frontier (EPF) demands the MVP, ORP to be constructed and other
portfolios along this non-linear frontier. Consequently a comparison can be made to the capital
market line (CML) which is based upon the risk-free rate (Y-intercept) and adding the market excess
return (rm - rf) in relation to units of risk, i.e. standard deviation. Thus a visual analysis easily reveals
that the constructed optimal risky portfolio outperforms the benchmarked index in terms of return
per unit risk. However this is not true for MVP making it unattractive to any rational investor.
Financial ratios Gold Silver Dow Oil JPY CHF GBP Wheat S&P UST (ORP) (MVP)
Avg. return 0,9% 1,1% 0,6% 1,1% 0,3% 0,4% 0,1% 0,5% 0,7% 0,5% 1,9% 0,3%
Variance 0,4% 1,0% 0,2% 0,9% 0,1% 0,1% 0,1% 0,4% 0,2% 0,0% 1,0% 0,0%
Stddev 6,2% 9,9% 4,5% 9,6% 2,7% 2,9% 2,5% 6,3% 4,5% 0,3% 10,0% 1,0%
Beta -0,01 0,23 0,90 0,06 -0,03 -0,08 -0,02 0,01 1,00 0,00 1,06 0,04
Table 6: Average return, Variance, Standard deviation and beta for all assets, benchmark, risk-free and portfolios
Calculating the average returns for both portfolio the visual analysis made in figure 8 is confirmed,
the optimal risky portfolio has a, on average, monthly return of 1,9% with a risk level at 10% standard
deviation and has a beta value slight above one (1.06) indicating a strong positive relationship to the
Figure 9: Efficient Portfolio Frontier & Capital Market Line (S&P 500)
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benchmarked index. The MVP shows low return but the lowest risk level and a beta value of 0,04
indicating very little linkage to the benchmarked index.
Gold Silver Dow Oil JPY CHF GBP Wheat S&P UST (ORP) (MVP)
Sharpe Ratio 7,8% 6,3% 4,3% 6,2% -4,3% -3,6% -13,9% 0,9% 4,3% 0,0% 14,4 % -16,7%
Treynor Ratio -59,0% 2,7% 0,2% 9,3% 4,1% 1,3% 13,8% 6,2% 0,2% 0,0% 1,4% -4,4%
E( r ) CAPM 0,9% 1,1% 0,8% 1,1% 0,3% 0,3% 0,1% 0,5% 0,8% 0,5% 2,1% 0,3%
Jens. Alpha 0,5% 0,6% 0,0% 0,6% -0,1% -0,1% -0,3% 0,1% 0,0% 0,0% 1,2% -0,2%
Table 7: Performance indicators for all assets, benchmark, risk-free and portfolios
From table 6 further calculations are made in order to deepen the evaluation of each asset and
portfolio, resulting in the financial ratios depicted in table 7. Excess return in terms of risk is
dominated by the ORP, followed by precious metals and oil. MVP and GBP underperform heavily and
indicate negative risk-return relations (<-10%).
Comparing the assets and portfolios on a Treynor basis, i.e. excess return in terms of beta value
(benchmarked market relationship) depicts that gold has a tiny negative beta value and as such is
hardly influenced by the markets, thus making the Treynor ratio somewhat inconclusive (-59%).
Likewise is the situation for British Pound Sterling, which has a negative excess return and a tiny
negative beta, thus also resulting in an inconclusive Treynor ratio (+13,8%).
Regarding alpha profits, measured by Jensen’s alpha on average every month, findings are that the
ORP provides a vast 1,2% alpha profits, followed by oil and silver at 0,6 %. Fourth in order comes gold
with a profit of 0,5%. Worst performers are GBP at -0,3% and MVP at a slightly higher -0,2%. Wheat
however provides a tiny 0,1% alpha return. None of the foreign exchange rates provide alpha profits,
neither does UST nor Dow Jones.
Finally, in figure 9 and 10, is a graphic representation of average returns and risk levels for all assets
and both portfolios constructed within this study.
Figure 10: Monthly risk levels, portfolios included Figure 11: Average monthly return, portfolios included
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Summary and conclusions
Summary With global markets flooded with fiat currencies and credit since the end of Bretton Woods by Nixon
in 1971 we have searched for the optimal alternative portfolio by applying the CAPM model and
statistical measurements. Thus constructing an alternative portfolio consisting of precious metals,
foreign exchange rates, commodities and stock indices.
By constructing the optimal portfolio we seek to find the portfolio allocations for an arbitrary passive
rational U.S. investor with hindsight, given the plethora of investment opportunities available today.
The minimum variance portfolio and the optimal risky portfolio, was calculated by using CAPM. Thus
finding the efficient portfolio frontier, which was compared to our benchmark - the S&P 500 index
through the capital market line.
Results obtained through spreadsheet programming showed that precious metals and oil showed
high returns on average, with high levels of risk, while wheat displayed less return but high risk. The
stock index Dow Jones Industrial Average provided a relatively high return in terms of risk, though
very highly correlated to the benchmark S&P 500 index. The assumed risk free asset provided low
return and displayed very little volatility over the time period. Foreign exchange rates did not offer
much return and all displayed a volatility level around three per cent. Interestingly the Swiss Franc
exchange rate displayed high positive correlation to the Japanese Yen but even higher negative
correlation to the British Pound Sterling. Highest alpha values were obtained in long positions in the
optimal risky portfolio followed by oil and then silver.
Conclusions The constructed optimal risky portfolio outperformed all other assets on a risk-return relationship
basis with a return of almost two percent per month on average with a volatility of ten percent.
Similarly it provides the highest alpha value of all securities. However, the minimum variance
portfolio with the highest diversification underperformed below the set capital market line, thus
making it an irrational investment choice. Therefore, any arbitrary risk-willing rational U.S. investor
would choose to invest in the optimal risky portfolio, granted hindsight.
Precious metals and oil all provided an investor with a high monthly return but with high risk.
Especially silver displayed high volatility over time. Most probably this is associated to leveraged
speculation in the precious metals market, with very little physical liquidity and very few contracts
being delivered. Also these monthly returns could be related to weakening of the purchasing power
of the dollar due to e.g. increases in money supply or perhaps a loss of confidence.
Wheat displayed a mediocre return but exhibited as high risk as precious metals thus making this
asset an irrational investment choice. An investor could however benefit from the diversification
effect of investing in wheat as it has very little correlation to other assets in the portfolio set.
21
Foreign exchange rates offer very small but stable returns with quite low risk levels. The Dollar has
thus likely lost much purchasing power over time in the international market, when accounting for 40
years of compounding positive returns of foreign exchange rates. This could also depend on a larger
money supply in the U.S. Especially the Swiss Franc has appreciated against the U.S. Dollar, gaining
on average almost a half percent each month since 1971, for a total averaged return of 549% as of
January 2011.
The high positive correlation between the exchange rates for the Swiss Franc and the Japanese Yen
could perhaps be a result from exploits of arbitrage possibilities in the interest rate and foreign
exchange markets. Also, as the British Pound Sterling is not gaining as much, perhaps due to similar
market conditions in the U.K. and the U.S. regarding money supply, deficits etc. Consequently, the
returns could be a sign of weakening of the dollar.
Furthermore when Swiss Francs and Japanese Yen appreciate against the U.S. Dollar the correlations
reveal that British Pound Sterling depreciates. Also, not surprisingly, we found very high positive
correlation between the benchmark S&P 500 index and the Dow Jones Industrial average, as both are
major stock indices in the same market.
However, reflecting over historical returns, they seem to vary in intensity and frequency over time.
Hence it is questionable if the returns truly are mean variance efficient.
Suggestions for further studies
Analogous study but account for inflationary effects and measure changes in purchasing
power i.e. trying to account for money supply changes.
Deeper analysis of the efficient mean variance - has correlation been the same throughout
the whole period? Also comparing changes in returns over time in order to analyze the
efficient market theorem and assess if markets have improved efficiencies over time.
Construct a currency neutral portfolio, perhaps priced in a precious metal and perform
similar evaluations.
Perform a study on selected data set and use the Arbitrage Pricing Theory instead of CAPM
and thus search for statistically significant factors when constructing the portfolio, factors of
interest could include foreign exchanges, money supply, trade deficits etc.
22
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Tables Tables with calculations and data are available online at:
Thesis Calculations and Data Microsoft Excel Spreadsheet
(http://actarius.se/storage/THESIS_PBW_VR_RN.xlsx)