The Problem with Modern Portfolio Theory

The world of finance is replete with theories and models. Yet, few concepts have left as indelible a mark as Modern Portfolio Theory (MPT).

Proposed by Harry Markowitz in his 1952 paper, “Portfolio Selection,” MPT has been both a guiding light and a point of contention among investors for decades.

What is Modern Portfolio Theory?

An example of the relationship between different asset classes

Modern Portfolio Theory revolves around the idea of diversification — essentially, the age-old wisdom of not putting all your eggs in one basket.

At its core, MPT posits that the risk and return of an overall portfolio are more important than the risk and return of individual assets.

Foundational Concepts:

• Expected Returns: An anticipated value for the return on an investment, often calculated based on historical data.
• Portfolio Volatility: The standard deviation of portfolio returns, representing the total risk of the portfolio.
• Efficient Frontier: A curve that defines the portfolios offering the highest expected return for a given level of risk (or the lowest risk for a given level of return).

The primary tenet is that a diversified portfolio can be constructed to optimize returns for a given level of risk or, conversely, minimize risks for a desired return. This optimization process leads to the efficient frontier, where portfolios lie on a curve representing the best risk-return trade-offs.

How Does Modern Portfolio Theory Work?

source: Wikipedia

Imagine you have multiple assets to invest in, each with its respective return and risk profile. According to MPT:

1. Diversification Benefits: By diversifying across assets that are not perfectly correlated, you can reduce the portfolio’s overall risk without necessarily sacrificing returns. This happens because individual asset volatilities can offset each other.
2. Optimal Portfolio Creation: For a given risk level, there exists an optimal combination of assets that will offer the highest possible return. This combination forms the efficient frontier.
3. Risk-Free Assets & Capital Allocation Line: Introducing a risk-free asset (like a treasury bill) allows for a combination of the risk-free asset and a portfolio on the efficient frontier, leading to a straight line known as the capital allocation line. The point where this line is tangent to the efficient frontier is the market portfolio.

This approach emphasizes the collective behavior of assets, acknowledging that individual asset behavior isn’t as crucial when viewed within the context of an entire portfolio.

The Problem with Modern Portfolio Theory

While Modern Portfolio Theory (MPT) has been a pillar of finance for decades, as financial markets have evolved and academic research has delved deeper into investor behavior and market dynamics, various criticisms have emerged.

Asset correlations can rapidly change during a bear market

1. Assumption of Rationality:

• Description: Central to MPT is the assumption that investors are rational actors who aim to maximize their utility (usually represented by expected returns) for a given level of risk.
• Reality and Examples: Behavioral finance has identified numerous instances where investors deviate from rationality. For example, during the Japanese asset price bubble in the late 1980s, investors continued to buy into skyrocketing real estate and stock prices, even when fundamental valuations could not justify such prices¹.
• Academic Insight: Kahneman and Tversky’s Prospect Theory illustrates how people make decisions involving probabilities. They found that investors often overvalue potential losses compared to potential gains, leading to irrational decision-making².

2. Dependence on Historical Data:

• Description: MPT models rely heavily on historical data to estimate expected returns, variances, and covariances.
• Reality and Examples: Markets, economies, and geopolitical scenarios evolve. The historical data from emerging markets, such as Brazil’s stock market in the 1990s, may not capture the entire range of potential future outcomes, given the rapidly changing economic environment and institutional reforms during that period³.
• Academic Insight: Ibbotson and Kaplan, in their 2000 study, discussed how historical returns are often poor predictors of future returns due to changing market conditions⁴.

3. Static Correlations:

• Description: MPT assumes that correlations between assets remain consistent.
• Reality and Examples: The 1997 Asian Financial Crisis saw previously uncorrelated economies and markets fall in tandem. For instance, while South Korea and Thailand had different economic structures, both faced massive capital outflows and devaluations, demonstrating converging correlations during crises⁵.
• Academic Insight: Longin and Solnik’s study on international equity markets showed that correlations between markets increase in volatile conditions, challenging MPT’s static correlation assumption⁶.

4. Over-reliance on Quantitative Analysis:

• Description: MPT is rooted in quantitative data, potentially sidelining qualitative factors.
• Reality and Examples: The downfall of Long-Term Capital Management (LTCM) in 1998 is a case in point. Despite the hedge fund being run by two Nobel Prize-winning economists and employing sophisticated models, they overlooked political and operational risks during the Russian financial crisis⁷.
• Academic Insight: Daniel and Titman’s 1997 study illustrated that stock returns were more closely linked to firm characteristics than to their beta coefficients, emphasizing the importance of qualitative factors⁸.

5. Over-Simplification of Investor Goals and Constraints:

• Description: MPT is rooted in the idea that risk and return are the primary considerations for investors.
• Reality and Examples: In the late 2000s, many pension funds across Europe shifted to more conservative assets, not just due to risk-return trade-offs, but due to regulatory pressures, liquidity needs, and long-term liabilities⁹.
• Academic Insight: A study by Ang, Papanikolaou, and Westerfield highlighted how investor objectives and constraints, such as labor income risks, can influence portfolio decisions beyond mere risk-return considerations¹⁰.

MPT Failures in the US Stock Market

1. The 1987 Stock Market Crash (Black Monday):

• What Happened: On October 19, 1987, U.S. stock markets witnessed their most significant one-day percentage drop in history, with the Dow Jones Industrial Average plummeting by 22.6%.
• MPT Shortcoming: MPT assumes a normal distribution of asset returns, but Black Monday defied this assumption, representing a multi-standard deviation event that was considered nearly impossible based on traditional models.

2. The Tech Bubble Burst (2000-2002):

• What Happened: At the turn of the millennium, the dot-com bubble, characterized by exuberantly valued tech stocks, burst. Between 2000 and 2002, the NASDAQ Composite, which had many of these tech stocks, lost 78% of its value.
• MPT Shortcoming: The bursting of the bubble showed that diversifying across sectors isn’t always sufficient. Many investors, believing they were adequately diversified, still faced substantial losses because various sectors were indirectly influenced by the tech sector’s downturn.

3. The 2008 Financial Crisis:

• What Happened: Triggered by the collapse of large financial institutions due to exposure to subprime mortgages, it resulted in sharp declines in consumer wealth, severe disruptions in financial markets, and the onset of a deep recession.
• MPT Shortcoming: Asset correlations, which are central to MPT, converged during the crisis. Diversification benefits diminished as a wide variety of assets, from stocks to real estate, all fell in tandem, challenging MPT’s foundational premise.

4. Long-Term Capital Management (LTCM) Crisis (1998):

• What Happened: LTCM, a hedge fund managed by two Nobel Prize-winning economists who heavily relied on advanced financial models, faced catastrophic losses during the Russian financial crisis.
• MPT Shortcoming: Despite the use of sophisticated models rooted in MPT principles, LTCM’s strategies did not account for “Black Swan” events or extreme market moves. Over-reliance on quantification and undervaluing of qualitative factors, like geopolitical risks, led to its downfall.

5. Growth vs. Value Dichotomy (2010s):

• What Happened: Throughout much of the 2010s, growth stocks (particularly in technology) significantly outperformed value stocks, contrary to the historical premium associated with value investing.
• MPT Shortcoming: MPT posits that higher risks are associated with higher expected returns. However, many growth stocks offered both higher returns and lower volatility than their value counterparts during this period, challenging traditional risk-return dynamics postulated by MPT.

Each of these events underscores the importance of understanding the assumptions and limitations of MPT.

In summary, while MPT offers a foundational framework for understanding risk and return in portfolios, evolving market dynamics, and continued academic inquiry suggest that it should be applied with caution and complemented with other financial tools and insights.

Sources:

The 10-Year vs. 2-Year Treasury Bond Spread: Implications for the Economy and the U.S. Stock Market

In the vast spectrum of economic indicators, one stands out for its uncanny ability to foreshadow potential downturns — the 10-Year vs. 2-Year Treasury Bond Spread.

Known as the “10-2 spread”, this measure has emerged as a focal point for policymakers, investors, and market watchers.

But what makes this spread so significant? Can it really be a trusted herald of impending recessions?

Relationship between the 2 Year and 10 Year Treasury Bond

An Introduction to the 10-2 Spread

At the foundation, the 10-2 spread is the difference in yields between the U.S. 10-year Treasury bond and the 2-year Treasury bond (1).

A bond’s yield, to put it simply, is the return an investor anticipates when buying a bond. When you map out the yields of Treasury bonds of varying maturities, the resulting graph is the yield curve.

Typically, a 10-year bond offers a higher yield than a 2-year bond because of the longer time frame and associated risks, especially inflation.

Factors Influencing the Spread

1. Monetary Policy Decisions: The actions of central banks, especially the Federal Reserve in the U.S., play a pivotal role in molding short-term interest rates.

For example, to counteract inflation or an overheating economy, the Fed might decide to hike rates, a move that would likely boost short-term bond yields (2).

2. Investor Sentiment and Behavior: Yields on long-term bonds, such as the 10-year bond, are influenced significantly by investor sentiment about the future.

If the collective market foresees economic headwinds, there might be a rush towards the relative safety of long-term bonds, driving up their prices and consequently pushing down their yields (3).

Source: CNN

3. Expectations of Future Inflation: The prospect of rising inflation in the future can make investors wary, leading to a demand for higher yields on long-term bonds to compensate for anticipated value erosion (4).

4. Global Economic Dynamics: We live in an interconnected global economy where events in one region can ripple across financial systems worldwide.

Factors like discrepancies in international bond yields and major shifts in global economies can also weigh on the U.S. yield curve.

Source Koyfin

Does the10-Year vs. 2-Year Treasury Bond Spread Predict Recessions?

Historical data provides some compelling evidence. An inverted yield curve—where the yield on the 2-year Treasury bond exceeds the 10-year yield—has been observed before every U.S. recession since the 1960s (5).

This inversion suggests a peculiar phenomenon: investors demonstrate more confidence in the economic outlook over a 2-year horizon than a 10-year one.

However, while history offers insights, it’s imperative to approach the spread with a nuanced perspective:

1. Time Lags and Variances: Post-inversion, the lead time to a recession can be wildly inconsistent. Historical trends show that after the yield curve inverts, it could be anywhere from a few months to over two years before a recession hits (6).

2. Potential False Alarms: Relying solely on the 10-2 spread as a deterministic recession predictor can be a precarious strategy. While it’s a robust indicator, it’s not infallible.

3. A Changing Global Landscape: Modern economies are complex, interconnected webs. International events, from Brexit to the economic policies of major players like China, can impact the U.S. yield curve (7).

4. Structural Market Changes: Over time, market structures evolve, influenced by regulations, technology, and financial innovations. These transformations can sometimes affect how traditional indicators, including the yield curve, behave and should be interpreted.

Implications for Various Stakeholders

For Policymakers

The yield curve, specifically the 10-Year vs. 2-Year Treasury Bond spread, provides policymakers with valuable feedback on the efficacy of their strategies (8).

An inverted curve might suggest that monetary policies need to be re-evaluated.

For Investors

The 10-2 spread is more than just a metric—it’s a sentiment barometer. Active portfolio adjustments in anticipation of potential economic shifts can be informed by movements in this spread (9).

For Economists

The spread offers a treasure trove of data, acting as a litmus test for the economy’s health and providing insights into the interplay of various macroeconomic factors.

Conclusions and Forward Outlook

In the sophisticated dance of economic indicators, the 10-2 Treasury bond spread certainly plays a pivotal role. While its historical track record is impressive, relying solely on it for economic forecasting can be misleading.

A holistic approach—one that takes into account myriad factors, both domestic and global, and understands the intricacies and potential anomalies of the spread—is the optimal strategy.

In this intricate game of economic prediction, the 10-Year vs. 2-Year Treasury Bond Spread is undeniably a powerful player, but it’s crucial to remember that it’s just one of many on the field.

Sources

1: U.S. Department of the Treasury. “Daily Treasury Yield Curve Rates.”
2: Federal Reserve Bank of St. Louis. “The Role of Monetary Policy in Interest Rate Determination.”
3: Investopedia. “Determinants of Interest Rates and Bond Yields.”
4: Federal Reserve Bank of Cleveland. “Analyzing Inflation’s Impact on Bond Yields.”
5: National Bureau of Economic Research. “Linking Yield Curve Inversions and Economic Downturns.”
6: The Financial Times. “The Complex Relationship Between Yield Curve Inversions and Economic Recessions.”
7: The Wall Street Journal. “Global Factors Affecting U.S. Yield Curves.”
8: Brookings Institution. “The Yield Curve and Its Policy Implications.”
9: J.P. Morgan Asset Management. “Investing in the Shadow of the Yield Curve.”

The 10-Year vs. 2-Year Treasury Bond Spread: Implications for the Economy and the U.S. Stock Market

Understanding Fibonacci Retracement Levels

Technical analysis remains a widely used method for forecasting the future price movements of financial assets. Among its myriad tools and techniques, the concept of Fibonacci Retracement levels stands out due to its historical significance and ubiquitous application.

Historical Origins

The inception of the Fibonacci sequence can be traced back to Leonardo of Pisa, an Italian mathematician from the 13th century, popularly known as Fibonacci.

In his book, Liber Abaci, he introduced a sequence of numbers to the Western world that later came to be known as the Fibonacci sequence. It begins as 0, 1, 1, 2, 3, 5, 8, 13, and so on. Each number is the sum of the preceding two.

Source: Math Images

Interestingly, this sequence isn’t just a numerical marvel; it manifests in various natural phenomena, including the arrangement of leaves on plants, the spiral of galaxies, and even the proportioning of features in human faces.

Translating Fibonacci to Financial Markets

In technical analysis, Fibonacci retracement levels are horizontal lines that indicate potential support or resistance levels. These levels are calculated by taking the difference between a major peak and trough and multiplying this distance by the key Fibonacci ratios, which are 23.6%, 38.2%, 50%, 61.8%, and 78.6%.

The Decline in the S&P 500 during 2022, stopped at its 50% retracement level from the COVID lows of 2020

For instance, if a stock price climbs from $10 to $20, then retraces to $15, it has retraced 50% of its move. Fibonacci retracement levels would plot potential support or resistance at a few distinct percentages of that move.

Applications in Trading

1. Identifying Support and Resistance: Traders employ these levels to identify potential price zones where an asset might reverse direction. For instance, if the price of an asset starts declining after a rise, it might find support at one of the Fibonacci levels.
2. Setting Stop-Loss and Take-Profit Points: Knowing potential reversal areas allows traders to set logical stop-loss or take-profit points, minimizing the emotional aspect of trading.
3. Combining with Other Tools: The accuracy of Fibonacci retracements increases when combined with other indicators like moving averages, RSI, or candlestick patterns.

Practical Implications

1. No Guarantees: Like all trading tools, Fibonacci retracements do not guarantee success. They provide a framework, but market psychology and external news can often drive prices.
2. More is Better: A principle that many traders follow is that the more the price respects a certain Fibonacci level in the past, the more likely it is to have significance in the future.
3. Depth of Retracement: While the 50% level isn’t a “true” Fibonacci number, it’s often included because assets frequently retrace about half of a significant move before resuming their trend.

Criticisms

Despite their popularity, Fibonacci retracement levels aren’t without detractors. Critics argue that:

1. Self-fulfilling Prophecy: The levels might work because many traders use them, not necessarily because they have any inherent predictive power.
2. Ambiguity: In trending markets, pinpointing the ‘right’ high and low for drawing retracements can be subjective.
3. Over-reliance: Solely depending on Fibonacci retracements without considering other market factors can lead to flawed decision-making.

Different technicians may use different starting points in their analysis

The allure of the Fibonacci sequence and its relevance in nature inevitably piques curiosity when it’s applied to financial markets. However, the key is understanding that Fibonacci retracement levels, while useful, are just one of many tools in a trader’s toolkit. They are best used in conjunction with a comprehensive trading strategy and a disciplined approach.

Sources:

1. Fibonacci, L. (1202). Liber Abaci.
2. Kirkpatrick, C. D., & Dahlquist, J. (2010). Technical Analysis: The Complete Resource for Financial Market Technicians. FT Press.
3. Pring, M. J. (2002). Technical Analysis Explained: The Successful Investor’s Guide to Spotting Investment Trends and Turning Points. McGraw Hill Professional.

Job Growth in the US Slows to a More Sustainable Pace

The latest job report shows a slowing in U.S. job growth, with employers adding 187,000 jobs in July, which falls short of economists’ forecast of 200,000 positions.

Source: US Dept of Labor

However, there is no cause for alarm as this indicates a shift towards a more sustainable pace, and it still surpasses the average monthly jobs gained pre-pandemic. In fact, the July figure is slightly higher than June’s revised payroll gain of 185,000.

A Move Towards Sustainable Growth

Although the headline number might be disappointing, it is important to understand that the reduced pace is actually a sign of stability rather than weakness.

This growth rate aligns with what some economists consider the long-term capacity of the U.S. labor force, ensuring that growth remains steady and manageable. It represents a maturing recovery where employers are strategically aligning job growth with underlying economic fundamentals.

Unemployment Rate and a Tight Labor Market

Another positive development is the decrease in the unemployment rate from 3.6% in June to 3.5% in July, against economists’ expectations of no change. Despite an aggressive Fed, the Unemployment Rate remains at historical lows.

Source: Federal Reserve Bank of St Louis

This decline reinforces the idea of a tight labor market, where employers are competing for workers, potentially leading to wage growth. Such a scenario can boost consumer spending and confidence, thereby driving further economic expansion.

Implications for the Federal Reserve and Interest Rates

Source: CME Group

Given the numbers, it doesn’t seem likely that the Federal Reserve will rule out future interest-rate hikes. Despite slower job growth, the continuous expansion and tight labor market may push the Fed to stick with gradual monetary tightening. This can have various effects on investors’ portfolios, especially in sectors sensitive to interest rates, like bonds and financials.

When making investment decisions, keep in mind:

Bonds: Given the possibility of future interest rate hikes, it may be wise to consider short-duration securities with less sensitivity to rate changes.
Equities: Sectors benefiting from economic growth and consumer spending can remain appealing, particularly with potential wage growth driving increased spending.
Dollar & Commodities: Stay vigilant on the U.S. dollar and commodities as their valuations can be impacted by interest rate decisions.

Although July’s job growth might seem underwhelming at first glance, further analysis reveals a trend toward sustainable growth and a tight labor market. Take these dynamics into account when determining your asset allocation and investment strategy, and stay tuned for any interest rate moves by the Federal Reserve.

____

John Rothe, CMT

The Big Mac Index: Exploring Currency Valuation and Inflation

The Big Mac Index is an intriguing economic indicator that offers a unique perspective on the world’s currencies and purchasing power parity (PPP). Created by The Economist in 1986, it has since gained significant recognition as a tool for illustrating currency value comparisons.

What is the Big Mac Index?

Taking a common product available in numerous countries—the McDonald’s Big Mac—the index compares its prices across different currencies. This provides insight into whether a currency is overvalued or undervalued in relation to the US Dollar.

Understanding Purchasing Power Parity (PPP)

The theory underlying the Big Mac Index is purchasing power parity (PPP). According to PPP, identical goods should have the same price in different countries, assuming no transaction costs or trade barriers, when expressed in a common currency.

Sources: The Economist, McDonald’s; Refinitiv Datastream;

IMF; Eurostat; LebaneseLira.org; Banque du Liban; 

The EconomistNote: All prices include tax

The idea is that if a currency is overvalued, local goods will seem more expensive to foreigners, and foreign goods will appear cheaper to locals. Conversely, an undervalued currency will make local goods seem cheaper to foreigners and foreign goods more expensive to locals.

How it Relates to Inflation

While the Big Mac Index does not directly measure inflation, it can indirectly provide insights into inflationary pressures. If the price of a Big Mac rises more quickly in one country compared to others, it may indicate a higher local inflation rate.

This shift in the Big Mac Index can serve as an indicator that the currency might be overvalued.

Unveiling the Unique Perspective of The Big Mac Index

The Big Mac Index offers a captivating and informal lens through which to view currency valuation and inflation trends. Its analysis of relative currency values provides valuable information for economic considerations and global financial insights.

Limitations:

While the Big Mac Index is a simplified measure, it is not without limitations. Factors that can affect its accuracy include:

Taxes and Import Duties: Varying taxation policies across countries can impact the retail price of a Big Mac.

Cost of Living and Labor Costs: These factors differ significantly between countries and can influence the final price.

Local Preferences and Competition: McDonald’s may adjust the Big Mac price differently based on competitive pressures or consumer preferences in certain regions.

Using the Big Mac Index:

Despite its simplicity, the Big Mac Index serves as a powerful tool for visualizing the complex concept of Purchasing Power Parity (PPP). It is instrumental in:

Gauging Relative Currency Valuation: Economists and investors rely on the Big Mac Index to assess the relative valuation of currencies and identify potential mismatches between market exchange rates and PPP.

Enhancing Learning: The index is an effective teaching tool, providing accessibility in explaining concepts like exchange rates and PPP.

The Big Mac Index is a novel and engaging economic indicator that offers insights into currency valuation based on the theory of PPP. While it does not directly measure inflation, its intuitive nature makes it a valuable resource for understanding fundamental economic concepts.

However, it is crucial to consider its limitations and underlying assumptions. The index should be used as a supplementary tool or for educational purposes rather than as the sole basis for economic decision-making.

Value at Risk (VaR): Uses and Controversies

In today’s investment landscape, where uncertainties abound, the ability to quantify and effectively manage risk can be a game-changer. Value at Risk (VaR) is a powerful tool that has gained substantial traction over the years due to its straightforward approach to estimating financial risk.

Value at risk Definition and Formula:

Value at Risk (VaR) is a statistical measure that quantifies the level of financial risk within a firm or investment portfolio over a defined time frame. It facilitates the calculation of the maximum potential loss a portfolio could incur with a given confidence level.

Typically, VaR is assessed over a one-day or ten-day period (Jorion, 2007).

The general formula for VaR is as follows:
VaR = Portfolio Value x Z-score x Portfolio’s Standard Deviation

Here, the Z-score indicates the number of standard deviations away from the mean in a normal distribution, corresponding to the desired confidence level.

For instance, a Z-score of 1.65 is employed for a 95% confidence level, while a Z-score of 2.33 represents a 99% confidence level. The portfolio’s standard deviation reflects the volatility of the portfolio (Linsmeier & Pearson, 2000).

Functionality of VaR:

Source: Investopedia

Let’s bolster our understanding of VaR through an example.

Consider a portfolio valued at $1 million, characterized by a standard deviation (volatility) of 5%. If we intend to calculate the one-day 95% VaR, we would utilize a Z-score of 1.65.

Applying the VaR formula, we find VaR = $1,000,000 x 1.65 x 5% = $82,500.

This implies that over a one-day period, you can be 95% confident that your losses will not exceed $82,500.

Pitfalls and Controversies Surrounding VaR

Despite its widespread use, Value at Risk (VaR) is subject to several criticisms, which are crucial to understand when utilizing this risk measurement tool in the financial industry.

Assumption of Normal Distribution:

VaR commonly assumes that financial returns follow a normal distribution, which is often an inaccurate representation in real-world financial markets.

Many financial returns exhibit skewness and kurtosis, indicating fat tails and asymmetry. Consequently, VaR may underestimate the probability of extreme losses, which frequently occur in the tail regions of the distribution (Taleb, 2007).

Failure to Specify Loss Beyond the VaR:

While VaR provides an estimate of the maximum loss at a given confidence level, it fails to indicate the magnitude of potential losses if events surpass the VaR threshold. This limitation becomes particularly problematic during severe market downturns.

In such situations, an alternative measure called Conditional VaR (CVaR) proves more useful as it calculates the expected loss given that the VaR threshold has been exceeded, offering a more comprehensive risk estimate (Acerbi & Tasche, 2002).

Lack of Subadditivity:

Ideally, the combined VaR of two portfolios should be equal to or less than the sum of their individual VaRs, assuming diversification benefits. However, VaR lacks the property of subadditivity, which diminishes its reliability as a risk measurement tool for diversified portfolios (Artzner et al., 1999).

2008 Financial Crisis

The 2008 financial crisis served as a wake-up call, exposing the limitations of VaR. Many financial institutions relying on VaR models were ill-prepared for the extreme events of the crisis. With VaR models predominantly leveraging short-term historical data, they failed to accurately predict and account for the severity of the crisis, leading to substantial losses (Danielsson, 2011).

By recognizing and understanding the pitfalls and controversies associated with VaR, financial professionals can better assess its limitations and explore alternative risk measurement approaches to ensure more accurate risk management.

References

1. Acerbi, C., & Tasche, D. (2002). On the coherence of Expected Shortfall. Journal of Banking & Finance, 26(7), 1487-1503.
2. Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999). Coherent measures of risk. Mathematical finance, 9(3), 203-228.
3. Danielsson, J. (2011). Financial risk forecasting: The theory and practice of forecasting market risk with implementation in R and Matlab. John Wiley & Sons.
4. Jorion, P. (2007). Value at Risk: the new benchmark for managing financial risk. McGraw-Hill.
5. Linsmeier, T. J., & Pearson, N. D. (2000). Value at Risk. Financial Analysts Journal, 56(2), 47-67.
6. Taleb, N. N. (2007). The black swan: The impact of the highly improbable (Vol. 2). Random House.