Basic Quantitative Concepts: Risk and Return
Abstract
Explore the shift from chance to mathematical precision in investing. Learn the importance of quantitative concepts, application in real scenarios, and the use of analytical tools like histograms. Understand returns analysis implications, the temporal dynamics of returns, and the multi-modal nature of returns. Delve into individual asset returns, correlation significance, and best practices for managing correlation. Gain insights into risk and asset classes, ensuring a comprehensive understanding of investment dynamics for informed decision-making.
Time to read: 9 to 10 minutes.
Level: Fundamental.
Category: Education Note.
Transition from chance to mathematical symbolism: In the world of investing, the way to describe return and risk has been symbolized, replacing chance with something more precise than mathematical symbolism. This has made it possible to conceptualize and lead to a frame of reference on which to build concepts and reach a greater understanding of what is happening.
Importance of quantitative concepts in investments: Quantitative concepts are critical for describing and analyzing the return and risk of an investment, as well as for making rational, evidence-based investment decisions.
Application of quantitative concepts in real scenarios: Quantitative concepts can be applied in real investment scenarios, using tools such as the histogram and other analysis methods, as long as their advantages and limitations are known, and the results are correctly interpreted.
Numerical Fundamentals
Basic knowledge of numerical aspects: To understand the fundamentals behind the investment process, it is necessary to have a basic knowledge of the numerical aspects related to the performance, risk of an investment and the time frame in which the description of the past is made.
Interpretation of numbers: An investor does not have to know how to perform these numerical calculations. What he or she needs to know is how to interpret the returns, risks, and characteristics of the calculation time frame.
Analytical Tools
Histogram: A histogram can be a useful tool to describe the risk-return characteristics of an asset or investment. A histogram is a graphical representation of the distribution of a set of data, where the data are grouped into intervals or bins and the height of each bin represents the frequency or count of data points that are found within that interval.
How to read and interpret a histogram: To read and interpret a histogram, the following aspects must be taken into account:
Shape: The shape of the histogram can indicate the type of distribution of the data, whether it is symmetrical, asymmetrical, unimodal, bimodal, etc.
Center: The center of the histogram can represent the measure of central tendency of the data, such as the mean, median or mode.
Dispersion: The dispersion of the histogram can reflect the variability or range of the data, as well as the measure of dispersion, such as the standard deviation or the coefficient of variation.
Tails: The tails of the histogram can show the presence of extreme or outlier values, as well as the measure of skewness or kurtosis of the data.
Complementary Analysis
Use of the histogram together with other methods of analysis: In general, a histogram should be used together with other methods of analysis, such as calculating the mean, standard deviation or other statistical measures of risk and return. These methods can complement the information provided by the histogram and help to make more informed decisions about the investment.
Limitations and errors of the histogram: It is important to know the limitations of aggregating information in one or several heuristics and the error that this can lead to if it is used as the only element to project future behaviors. The histogram cannot capture all the complexity of the data nor predict with certainty the future behavior of the investment. Therefore, it should be used with caution and critical sense.
Returns Analysis and its Implication for Investors
Analyzing and comprehending returns involves employing a time-based measure and constructing a histogram. This entails observing the ranges within which returns have fluctuated over a specific period throughout a historical timeframe.
When using daily returns over extended historical periods, a histogram emerges, and when broken down into smaller measurement intervals, it forms a distribution. In the realm of financial returns, the distribution of daily returns over extended measurement periods closely resembles a normal distribution. Employing a normal distribution significantly simplifies the calculations and mathematics behind investment portfolio management. This simplification arises from the ease of determining two key distribution moments – the mean and the standard deviation – to describe the behavior of the underlying series.
However, a challenge arises as current knowledge indicates that stock market returns do not follow a normal distribution. While some asset returns may conform to a normal distribution, many exhibit features such as "fat tails" (where extreme events occur more frequently than expected in a normal distribution) and asymmetry.
Eugene F. Fama's influential 1965 article, "The Distribution of Stock Price Returns," published in the Journal of Business, is often cited on this subject. Fama's analysis of daily stock returns reveals evidence of "fat tails" and asymmetry, challenging the assumption of a normal distribution. He concludes that stock returns follow a more leptokurtic distribution (indicating "fat tails") and are negatively skewed compared to a normal distribution.
Numerous studies have explored return distributions across various asset types, including currencies, commodities, and bonds, consistently finding evidence of non-normality. This empirical evidence strongly supports the idea that return distributions are not normal, carrying significant implications for risk management and asset pricing.
Implications for Investors:
Difficulty in Predicting Returns: Non-normal distributions can be harder to predict than normal distributions, posing challenges for investors in estimating expected returns and making informed portfolio allocation decisions.
Increased Risk: Non-normal distributions often exhibit "fat tails," increasing the likelihood of extreme events. This heightened risk requires investors to be vigilant, as unexpected events can significantly impact their portfolios.
Significance of Diversification: While diversification is crucial for all investors, it becomes even more critical in non-normal distributions. By spreading investments across various assets, investors can mitigate the impact of extreme events and potentially achieve more stable returns over time.
Risk Management Importance: Due to the elevated risk associated with non-normal distributions, investors need robust risk management strategies. This might involve implementing "stop-loss" orders, using options to hedge against downside risk, or employing other risk management techniques.
Long-Term Focus: Given the potential for short-term volatility in non-normal distributions, maintaining a long-term perspective is essential. Focusing on long-term investment goals helps investors withstand short-term fluctuations and potentially achieve more stable returns over time.
Temporal Dynamics of Returns:
The impact of time on financial returns is significant and depends on the investment type, financial asset, and other macroeconomic and market factors. Several examples illustrate how time affects returns:
Long-Term Trends: Returns may exhibit long-term trends, such as a bullish or bearish trend in the stock market, driven by fundamental economic factors like economic growth, inflation, and government policies.
Short-Term Volatility: Short-term returns can be more volatile due to market condition changes, such as political uncertainty, commodity price fluctuations, or currency fluctuations.
Economic Cycles: Economic cycles can influence financial returns as economic conditions, fiscal policies, and monetary policies change over time. During economic growth periods, returns may be higher, while recessions may lead to lower returns.
Diversification Across Time and Multi-Modal Nature of Returns
Diversifying investments across different asset classes and over time can reduce portfolio volatility and risk, potentially enhancing long-term returns.
Returns exhibit a multi-modal nature over different time frames, challenging the assumption of a unimodal distribution as predicted by the normal distribution. Examining the S&P 500 annual returns illustrates this:
At an annual scale, market returns appear random and align relatively well with a normal distribution.
Over a 3-year period, returns become bimodal, indicating distinct market regimes.
Extending the time horizon to 10 years reveals a trimodal distribution, reflecting different market conditions.
a. The first mode corresponds to decades in a secular bear market, yielding modest single-digit returns.
b. The second mode represents periods with both pronounced bull and bear markets, resulting in varied returns.
c. The third mode on the right signifies a secular bull market with sustained upward trends.
Individual Asset Returns:
Investing in individual assets involves calculating returns as the difference between the investment's initial and realized values. This calculation includes income from the investment (dividends or interest) and deducts investment-related expenses (commissions and taxes).
Risk and the Asset Classes
Returns alone do not provide investors with all the necessary information about their investments, as they do not quantify the amount of risk incurred to achieve them. On the other hand, the individual risk of a financial instrument also does not convey much information when managing a diversified investment portfolio; in this context, the impact of the risk of each instrument tends to diminish, especially when a good level of diversification has been achieved.
As discussed in previous sections, risk can be defined as the uncertainty associated with an investment earning the expected rate of return. In the context of investment portfolios, when the number of financial instruments is large and the proportion invested in each of them is similar, the individual risk of each instrument becomes relatively insignificant compared to the portfolio risk. In this scenario, the critical aspect becomes the relationship or degree of association among the assets. Statistically, this degree of association is calculated as the correlation between pairs of assets.
Understanding correlation is essential for successful investment, as it aids investors in managing risk, diversifying their portfolios, and optimizing returns. This chapter will delve into the concept of correlation, its significance in investment, and best practices for managing correlation.
Correlation is a statistical measure of the relationship between two variables, X and Y. For example, it can gauge the extent to which temperature (X) is related to ice cream production (Y). One would expect higher temperatures to correspond with increased ice cream production.
There are different types of correlation depending on how the two variables move in relation to each other. A positive correlation means that as one variable increases, the other variable also increases. For instance, there is a positive correlation between height and weight: taller people tend to weigh more than shorter people. A negative correlation means that as one variable increases, the other variable decreases. For example, there is a negative correlation between smoking and life expectancy: smokers tend to have shorter lifespans than non-smokers. A zero or null correlation means that there is no relationship between the two variables. For instance, there is no correlation between hair color and IQ: blondes are neither smarter nor less intelligent than brunettes.
To measure the magnitude and direction of the correlation between two variables, a numerical value called the correlation coefficient is used. The most common type of correlation coefficient is the Pearson correlation coefficient, which ranges from -1 to +1. A +1 value indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation at all. The closer the value is to +1 or -1, the stronger the correlation; the closer it is to 0, the weaker the correlation.
Correlation coefficients can be utilized to analyze the relationship between different pairs of investments. For example, suppose you want to diversify your portfolio by investing in both stocks and bonds. You can calculate the correlation coefficient between the returns of stocks and bonds to see how they move together. If the correlation coefficient is positive, it means that stocks and bonds tend to rise and fall together; if it is negative, it means that stocks and bonds tend to move in opposite directions; if it is zero, it means that stocks and bonds are independent of each other.
However, caution must be exercised when considering correlation coefficients, as they depend on many variables. Here are some examples of correlation coefficients for different pairs of investments:
Stocks and bonds:
The correlation between stocks and bonds depends on various factors, such as interest rate levels, inflation expectations, economic growth, risk appetite, and market sentiment. In general, stocks and bonds have a low or negative correlation when interest rates are rising, inflation is high, or the economy is slowing down. This is because stocks are more sensitive to these factors than bonds, and investors may seek the safety and stability of bonds amid the uncertainty and volatility of stocks. Conversely, stocks and bonds have a high or positive correlation when interest rates are falling, inflation is low, or the economy is growing. This is because both stocks and bonds benefit from these factors, and investors may prefer the higher returns and growth potential of stocks over the lower yields and fixed income of bonds.
The source of the correlation between stocks and bonds can be traced to the underlying cash flows and risks of each asset class. Stock returns depend on growth prospects, while bond returns depend on interest rate levels, liquidity, and solvency. The correlation between stocks and bonds reflects how these cash flows and risks are affected by changes in the macroeconomic environment and investor expectations.
Gold and oil:
The correlation between oil and gold prices is complex and dynamic, depending on various factors such as supply and demand, inflation, currency fluctuations, market sentiment, and political events.
One way to measure the correlation between oil and gold prices is to use the gold-to-oil ratio, which divides the price of gold by the price of oil. This ratio indicates how many barrels of oil can be bought with one ounce of gold. A high ratio means that gold is relatively expensive compared to oil, while a low ratio means that oil is relatively expensive compared to gold.
The source of oil and gold prices is determined by different markets and mechanisms. Oil prices are primarily influenced by the Organization of the Petroleum Exporting Countries (OPEC), which controls around 40% of global oil production and can adjust its production to impact the supply-demand balance. Oil prices are also affected by geopolitical events, such as wars, sanctions, and natural disasters, which can disrupt oil production or transportation.
Gold prices are mainly influenced by demand for gold as a store of value, hedge against inflation, and safe-haven asset in times of uncertainty. Gold prices are also affected by the supply of gold from mining and recycling activities, as well as central bank policies that can increase or decrease their gold reserves. Gold prices are often quoted in US dollars, so changes in the value of the dollar can also affect the price of gold.
US dollar and euro:
The exchange rate between USD and EUR is determined by the supply and demand for both currencies in the foreign exchange market. Supply and demand are influenced by various factors, such as interest rate differentials, economic growth prospects, fiscal and current account positions, geopolitical events, market expectations, and speculation. The exchange rate fluctuates constantly, sometimes with high volatility.
Bitcoin and Ethereum:
The correlation between bitcoin and ethereum can be influenced by various factors, such as market sentiment, regulatory developments, technological innovations, and network events. For example, if both cryptocurrencies experience increased demand due to positive news or greater adoption, their prices may rise together, increasing their correlation. Conversely, if one cryptocurrency faces a technical issue or a security breach affecting its functionality or credibility, its price may fall while the other remains stable or rises, reducing their correlation.
A potential source of correlation between bitcoin and ethereum is their shared history and common origin. Both cryptocurrencies were inspired by the vision of Satoshi Nakamoto, the anonymous creator of bitcoin, who published a whitepaper in 2008 proposing a new form of digital money that would operate without intermediaries or central authorities. Ethereum was launched in 2015 by Vitalik Buterin and other founders who aimed to expand the capabilities of bitcoin by creating a platform that supported smart contracts and decentralized applications. Thus, both cryptocurrencies share some fundamental values and principles, such as decentralization, transparency, and innovation.
Another potential source of correlation between bitcoin and ethereum is their mutual influence and interdependence. Both cryptocurrencies are part of a broader ecosystem of blockchain projects and platforms that interact and collaborate. For example, some projects use both bitcoin and ethereum as collateral or payment methods for their services or products. Some projects also bridge the gap between the two networks by enabling cross-chain transactions or interoperability. Therefore, both cryptocurrencies can benefit from the growth and development of the blockchain industry as a whole.
As seen from these examples, correlation, far from being a causal variable, is merely a limited measure of the consequences of a large number of interdependencies that must be considered when analyzing any correlation value.
Ignoring the correlation between variables can lead to serious errors and misleading conclusions in data analysis. Correlation measures the magnitude and direction of the linear relationship between two variables and can help identify possible causal effects, confounding factors, or spurious associations. If correlation is ignored, one might mistakenly assume that there is no relationship between variables, or that the relationship is stronger or weaker than it really is. This can result in inaccurate predictions, invalid inferences, or missed opportunities for intervention or improvement. Therefore, it is crucial to always verify the correlation between variables before proceeding with any further analysis or drawing conclusions from the data.
There are two complex ideas we want to briefly mention to illustrate the concept of risk further. Firstly, the variability of the expected average return of an investment portfolio tends to decrease over time, making the average return to be aspired to more certain. Secondly, total risk decreases with diversification, meaning the range of possible outcomes for the expected average return during and at the end of the investment horizon is smaller in a highly diversified portfolio compared to one that is not.
References:
Williams, Edward E., and John A. Dobelman. Quantitative Financial Analytics: The Path To Investment Profits. Hoboken, NJ: John Wiley & Sons, 2017.
Miller, Michael B. Quantitative Financial Risk Management. Hoboken, NJ: John Wiley & Sons, 2018.
