The Math of Investing

This is an essay I originally submitted for an Information Systems class in college, and I think y’all might get a kick out of it. Enjoy!

Investors have 2 goals: maximize returns and minimize risk. However, with tens of thousands of companies around the world and dozens of asset classes, this seemingly simple directive is extremely challenging without the help of mathematical models and computer software. Thanks to the contributions of Nobel Prize winner Harry Markowitz in 1952 and continual supplements to his original work, even novice investors can achieve returns comparable to institutional management firms. Operations research professionals are utilized to improve portfolios with Markowitz’s Modern Portfolio Theory, and computer analysists gather and model overwhelming amounts of data on a daily basis. In this way, math is used to increase returns and minimize risk.

Harry Markowitz revolutionized the connection between math and investing in a paper called Portfolio Selection, which was published in the Journal of Finance in 1952. His work, now referred to as Modern Portfolio Theory, rests on a couple underlying assumptions. First and foremost, it assumes that an investment’s annual rate of return resembles a standard normal distribution. Second, the theory expects future market behavior to resemble past occurrences. Even though neither of these assumptions is correct, they are accurate enough to create a useful model. With that foundation, mathematical concepts can be applied to each individual stock, bond, fund, etc., and they can be combined to form optimized portfolios.

The mean of an asset’s standard normal distribution is its expected annual return. This number is the percentage its value is expected to increase over the course of a year. However, performances during each individual year will vary significantly as measured by the standard deviation. Because of the assumption that future performance will resemble previous results, Modern Portfolio Theory expects each asset class to increase in value over the long run. For this reason, risk is tied to volatility, or the size of fluctuations in value over time; the model does not consider the potential for an investment to go belly up. Since the ultimate goal is to build a portfolio, or collection of several different assets, they must be rated on their correlation to one another.

To evaluate the benefit of diversification, mathematicians use the covariance of 2 different investments. This measures the likelihood of their deviations resembling one another. The covariance between 2 identical assets will be 1, somewhat similar investments will have a covariance between 0 and 1, completely uncorrelated investments will have a value of 0, and inversely related assets will have a value between 0 and -1. On a large scale, every single asset has a covariance with every other asset in the market. The goal in building a portfolio is to combine several different investments that move up and down at different times. This produces a nice net gain, but more importantly, the portfolio’s overall value will increase at a smooth level without drastic spikes and drops. Markowitz’s research includes a tool to optimize these possible combinations.

Based on Modern Portfolio Theory, each given pool of investments has 1 combination that maximizes the expected return at any given level of expected risk. If each possible combination was plotted on a graph with axes for risk and standard deviation, the points would form a distinct curve. This is called the efficient frontier. Any combination that produces a point inside of the curve is inefficient because either the return could be increased without raising risk, or the risk could be lowered without dropping expected return. However, points outside of the curve are unattainable; no investment can provide that combination of risk and return. I have created the following illustration to explain this topic.

Efficient Frontier

Modern Portfolio Theory provides an excellent understanding of investments at a basic level, but it has two glaring problems. First, as the disclaimer for virtually every adviser states, past performance does not guarantee future results. Secular market trends can last for decades, only to reverse completely. Take interest rates for example: they’ve been gradually decreasing for almost 35 years, but after reaching zero, they’ll probably start going back up. This means that the future performance of bond funds will be significantly lower than in past years, which can impact portfolio construction. The second problem with Modern Portfolio Theory is the lack of available data. Business cycles last an average of about 6 years, and to properly measure the performance and volatility of an investment, economists need information from several cycles. This means that a good model requires at least 20 or 30 years of data, but many companies and funds haven’t been around that long. For these reasons, investment firms are using computer analysis to create performance forecasts for the real world.

With almost $4 trillion under management, Vanguard is one of the premier investment companies, and in order to help its clients make informed decisions, it has created the Vanguard Global Capital Markets Model (abbreviated VCMM). The model is “used as the data-generating process in combination with Monte Carlo methods to derive both expectations and distributional properties of asset returns.” Vanguard’s forecasting has 4 main modules:

  1. The core module examines macroeconomic risk factors and drivers.
  2. The local attribution module links local asset-class returns with regional risk factors.
  3. The global attribution module forecasts global asset-class returns based on worldwide risk factors and foreign currencies.
  4. The simulation module combines these relationships with regression-based Monte Carlo methods to generate a range of potential future returns based on 10,000 simulations.

By utilizing advanced computer analysis techniques, the VCMM can create forecasts with far greater accuracy than traditional approaches. While the Modern Portfolio Theory simply plugs historical data into a standard normal distribution, Vanguard’s team of economists and mathematicians can incorporate a variety of factors into its model. These types of models are unique for several reasons:

  • They can account for the way current market conditions drive future results.
  • They quantify the impact of factors such as changing interest rates, inflation shocks, and economic growth.
  • They are forward looking, considering the possibility of extreme risk events that haven’t occurred in the past.
  • By focusing on ranges of outcomes instead of point forecasts, they can better incorporate statistical uncertainty.

Basic models, advanced simulation software, and everything in between would be impossible without the big data capabilities of computer analysts. If you get onto the internet right now, you can find the price of any fund or stock on each individual day of its existence. We also have records for inflation, housing prices, currency relationships, GDP’s, tax rates, and trade deals. It is only because of this mind boggling quantity of information that economists can predict the performance of asset classes and perspective portfolio allocations. The technical expertise provided by computer analysts is vital in collecting and applying this information in an easy to understand manner that the average investor can interpret and use in the decision-making process.

For decades, math has been improving returns and decreasing risk by helping investors make decisions. It started with basic statistics: modeling individual assets in standard normal distributions and then combining them to form a portfolio with one mean and standard deviation based on their covariances. Harry Markowitz’s approach is still used regularly today, but because of its simplified premises and long-range data requirements, the results have limited accuracy. With advanced software and simulation techniques, mathematicians can now look at the impact of current conditions on possible future returns and include the potential for new risk factors in their predictions. All of this work is made possible by the work computer analysts do to interpret and apply big data in a way that ordinary people can study and use in investment decisions. In these ways, math is being used to improve returns and decrease investment risk.

I hope you enjoyed the essay! I find it fascinating the way math and investing are connected. If you enjoyed reading this, then scroll down, click on the Facebook logo, and like The Declassified Dollar page to get great content sent straight to your feed. Special thanks to Investopedia, BNY Mellon, Vanguard, and Trading Economics, which I used as sources for the paper.

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