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Frederick ”Forrest” Miller Worcester Polytechnic Institute |
Farzin Ahmadi Johns Hopkins University | |
It is with profound sorrow that we commemorate the recent loss of Harry Markowitz, an eminent figure in finance. The mathematical treatment of finance was pioneered by Harry Markowitz’s 1952 Paper "Portfolio Selection." This ground-breaking work established modern portfolio theory and his work continues to serve as an inspiration for researchers in finance. His Nobel Prize winning approach serves as a major benchmark for any portfolio allocation strategy.
The mathematical and scientific theory of finance developed immensely into the field that it is today, primarily in the 20th century. The field gained major recognition in the 1990 Nobel Prizes in Economic Sciences, awarded to Harry Markowitz, Merton Miller, and William F. Sharpe “for their pioneering work in the theory of financial economics” Markowitz et al. (2023). In 1997, the theory was expanded upon by Robert C. Merton and Myron Scholes “for a new method to determine the value of derivatives” Merton and Scholes (2023). Their work in the field has developed the two major arms of mathematical finance, known as the P world and the Q world. In the P world, the goal is to model the future, while the Q world’s goal is to extrapolate the present. Problems in the P world are focused on portfolio and risk management, while Q world problems are often option pricing problems. Quantitative Finance is now a huge area of research, being featured in many top journals such as Management Science and Operations Research. Researchers investigate new ways to describe market behavior, prescriptive methods for portfolio management, asset pricing, and more. In this article, we describe how prescriptive analytic methods can form the bedrock of a mathematical approach to portfolio management.
Introduction
The problem of portfolio management is defined as finding a set and quantity of stock shares to buy to achieve an optimal return. There is a need to prescribe a portfolio P that is “optimal” for the investor and their goals. Mathematically, we have that stocks (or assets more generally) S1 , ..., Sn yield returns R1 , ..., Rn over a certain investment time period (see Ruppert, 2004, Chapter 3 and 5). Based on these returns, the goal of portfolio managers is to find a linear combination
w1R1 +···+wnRn =R (1)
such that portfolio managers are “happy”, meeting their investment goals. While investors buy assets, it is much more common to think about the problem in terms of total returns, as shown in Equation (1). The manager’s “happiness” is typically achieved through yielding a certain expected return μp at a given risk level σp. Here, wi represents the weight of the ith asset. ωi > 0 translates to being long in the asset, ωi = 0 means that the investor does not own the asset and ωi < 0 is equivalent to being short in the asset. Here, we assume that a portfolio manager wishes to invest all of their funds into stocks. Portfolio managers then have the constraint that
Σ wi =1 (2)
In other words, this constraint says an investor must invest 100% of their funds into stocks. In times of economic growth, this can be achieved by selecting a portfolio that is similar to the entire market. However, during more volatile market conditions, managers aim to have more robust portfolios. With this goal in mind, the Modern Portfolio Theory (MPT), the basis for almost all quantitative portfolio management, was developed.
Markowitz’s Modern Portfolio Theory
The goal of an investor is to make a profit, which is achieved through portfolio management. A portfolio can be viewed as a linear combination of assets as seen in Equation (1). Typically, R1, ..., Rn are perceived as random variables that provide a certain return upon each realization. Therefore, it is common for portfolio managers to consider the expected return from their assets. Portfolio managers often aim to minimize risk while targeting a specific return μp.
Letting w be the vector of all weights, the variance of Rp can be derived as follows:
For the optimization, it is key to note that Ω is a symmetric, positive semidefinite matrix as cov(Ri, Rj ) = cov(Rj, Ri).
This creates the following optimization problem.
Which can be solved analytically using classical optimization theory (Ruppert, 2004, Section 5.5). Since μp is a hyperparameter, modifying μp creates a set of risk minimal weights for each target return. This creates what is known as the efficient frontier as seen in Figure 1. Note that the lower half of the parabola is marked in red. This is because investing in this half of the efficient frontier is inefficient, as we can achieve higher returns for the same level of risk shown in blue.
Figure 1: An example of efficient frontier
Restraining the weights further
In the original Markowitz portfolio problem, we require that the sum of the weights is 1. This means it is possible for some ωi to be negative, if there are other assets that balance this out. This can create w weights where there is extreme shorting present. Due to regulations known as margin requirements InteractiveBrokers (2023), the Markowitz problem is often augmented with the following constraint set:
li≤ wi ≤ ui for all i in {1,..,n}
Here, li is the lower limit, and ui is the upper limit on how much to invest into a certain asset. When li = 0 for every asset, this prevents shorting under the Markowitz framework.
The Risk-free Asset and Capital Asset Pricing Model
After the creation of the Markowitz framework, William Sharpe made a modification to the model based upon the existence of a risk free asset Sharpe (1964). A risk free asset is typically represented by a money market account offered by a large bank. The risk free asset yields a risk free rate r that is greater than 0. Thus, the set of portfolios of interest consists of portfolios that have a y intercept at r and are tangent to the frontier, as shown in Figure 2. The line created is known as the capital asset line. This requires modifying Problem 5 by removing the constraint Σ wi = 1 and having 1 − Σ wi invested into the risk free asset i=1 i=1 (see Ruppert, 2004, Chapter 5). The derivation of the tangency portfolio can be found in Section 5.5.4 of Ruppert (2004).
Figure 2: The Efficient Frontier with a Risk Free Asset. Source: Wikimedia (2005)
(Post) Modern Portfolio Theory
MPT is primarily used as a benchmark for institutions when developing and testing more complex trading strategies McClure (2022). In fact, the CFA (Certified Financial Advisor) Institute claims that “(MPT) has now become the prominent paradigm for communicating and applying principles of risk and return in portfolio management” CFA (2023a).
In the following years, much work has been done to relax the assumptions made by Markowitz and Sharpe to develop more realistic portfolio models. This has led to the development of post modern portfolio theory (PMPT). One aspect explored by PMPT is finding new measures of risk beyond the variance of the asset Geambasu et al. (2013); CFA (2023b). For further information on recent developments in portfolio management, we refer the reader to Gunjan and Bhattacharyya (2022). In particular, some popular risk measures include value at risk and expected shortfall. Value at Risk (VaR) is the lower quantile of the loss distribution of the profit and loss of X.
Although, value at risk is a very popular risk measure despite theoretical drawbacks, such as a lack of subadditivity Tangpi (2022); Föllmer and Schied (2016). Expected shortfall (also known as Conditional value at risk, CVaR) is defined as the expected loss beyond the VaR Tangpi (2022). Expected Shortfall is more theoretically powerful, as it is convex.
The Success of OR in Finance
Operations Research techniques have been particularly useful and successfully implemented in finance applications. Among different operations research techniques, mathematical programming is probably the most widely used in financial markets,
while other techniques like Monte Carlo Simulation, game theory, and network models have also seen success. The primary drivers of success in the financial markets are due to generally well-defined problems with clear quantifiable objectives and constraints that lead to implementable solutions. Additionally, unlike many other operations research applications, the historical data required for applying operations research techniques to financial markets are usually available Board et al. (2003).
Operations Research techniques have the potential to be widely successful for firms that apply them. For example, Gurobi has been used for successful portfolio management by some hedge funds Gurobi (2022a,b). Due to the nature of how quantitative the markets are, it is likely that firms are utilizing operations research for their investment strategies. Some of the most quantitative firms have reported 40% returns in 2022, and others are not far behind them Lee and Kumar (2022). The average quantitative hedge fund performance in 2022 was 8.5% AURUM (2023), while the S&P 500 dipped 19% in the same time frame.
When Things go ”wrong”
When we have a weight wi that is negative, this means that we are shorting the asset. This means that the investor makes money when the asset decreases in value. Shorting comes with many risks and restrictions, and is typically quite difficult to implement in practice. This is because shorting introduces unbounded risk for certain assets Chen (2023).
Leading up to the 2008 Recession, value at risk was an extremely popular risk measure despite its theoretical drawbacks. Thus, some blame the overreliance on VaR as one of the many root causes of the disaster Oxford (2016); Nocera (2009). Additionally, risk is often benchmarked against historical events Nocera (2009). This type of erroneous data can lead models to draw inaccurate conclusions StuartReid (2015).
As (P)MPT will give optimal solutions to the problem it is given, extreme short positions could potentially be undertaken without considering the specific firm being shorted against. This can result in significant premiums in insurance costs for such shorts. This happened to Michael Burry’s short position in the housing market Lewis (2010).
Conclusion
Operations Research and prescriptive analytic techniques will only continue to become more popular in the financial services industry. As other management strategies find themselves more exposed to the market’s swings, demand for robust portfolio management will remain strong. PMPT and MPT will continue to serve as a backbone and key benchmark for firms deciding on their investment strategies.
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