**Highlights:**

**Highlights:**

- Quantitative finance refers to the development and testing of mathematically based investment strategies.

- One of the most powerful tools in a Quant’s tool box is the Monte Carlo simulation.

- Monte Carlo simulations can be used to forecast the risk and return of rules-based investment strategies.

*On my honeymoon I traveled out west. When I visited the casino and saw all these smart well-dressed people participating in a game with the odds against them, it was then that I realized I won’t have a problem getting rich!*

*– Warren Buffett*

Stanislaw Marcin Ulam was a Polish mathematician who immigrated to America in 1939. In 1940, he became an assistant professor at the University of Wisconsin–Madison (Go Badgers!), and in 1943 he accepted an invitation to work on the Manhattan Project, the research program that developed the first nuclear weapons. After the war, Stan suffered an acute attack of encephalitis, which required brain surgery.

During his recovery Stan passed the time by playing solitaire. He began to wonder if a computer program could be written to simulate the possible outcomes of a game of solitaire and thereby determine the probability of the outcome in advance. He shared his idea with two of his colleagues from the Manhattan Project. Together, the three mathematicians developed a computerized method to estimate the probability of future uncertain outcomes, which they named a Monte Carlo simulation.

As every investor should recognize, just like solitaire the future outcome of investment strategies are uncertain because markets behave unpredictably at times. As I noted here and here, it is this uncertainty that can cause individual investors and investment advisors alike to make rash and regrettable decisions when markets are behaving badly. Accordingly, it is very important for investors to have a solid understanding of how the value of their portfolio might be impacted by a bear market or by a disaster, war or similar dramatic occurrence (referred to as a “black swan” event), because that knowledge can provide the confidence needed to stay the course during a storm.

Quantitative finance is the name given to the development and testing of mathematically based investment strategies, and the use of math to give valuable insights as to the impact a recession or black swan event have on an investment portfolio. Quantitative finance professionals (commonly called “quants”) use software such as R and Python to create computer-run algorithms that have the potential to forecast returns and calculate the level of risk that can reasonably be expected from an investment strategy over a given time period.

One of the most powerful tools in a quant’s tool box is the Monte Carlo simulation. A Monte Carlo simulation (also known as probabilistic modeling) uses a large number (typically 10,000 or more) of computer-generated hypothetical results (simulations) to develop projections of future possible outcomes. The results of these hypothetical situations can be used to provide estimates (for a given degree of confidence) of an investment approach’s risk and return over various time periods and in a variety of market conditions.

“Tail Risk” is the term quants have coined to describe the largest decline a portfolio might experience as a result of a bear market or losses due to unfamiliar market conditions, such as a disaster, war or similar dramatic event.[1] Investors and portfolio managers can (and should) use Monte Carlo simulations to calculate a variety of statistical measures for insight on an investment strategy’s tail risk.

**Drawdown:** Refers to a decline in a portfolio’s value from the previous high to its lowest value prior to regaining the former high. As such, it is a good indication of downside risk over a specified period of time.

**Recovery Period**: The time it takes for a portfolio to return to its previous high after a drawdown.

**Value-at-Risk (VaR):** Value-at-Risk attempts to estimate, for a specific confidence level, the point to which the value of a portfolio might decline for a given time period. We calculate VaR for the 1-month, 3-month, 6-month, and 9-month time frames based on Monte Carlo simulations.

**Expected Shortfall (ES):** Expected Shortfall attempts to estimate the extent of a decline in a portfolio’s value that could theoretically occur if losses exceed VaR estimates. It does so by calculating a weighted average return of the worst returns. I prefer calculating ES for 1-month, 3-month, 6-month, and 9-month time frames.

**Worst Return (WR):** This method uses Monte Carlo simulations to calculate the lowest rolling returns generated by Monte Carlo simulations to estimate the worst returns for 1-year, 3-year, 5-year, and 10-year time frames. Estimated Shortfall’s predictive power weakens for longer time frames. As a result, I use the Worst Return method to estimate the worst-case scenario for portfolio returns for periods of 1 year and beyond.

The following chart gives a visual representation of Value-at-Risk and Expected Shortfall. The *x*-axis (Returns) shows the performance for a hypothetical portfolio. The *y*-axis (Frequency) shows the probability of the returns — the higher the green bar, the more likely a portfolio will experience the corresponding level of return on the *x*-axis.

The red line represents the VaR for a given confidence level. Expected Shortfall, represented by the orange line, is the weighted average of the returns that are beyond (worse than) the VaR.

Banks, insurance companies, and pension funds have long recognized the benefits of using quantitative analysis to estimate the impact recessions and black swans can have on their portfolios. Unfortunately, individual investors and financial advisors have been slow to adopt quantitative finance. Perhaps this is because many advisors are not comfortable with mathematics, or because ad hoc investment approaches that are based on predictions (i.e. guesses) or beliefs (superstition) do not lend themselves to quantitative analysis. As followers of this blog already know from my earlier posts, I am a firm believer in using rules-based investment strategies that are sensible, academically sound, and perhaps most – importantly, can be vigorously analyzed.

There’s an old saying that in every bet there is a fool and a thief. It refers to the fact that unless the odds of winning or losing are 50/50, one person – the thief – is going to have an edge. Monte Carlo simulations can help tilt the odds of dealing your portfolio a winning hand.

By comparison, the poor guys in “The Hangover” never seem to have the odds in their favor.

Thank you for reading,

Mr. Market Commentator

[1] “Tails” refer to the narrow ends of the bell shape associated with a graphic illustration of a “normal” distribution of possible investment returns. While traditional portfolio strategies tend to assume that returns will be distributed in a normal fashion, in reality markets frequently perform abnormally.

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