I wish I could explain my love for calculus. Despite the obscurity of Greek letters, the surplus of intricate theorems, and the complexity of equations that resemble something more like a cryptic alphabet than a string of numbers, there’s something so rewarding about simplifying the most nightmarish equation into y = sin(x). But y = sin(x) isn’t something immediately concrete: what follows, after you’ve graphed your sine wave? When you’re drowning in a challenging course, it’s understandably hard to remain curious about something so abstract and disconnected from real life between gasps of air. This is my attempt at justifying why NMM 2270, applied mathematics, will be a highlight in my second year—and how any math course could be yours too—because of its critical applications in business.
Unravelling the rhythms of stock charts, the behaviour of volatility clusters, and the intricate patterns in trading’s technical analysis, you find something critical but overlooked: math. Every financial model you can think of, from Malliavin calculus for evaluating sensitivity to the Black-Scholes Model for estimating option prices, calculus remains stubbornly relevant no matter how much you try to avoid STEM (sorry business majors). Algorithmic trading, for instance, applies derivatives to the premise of concavity to understand stock graphs, driving decisions on when to enter or exit a trade; good economic decisions cannot be made without the help of cost-minimization formulae. Even determining the optimal range to price a product, unfortunately, requires a mathematical background. However, underlying the exciting and flashy facets of business, whether it be global trading, credit lending, or quantitative research, calculus is applied in a less glorious but arguably, more critical way: in risk management.
As with every investment, there’s risk: day traders could succumb to fluctuations in the market, startup products could fail to catch on, and tech firms could lose their battles against AI. On a larger scale, financial institutions face a level of risk that increases proportionally to the value of their assets and the size of their clients. For banks, the mismanagement of risk cannot simply be followed by calling it quits and cutting losses; it means losing the trust and business of their clients (but gaining the legal fees!). That’s why risk management, despite lacking a reputation as business’s most exciting sector, is the armour of financial institutions.
First, risk needs to be quantified, which already poses a challenge. As Thomas Mayer, PhD, CFA, had articulated, “Methods to calculate possible results at, say, the roulette table, were of little use in determining the products of another European way or the future price of copper [...] or discount for the social status of property owners decades later. These possibilities were simply not calculable.” Calculus, in this sense, has become more of an art than a science, where different streams of business have adopted and molded it to produce a risk-evaluating metric. A familiar example is your credit score, the product of a mathematical algorithm that determines how much interest you pay and what the cap on your credit limit is. Similarly, credit risk analysis comes from scrutinizing historical data, like from S&P Global Ratings, while institutional traders would likely seek risk models compiled with data from stock exchanges. Having done risk analysis, the institution would evaluate it against their risk appetite, weighing factors like the potential profit margin as well as the probability of defaulting. Calculus-infused financial models do some of the heavy lifting, helping risk management departments determine exactly how much to lend, invest, or insure, depending on the scenario.
After talking to experts in the field, one quote stood out: the purpose of risk management is not necessarily to eliminate the risk, but rather, to optimize it. The motto “high risk, high reward” is remarkably applicable, where risk optimization retains the potential for high returns. As a result, what she found most rewarding about her profession is her team's power to influence change for big clients. For instance, if a company uses a loan to launch an exciting new product line, her team could rightfully take credit for approving the financing that made it possible. Considering the critical role of risk management as a line of defence for financial institutions, it does not get the exposure it deserves.
However, if risk management piques your interest, you can expect to work on projects and develop strategies that reduce risk, such as reviewing data and brainstorming with modelling teams. Having a thorough understanding of statistics, Excel, and of course, calculus, is necessary, with a baseline knowledge of coding languages helpful as well. You’ll likely be working on consolidating and analyzing data, sometimes forecasting statistic models to project losses given a default. Having strong interpersonal skills should also not be overlooked, despite the sector being more data-driven; it is crucial to understand the client and their motivation to assess risk from a qualitative standpoint.
Rest assured, calculus is integral to every sector where business, especially, is a goldmine for number-crunching opportunities. So, the next time you’re sitting in class, wondering when you’ll ever use the material you’re learning, you can be comforted by the fact that a career in risk management is just waiting to be explored.