Mark Greenfield, finishing Math PhD, shared some of his thoughts and experiences about his process of figuring out his post-PhD next steps, finding an internship in finance. Please check out his inspiring story!
This will be partly autobiographical and partly general advice. Each section has “Takeaways” which you can focus on if you just want to read some general suggestions and skip over the story parts. Everyone has a unique career path, and mine is just starting to unfold as I reach the final stage of my graduate school career. I hope my story and thoughts will be helpful to some of you. Feel free to email me at firstname.lastname@example.org if you’d like to chat. I would also be happy to help with interview practice.
From professor plans to considering alternatives
Like most pure math PhD students, I came in planning to be a professor in math academia. I enjoyed my undergraduate research projects and my early graduate school coursework. I always had in mind that the job market for academic math is very difficult, and that the fancy positions in nice cities are excessively hard to come by, but I felt that I would still be interested in positions at smaller schools or less prestigious institutions. I knew there always remained a possibility of leaving academia, and at some point I began to wonder about what alternatives exist and what kinds of skills are sought by employers.
Starting in my third year, I began asking around to learn about different kinds of career pathways. I attended some career seminars, wandered around the annual math department career fair, searched online, and so on. There are also different kinds of jobs within academia, such as primarily teaching-focused positions (Lecturer at a big school like UM, or professor at a liberal arts college, or teaching at the pre-college level). I have friends and colleagues in many of these positions, and I reached out to hear about their experiences. In the end, I realized that industry is probably the right direction for me.
My biggest concern was that my skills were not useful and it is too late to learn what I needed. Many friends around me who transitioned to industry had an undergraduate major in CS or applied math, and this seemed like a huge advantage. While it was certainly helpful for those friends, it is by no means necessary.
- Keep an open mind and know that your math skills are useful in many places.
- Don’t be afraid to reach out to people, even if you don’t already know them.
- There are many different career experiences – ask people about theirs!
Starting to prepare for industry
In my fourth year, I began more seriously asking around for different suggestions on ways to improve my skills for an industry career. It became clear to me that computing skills were the most important area in which I was relatively lacking. I had a brief “introduction to programming” course in 2011, but nothing since then. It was an important realization that there are in fact different departments at UM, and you can take classes outside the math department even if you have already completed your cognate! (My cognate was Physics 535: General Relativity, which has not been useful for career purposes, but enriched my understanding and appreciation of the universe, and I am very glad I took it.)
After seeking suggestions from several people, I decided to enroll in EECS402: Programming for Scientists and Engineers in Winter 2019. This is a great course to take and I strongly recommend it for anyone looking to develop some computing skills, even if they plan to stay in academia. This course assumes no programming experience and quickly teaches a broad, yet still deep, introduction to C++ programming, aimed primarily at graduate students in STEM. C++ is still used across industry, and other popular languages such as Python are easy to pick up after getting comfortable with C++. After speaking with some undergraduate students, it seems that EECS402 covers most of the material in 2 or 3 semesters of the CS major!
During that year, I discovered one additional factor which I realized to be perhaps the most important in deciding to focus on industry careers. I found myself enjoying my programming coursework at least as much as my math research work, and more motivated to learn how to solve real-world problems than to continue going deeper into abstract mathematics.
After taking EECS402 in Winter 2019, I completed three Coursera courses over the summer which were extremely helpful. These are the Applied Data Science with Python courses offered through University of Michigan’s Coursera platform, which are free for U of M students. They go through some major tools that are very similar to what is used in many industry jobs for mathematicians. The first three courses cover Pandas (data manipulation), MatPlotLib (complex plotting), and Sci-Kit Learn (Machine Learning). These three packages are extremely popular and almost any data-focused position will use these tools, or something very similar.
- Ask around for (1) what skills are useful for careers that interest you and (2) how to develop those skills. There are many great classes, some for credit and some online.
- Think carefully about what it is you enjoy about possible career directions and why you want to go in a certain direction. Other directions may be more tractable yet retain most of what you are seeking! (e.g. do you specifically want to study hyperbolic geometry, or do you more generally want interesting intellectual challenges?)
- On top of mathematics, computing skills are pretty much necessary and sufficient for getting into industry careers. Fortunately, you can very quickly pick up enough to get your foot in the door (and maybe even pass an interview process).
- In industry, you need a more broad set of skills – having any graduate training in mathematics at all is enough depth for most jobs.
- Take U of M’s Applied Data Science with Python courses. As a mathematician you may find them tedious at times, but the skills I gained from them very likely made the difference in my getting the offer in the end!
Applying for the internship
Starting in Fall 2019 (my fifth year), Prof. Karen Smith in collaboration with the Erdős Institute (a relatively new group of mathematicians organizing seminars and workshops to build connections and prepare grad students for industry jobs) started the Invitations to Industry seminar in our department. Every few weeks, a mathematics PhD holder comes to speak about their job in industry. I attended these, and found them all quite interesting.
One that caught my attention more than the others was the presentation by Dr. Chris Hammond (UM Math PhD, 2009) from Susquehanna International Group (SIG), a market maker and quantitative trading firm which focuses on financial derivatives such as stock options. At the time, I did not know what any of that meant, but I attended the talk and was intrigued. After the talk, I stuck around for some snacks, and spoke with Chris as well as Joey Thompson, a SIG recruiter (and former trader, with a mathematics background), about the position. Joey suggested that I apply for the SIG internship, a 10-week program designed to introduce STEM PhD students to the job of being a “Quant” (quantitative finance researcher). I had never before considered doing something like that, but the timing was just right and it seemed like an exciting opportunity.
That night, I emailed Joey asking a few more questions about the internship. He answered my questions quickly and invited me to start the interview process! He had apparently found my personal webpage and CV and had all he needed to put me on the list of applicants. My website and CV were not at all tailored for industry, but SIG (among other companies) is happy to recruit academics. So, I never really had to apply! I decided, sure, I have nothing to lose by trying, so I requested to begin the interview process two weeks later.
In terms of timing: job openings (including this internship) are very different from academic positions. The positions are filled on a rolling basis, so it may be more competitive if you wait. I began my interview process in October and was finished by late November. Some positions were filled several months later.
Thus began my preparation for the interviews! I was told to expect mostly math questions (especially probability, statistics, calculus, and linear algebra), along with possible computation questions, but no finance knowledge was required. Even if I am comfortable with those topics, answering them in an interview setting is a bit different from using them in the context of research, so I knew it would behoove me to practice and review. I’ll discuss this in the next section.
- Go to the Invitations to Industry talks, even if you are primarily focusing on academic careers. You might make some important connections! And, there is a lot of interesting work being done outside of academia.
- HAVE A WEBSITE AND CV AVAILABLE ONLINE, even way before you think you’ll need it. It can (and probably should) be very simple and clean, and needs only to introduce you as a mathematician (and possible future industry job applicant).
- If you are at all intrigued by a position, just try for the interview process. There are hundreds of great places to work, so if one goes badly, you’ll just try again with more experience.
- Apply early! It is advantageous to start earlier when it may be less competitive, and when you have more time to apply to additional positions if the first applications don’t pan out.
The interview process
At SIG, they generally seem to treat their graduate research interns as prospective full-time employees, so the interview process for the internship is identical to that for the full position. I’ll discuss the steps in the process as well as how I prepared at each stage. Of course, I am not at liberty to give specific questions, but I can be transparent about the steps in the process.
- Initial application: I did not have to do this stage – Joey found my website and CV online and invited me to do an interview! In general, you’ll just need to fill out some very basic information, as well as a CV and possibly something like a cover letter (depending on what company). The only preparation here is to ensure you have a clean and clear professional webpage, and a CV ready to send out. There are usually no reference letters needed.
- THREE phone interviews: these focused on math questions, mostly probability and statistics, but nothing too advanced (no measure theory, no stochastic differential equations). Some were trickier than others. In addition to solving math problems over the phone, you need to show interest in the industry and the company as well as an ability to communicate. For me, the interviews were spread to about one per week throughout October 2019. As for preparation, I found a few books (Heard On The Street, and 150 Most Frequently Asked Questions on Quant Interviews, but there are others) and worked through most of the relevant sections. I also found some old probability and statistics books from my undergraduate studies and practiced some of the basics. No individual question is particularly hard, but the context and pressure to solve it quickly can be daunting. I needed hints on several questions, but this did not disqualify me.
- Take-home data exercise: After I finished the third phone interview, SIG sent me some data and a prompt. I spent several hours per day throughout the following week working out my attempt, first at the library to study relevant methods, later on the computer writing some code. I found my background from the U of M Coursera courses extremely helpful for this part!
- In-person interview day: I was lucky enough to have the opportunity to visit the facility in person back in November of 2019. My interview day consisted of about 5 hours of additional interviews with 5 people and informal discussions with two others. It was mostly additional math questions, but there were also several questions about computing and some logic puzzles, as well as some questions to gauge my interest in the company and the work (as one should expect at any interview). I prepared mostly by continuing the same math practice, but I also did some extra practice on the non-technical side (have someone ask you: “Why are you interested in this position?” and other standard interview questions). I also read most of the pages on the company website. These “soft” skills are far too often overlooked.
Other companies will have different processes, but from discussing with others, I will summarize below some common points to be ready for in terms of interviewing.
- Prepare for the interviews. Do your homework: practice probability, read about the company, prepare questions to ask your interviewers (about the company, their career path, etc.), practice interviewing with friends.
- Don’t overlook the soft skills.
- Get excited about the company and the work! If you can’t convince yourself to be interested in the work, it is probably not the right place for you to apply.
- Some companies/interviewers might ask you to write code on the board. SIG does not require its research applicants to be as qualified in computing as software engineers, but you should have some nonzero computing background.
- Some companies/interviewers might ask you to discuss your research work. Even if you do not have your own results and publications, be ready to talk about some research-level topics with a broad mathematical or non-mathematical audience. This could be in a formal presentation or just some brief questions.
Preparing for the internship
While I was told there is no need to do additional preparation, I went ahead and did a bit of extra study as the internship approached. SIG’s internship, like many others, ends with the possibility of a full-time job offer pending graduation, so if you have any interest in going in this direction, it is crucial to do your best work and make the best impression. Here is a list of things I did. Of course, different internships will require different skills, and what worked for me may not work for you.
- Read through some online resources (e.g. Morningstar Finance) to learn about the very basics of stocks, stock markets, personal finance, and investing. See next point for why this is relevant.
- Read most of K. Pilbeam’s Finance and Financial Markets. This gave me a broad, non-technical introduction to the markets I was studying. While the work itself is mostly mathematical, as we all should know from math research, understanding the motivation, big picture, and even sometimes the history of mathematical ideas can help with the work. Hopefully, you will also find it interesting on its own (if not, maybe a different industry would be a better fit).
- Finish the Applied Data Science with Python Coursera sequence. While the tools in the last two courses were less useful for my specific tasks than the first three courses (mentioned earlier), it was good practice with some data science skills. Perhaps those other tools will be more useful later!
- Work through A. Ng’s Machine Learning Coursera course (taught through Stanford). This course is a little older and uses Matlab, so it may have been more relevant to try some other options. A friend recommended just finding datasets on Kaggle and working on individual projects. Kaggle may give more practice on using the tools, but less intuition for how the math works. For certain parts of industry though, either one may be more useful.
- Download the course materials for an old section of Math 525: Introduction to Probability and work through the homework problems on my own. You don’t really need 625-level probability for industry, at least as far as I have seen. But, you should be comfortable with the basics.
- Take Math 526: Discrete-state Stochastic Processes in Winter 2020. I chose this one after reaching out to Molly Bannow, the coordinator of the Quant Master’s program at UM. She had several helpful suggestions about the kinds of courses students in that program take to prepare for similar jobs in industry. Math 526 was my first probability course since 2012. There was not as much deep abstraction as I have gotten used to, but this was refreshing as well as helpful for my internship! Some of the concepts in this class would also have made my interviews easier….
- Do some investing. Even with a very small amount of money, you can open an account (e.g. with Vanguard, Robinhood, etc.) and learn about the process of buying shares of stock, or even options if that is reasonable for you. I’m happy to chat about any of these. Even if it is not feasible to invest, you can find some way to keep a fantasy portfolio.
- I should have done more programming practice, although everything still went alright. I would imagine internships at technology companies would require a higher level of computational skills, but perhaps the interviews at such places would weed out those who aren’t ready for that (such as myself!) anyway. I think in some positions a deeper knowledge of algorithms would be helpful.
- Industry jobs require a different skill set from academia. There are numerous opportunities to learn (mostly for free, since we are at a university) the relevant skills
- If you are planning on an industry career, your research does not need to be your only focus. Industry jobs care that you have some research experience, but once you’re interviewing or at an internship, you need to be competent at the work the company does.
- Don’t just focus on the math parts of your industry interests. If you can confidently and competently talk about the industry and show you understand the big-picture idea of what the company does, you will stand out. Everyone in these positions has graduate degrees in math/physics/CS/STEM, but not everyone is showing genuine interest in the work and knowledge of the industry!
The internship itself is a chance to get to know what it is like to work at the company. Due to the pandemic, I was participating remotely. I am extremely grateful that SIG was able to quickly redesign the internship program and make things work in a virtual setting. Regardless of the format, do not expect to have any time at all to work on your research! You will have nights and weekends off, but you should focus your energy and efforts on the internship, and use the down time to rest and recharge rather than work extra hours.
SIG requires employees (including interns) to be working from 8am – 5pm, Monday through Friday (in your time zone). From my apartment in Ann Arbor, I logged into a (very powerful) computer at SIG to do my work. A typical day in the internship consists of 1-3 hours of seminars or lectures, 30-60 minutes of meeting with your mentor, and the rest of the time working on your project. There is a lot going on, with plenty of opportunities to develop your skills and build connections.
The seminars and lectures are a mix of topics: some are for all interns (including those on trading, technology, and business teams) and some are just for the research interns. There were seminars about the finance industry in general, about different teams at SIG, about mathematical finance topics, and so on. There was even an ongoing competition about predicting possible news and stock market events. I imagine other companies would have some educational component, but one aspect that attracted me to SIG is how much time they are willing to invest in educating their recruits (as well as experienced employees), and the strong emphasis on continued learning. I even reached out to several of the presenters afterwards with additional questions and comments, and it was clear to me that they enjoyed discussing the material further.
Everyone is assigned a mentor during the internship, and you meet with them most days. I was impressed with how much time my mentor was willing and able to meet with me. Other programs I have heard of may have less time spent with a mentor. As we often hear about in academia, good mentoring makes a world of difference in one’s career at any stage, and so in my opinion, one aspect of a good internship program is the presence of effective mentors. SIG in particular not only assigns everyone a mentor, but continually emphasizes that we should feel welcome to reach out to others at the company for any kind of advice, be it technical advice or career advice or otherwise. I took advantage of this many times and learned a lot more about the company and the industry than if I had merely passively sat through the lectures.
Finally, at SIG (and I believe in most other similar internships) you will be assigned a project. This is the most important part of the internship, and you can expect to work hard on this throughout the summer. For me, I primarily worked in Jupyter notebooks using Python and a lot of data. I am not a software developer, so I did not need to focus on optimizing the code or writing something to be deployed – as a researcher I am trying to understand the data rather than write and implement the actual final products. In order to make a good impression, you will need to work hard and make progress on any topic they give you. My impression is that as a full-time employee, over time you gain more control over what you work on and what you research – the only limitation is that it is relevant to the company’s bottom line.
Beyond the typical days, there are several other noteworthy items. A few times during the summer, I had to give a presentation. The first few were to small groups of around a dozen people, but the final presentation had around 80 people in the audience, including several of the company founders and the majority of the researchers at SIG. While certainly intimidating, it impressed me how much everyone at the company values the internship program and our projects. There are also several social events for the interns, most of which have nothing to do with working at a finance company. For the remote internship, there was painting, cooking, tie-dye, and a few others over Zoom. For the usual in-person internship, I heard there are plenty of fun dinners (some formal, some informal), attendance at a baseball game, and more. It seems that I missed out on some open bars, as well!
Overall, it was a very busy summer and I worked hard throughout the internship. It was also a very rewarding experience and I learned a great deal about working in industry and within finance in particular. It was enough for me to decide that even if SIG didn’t work out in the long-term, I will proceed in this direction for my career. Fortunately, I was lucky enough to get the return offer, which I accepted. The job will start in June of 2021, pending graduation! Less than a year after hearing about the company for the first time, I am now very excited to move to Philadelphia and begin my career at SIG after I finish my degree.
- The internship is a real job: take it seriously, come in with a positive and productive attitude, and work hard!
- Take advantage of every opportunity the internship presents. Not only will you get more out of the internship itself, but you will look impressive (at least, if the interest is genuine!)
- Put your best effort into every aspect of the internship. Your project is the most important, but it is not the only part.
- Reach out to people in all different parts of the internship. Build connections, learn about different aspects of the company, make new friends – there are plenty of reasons to just ask, some strategic and some just for fun. This is related to the previous points!
Miscellaneous additional thoughts
- Perhaps the hardest and most important part of everything here is simply being willing to reach out and talk to people. When I spoke with Joey after the Invitations to Industry seminar, I had no intention of applying for a job that day. It turned out to be one of the most important interactions throughout my entire graduate school career.
- Many people have a very negative view of the finance world for a variety of reasons. Like most things in life, it is never so simple. The finance industry is massive and complex. There are parts which do amazing and interesting work that is positively impacting the world, and there are parts which are more prone to corruption (perhaps the latter is an understatement). This is true in any large institution, including universities and governments. Prof. Becca Winarski (past UM postdoc, now faculty at College of the Holy Cross) passed on some advice to me: you can do more good at improving an institution from the inside than from the outside.
- That said, I found the environment at SIG to be extremely welcoming. I will fully admit that I am coming from a place of privilege and my experience may not reflect that of others. During my brief virtual time there, I did not observe any toxic or problematic culture which is prevalent in other spaces, and there was significant emphasis on a positive and healthy workplace culture.
- If you enjoy teaching, in industry you may have the opportunity to give lectures, as well as the opportunity to host interns once you gain more experience. This is like teaching without having to do much lesson planning or grading!
- Working in some of these big industries provides some important “life” advantages. You have more power over what city you live in, while in academia you land wherever you get the offer. Further, for most academic positions, the salary is lower to the point where there are many additional stressors which are less present for some industry positions. For example, having a family, owning a house, and paying for several college educations can be incredibly difficult, even on a $100,000/year salary. Further, paying for life expenses and healthcare in retirement will be nearly impossible without a well-funded retirement plan, and it is unwise to assume that you will be healthy and happy enough to keep working full-time well into your senior years (a lot can happen in half a year… let alone half a century). This is not to say that academic careers cannot do these things, but it is more challenging and more sacrifices will have to be made, some of which are difficult to plan for in advance. For many people, the rewards of an academic career are more than worth it. Do what makes you happy!
—Mark Greenfield, UM Math PhD 2021 (expected)