Finance algebra
With the following variables the periodic (annual) repayment is given by The recurrence equation for the balance b at the end of month x is derived from b[x + 1] = b[x] (1 + r) - d where b[0] = s giving The interest portion of the final payment is b[n - 1] r and the total principal repaid at the end of period n - 1 is s - b[n - 1] Solving simultaneously n = 8.9998 and s = 7240 The principal repaid at the end of the first period is s - b[1] or d - r s = 479.74
According to my calculations, you always lose money on group B.  x = average monthly balance  Income for a year = 0.015 * (12 * x) = 0.18 * x  Cost of funds for one month = 0.04 * x  Cost of funds for one year = 12 * (0.04 * x) = 0.48 * x  Profit? at end of year   = income_for_year - cost_of_funds_for_one_year  = (0.18 * x) - (0.48 * x)  = forever loss There are really three routes in academic finance.   1) Corporate finance - you probably have enough math for this  2) Market forecasting and modeling - this relies more heavily on econometrics, or more basic math  3) Quantitative valuations - most of the work here is in exotic asymmetric derivatives, which is almost entirely differential equations. Finance at the PhD level requires a pretty complete understanding of linear algebra, multivariate calc (cIII), vector calc, stochastic calc, diff eq, and numerical analysis.  As someone who had math for one of his majors for undergrad, it's not about the intuitive understanding but the abstract as well.  Most of the finance professors that I had and researched under were originally undergrad for math.  Very few of them actually started out in business roles/degrees.... Shareholders are the equity holders. They mean the same thing.  A simplified formula for the total value of a company is the value of its equity, plus the value of its debt, less its cash (for reasons I won't get into). There are usually other things to add or subtract, but that's the basic formula. I work in asset pricing/market microstructure research so I do come across a decent amount of topics at work that utilize my applied math degree. For the topics I'm working on right now on it's usually time series or econometrics stuff like vector autoregression, principal components, regressions, etc. Some of my coworkers are working on papers that use stochastic calculus and other aspects of continuous time finance. The accounting equation, in short, is: This can be further broken down into: Which can be further broken down into: The GnuCash equation is right, though I would substitute the word equity in that equation with a more-specific paid-in capital. Equity is (simply put) made up of 2 parts: shareholders' equity and retained earnings. Shareholders' equity is the amount invested by shareholders. Retained earnings is the amount earned by the business on behalf of the shareholders. Retained earnings is directly affected by your net income (which is income minus expenses). An increase in income will result in an increase in retained earnings. This must be balanced somewhere. Usually an increase in an asset. It may also be balanced by a decrease in equity. Likewise, increase in expenses will result in a decrease in retained earnings, which must also be balanced. Something like cost = a × avg_spreadb + c × volatilityd × (order_size/avg_volume)e. Different brokers have different formulas, and different trading patterns will have different coefficients. I'm a third-year PhD student in Finance at a state university in the Southeast.  Accordingly, I do not have time write a detailed answer for you, because I am studying for a test.  You can PM me sometime after Thursday, and I can give you my perspective. The three places you want to focus on are the income statement, the balance sheet, and cash flow statement. The standard measure for multiple of income is the P/E or price earnings ratio  For the balance sheet, the debt to equity or debt to capital (debt+equity) ratio. For cash generation, price to cash flow, or price to free cash flow. (The lower the better, all other things being equal, for all three ratios.) Assuming you were immersed in math with your CS degree, the book **'A Non-Random Walk Down Wall Street' by Andrew Lo** is a very interesting book about the random walk hypothesis and it's application to financial markets and how efficient markets might not necessarily imply complete randomness.    Lots of higher level concepts in the book but it's an interesting topic if you are trying to branch out into the quant world.  The book isn't very specific towards algorithmic trading but it's good for concept and ideas.  Especially for general finance, that will give you a good run down about markets and the way we tackle modern finance.    **A Random Walk Down Wall Street** (which the book above is named after) by **Burton Malkiel** is also supposed to be a good read and many have suggested reading it before the one I listed above, but there really isn't a need to do so.    For investing specifically, many mention **'The Intelligent Investor' by Benjamin Graham** who is the role model for the infamous Warren Buffet.  It's an older book and really dry and I think kind of out dated but mostly still relevant.  It's more specifically about individual trading rather than markets as a whole or general markets.  It sounds like you want to learn more about markets and finance rather than simply trading or buying stocks.  So I'd stick to the Andrew Lo book first.    ---  Also, since you might not know, it would be a good idea to understand the capital asset pricing model, free cash flow models, and maybe some dividend discount models, the last of which isn't so much relevant but good foundations for your finance knowledge.  They are models using various financial concepts (TVM is almost used in every case) and utilizing them in various ways to model certain concepts.  You'd most likely be immersed in many of these topics by reading a math-oriented Finance book.  Try to stay away from those penny stock trading books, I don't think I need to tell a math major (who is probably much smarter than I am) that you don't need to be engaging in penny stocks, but do your DD and come to a conclusion yourself if you'd like.  I'm not sure what career path you're trying to go down (personal trading, quant firm analyst, regular analyst, etc etc) but it sounds like you have the credentials to be doing quant trading.    ---  Check out www.quantopian.com.  It's a website with a python engine that has all the necessary libraries installed into the website which means you don't have to go through the trouble yourself (and yes, it is fucking trouble--you need a very outdated OS to run one of the libraries).  It has a lot of resources to get into algorithmic trading and you can begin coding immediately.  You'd need to learn a little bit of python to get into this but most of it will be using matplotlib, pandas, or some other library and its own personal syntax. Learning about alpha factors and the Pipeline API is also moderately difficult to get down but entirely possible within a short amount of dedicated time.     Also, if you want to get into algorithmic trading, check out Sentdex on youtube.  He's a python programmer who does a lot of videos on this very topic and has his own tool on quantopian called 'Sentiment Analyzer' (or something like that) which basically quantifies sentiment around any given security using web scrapers to scrape various news and media outlets.  Crazy cool stuff being developed over there and if you're good, you can even be partnered with investors at quantopian and share in profits.    You can also deploy your algorithms through the website onto various trading platforms such as Robinhood and another broker and run your algorithms yourself.    Lots of cool stuff being developed in the finance sector right now.  Modern corporate finance and investment knowledge is built on quite old theorems and insights so expect a lot of things to change in today's world.  ---  With a math degree, finance should be like algebra I back in the day.  You just gotta get familiar with all of the different rules and ideas and concepts.  There isn't that much difficult math until you begin getting into higher level finance and theory, which mostly deals with statistics anyways like covariance and regression and other statistic-related concepts.  Any other math is simple arithmetic. The quant aspect '''''. This is the kind of math I was wondering if it existed, but now it sounds like it is much more  complex in reality then optimizing by evaluating different cost of capital. Thank you for sharing I appreciate it. I didn't operate under selling the asset year five but other than that I followed this example. I appreciate the help. These assignments are just poorly laid out. Financial management also plays on different calculation interactions  so it is difficult for me to easily identify the intent at times. Thanks again. I ended up writing a simulation in R. Here is my code: It produces a plot like this:  This code assumes you have a lump sum and either wish to pay down a loan or invest it all immediately. Feedback welcome. I'm not too worried about the math. I can't speak for the calculus, but I seriously doubt the statistics I will use will be more complicated than what I used in Econometrics.   Where will calculus come into play? I haven't heard any of my finance major friends talk about using anything more than simple derivation. Average rates of return usually assume compounding, so your formula would be for annual compounding ,or for continuous compounding.