Financial mathematics classes in English, online
Looking for financial mathematics classes in English while you study in Spain? This is the subject I am asked about most. Compound interest, annuities, loan amortisation schedules and effective rates, taught one to one, online, at the time that suits you.
Financial maths has a reputation for being the module that separates the year group, and it is usually for the wrong reason. It is not that the maths is hard. It is that the notation changes from textbook to textbook and nobody tells you what the formula is actually doing. Once you can picture the cash flows on a timeline, most of the exercises stop being formula-hunting.
Simple and compound interest, and the rates that hide behind them
We start with the difference that everything else rests on: with simple interest you earn on the principal only, with compound interest you earn on the interest as well. From there we go into nominal against effective rates, equivalent rates for different compounding periods, and the conversion between them. This is where most people lose marks, not because the arithmetic is hard but because they take a nominal annual rate quoted with monthly compounding and use it as if it were annual. We work on that until the conversion is automatic.
Discounting: commercial against rational
Two methods with similar names and different answers, which is exactly why exams like them. Rational discounting asks what present amount, invested at the given rate, would grow into the future sum. Commercial discounting takes the interest off the face value directly, which is how bills of exchange are actually settled in practice. We work both, and more importantly we work out how to tell from the question wording which one is being asked for.
Annuities: constant, growing, deferred and perpetual
An annuity is simply a series of payments, and almost every applied problem in the subject is an annuity in disguise. Ordinary and annuity-due, present and future value, deferred annuities where the payments start later, growing annuities where each payment rises by a fixed percentage, and perpetuities. We build each formula from the geometric series rather than handing it to you, because in the exam you will meet a variant nobody showed you and you will need to derive it.
Loans, amortisation schedules and bond issues
The heart of the syllabus. The French method with constant instalments, which is what a Spanish mortgage actually uses, the constant principal method, and the American method where you pay interest only and repay the principal at the end. We build full amortisation schedules by hand, work out outstanding principal at any point, and handle early repayment. We also cover bond issues, where a company borrows from many lenders at once and redeems the bonds by draw.
APR and the rate the exam really wants
The annual percentage rate exists so that two loans with different fees and payment frequencies can be compared. It includes charges and it annualises the actual compounding. In practice that means the APR is never below the nominal rate: the two coincide only when there are no fees and payments are annual, and as soon as there are charges or monthly instalments it moves above. We calculate it properly, including the internal rate of return approach when the cash flows are irregular.
If your module is called something else
Financial mathematics travels under several names, and in many programmes it is not a module of its own but a block inside a finance course. If any of these is what you have in front of you, the syllabus is this one:
- Mathematics of Finance — the same name with the words the other way round; identical content.
- Financial Operations — the literal translation of the Spanish operaciones financieras, usual in degrees that split it from the maths module.
- Time Value of Money — the chapter name, when compound interest, annuities and discounting arrive inside a wider finance module.
- Interest Theory — the actuarial naming for the same compound interest, annuities and loan schedules.
- Loans and Annuities — the applied half on its own: amortisation schedules, the French system and bond issues.
- Matemáticas Financieras — the same syllabus taught in Spanish, if you would rather have the lessons in the language of your exam.
The content is the same in all of them: simple and compound interest, equivalent and nominal rates, discounting, annuities in their four shapes, loan schedules and APR. What changes is the notation each lecturer uses for the rate, and that is worth settling in the first session.
How the lessons work
One to one, by video call, on a shared whiteboard where we draw the timeline for every problem before touching a formula. No fixed schedule, no packages, no minimum number of sessions. Send me your notes and past papers first and we work on your lecturer's style of question. In English, in Spanish, or moving between the two as suits you.
Where almost everyone gets stuck
Using a nominal rate as if it were effective
A nominal annual rate of 12% with monthly compounding is not 12% a year. The monthly rate is 1%, and the effective annual rate is 1.01 to the twelfth minus one, which is 12.68%. Mixing the two is the single most expensive mistake in this subject, because it makes every subsequent line of the exercise wrong.
Putting the valuation date in the wrong place
Every annuity formula values the payments at one specific moment: the ordinary annuity formula gives you the value one period before the first payment. If the first payment is deferred, that result is not yet the present value and you still have to discount it back. Draw the timeline and mark where the formula lands you.
Building a schedule without checking that the principal closes at zero
An amortisation schedule has a built-in check: the principal repayments must add up to the amount borrowed, and the outstanding balance must be zero after the final instalment. If it is not, you have an error somewhere above. Rounding will leave a cent or two, which is fine. Anything larger is a mistake, and it is worth finding before you hand the paper in.
Worked example: French amortisation schedule, 10,000 euros at 5% over 4 years
You borrow 10,000 euros, to be repaid in four equal annual instalments, at an effective annual interest rate of 5%. Calculate the instalment and build the amortisation schedule, checking that the outstanding principal closes at zero after the fourth year.
- Constant instalment (ordinary annuity): a = 10,000 x 0.05 / (1 - 1.05^-4). Since 1.05^4 = 1.21550625, then 1.05^-4 = 0.82270247 and the denominator is 0.17729753. So a = 500 / 0.17729753 = 2,820.12 euros.
- Year 1: interest = 5% of 10,000 = 500.00 euros. Principal repaid = 2,820.12 - 500.00 = 2,320.12 euros. Outstanding = 10,000 - 2,320.12 = 7,679.88 euros.
- Year 2: interest = 5% of 7,679.88 = 383.99 euros. Principal repaid = 2,820.12 - 383.99 = 2,436.13 euros. Outstanding = 7,679.88 - 2,436.13 = 5,243.75 euros.
- Year 3: interest = 5% of 5,243.75 = 262.19 euros. Principal repaid = 2,820.12 - 262.19 = 2,557.93 euros. Outstanding = 5,243.75 - 2,557.93 = 2,685.82 euros.
- Year 4: interest = 5% of 2,685.82 = 134.29 euros. The final principal repayment is exactly the outstanding balance, 2,685.82 euros, so the last instalment comes to 2,820.11 euros, one cent lower because of rounding, and the balance closes at 0.00.
SolutionThe constant instalment is 2,820.12 euros a year. The four principal repayments (2,320.12 + 2,436.13 + 2,557.93 + 2,685.82) add up to exactly the 10,000 euros borrowed, total interest is 500.00 + 383.99 + 262.19 + 134.29 = 1,280.47 euros, and the total repaid is 11,280.47 euros.
About these lessons in particular
My course is called financial math, not financial mathematics. Is it the same thing?
Almost certainly yes. Spanish programmes call it matematicas financieras and English-taught ones use financial mathematics, financial math or mathematics of finance. Send me the syllabus and I will confirm before we book anything.
Do I need a financial calculator?
No. Everything in this course can be done with a scientific calculator, and I would rather you understood the formula than pressed a key that hides it. If your university requires a specific model, we can work with that too.
Can we work from my own past papers?
That is how I prefer to do it. Send me your lecturer's problem sets and any past exams before the first lesson and we build the sessions around them.
What is the difference between the nominal rate and the effective rate, in one sentence?
The nominal rate is quoted per year but charged in fractions of a year without compounding inside, and the effective rate is what you actually end up paying once that compounding is counted. Dividing a nominal annual rate by twelve gives you the monthly rate; raising that to the twelfth power and subtracting one gives you the effective annual rate, and those two numbers are not the same.
Why does my amortisation schedule stop matching after a few rows?
Almost always because the rate and the period do not match, or because the interest is being charged on the wrong balance. Interest in each row is the rate for the period times the balance outstanding at the start of that row, never the original amount. Once the first two rows are right, the rest follows.
Can we do just the loans block before the exam?
Yes, that is the most requested block. Amortisation schedules, the French system and the outstanding balance at a given date are self-contained, and one or two sessions on them is a normal way to use these lessons.
Which APR does the exam want, the one in the contract or the one I calculate?
The one you calculate from the actual cash flows, including fees and charges, which is usually not the rate printed in the statement. We practise the distinction explicitly, because the question is nearly always designed around it.
Where can I find a financial mathematics tutor in English for a Spanish degree?
Here, and online is the right format for this one in particular: compounding, discounting, the APR that your syllabus calls TAE, annuities and loan amortisation schedules are far easier to follow on a shared spreadsheet than on a whiteboard. I work with students across Spain in Business Administration, Economics and Finance degrees, including when the module is called Mathematics of Finance, Financial Operations, Interest Theory or Matematicas Financieras.
I also teach
Shall we work on it together?
Tell me where you are, which university you are at and when the exam is. I will get back to you as soon as I can.