ECON5005 · Quantitative Tools for Economics
Financial Mathematics
Financial mathematics is one of the most calculation-heavy strands of ECON5005 Quantitative Tools for Economics at the University of Sydney. It applies the exponential and logarithm tools of Week 3 to money over time: compound interest, geometric series, present value, annuities, mortgages and perpetuities. The unifying idea is the time value of money — compounding moves a value forward in time, discounting pulls a future value back to today — and mastering the six core formulas turns most exam questions into careful substitution.
What this chapter covers
- 01Compound interest: grow a principal forward with S = P(1 + r/m)^(mt), including monthly and continuous compounding
- 02Why more frequent compounding raises the future value, and where the effect flattens out
- 03Solve for time to a target (including doubling time) by taking logs of the future-value equation
- 04Sum a finite geometric series and recognise it as the engine behind every cash-flow stream
- 05Present value: discount a single future amount back to today with PV = FV/(1 + r)^n
- 06Value an ordinary annuity as a geometric series of discounted equal payments
- 07Rearrange the annuity present value into the mortgage repayment formula and read off total interest
- 08Price a perpetuity as PV = C/r and see why a higher discount rate lowers present value
- 09Distinguish ordinary (end-of-period) timing from an annuity-due (start-of-period) and adjust by (1 + r)
Monthly repayment on a 25-year mortgage
- +1Set up the per-period figures: monthly rate r = 0.054/12 = 0.0045; number of payments n = 12 x 25 = 300.
- +1Compute the discount factor: (1.0045)^300 = e^(300 ln 1.0045) = e^1.3470 which is about 3.8458, so (1.0045)^-300 is about 0.26003 and 1 - (1.0045)^-300 is about 0.73997.
- +1Apply the payment formula x = r x PV / [1 - (1 + r)^-n] = (0.0045 x 250,000) / 0.73997 = 1,125 / 0.73997.
- +1Monthly payment: x is approximately $1,520.32 per month.
- +1Total interest: total repaid = 1,520.32 x 300 = $456,096; interest = 456,096 - 250,000 = $206,096.
Key terms
- Compound interest
- Interest that is added to the balance each period and then itself earns interest; future value S = P(1 + r/m)^(mt) for principal P, nominal annual rate r, m compounds per year and t years.
- Present value (PV)
- The value today of a future cash flow, found by discounting: PV = FV/(1 + r)^n. It is compounding run in reverse and is always smaller than the future amount.
- Geometric series
- A sum whose terms multiply by a common ratio r each step; the finite sum is a(r^n - 1)/(r - 1) for r not equal to 1. It underlies annuities, savings plans and perpetuities.
- Annuity (ordinary)
- A finite stream of n equal payments made at the end of each period; its present value is PV = x[1 - (1 + r)^-n]/r, a geometric series of discounted payments.
- Mortgage payment
- The equal instalment that repays a loan of present value PV over n periods: x = r x PV / [1 - (1 + r)^-n], obtained by rearranging the annuity present-value formula.
- Perpetuity
- A level payment C made forever, first payment at the end of year 1; its present value collapses to PV = C/r. A higher discount rate r lowers this value.
- Time value of money
- The principle that a dollar today is worth more than a dollar later because it can earn interest; cash flows at different dates must be moved to a common date before they are compared or added.
- Continuous compounding
- The limit of compound interest as the number of compounds per year grows without bound: S = Pe^(rt). For a fixed nominal rate it gives the largest future value.
Financial Mathematics FAQ
Do I need a formula sheet or calculator in the ECON5005 exams?
The mid-semester and final are both in-person written exams (60 minutes and 2 hours respectively). Whether they are open or closed book, and the calculator or formula-sheet policy, are not stated consistently in the general course materials, so confirm these on your Canvas Assessment page before exam day. Either way, practise writing the six core formulas from memory so you are never dependent on a sheet.
What is the difference between an ordinary annuity and an annuity-due?
An ordinary annuity pays at the end of each period (the default the boxed formulas assume); an annuity-due pays at the start of each period. Because every payment in an annuity-due arrives one period earlier, its present value is the ordinary value multiplied by (1 + r). Always state which timing you are using — markers look for that assumption.
Can AI help me with financial mathematics in ECON5005?
Yes, used the right way. Sia is an AI tutor that explains concepts step by step — you can ask it to walk through why more frequent compounding raises the future value, how the mortgage payment formula is derived from a geometric series, or where an off-by-one in n creeps in, and it will show the working so you can learn the method. It will not sit your quiz or exam or hand you graded answers; the value is in understanding each step so you can reproduce it yourself under exam conditions.
Studying with AI? Sia — free AI economics tutor works through ECON5005 step by step.
Exam move
Treat financial mathematics as pattern recognition on top of one principle: compounding pushes money forward, discounting pulls it back. Memorise the six core formulas (compound future value, present value, finite geometric series, annuity present value, mortgage payment, perpetuity) and, for each, practise identifying r, n and the correct first cash flow before substituting anything. Drill the two directions until they are automatic — solving for the future value, and solving for time with logs — and always run a sanity check (a present value must be smaller than the future amount; total mortgage repayments must exceed the loan). Show the formula and substituted numbers on every question so method marks survive an arithmetic slip, and budget roughly two minutes per mark in the mid-semester exam.
