TheIBAxolotl
Regular payments
Many real financial problems involve a payment every period, not a single deposit. A loan (mortgage, car finance) is repaid by equal instalments; each one first covers the interest owed for that period and the rest reduces the debt. This gradual paying-off is called amortisation. An annuity is the reverse direction: a stream of equal payments, either paid into a savings plan or received as income from a lump sum.
How a loan balance moves
In each period: new balance = old balance × (1 + period rate) − payment. Early on most of the payment is interest; near the end most of it is principal. That is why the total interest on a long loan can be very large.
The finance solver does the work
In this course you are expected to use the GDC's TVM solver rather than a formula. The key is the sign convention: from the borrower's point of view, money received is positive and money paid out is negative.
| Situation | PV | PMT | FV |
|---|---|---|---|
| Loan taken out and fully repaid | + loan | − instalment | 0 |
| Saving monthly from nothing | 0 | − deposit | + final amount |
| Lump sum paying an income until empty | − lump sum | + income | 0 |
Set P/Y and C/Y to the number of payments and compounding periods per year; they are usually equal.
Worked example
Sofia borrows USD 18 000 at a nominal 6% per year compounded monthly, to be repaid in equal monthly instalments over 5 years.
- Solver: N = 60, I% = 6, PV = 18 000, FV = 0, P/Y = C/Y = 12. Solve PMT: \(\approx -347.99\), so she pays USD 347.99 per month.
- Total repaid: $60 \times 347.99 = $ USD 20 879.40.
- Total interest: $20\,879.40 - 18\,000 = $ USD 2879.40.
(Behind the scenes the solver is using \(PMT = \dfrac{PV \cdot i}{1 - (1+i)^{-N}}\) with \(i = 0.005\); you do not need to memorise this.)
Savings version. Depositing USD 200 at the end of every month for 10 years at 4% compounded monthly: N = 120, I% = 4, PV = 0, PMT = −200, solve FV \(\approx\) USD 29 449.96, of which USD 24 000 was deposited.
Balance after some payments
To find how much is still owed after, say, 2 years of Sofia's loan, keep PV, PMT and I% but set N = 24 and solve for FV. The result (negative from the solver's view) is the outstanding debt, about USD 11 439. The principal repaid so far is the original loan minus this balance.
Comparing options
Compare loans by the monthly payment, the total amount repaid, and the total interest. A longer term lowers the monthly payment but raises the total interest. Questions may also ask you to find the rate (solve I%) or the number of months needed (solve N, then round up).
Common mistakes
- Giving PV and PMT the same sign for a loan.
- Entering the monthly rate (0.5) into I% instead of the annual rate (6).
- Reporting a non-whole N without rounding up; a final smaller payment completes the loan.
- Forgetting that total interest = total paid − amount borrowed.
Exam tips
- Always list your solver inputs; a correct list earns method marks even with a slip.
- Give money to 2 decimal places.
- Explain in words when asked which option is "better": say better for whom and by which measure.