1.7 Amortization and annuitiesIB Maths: Applications and Interpretation SL: Revision notes

A loan is amortized when it is repaid by a series of equal payments, each covering the interest due plus part of the principal (the amount borrowed). Early payments are mostly interest; later payments are mostly principal. In IB exams payments are made at the end of each period, and interest is compounded at the same frequency as payments, unless stated. Useful quantities:

• total repaid $=$ number of payments $\times$ payment;
• total interest $=$ total repaid $-$ amount borrowed;
• balance $=$ amount still owed after a given number of payments. Example: 12 000 USD at a nominal rate of 6% compounded monthly, repaid over 4 years: $N=48$ and the payment is $281.82$ USD, so total interest is $48\times281.82-12\,000=1527.38$ USD.

An annuity is a sequence of equal payments made at regular intervals. For savings, regular deposits earn compound interest and the balance grows. Example: depositing 200 EUR at the end of each month at 3.6% compounded monthly for 5 years gives a balance of $13\,126.32$ EUR. The deposits total $12\,000$ EUR, so interest earned is $1126.32$ EUR. Loans and savings use the same financial tools. The difference is the direction of the cash flow: a loan starts with a positive $PV$ and the payments reduce the balance to $0$; savings start from $PV=0$ and the deposits build up to a positive $FV$.

The financial solver (TVM solver) has these variables:

• $N$: total number of payments (years $\times$ payments per year);
• $I\%$: nominal annual interest rate;
• $PV$: present value (amount borrowed or initial deposit);
• $PMT$: payment each period;
• $FV$: future value (balance at the end);
• $P/Y$ and $C/Y$: payments per year and compounding periods per year. Enter four or five values and solve for the unknown. Money paid out is entered as a negative number and money received as positive (check your model's convention). For a loan fully repaid, $FV=0$. Write down the values you enter ($N=48$, $I\%=6$, $PV=12\,000$, $FV=0$, $P/Y=C/Y=12$) because the method marks depend on them. No formula is required: the annuity formula is not examined.

Solving for $N$ often gives a non-integer. Payments are whole, so round up for a loan to be fully repaid: $N=56.9$ means $57$ payments. The final payment is then smaller than the others. For savings, round up to the first month in which the target is exceeded. To find the balance after $k$ payments, set $N=k$ with the same $I\%$, $PV$ and $PMT$ and solve for $FV$. For 150 000 AED at 4.5% over 20 years, the monthly payment is $948.97$ AED and the balance after 60 payments is $124\,049.99$ AED. Shortening the term raises the payment but lowers the total interest.

Compound interest is exponential growth: after $n$ periods with rate $r$ per period, a single deposit $P$ grows to $P(1+r)^{n}$. A loan balance combines this growth with repeated payments, which is why the balance falls slowly at first. Here $r=\frac{\text{nominal rate}}{\text{compounding periods per year}}$, so 6% compounded monthly uses $r=0.005$ each month. You need not use the annuity formula. Knowing where the growth factor comes from helps you check that a GDC answer is sensible: deposits totalling $12\,000$ EUR should give a balance a little above $12\,000$ EUR.