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Annuities and Amortisation

≈ 30 min
0/3 practice checks

Paying in versus paying off

An annuity is a series of equal payments xx made at equal intervals at a rate ii per period for nn periods.

Future value (saving up). Money you deposit now keeps growing, so the total at the end is

F=x[(1+i)n−1]iF = \frac{x\left[(1+i)^n - 1\right]}{i}

Present value (paying off). A loan today equals the value now of all the future repayments:

P=x[1−(1+i)−n]iP = \frac{x\left[1 - (1+i)^{-n}\right]}{i}

Always convert the quoted annual rate: for a nominal 9% p.a. compounded monthly, i=0,0912=0,0075i = \dfrac{0{,}09}{12} = 0{,}0075 and nn counts months.

Worked example: a retirement annuity

Sipho pays R1 500 at the end of every month into a fund earning 9% p.a. compounded monthly for 5 years, so i=0,0075i = 0{,}0075 and n=60n = 60:

F=1500[(1,0075)60−1]0,0075=1500(0,565681)0,0075≈R113 136,20F = \frac{1500\left[(1{,}0075)^{60} - 1\right]}{0{,}0075} = \frac{1500(0{,}565681)}{0{,}0075} \approx \text{R}113\,136{,}20

He paid in 1500×60=R90 0001500 \times 60 = \text{R}90\,000, so about R23 136 is interest.

Amortisation and the outstanding balance

A loan is amortised when each instalment pays the period's interest first and the rest reduces the capital. Immediately after the kk-th of nn payments, the balance outstanding is the present value of the payments that are still to come:

Balance=x[1−(1+i)−(n−k)]i\text{Balance} = \frac{x\left[1 - (1+i)^{-(n-k)}\right]}{i}

Thandi deposits R1 500 at the end of every month into a savings account earning 9% p.a. compounded monthly. How much (in rand) is in the account immediately after her last deposit, 5 years later? Round to the nearest rand.

The Dlamini family takes a home loan of R850 000 over 20 years at 11,25% p.a. compounded monthly. The first repayment is made one month after the loan is granted. Calculate the monthly repayment in rand, correct to the nearest rand.

A vehicle loan is repaid with 240 equal monthly instalments of xx rand at ii per month. Which expression gives the balance still outstanding immediately after the 60th payment?