Therefore, the present value of all outstanding future payments equal the present amount outstanding. This is the prospective method for calculating capital outstanding.
Let's return to a previous example. Recall the case where we were trying to repay a loan of R200 000 over 20 years. A R10 000 deposit was put down, so the amount being payed off was R190 000. At an interest rate of 9% compounded monthly, the monthly repayment was R1 709,48. In
[link] , we can see that after 12 months, the amount outstanding was R186 441,84. Let's try to work this out using the the prospective method.
After time 12, there are still
$19\times 12=228$ repayments left of R1 709,48 each. The present value is:
$$\begin{array}{ccc}\hfill n& =& 228\hfill \\ \hfill i& =& 0,75\%\hfill \\ \hfill Y& =& R1\phantom{\rule{3.33333pt}{0ex}}709,48\times \frac{1-1,{0075}^{-228}}{0,0075}\hfill \\ & =& R186\phantom{\rule{3.33333pt}{0ex}}441,92\hfill \end{array}$$
Oops! This seems to be almost right, but not quite. We should have got R186 441,84. We are 8 cents out. However, this is in fact not a mistake. Remember that when we worked out the monthly repayments, we rounded to the nearest cents and arrived at R1 709,48. This was because one cannot make a payment for a fraction of a cent. Therefore, the rounding off error was carried through. That's why the two figures don't match exactly. In financial mathematics, this is largely unavoidable.
As an easy reference, here are the key formulae that we derived and used during this chapter. While memorising them is nice (there are not many), it is the application that is useful. Financial experts are not paid a salary in order to recite formulae, they are paid a salary to use the right methods to solve financial problems.
Definitions
$P$ |
Principal (the amount of money at the starting point of the calculation) |
$i$ |
interest rate, normally the effective rate per annum |
$n$ |
period for which the investment is made |
$iT$ |
the interest rate paid
$T$ times per annum, i.e.
$iT=\frac{\mathrm{Nominal}\mathrm{Interest}\mathrm{Rate}}{T}$ |
Equations
$$\left(\begin{array}{c}\mathrm{Present}\mathrm{Value}-\mathrm{simple}\hfill \\ \mathrm{Future}\mathrm{Value}-\mathrm{simple}\hfill \\ \mathrm{Solve}\mathrm{for}\mathrm{i}\hfill \\ \mathrm{Solve}\mathrm{for}\mathrm{n}\hfill \end{array}\right\}=P(1+i\xb7n)$$
$$\left(\begin{array}{c}\mathrm{Present}\mathrm{Value}-\mathrm{compound}\hfill \\ \mathrm{Future}\mathrm{Value}-\mathrm{compound}\hfill \\ \mathrm{Solve}\mathrm{for}\mathrm{i}\hfill \\ \mathrm{Solve}\mathrm{for}\mathrm{n}\hfill \end{array}\right\}=P{(1+i)}^{n}$$
Always keep the interest and the time period in the same units of time (e.g. both in years, or both in months etc.).
End of chapter exercises
- Thabo is about to invest his R8 500 bonus in a special banking product which will pay 1% per annum for 1 month, then 2% per annum for the next 2 months, then 3% per annum for the next 3 months, 4% per annum for the next 4 months, and 0% for the rest of the year. The are going to charge him R100 to set up the account. How much can he expect to get back at the end of the period?
- A special bank account pays simple interest of 8% per annum. Calculate the opening balance required to generate a closing balance of R5 000 after 2 years.
- A different bank account pays compound interest of 8% per annum. Calculate the opening balance required to generate a closing balance of R5 000 after 2 years.
- Which of the two answers above is lower, and why?
- 7 Months after an initial deposit, the value of a bank account which pays compound interest of 7,5% per annum is R3 650,81. What was the value of the initial deposit?
- Thabani and Lungelo are both using UKZN Bank for their saving. Suppose Lungelo makes a deposit of
$X$ today at interest rate of
$i$ for six years. Thabani makes a deposit of
$3X$ at an interest rate of
$0.05$ . Thabani made his deposit 3 years after Lungelo made his first deposit. If after 6 years, their investments are equal, calculate the value of
$i$ and find
$X$ . If the sum of their investment is R20 000, use
$X$ you got to find out how much Thabani got in 6 years.
- Sipho invests R500 at an interest rate of
$log(1,12)$ for 5 years. Themba, Sipho's sister invested R200 at interest rate
$i$ for 10 years on the same date that her brother made his first deposit. If after 5 years, Themba's accumulation equals Sipho's, find the interest rate
$i$ and find out whether Themba will be able to buy her favorite cell phone after 10 years which costs R2 000.
- Calculate the real cost of a loan of R10 000 for 5 years at 5% capitalised monthly. Repeat this for the case where it is capitalised half yearly i.e. Every 6 months.
- Determine how long, in years, it will take for the value of a motor vehicle to decrease
to 25% of its original value if the rate of depreciation, based on the reducing-balancemethod, is 21% per annum.