When a loan is provided with discounted interest, the borrower receives the principal at the beginning of the loan period minus the interest that the principal would otherwise accrue during the loan period. This method may apply to certain types of financial instruments, such as bills of exchange. For example, if you were to borrow £100 for one year at a discount rate of 4%, you would receive £96 at the outset and after a year, you would repay £100.
However, as the borrower will receive £96 instead of £100, the true interest rate on the loan will be higher than the discount rate. The true interest rate can be calculated using the present value formula:
\(PV=\frac{FV}{1\:+\:in}\)
where:\(FV=future\: value,PV= present \: value,i= interest \:rate,n= length \:of\: investment.\)
This can be turned around to give
\(i=\frac{FV}{PV}\:1/n\:-1\)
So in the above example,
\(i=\frac{100}{96}\:-\:1=0.0417\)
Therefore the true interest cost of this particular investment is 4.17%.
The concept of discounting can also be used to determine the present value of a sum that is expected in the future. For example, if you know that you will receive £500 in two years’ time, and the annual interest rate is 5%, you can calculate the present value of this sum using:
\(PV=\frac{FV}{1\:+\:i\:^n}\)
In this case,
\(PV=\frac{500}{1\:+\:0.05\:^2}=£453.51\)
Another way of presenting this is to say that the future value can be multiplied by the discount factor,\( \frac{1}{1 \:+ \:in}\),in order to obtain the present value. In the above example, the discount factor is:
\(\frac{1}{1 \: + \:0.05\:^2}\: = 0.9070295\)
Multiplying the future value (£500) by the discount factor gives the present value as £453.51.
It may be necessary to determine the present value of several different future cash flows. For example, a company may expect to receive £350 divided into three payments – P1,P2 and P3 to be paid after one, two and three years respectively. Let us assume the value of p1 is £100, the value of P2 is £110 and the value of P3 is £140. The annual interest rate is 4%. In order to determine the present value of these, the formula can be adapted to give:
PV = \(\frac{p1}{1\:+\:i} + \frac{p2}{1\:+\:i^2} + \frac{p3}{1\:+\:i^3}\)
= \(\frac{£100}{1.04}\: + \: \frac{£110}{1.08} \:+ \:\frac{£140}{1.12}\)
= \( £ \:323.00\)
The present value of the combined cash flows is therefore £323.
