For investments made over longer periods, it is normal for coupons or interest payments to be paid. These payments can themselves be invested for the remaining term of the investment. This ‘compounds’ the return. So while the proceeds of this investment are strictly only from these coupons or interest payments, the reinvestment of the monies received produces an additional return which is calculated as follows:

#### Total proceeds of a long-term investment for n years

\( = principal \times 1 + interest \: rate^n\)

This assumes that the interest is paid annually and can be re-invested at the same rate as the original investment.

To give a worked example, consider a sum of €100, invested at 3% for ten years. Assuming that any interest is reinvested at 3% a year, the total proceeds will be:

\(100 \times 1 + 0.03^{10} =€134.39\)

##### Conventional calculator

To perform the above calculation with a conventional calculator, you need to press the following buttons:

\(1 \: + \: 0.03 \: = \: (this \: gives \: the \: result \: 1.03)\)

\(M \:+\:(this \: saves \: the \: above \: result \: in \: the \: memory)\)

\(C\:(this\: clears\: the\: display,\: but\: not\: the\: memory)\)

\(MR\:^*\:MR\:^*\:MR\:^*\:MR\:^*\:MR\:^*\:MR\:^*\:MR\:^*\:MR\:^*\:MR\:^*\:MR\:=\)

\(^* \: 100\:=\:\)

This gives the result 134.39

##### Scientific calculator

Using the scientific calculator on a Windows computer, you need to press the following buttons:

(Start/Programs/Accessories/Calculator/View/Scientific)

\(1\: +\: 0.03 \:x^y \:10\: = \:*\: 100 =\)

This gives the result 134.39

##### HP12C

Using an HP12C or equivalent calculator using Reverse Polish Notation:

\(.03\:\: ENTER\)

\(1\: +\)

\(ENTER \:10\: y^x\)

\(100 \: ×\)

\(This\: gives\: the \:result\: 134.39\)

This calculation has assumed an annual interest payment, but it can also be used for more frequent payments. For example, if the interest is paid half-yearly, the interest rate becomes 1.5% (3% per annum, paid over half a year) and n becomes 20, giving us the answer: €134.69. This enables the additional benefit of more frequent interest payments to be estimated.