Simple interest
A bank account offering simple interest on a deposit will reward the investor with a fixed amount every period, the size of which depends on the size of the original deposit.
If an investor deposits an amount C into a bank account paying simple interest at rate i, then in one year's time the bank account will have an amount C(1+i). This consists of the initial deposit, C, and the interest earned, i·C. During the second year however, the interest itself does not earn interest and once again i·C is earned on the initial deposit.
If after n years the investor closes the account he will have a sum of C·(1+n·i). This is made up of the initial deposit C plus n years of interest on the initial deposit n·i·C.
Compound interest
The fundamental feature of compound interest is that any interest itself earns interest.
An investor deposits an amount C into a bank account offering compound interest at an effective rate i per annum.
After one year the amount in the bank account would be equal to C·(1+i)
If after n years the investor closes the account he will have a sum of C·(1+ni). This is made up of the initial deposit C plus n years of interest on the initial deposit n·i·C.
A1 = C(1+i)
During the second year however, the interest itself will earn interest so at the end of the second year there will be C(1+i)2
A2 = A1(1+i) = C(1+i)(1+i) = C(1+i)2
This time, if the investor closes the account after n years he will have a sum of C(1+i)n consisting of the initial investment C and the cumulative interest earned on the initial investment, C((1+i)n - 1).
The following graph shows how £1 invested in an account grows if the account offered 10% simple interest and if the account offered compound interest.
This leads naturally onto the concept of the time value of money.
The Time Value of Money
In the previous section we saw that an amount C deposited today grew to C(1+i)n by time n. We can look at this in another way by suggesting that a sum of money C(1+i)n in n years' time is worth only C today.
That is to say, an investment of C/(1+i)n will produce a return of C at time n. We say that the discounted or present value at time 0 of C due at time n is equal to C/(1+i)n.