### Time Value of Money: Deposits to Reach a Target Sum & Loan Amortisation

Wednesday, March 14th, 2012

In Time Value of Money: Present & Future and Time Value of Money: Financial Tables we discussed fixed amounts and moved onto Time Value of Money: Ordinary Annuities, Time Value of Money: Annuities Due and Time Value of Money: Mixed Streams. Then we discussed Time Value of Money: Compound Interest (redux) and Time Value of Money: Nominal versus Effective Interest Rates. It’s time to discuss special applications like how to work out the deposits needed to reach a target sum and loan amortisation.

### Deposits to Reach a Target Sum

If we had to work out the size of deposits needed to reach a target sum of $50,000 at 4 years with 6 per cent interest we can revisit the Future Value of an Ordinary Annuity (with the help of Time Value of Money Financial Tables) where PMT is the size of the periodic payments, *i* represents the interest rate and *n* is the number of years:

Future Value of an Ordinary Annuity_{n} = PMT * (FVIFA_{i,n})

Shuffling this formula around gives us another formula to discover the size of payments given the same information:

PMT = Future Value of an Ordinary Annuity_{n} / FVIFA_{i,n}

This becomes:

- PMT = Future Value of an Ordinary Annuity
_{4}/ FVIFA_{0.06,4} - PMT = $50,000 / 4.375
- PMT = $11,428.57

This means that four payments of $11,428.57 at the end of each year at 6 per cent interest will result in the target sum of $50,000 in the account.

### Loan Amortisation

Amortisation means to *pay off* or to *decrease over a period*. So loan amortisation is the paying off of a loan at a given percent interest rate over a set period of time with equal payments being made at the end of each year.

If we take out a loan of $30,000 at 9 per cent interest over 5 years we can revisit the Present Value of an Ordinary Annuity:

Present Value of an Ordinary Annuity_{n} = PMT * (PVIFA_{i,n})

Shuffling this formula around we can once again identify for the size of payments:

PMT = Present Value of an Ordinary Annuity_{n} / PVIFA_{i,n}