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

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}

The formula works through as:

- PMT = Present Value of an Ordinary Annuity
_{5}/ PVIFA_{0.09,5} - PMT = $30,000 / 3.890
- PMT = $7,712.08

This means that five annual payments of $7,712.08 at the end of each year at 9 per cent interest will pay off the $30,000 loan.

If you are a little confused about the two concepts then consider that a deposit to reach a target future sum is solved for a Future Value because the sum itself is in the future. Whereas, a loan is taken out in the present and needs to be solved for a Present Value as it is paid off over time (amortised).

### A Loan Amortisation Schedule

When you take out a loan the bank will create an Amortisation Schedule that explains the breakdown of principal and interest as it is paid off over the term of the loan.

### The Next Step: Calculate the Number of Time Periods

Calculating the deposits needed to accrue a future target sum and working out loan amortisation are handy skills for business managers. In the next article we will look at the last Time Value of Money application to be discussed in this short series – using Financial Tables to calculate the number of time periods to reach a specified future sum.

I would also advise you to pick up any decent copy of a managerial finance textbook to underpin these articles and to complete the exercises that will cement this understanding at the end of each chapter. The textbook will also provide more precise context important to your understanding.

### Time Value of Money 101 Series

- Time Value of Money: Present & Future
- Time Value of Money: Financial Tables
- Time Value of Money: Ordinary Annuities
- Time Value of Money: Annuities Due
- Time Value of Money: Mixed Streams
- Time Value of Money: Compound Interest (redux)
- Time Value of Money: Nominal versus Effective Interest Rates
- Time Value of Money: Accumulation of a Target Sum & Loan Amortisation
- Time Value of Money: Time Periods to Reach a Sum & Growth Rates