### Time Value of Money: Time Period to Reach a Sum & Growth Rates

Friday, March 16th, 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), Time Value of Money: Nominal versus Effective Interest Rates and Time Value of Money: Deposits to Reach a Target Sum & Loan Amortisation. This article explains how to discover the number of time periods to reach a target sum and how to quickly work out growth (or effective interest) rates.

### Time Periods to Reach a Target Sum (Single Amount)

Given a need to determine how many years it would take for a $10,000 investment at 7 per cent to grow to $16,000 we could treat *n* as the number of years and *i* as the interest rate.

The first step is to divide the first payment (Present Value) by the amount received at the end (Future Value):

$10,000 / $16,000 = 0.6250

The second step is to place that value into the Present Value Interest Factor:

Present Value Interest Factor_{0.07,n} = 0.6250

The final step is to look in the Present Value Interest Factor (for a single amount) Time Value of Money Chart in the column for 7 per cent. Our number (0.6250) falls just short of 7 years with 0.623. Therefore, an initial deposit of $10,000 at 7 per cent will take nearly 7 years to mature into $16,000.

### Time Periods to Reach a Target Sum (Annuities)

In a similar fashion it is possible to calculate the unknown term of an annuity. In this case we might want to work out how long it would take to repay a $16,000 loan at 7 per cent with equal end of year payments of $3,000.

The first step is to divide the size of the loan by the size of the payments:

$16,000 / $3,000 = 5.3333

The second step is to place that value into the Present Value Interest Factor of an Ordinary Annuity:

Present Value Interest Factor of an Ordinary Annuity_{0.07,n} = 5.3333

The final step is to look in the Present Value Interest Factor for an Ordinary Annuity Time Value of Money Chart in the column for 7 per cent. Our number (5.3333) is just short of the number corresponding in that chart for 7 years (0.5.389). Therefore, our loan of $16,000 at 7 per cent with payments of $3,000 will take just under 7 years to complete.