# StevenClark.com.au ## Time Value of Money: Time Period to Reach a Sum & Growth Rates

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 Factor0.07,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 Annuity0.07,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.

### Growth Rate (Single Amount)

A similar problem is working out the growth rate, or the compound annual interest rate, across a number of cash flows. The example we will use is:

• 2011 – \$18,935
• 2010 – \$18,154
• 2009 – \$17,406
• 2008 – \$16,688
• 2007 – \$16,000

Note that there are five cash flow payments and we are interested in the four growth rate spaces (represented in calculations as i) that fall between them.

The first step is to divide the earliest payment (Present Value) by the latest payment (Future Value):

\$16,000 / \$18,935 = 0.845

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

Present Value Interest Factori,4 = 0.845

The final step is to look in the Present Value Interest Factor (for a single amount) Time Value of Money Chart in the row for four years. Our number (0.845) falls closest to the value for 4 per cent (0.855). Therefore, five annual payments with equal growth from \$16,000 to \$18,935 would have a growth rate just over 4 per cent.

### Growth Rate (Annuities)

We can also work out the interest rate on an annuity. For example, if we took out a business loan of \$16,000 with equal end of year repayments of \$4,300 over the next four years we could work out the effective (after compounding) interest rate:

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

\$16,000 / \$4,300 = 3.7209

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

Present Value Interest Factor of an Ordinary Annuityi,4 = 3.7209

The final step is to look in the Present Value Interest Factor for an Ordinary Annuity Time Value of Money Chart in the row for four years. Our number (3.7209) falls closest to the value for 3 per cent (3.717). Therefore, a loan of \$16,000 with four equal payments of \$4,300 over four years has a growth rate of around 3 per cent.

### Conclusion: The Idea of Time Value of Money

One of the most important things to understand about administering your business is that money has a time value. The longer money pools in your account and the more effectively it is utilised in that process the better it will always be for your business. General rules like pay your bills at the end of the invoice period and collect as early as possible are healthy ways to operate.

But, even more important, I hope this short series on Time Value of Money 101 has given you some basic formulas that can improve your business decision making. None of this is rocket science, but if you expect to compete in the hyper-competitive modern business environment over the long-term it’s exactly the sort of skill you need to be bringing on board.

While money isn’t the reason you should be in business… it should be to serve a public need not met by government and a by-product of doing that well means people will give you money… the simple fact is that money is the blood of any business. If you can’t meet your current liabilities (short-term debt) as they fall due then profitability means squat, you’re done. It’s over.

If you really want to do well in business you’ll try to learn everything about money that you can get your hands on… and then some. Enjoy.

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