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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.

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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 Annuity₄ / 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

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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). At this point it’s important to understand nominal versus effective annual rates of interest.

Nominal & Effective Rates of Interest

It is important to be able to differentiate between the interest that you sign up for in a contract and the amount of interest that actually accrues in the account.

The Nominal Interest Rate (NIR) is the interest rate stated in the contract. While, the Effective Interest Rate (EIR) is the rate of interest that is paid or accrues and this takes into account the effect of interest compounding over time. If compounding of interest is annual then the NIR and the EIR will be the same.

Calculating the Effective Interest Rate

The formula to calculate the EAR, where i represents the NIR and m represents the compounding frequency:

EAR = (1 + (i / m))^m – 1

For a NIR of 12 per cent we can look at the calculations for annual and quarterly EIR:

• Effective Interest Rate = (1 + (0.12 / 1))¹ – 1
• Effective Interest Rate = 1.12 – 1
• Effective Interest Rate = 0.12

This demonstrates that the EIR of 12 per cent does equal the NIR of 12 per cent when compounding annually. However, this is not the case for quarterly compounding (or any compounding shorter than a year):

• Effective Interest Rate = (1 + (0.12 / 4))⁴ – 1
• Effective Interest Rate = (1.03)⁴ – 1
• Effective Interest Rate = 1.1255 – 1
• Effective Interest Rate = 0.1255

Quarterly compounding produces an EIR of 12.55 per cent from a NIR of 12 per cent. The shorter the compounding period the larger the EIR becomes.

The Next Step: More Applications of Time Value of Money

The difference between NIR and EIR is one of those small but fundamental pieces of knowledge that you need to make effective financial decisions. In the next article we will look at other applications of the Time Value of Money.

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