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Archive for March, 2012

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 Annuityn = PMT * (FVIFAi,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 Annuityn / FVIFAi,n

This becomes:

  • PMT = Future Value of an Ordinary Annuity4 / FVIFA0.06,4
  • 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 Annuityn = PMT * (PVIFAi,n)

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

PMT = Present Value of an Ordinary Annuityn / PVIFAi,n

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Time Value of Money: Nominal versus Effective Interest Rates

Sunday, March 11th, 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). 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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Time Value of Money: Compound Interest (redux)

Saturday, March 10th, 2012

The discussion in Time Value of Money: Present & Future about fixed amounts led us to Time Value of Money: Financial Tables and then onto Time Value of Money: Ordinary Annuities and Time Value of Money: Annuities Due. The last article took us into Time Value of Money: Mixed Streams. And in this article we revisit compound interest.

Semi-Annual and Quarterly Compounding

You will find that interest is going to be compounded by financial institutions in all sorts of increments – annually, monthly, weekly or even daily. So we need a fast and effective formula to calculate the effect of that compounding; where i is the interest rate, m is the number of terms in the year and n indicates the number of years:

Future Valuen = Present Value * (1 + (i / m))m*n

Solving for a quarterly compounding of $2,000 at an interest rate of 8 per cent over 2 years:

  • Future Value2 = $2,000 * (1 + (0.08 / 4))4 * 2
  • Future Value2 = $2,000 * (1 + 0.02)8
  • Future Value2 = $2,000 * (1.02)8
  • Future Value2 = $2,000 * 1.1717
  • Future Value2 = $2,343.40

Skinning the Future Value Cat another way

The example I chose was specifically taken to show the relationship between the Future Value result of compounding over those eight time periods ($2,343,40) with the equivalent result provided by the Financial Tables for Future Value of a fixed amount ($2,332).

  • Future Value = PMT * FVIVi,n
  • Future Value = $2,000 * FVIV0.08,2
  • Future Value = $2,000 * 1.166
  • Future Value = $2,332

Had the interest rate been 7 per cent and the quarterly interest over 3 years this would have been more difficult to prove on the Financial Tables because the term for each period’s interest rate would have reduced to 0.0175 per cent across 12 compounding periods. Understanding this relationship, it would be advisable to stick with the formula created specifically to calculate compound interest more frequently than a year.

Continuous Compounding

Continuous compounding is where interest is immediately and continuously earned and compounds interest on itself. The compounding periods become extremely small. This is the highest rate of interest that can be earned as compound interest.

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About the Author

Steven Clark Steven Clark - the stand up guy on this site

My name is Steven Clark (aka nortypig) and I live in Southern Tasmania. I have an MBA (Specialisation) and a Bachelor of Computing from the University of Tasmania. I'm a photographer making pictures with film. A web developer for money. A business consultant for fun. A journalist on paper. Dreams of owning the World. Idea champion. Paradox. Life partner to Megan.

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