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Archive for the 'TVM 101' Category

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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Time Value of Money: Mixed Streams

Friday, March 9th, 2012

Fixed amounts were the first calculations discussed in Time Value of Money: Present & Future and we filled out that knowledge with Time Value of Money: Financial Tables. Then we moved into Time Value of Money: Ordinary Annuities and Time Value of Money: Annuities Due. The next step is to understand Mixed Streams of cash inflows and outflows in your business.

Future Value of Mixed Streams of Cash Flow

Unlike a regular periodic cash flow like an annuity, there are often cases where you need to be able to evaluate the Future Value and Present Value of a mixed stream of cash inflows / outflows.

To calculate the Future Value of a Mixed Stream of cash flows you can simply work out the individual values using Financial Tables and find their total. For example, you may expect to receive a series of cash flows over the next 4 years of $10,000, $12,000, $8,000 and $21,000 and you expect that a reasonable amount to earn on the investment is 7 per cent interest.

To follow those calculations through:

  • Future Value(1) = Present Value * (FVIFi,n)
  • Future Value(1) = $10,000 * (FVIF0.07,3)
  • Future Value(1) = $10,000 * 1.225
  • Future Value(1) = $12,250
  • Future Value(2) = Present Value * (FVIFi,n)
  • Future Value(2) = $12,000 * (FVIF0.07,2)
  • Future Value(2) = $12,000 * 1.145
  • Future Value(2) = $13,740
  • Future Value(3) = Present Value * (FVIFi,n)
  • Future Value(3) = $8,000 * (FVIF0.07,1)
  • Future Value(3) = $10,000 * 1.070
  • Future Value(3) = $8,560

The Future Value(4) of the $21,000 remains at $21,000 because it accumulates zero interest.

Adding the Future Value(1) + Future Value(2) + Future Value(3) + Future Value(4) gives us the Future Value of that Mixed Stream: $55,550.

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Time Value of Money: Annuities Due

Thursday, March 8th, 2012

In Time Value of Money: Present & Future we looked at fixed amounts and expanded that understanding into Time Value of Money: Financial Tables and Time Value of Money: Ordinary Annuities. That brings us to Annuities Due.

Future Value Interest Factor of an Annuity Due

Cash flows of an Annuity Due are realised at the beginning of each period. In contrast, the cash flows of an Ordinary Annuity are at the end of each period.

The formula to calculate the Future Value of an Annuity Due it is necessary to adjust the Future Value Interest Factor of an Annuity to include one extra period of accrued interest:

FVIFAi,n (annuity due) = FVIFAi,n * (1 + i)

Running with the previous example of $2,000 payments at five per cent interest over four years we can plug in the numbers and use Financial Tables to solve for the Future Value Interest Factor of an Annuity Due:

  • FVIFA0.05,4 (annuity due) = FVIFA0.05,4 * (1 + i)
  • FVIFA0.05,4 (annuity due) = FVIFA0.05,4 * 1.05
  • FVIFA0.05,4 (annuity due) = 4.310 * 1.05
  • FVIFA0.05,4 (annuity due) = 4.5255

Having resolved the FVIFAi,n our Future Value of the Annuity Due equation becomes:

  • Future Value of the Annuity Due = PMT * FVIFAi,n (annuity due)
  • Future Value of the Annuity Due = $2,000 * FVIFA0.05,4 (annuity due)
  • Future Value of the Annuity Due = $2,000 * 4.5255
  • Future Value of the Annuity Due = $9,051

Time line for a Future Value of an Annuity Due of $2,000 cash outflow at 5% for 4 years

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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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