Handout 1 Some additional notes

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Handout 1 (Some additional notes)

• Chapter 4

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Deriving the annuity and perpetuity formula

• The annuity formula and perpetuity formula are
  derived from a simple application of the sum of
  geometric series.

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• Consider you would like a simplified formula for
  the following summation. (The sequence of the
  number in the form is called geometric series)

• To simplify the formula, first consider
  multiplying both side of (1) by s. Then, you will
  get

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• Now consider taking a difference X-sX. You can
  see that many terms will cancel out

Therefore, we have
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Deriving annuity formula

• To derive the annuity formula, simply plug aC
  and s1/(1r). Then you will have the annuity
  formula

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Deriving the perpetuity formula

• Perpetuity is the case where T becomes extremely
  large. (T approaches to infinity).
• Now go back to equation (3). You can notice that,
  if 0ltslt1, then sT 0 as T becomes very large.
• This means that if 0ltslt1, we have

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Deriving the perpetuity formula

• To derive the perpetuity formula, simply plug in
  aC, and s1/(1r) in equation (4).
• (Notice that 1/(1r) is always strictly
  between 0 and 1, the formula applies.)