Ray's New Higher Arithmetic · Arts. 339–341 · Unit 22: Percentage with Time: Compound Interest and Annuities

98. Annuities and perpetuities

Goal: Your student names the kinds of annuities and finds the value of a perpetuity, immediate or deferred.

⏱ About 40 minutes · 7 steps

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You'll need

  • The book
  • A slate, small whiteboard or scrap paper
  • A notebook and pencil
  • A calculator, for checking only (optional)

What's in the book

Section IX, Annuities: definitions (perpetual, limited, certain, contingent, immediate, deferred, forborne, final and present value), uses, a remark on when an annuity begins, Case I (perpetuity) and Case II (deferred perpetuity), with examples. Answers are printed.

The lesson, step by step

  1. Warm-up

    Ask: A piece of land rents for $250 a year forever. If money earns 6%, what is the land worth? Let your student think: what sum earns $250 a year?

  2. The idea

    Read:

    An Annuity is a sum of money payable at yearly or other regular intervals.

    A Perpetual Annuity, or a Perpetuity, is one that continues forever.

    A Certain Annuity begins and ends at fixed times.

    A Contingent Annuity begins or ends with the happening of a contingent event as the birth or the death of a person.

  3. Where annuities show up

    Read:

    leases, life-estates, rents, dowers, pensions, reversions, salaries, life insurance, etc.

  4. Case I

    Read:

    Divide the given payment by the interest of $1 for one interval at the proposed rate.

    What is the initial value of a perpetual lease of $250 a year, allowing 6% interest?

    $250 ÷ .06 = $4166.67.

  5. Case II: deferred

    Find the present value of a perpetuity of $250 a year, deferred 8 yr., allowing 6% interest.

    Its value 8 years from now is $4166.67. Today's value is the present worth of that: $4166.67 ÷ 1.5938481 = $2614.22 (the divisor is from the compound interest table).

  6. Practice

    Your student works examples such as:

    What must I pay for a perpetual lease of $756.40 a year, to secure 8% interest?

    Find the present value of a perpetuity of $780 a year, to commence in 12 yr., int. 5%.

  7. Wrap-up

    Ask: Why is a perpetuity worth less if it starts 8 years from now? (You wait 8 years before the first payment.)

Tip

A footnote in the book says families may leave this chapter until after Series (progressions). That works well too.

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