Ray's New Higher Arithmetic · Arts. 342–347 · Unit 22: Percentage with Time: Compound Interest and Annuities
99. Annuities certain: present value, final value, payment, time and rate
Goal: Your student finds the present and final value of an annuity that runs a set number of years, and uses the annuity table to find a payment, a time or a rate.
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
Case III (present value of an annuity certain), Case IV (final or forborne value), a table of present values of $1 a year, and Cases V to VII by the table (payment, time to run, rate), with examples. Most answers are printed.
The lesson, step by step
Warm-up
Ask: Is $250 a year for 8 years worth more or less than $250 a year forever? (Less.)
Case III
Read the rule:
Find the present value of two perpetuities having the given rate, payment, and interval, one of them commencing when the annuity commences, and the other when the annuity ends.
The difference between these values will be the present value of the annuity.
The book's example: $250 a year for 8 years at 6% is $4166.67 − $2614.22 = $1552.45.
Case IV
Read:
Consider the annuity a perpetuity, and find its initial value by Case I.
The compound interest of this sum, at the given rate for the time the annuity runs, will be the final or forborne value.
The book's example: $4166.67 × .5938481 = $2474.37.
A homely example
A pays $25 a year for tobacco: how much better off would he have been in 40 yr. if he had invested it at 10% per annum?
Let your student work it by Case IV.
By the table
Case V, the payment:
An immediate annuity running 11 yr., can be purchased for $6000; what is the payment, int. 6%?
The table says $1 a year for 11 years is worth $7.886875. $6000 ÷ 7.886875 = $760.76.
Case VI, a debt paid off by installments:
In what time will a debt of $10000, drawing interest at 6%, be paid by installments of $1000 a year?
Independent practice
Your student works one or two examples of each case. Case VII asks for a rate; the two examples there have no clear printed answers, so check with the answers below.
Wrap-up
Ask: What do a mortgage paid off by yearly installments and a savings plan have in common? (Both are annuities.)
Afterwards
Plan two or three sittings: Cases III and IV, then V and VI, then VII.
Answers to the book's problems
Case VII (rate of interest): 1) $9000 ÷ $750 = 12, the value of $1 a year for 20 years; this falls between the 5% and 6% columns, at about 5½% (5.45% worked closely). 2) $650 ÷ $80 = 8.125 for 14 years; this is a little over 8% (about 8¼%).
Worked out for this site from the scan. If a number in your copy differs, trust the book.