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Equations of value and the time line

Every FM problem is one equation: money in equals money out, with both sides valued at the same date.

The formulas

General form
Comparison date is free
Method of equated time

approximate single date replacing several payments

Where it comes from

  1. Value is additive and time-consistent: moving every term to a common date multiplies the whole equation by the same factor, so the SOLUTION is unaffected by the date you choose.
  2. That freedom is the technique — choose the date that makes the most terms collapse, usually the date of the unknown payment.

Worked example

A borrower repays a loan with 800 at the end of year 3 and 1,500 at the end of year 7. At an effective annual rate of 5%, what single payment at the end of year 5 is equivalent?

  1. Value everything at time 5, the date of the unknown payment.
  2. The 800 accumulates two years: .
  3. The 1{,}500 discounts two years: .
  4. The equivalent single payment is .

Answer: 2,242.54

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

  • Draw the time line first. Every question that goes wrong went wrong on the time line, not in the algebra.
  • Pick the comparison date that kills the most exponents.

Traps

  • Valuing the two sides at different dates.
  • Using the method of equated time as if it were exact — it is an approximation and always gives a date slightly later than the exact one.

Related

Drill this: the Exam FM question bank has original questions on this topic, and today’s free round is open to everyone.