ActuaryDeck

Home/Exam FM/Equations of value

Exam FMMeasurement of interestFree to read

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
∑kCk vtk=∑kDk vsk\sum_{k} C_k\, v^{t_k} = \sum_{k} D_k\, v^{s_k}
Comparison date is free
multiply both sides by (1+i)T\text{multiply both sides by } (1+i)^{T}
Method of equated time
tˉ=∑ktkCk∑kCk\bar t = \dfrac{\sum_k t_k C_k}{\sum_k C_k}

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: 800(1.05)2=882.00800(1.05)^{2} = 882.00.
  3. The 1{,}500 discounts two years: 1,500v2=1,500/(1.05)2=1,360.541{,}500 v^{2} = 1{,}500/(1.05)^{2} = 1{,}360.54.
  4. The equivalent single payment is 882.00+1,360.54=2,242.54882.00 + 1{,}360.54 = 2{,}242.54.

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.