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Present Value

level 32 course

What money arriving later is worth today, at a stated discount rate.

Pen and paper is fine · no calculator needed why?

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Opens at level 32.

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Builds on: Compound Growth (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

Present value is what money arriving later is worth today. It runs compounding backward: if $1,000 can grow 10% to $1,100 in a year, then $1,100 a year from now is worth $1,000 today.

The discount rate is the yearly return you assume you could earn instead. It is an assumption you state, not a fact about the future. To discount, divide by the growth factor once for each year. Never subtract the rate: $1,100 divided by 1.1 is $1,000, while taking 10% off gives $990.

The discount factor is what one future dollar is worth today.

Techniques

Divide by the growth factor

Bringing one future amount back to today.

  1. Write the growth factor: 1 plus the rate. 25% is 1.25.
  2. Divide by it once for each year. For two years, 1.2 × 1.2 = 1.44.
  3. Check: grow your answer forward and you land on the future amount.
worked example

Example: Assume a 20% annual discount rate. What is $1,440 received two years from now worth today?

  1. 1.2 × 1.2 = 1.44.
  2. $1,440 ÷ 1.44 = $1,000.
  3. Check: $1,000 grows to $1,200, then $1,440.

Answer: $1,000

Bring the later offer to today

Choosing between money now and money later.

  1. Discount the later amount to today.
  2. Compare it with the money on offer now.
  3. The bigger value today wins. Equal values are worth the same.
worked example

Example: Offer A: $1,000 now. Offer B: $1,150 in one year. Assume a 25% annual discount rate. What is Offer B worth today?

  1. $1,150 ÷ 1.25 = $920.
  2. That is less than $1,000, so Offer A is worth more today.

Answer: $920

The discount factor

What one future dollar is worth today.

  1. Multiply 1 by the growth factor once for each year.
  2. Divide 1 by the result.
  3. Round to 3 decimal places if it doesn't end sooner.
worked example

Example: Assume a 10% annual discount rate. What is the discount factor for money received in 2 years? Round to 3 decimal places.

  1. 1.1 × 1.1 = 1.21.
  2. 1 ÷ 1.21 ≈ 0.826.

Answer: 0.826

Tips by skill

  • TipPresent value, one year out: Divide the future amount by 1 plus the rate. Never subtract the rate from it.
  • TipPresent value, two years out: Divide by the growth factor twice, or once by the factor multiplied by itself.
  • TipMoney now or money later?: Discount the later offer to today, then compare it with the money now.
  • TipDiscount factor: Divide 1 by the growth factor for all the years. The answer is always below 1.

Watch out for

  • Subtracting the rate instead of dividing. At 10%, $1,100 a year out is worth $1,000 today, not $990.
  • Dividing by 1.2 for two years at 10%. Discounting compounds too: divide by 1.21.
  • Picking the bigger number without discounting it first.
  • Giving the growth factor when the question asks for the discount factor, which is 1 divided by it.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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