Present Value
What money arriving later is worth today, at a stated discount rate.
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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.
more of the idea
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
- Write the growth factor: 1 plus the rate. 25% is 1.25.
- Divide by it once for each year. For two years, 1.2 × 1.2 = 1.44.
- Check: grow your answer forward and you land on the future amount.
Assume a 20% annual discount rate. What is $1,440 received two years from now worth today?
- 1.2 × 1.2 = 1.44.
- $1,440 ÷ 1.44 = $1,000.
- Check: $1,000 grows to $1,200, then $1,440.
Answer: $1,000
Bring the later offer to today
- Discount the later amount to today.
- Compare it with the money on offer now.
- The bigger value today wins. Equal values are worth the same.
Offer A: $1,000 now. Offer B: $1,150 in one year. Assume a 25% annual discount rate. What is Offer B worth today?
- $1,150 ÷ 1.25 = $920.
- That is less than $1,000, so Offer A is worth more today.
Answer: $920
The discount factor
- Multiply 1 by the growth factor once for each year.
- Divide 1 by the result.
- Round to 3 decimal places if it doesn't end sooner.
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 = 1.21.
- 1 ÷ 1.21 ≈ 0.826.
Answer: 0.826
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.
practice
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Present value, one year out
worked example
Assume a 25% annual discount rate. What is $375 received one year from now worth today?
- Present value reverses growth: divide by 1 + the rate.
- $375 ÷ 1.25 = $300.
Answer: $300.00
Present value, two years out
worked example
Assume a 20% annual discount rate. What is $12,528 received two years from now worth today?
- Divide by 1.2 twice, or by 1.2² = 1.44 once.
- $12,528 ÷ 1.44 = $8,700.
Answer: $8,700.00
Money now or money later?
worked example
Offer A: $2,900 now. Offer B: $3,045 in one year. Assume a 5% annual discount rate. Which is worth more today?
- Offer B (money later)
- They're worth the same today
- You can't compare money at different dates
- Offer A (money now)
- Bring B back to today: $3,045 ÷ 1.05 = $2,900.
- That equals Offer A's $2,900, so they're worth the same today.
Answer: They're worth the same today
Discount factor
worked example
Assume a 100% annual discount rate. What is the discount factor for money received in 2 years? Round to 3 decimal places if needed.
- Factor = 1 ÷ 2² = 1 ÷ 4 = 0.25.
- Multiply any future amount by it to get today's value.
Answer: 0.25