Present Value: Less Than 1 Year
Sometimes cash arrives in 3, 6 or 9 months, not in whole years. To find its present value, express the time as a fraction of a year and use that fraction as the exponent when you discount at the annual rate.
Present value for less than a year, explained simply
To discount cash due in a few months, write the time as a fraction of a year and use it as the exponent: PV = FV ÷ (1 + r)^(months ÷ 12). In the example, $1,000 due in 9 months at 8% is worth about $943.91 today.
What this lesson on discounting months covers
- How to turn months into a fraction of a year for discounting
- The present value formula with a fractional exponent
- How the compound and simple interest answers compare
- How to avoid mixing annual and monthly rates
Present value for months, explained
- FV
- amount to be received
- r
- annual discount rate
- m
- number of months until payment
- r
- annual rate
- m
- number of months
How to discount 9 months step by step (worked example)
You will receive $1,000 in 9 months. The annual discount rate is 8%, compounded annually. What is it worth today?
- Convert the time: 9 months = 9 ÷ 12 = 0.75 years.
- Discount factor base: (1.08)^0.75 ≈ 1.05942.
- Divide: PV = $1,000 ÷ 1.05942 ≈ $943.91.
- For comparison, simple interest gives $1,000 ÷ (1 + 0.08 × 0.75) ≈ $943.40.
Answer: About $943.91 today with compound discounting (rounded to the cent).
Three ways to discount 9 months at 8%
| Method | Formula | PV of $1,000 |
|---|---|---|
| Annual compounding, fractional year | 1,000 ÷ 1.08^0.75 | $943.91 |
| Simple interest | 1,000 ÷ (1 + 0.08 × 0.75) | $943.40 |
| Monthly compounding | 1,000 ÷ (1 + 0.08 ÷ 12)⁹ | $941.95 |
Harder questions on discounting months
What if the 8% compounds monthly?
Use the monthly rate and 9 periods: $1,000 ÷ (1 + 0.08 ÷ 12)⁹ ≈ $941.95. Read the question carefully: the compounding rule decides the method.
Why do the three methods give different answers?
Each assumes interest builds up differently within the year. The gaps are small for short periods, but on an exam only the method the question implies gets full marks.
Common questions on discounting months
How do you calculate present value for less than one year?
Express the time as a fraction of a year and use it as the exponent: PV = FV ÷ (1 + r)^(months ÷ 12). For $1,000 due in 9 months at 8%, PV = $1,000 ÷ 1.08^0.75 ≈ $943.91.
Can I use a decimal number of years in the present value formula?
Yes. The exponent can be any number, including 0.25 or 0.75. It means you are discounting for part of a year at the annual rate. Use the power key on your calculator.
Is the present value for 6 months just half a year of interest?
Not exactly, under compound interest. At 8% a year, half a year of compounding is 1.08^0.5 ≈ 1.0392, not 1.04. The 1.04 figure is the simple interest version, which some short-term problems use.
What if the rate is compounded monthly instead?
Then use the monthly rate and count months: PV = FV ÷ (1 + r ÷ 12)^m. The fractional exponent method is for a rate quoted as compounded annually.