Present Value Compounding
When interest compounds more than once a year, you have to discount with the matching periodic rate and number of periods. More frequent compounding means a lower present value, because the money you set aside today would grow faster.
Present value with monthly compounding, explained simply
With compounding m times a year, discount at r ÷ m for m × n periods: PV = FV ÷ (1 + r ÷ m)^(m × n). In the example, $10,000 needed in 5 years at 6% compounded monthly means depositing about $7,413.72 today.
What this present value compounding lesson covers
- How to adjust the rate and periods for monthly or quarterly compounding
- The present value formula for any compounding frequency
- Continuous discounting with e^(−rt)
- Why more frequent compounding lowers present value
Present value compounding formulas, explained
- FV
- future amount
- r
- stated annual rate
- m
- compounding periods per year
- n
- number of years
- e
- about 2.71828
- r
- annual rate
- n
- number of years
How to calculate present value with monthly compounding (worked example)
You need $10,000 in 5 years. The bank pays a stated 6% a year, compounded monthly. How much must you deposit today?
- Periodic rate: 6% ÷ 12 = 0.5% = 0.005.
- Number of periods: 5 × 12 = 60.
- Discount: PV = $10,000 ÷ (1.005)⁶⁰ ≈ $7,413.72.
- For comparison, annual compounding gives $10,000 ÷ (1.06)⁵ ≈ $7,472.58.
Answer: About $7,413.72 today with monthly compounding (rounded to the cent).
Present value of $10,000 in 5 years at 6%, by compounding
| Compounding | Rate per period | Periods | Deposit today |
|---|---|---|---|
| Annual | 6% | 5 | $7,472.58 |
| Quarterly | 1.5% | 20 | $7,424.70 |
| Monthly | 0.5% | 60 | $7,413.72 |
| Continuous | — | — | $7,408.18 |
Harder present value compounding questions
Why does more frequent compounding lower the present value?
Money set aside today grows faster when interest compounds more often, so you need to put less aside to reach the same $10,000.
What is the continuous-compounding answer?
$10,000 × e^(−0.06 × 5) ≈ $7,408.18, the lowest of all, because continuous compounding grows money fastest.
Common present value compounding questions
How do you calculate present value with monthly compounding?
Divide the annual rate by 12 and multiply the years by 12, then discount: PV = FV ÷ (1 + r ÷ 12)^(12 × n). For $10,000 in 5 years at 6%, PV = $10,000 ÷ 1.005⁶⁰ ≈ $7,413.72.
Why does more frequent compounding lower present value?
Because money deposited today grows faster when interest compounds more often, so you need to set aside less to reach the same future amount. Monthly compounding gives a lower present value than annual compounding at the same stated rate.
What is the present value formula for continuous compounding?
PV = FV × e^(−r × n), which is the same as FV ÷ e^(r × n). For example, $10,000 in 5 years at 6% continuous is about $10,000 × e^(−0.3) ≈ $7,408.18.
How do I do quarterly compounding in a present value problem?
Use r ÷ 4 as the rate and 4 × n as the number of periods. On a financial calculator, set N to the total quarters and I/Y to the quarterly rate, or set P/Y and C/Y to 4.