Present Value Compounding

What you will learn

• 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

The formulas

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?

1. Periodic rate: 6% ÷ 12 = 0.5% = 0.005.
2. Number of periods: 5 × 12 = 60.
3. Discount: PV = $10,000 ÷ (1.005)⁶⁰ ≈ $7,413.72.
4. 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).

Common 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.