How do you derive the PV formula from the FV formula?

Start with the future value formula FV = PV × (1 + r)^n. Divide both sides by (1 + r)^n to isolate PV: PV = FV / (1 + r)^n. The term 1/(1+r)^n is called the discount factor.

What is a discount factor?

The discount factor is 1/(1+r)^n, a multiplier between 0 and 1 that converts a future value to a present value. A higher discount rate or longer time period results in a smaller discount factor and therefore a lower present value.

What is the PV formula for an annuity?

The present value of an annuity formula is PV = PMT × [(1 − (1+r)^(−n)) / r], where PMT is the periodic payment, r is the discount rate per period, and n is the number of periods. For annuity due, multiply by (1+r).

How do I calculate present value in Excel?

Use the =PV(rate, nper, pmt, [fv], [type]) function. For a lump sum of $50,000 in 10 years at 6%: =PV(0.06, 10, 0, -50000) returns $27,919.74. For an annuity: =PV(rate, nper, -pmt).

What is the difference between PV and FV formulas?

The FV formula compounds money forward in time: FV = PV × (1+r)^n. The PV formula discounts money backward: PV = FV / (1+r)^n. They are mathematical inverses — FV grows value, PV shrinks it, both using the same rate and time period.

How does the discount rate affect present value?

A higher discount rate reduces the present value because the denominator (1+r)^n becomes larger. This reflects greater opportunity cost — if you can earn more elsewhere, a future payment is worth less to you today. For example, $10,000 in 10 years: at 5% PV = $6,139; at 10% PV = $3,855.

Present Value Formula — PV Formula Explained with Examples & Calculator

Q: What is the present value formula?

The present value formula is PV = FV / (1 + r)^n, where FV is the future value, r is the discount rate per period, and n is the number of periods. It calculates what a future sum of money is worth in today's dollars.

The present value formula is the cornerstone of the time value of money. This comprehensive guide walks you through the PV formula for lump sums, annuities, and growing cash flows — with real-world examples, Excel instructions, and an interactive calculator.

💵 Quick PV Calculator

Future Value (FV)

Annual Discount Rate

Number of Years

years

What Is the Present Value Formula?

The present value (PV) formula calculates what a future sum of money is worth in today's dollars. It is the mathematical inverse of the future value formula, and it is based on one of the most important principles in finance: a dollar today is worth more than a dollar tomorrow.

The basic PV formula for a single lump sum is:

PV = FV ÷ (1 + r)ⁿ

Or equivalently:

PV = FV × [1 / (1 + r)ⁿ]

Where:

PV = Present Value (today's worth)
FV = Future Value (amount to be received in the future)
r = Discount rate per period (as a decimal)
n = Number of compounding periods

The term 1 / (1 + r)ⁿ is called the discount factor — it is always between 0 and 1, and it shrinks the future amount to reflect the opportunity cost of waiting.

PV Formula for a Lump Sum

The simplest application of the PV formula is discounting a single future payment (lump sum) to today's value. This is used in bond pricing, legal settlements, and savings goal planning.

Derivation from the FV Formula

The present value formula is derived directly from the compound interest (future value) formula. Starting with:

FV = PV × (1 + r)ⁿ

Dividing both sides by (1 + r)ⁿ:

PV = FV ÷ (1 + r)ⁿ

With Multiple Compounding Periods

When interest compounds m times per year, the formula adjusts to:

PV = FV ÷ (1 + r/m)^m×n

For continuous compounding:

PV = FV × e^−r×n

💡 Example: What is $100,000 received in 15 years worth today at 7% annual compounding?
PV = $100,000 ÷ (1.07)¹⁵ = $100,000 ÷ 2.7590 = $36,245

The discount factor is 1/2.7590 = 0.3624 — meaning each dollar received in 15 years is only worth 36.24 cents today.

PV Formula for an Annuity

When you receive a series of equal payments over time, the present value of an annuity formula sums the discounted value of each payment:

Ordinary Annuity (End-of-Period)

PV_annuity = PMT × [(1 − (1 + r)⁻ⁿ) / r]

Annuity Due (Beginning-of-Period)

PV_due = PMT × [(1 − (1 + r)⁻ⁿ) / r] × (1 + r)

The factor [(1 − (1+r)⁻ⁿ) / r] is called the Present Value Interest Factor of Annuity (PVIFA).

Perpetuity (Infinite Annuity)

When payments continue forever (n → ∞), the formula simplifies to:

PV_perpetuity = PMT ÷ r

This is widely used in stock valuation (the Gordon Growth Model) and real estate cap rates.

PV Formula for Growing Cash Flows

When cash flows grow at a constant rate g each period, you need the growing annuity or growing perpetuity present value formulas:

Growing Annuity

PV = PMT × [(1 − ((1+g)/(1+r))ⁿ) / (r − g)]

Growing Perpetuity (Gordon Growth Model)

PV = PMT ÷ (r − g)

This is the foundation of the Dividend Discount Model used to value stocks: Price = D₁ / (r − g), where D₁ is next year's dividend, r is the required return, and g is the dividend growth rate.

💡 Example: A stock pays a $2.00 dividend next year, growing at 4% annually. Your required return is 10%.
Price = $2.00 / (0.10 − 0.04) = $33.33

How to Calculate PV in Excel & Google Sheets

Both Excel and Google Sheets include a built-in =PV() function:

=PV(rate, nper, pmt, [fv], [type])

Excel PV Examples

Scenario	Formula	Result
Lump sum: $50,000 in 10 years at 6%	`=PV(0.06, 10, 0, -50000)`	$27,919.74
Annuity: $1,000/year for 20 years at 5%	`=PV(0.05, 20, -1000)`	$12,462.21
Monthly: $500/month for 10 years at 6%	`=PV(0.06/12, 120, -500)`	$44,955.04
Annuity due: $1,000/year, 20 yrs, 5%	`=PV(0.05, 20, -1000, 0, 1)`	$13,085.32
Uneven cash flows (NPV)	`=NPV(0.08, 5000,7000,10000)`	$18,786.60

Note: Excel's =PV() returns a negative value (representing a cash outflow to invest). Wrap in =ABS() if you prefer a positive number.

For Google Sheets, the syntax is identical. You can also use =NPV(rate, value1, value2, ...) for uneven cash flows.

Present Value vs. Future Value Formula — Key Differences

Aspect	Present Value (PV)	Future Value (FV)
Direction	Backward in time (discounting)	Forward in time (compounding)
Formula	PV = FV / (1+r)ⁿ	FV = PV × (1+r)ⁿ
Question	"What is future money worth today?"	"What will today's money be worth later?"
Key factor	Discount factor: 1/(1+r)ⁿ	Compound factor: (1+r)ⁿ
When rate rises	PV decreases (future worth less)	FV increases (money grows faster)
Common uses	Bond pricing, NPV, loan valuation	Savings goals, retirement projections

Discounting and compounding are mathematical inverses. If you invest $36,245 today at 7% for 15 years, it grows to $100,000 (FV). Conversely, $100,000 received in 15 years has a PV of $36,245 at 7%.

Real-World PV Calculation Examples

Example 1: College Savings Goal

You need $200,000 for college in 18 years. At a 7% return, how much should you invest today?

PV = $200,000 ÷ (1.07)¹⁸ = $200,000 ÷ 3.3799 = $59,172

Example 2: Pension Valuation

A pension pays $3,000/month for 25 years. At a 5% discount rate with monthly compounding:

PV = $3,000 × [(1 − (1 + 0.05/12)⁻³⁰⁰) / (0.05/12)] = $511,567

The $900,000 total payments ($3,000 × 300) are worth only $511,567 today.

Example 3: Business Investment (NPV)

A project costs $100,000 and generates $30,000/year for 5 years at a 12% discount rate:

PV of cash flows = $30,000 × [(1 − 1.12⁻⁵) / 0.12] = $108,143

NPV = $108,143 − $100,000 = +$8,143 → profitable investment.

Example 4: Discount Factor Table

Years	r = 3%	r = 5%	r = 7%	r = 10%
5	0.8626	0.7835	0.7130	0.6209
10	0.7441	0.6139	0.5083	0.3855
15	0.6419	0.4810	0.3624	0.2394
20	0.5537	0.3769	0.2584	0.1486
30	0.4120	0.2314	0.1314	0.0573

Reading the table: at 7% for 20 years, the discount factor is 0.2584 — meaning $1 received in 20 years is worth only 25.84 cents today.

Frequently Asked Questions

The present value formula is PV = FV / (1 + r)ⁿ, where FV is the future value, r is the discount rate per period, and n is the number of periods. It calculates what a future sum of money is worth in today's dollars based on the time value of money principle.
Start with the compound interest formula: FV = PV × (1 + r)ⁿ. To isolate PV, divide both sides by (1 + r)ⁿ: PV = FV / (1 + r)ⁿ. The term 1/(1+r)ⁿ is called the discount factor.
The discount factor is 1 / (1 + r)ⁿ, a multiplier between 0 and 1 that converts a future value to its present equivalent. For example, at 8% for 10 years, the discount factor is 0.4632 — meaning each future dollar is worth 46.32 cents today.
The PV of an annuity = PMT × [(1 − (1+r)⁻ⁿ) / r]. This sums the discounted value of each equal periodic payment. For annuity due (payments at beginning), multiply by (1+r). For a perpetuity (infinite payments), PV = PMT / r.
Use =PV(rate, nper, pmt, [fv], [type]). For a lump sum: =PV(0.06, 10, 0, -50000) returns $27,920. For an annuity: =PV(0.05, 20, -1000) returns $12,462. Use =NPV(rate, values) for uneven cash flows.
The FV formula compounds forward: FV = PV × (1+r)ⁿ. The PV formula discounts backward: PV = FV / (1+r)ⁿ. They are inverses — FV grows money over time, PV shrinks future money to today's dollars. Both use the same rate and period.
Higher discount rate = lower present value. The discount rate reflects opportunity cost and risk. At 5%, $10,000 in 10 years is worth $6,139 today. At 10%, it's only $3,855. This is because a higher rate means you need less money today to reach the same future amount.