What is the present value formula?

For a lump sum: PV = FV / (1 + r)^n. For an ordinary annuity (payments at end of period): PV = PMT x [1 - (1 + r)^(-n)] / r. For an annuity due (payments at beginning of period): multiply the ordinary annuity PV by (1 + r). Example: PV of $10,000 received in 5 years at 6%: PV = 10,000 / (1.06)^5 = $7,472.58.

What discount rate should I use for present value?

The discount rate should reflect the opportunity cost of capital: the return you could earn in the next best alternative at equivalent risk. For corporate projects: use WACC (8 to 12% for most large companies). For personal finance decisions: use a savings or investment rate you can reliably earn. For risk-free comparisons: use the current Treasury rate. The higher the risk, the higher the discount rate.

What is the difference between PV and NPV?

PV is the present value of a single future cash flow or a stream of equal payments. NPV (Net Present Value) is the sum of the present values of all cash flows in a project, including both inflows and outflows. NPV = Sum of PV of all inflows minus Sum of PV of all outflows. A positive NPV means the investment creates value above the required return rate.

What is an annuity due vs ordinary annuity?

An ordinary annuity has payments at the end of each period (most loans, mortgages, bond coupons). An annuity due has payments at the beginning of each period (rent, insurance premiums, lease payments). Because annuity due payments are made one period earlier, each payment is compounded for one additional period, making the PV of an annuity due always higher than an ordinary annuity by a factor of (1 + r).

Present Value Calculator

Two calculators: (1) present value of a future lump sum; (2) present value of periodic annuity payments. Both include full schedules and charts.

PV of Future Money

How much is a future lump sum worth in today's dollars?

Future Value (FV) ?

Number of Periods ?

Interest Rate / Period ?

Compounding Frequency

PV of Periodic Payments

Present value of a stream of equal payments - ordinary annuity or annuity due.

Number of Periods

Interest Rate / Period

Periodic Payment (PMT) ?

Compounding Frequency

Payment Timing

End (Ordinary) Beginning (Due)

PV of Future Money - Results

Lump Sum Discount

Present Value

$7,472.58

Future Value

$10,000

Discount (Total Interest)

$2,527.42

Periods

Present Value Decay Over Time

Discounting Schedule

Period	PV Factor	Discounted Value	Remaining Discount

PV of Periodic Payments - Results

Ordinary Annuity

Present Value

$736.01

Future Value

$1,318.08

Total Principal

$1,000

Total Interest

$318.08

$1.3K

future

Accumulated Deposits and Interest

Payment Schedule

Period	Deposit	Interest	Ending Balance	PV of PMT

What Is Present Value?

Present value (PV) is the current worth of a future sum of money, given a specific discount rate. It is based on the time value of money: a dollar available today is worth more than a dollar available in the future because today's dollar can be invested and earn returns.

PV answers the question: if I need $X at a future date, how much do I need today - invested at a given rate - to reach that amount? Or equivalently: given a future cash flow, what is its worth in today's dollars?

PV = FV / (1 + r)^n FV = future value · r = discount rate per period · n = number of periods Example: $10,000 in 5 years at 6% annual rate PV = 10,000 / (1.06)^5 = $7,472.58

This means you would need $7,472.58 today, invested at 6% per year, to have exactly $10,000 in 5 years. The $2,527.42 difference is the interest earned between now and then - money the investment generates on your behalf.

The Discount Rate

The discount rate reflects the opportunity cost of capital: the return you give up by choosing this investment over the next best alternative at the same risk level. A higher discount rate means future cash flows are worth less today, because you assume you could earn more with the money right now. A lower rate makes future cash flows more valuable.

Present Value of an Annuity

When a series of equal payments (PMT) is made over multiple periods at a fixed interest rate, the combined present value is the sum of the PV of each individual payment.

PV = PMT x [1 - (1 + r)^(-n)] / r Ordinary annuity: payments at end of each period Annuity due: PV = PMT x [1 - (1 + r)^(-n)] / r x (1 + r) Annuity due: payments at beginning of each period. Always higher by factor of (1 + r).

For 10 annual payments of $100 at 6%:

Ordinary annuity PV = 100 x [1 - (1.06)^(-10)] / 0.06 = $736.01
Annuity due PV = $736.01 x 1.06 = $780.17

The annuity due is $44.16 higher because each payment arrives one period earlier and therefore discounts for one fewer period.

Ordinary Annuity vs Annuity Due

Type	When Paid	Common Use	PV vs Ordinary
Ordinary Annuity	End of period	Mortgages, car loans, bond coupons	Baseline
Annuity Due	Beginning of period	Rent, insurance premiums, leases	Higher by (1+r)

PV vs NPV: The Key Distinction

Present value and net present value are closely related but answer different questions.

Present value (PV) is the current worth of one future cash flow or a stream of identical payments. It asks: what is this future amount worth today? There is no comparison to a cost - it is a pure discounting exercise.

Net present value (NPV) subtracts the initial cost from the present value of all expected future cash flows. It asks: after paying for this investment, how much value does it create (or destroy) in today's dollars?

NPV = -Initial Cost + PV of all future cash flows = -C_0 + CF_1/(1+r) + CF_2/(1+r)^2 + ... + CF_n/(1+r)^n Positive NPV: investment creates value. Negative NPV: investment destroys value at the given discount rate.

Example: you invest $5,000 today to receive $1,200 per year for 6 years at a 10% discount rate.

PV of annuity = 1,200 x [1 - (1.10)^(-6)] / 0.10 = $5,226.70
NPV = $5,226.70 minus $5,000 = +$226.70

The positive NPV means the investment earns slightly above the 10% required return. Had the discount rate been 11%, the NPV would turn negative and the investment would fail to meet the required return.

ℹ️

The IRR (Internal Rate of Return) is the discount rate at which NPV equals exactly zero. For the example above, the IRR is approximately 11.5%. If your required return is 10%, the project is acceptable. If it is 12%, the project should be rejected because it does not earn the required return.

How the Discount Rate Affects PV

Small changes in the discount rate produce large changes in present value, especially for cash flows far in the future. This is one of the most important and often underestimated aspects of discounted cash flow analysis.

PV of $10,000 Received in the Future at Different Rates

Years Away	At 3%	At 6%	At 10%	At 15%
1 year	$9,709	$9,434	$9,091	$8,696
5 years	$8,626	$7,473	$6,209	$4,972
10 years	$7,441	$5,584	$3,855	$2,472
20 years	$5,537	$3,118	$1,486	$611
30 years	$4,120	$1,741	$573	$151

At a 3% rate, $10,000 in 30 years is worth $4,120 today. At 15%, it is worth only $151. The same cash flow is 27 times more valuable under one assumption than the other. This sensitivity is why discount rate selection is the most consequential judgment in any DCF analysis.

⚠️

In real estate and infrastructure analysis, inflating the discount rate by even 1 percentage point can turn a viable project into an apparent loser. Always test your PV and NPV calculations with a range of discount rates (sensitivity analysis) rather than relying on a single point estimate.

Real-World Uses of Present Value

Mortgage and Loan Pricing

A mortgage is the lender's present value calculation in reverse. The lender gives you a lump sum today (the loan amount) and receives a stream of monthly payments over 15 or 30 years. The present value of those payments at the agreed interest rate exactly equals the loan amount on day one.

Bond Valuation

A bond's fair market price is the present value of all its future cash flows: each coupon payment and the face value at maturity, all discounted at the current market yield. When interest rates rise, the discount rate rises, the PV of future coupons falls, and bond prices fall. This is the direct mathematical reason for the inverse relationship between bond prices and interest rates.

Retirement Planning

The "lump sum needed at retirement" is a present value question. If you need $60,000 per year in retirement for 25 years and your portfolio earns 5%, the PV of that annuity is the amount you must accumulate by retirement day. PV = 60,000 x [1 - (1.05)^(-25)] / 0.05 = $845,566.

Business Valuation

The intrinsic value of a business in a discounted cash flow model is the present value of all projected free cash flows, discounted at the company's WACC. This includes a terminal value: the present value of all cash flows beyond the explicit forecast period, typically modeled as a growing perpetuity.

Court Settlements and Structured Payments

Lump-sum settlements are compared to structured payment streams using present value. A plaintiff choosing between $500,000 today or $60,000 per year for 10 years should use PV at their personal discount rate to determine which is worth more. At 5%: PV of annuity = $463,130 - so $500,000 today is better. At 2%: PV = $538,259 - so the annuity is better.

Perpetuities: PV of an Infinite Stream

A perpetuity is an annuity that pays forever. The present value formula simplifies dramatically because the sum of an infinite geometric series converges to a finite number when the discount rate exceeds zero.

PV of Perpetuity = PMT / r PMT = payment per period · r = discount rate per period Example: $1,000 per year forever at 5%: PV = $1,000 / 0.05 = $20,000

This is not just theoretical. UK government bonds called "consols" historically had no maturity date. More practically, the Gordon Growth Model uses the perpetuity formula to value stocks that pay dividends expected to grow at a constant rate:

PV of Growing Perpetuity = PMT / (r - g) g = constant growth rate in perpetual payments (must be less than r) Gordon Growth Model: Stock Price = Next Year Dividend / (Required Return - Dividend Growth Rate) Example: dividend of $2, required return 10%, growth 4%: Price = $2 / (0.10 - 0.04) = $33.33

The growing perpetuity formula also forms the basis of the terminal value in DCF models. After the explicit forecast period (typically 5 to 10 years), the analyst assumes cash flows grow at a constant long-run rate forever. The terminal value captures the present value of those perpetual cash flows and is often the largest single component of a DCF valuation.

Frequently Asked Questions

What is the difference between present value and future value?

Present value and future value are the two ends of the same time value of money equation. Future value answers "how much will this grow to?" Present value answers "how much is this future amount worth today?" They are inverses: FV = PV x (1 + r)^n and PV = FV / (1 + r)^n. Given the same interest rate and period, a higher FV produces a higher PV, and a longer time period reduces PV (more discounting).

How does compounding frequency affect present value?

More frequent compounding reduces present value for the same nominal annual rate, because more frequent compounding produces a higher effective annual rate, which means more discounting. The PV of $10,000 in 5 years at 6% nominal: annually = $7,472.58; semi-annually = $7,440.94; monthly = $7,413.72; daily = $7,408.18. The difference is modest but grows with longer horizons and higher rates.

Can present value be negative?

The PV of a single positive future cash flow is always positive. However, a project's NPV can be negative if the present value of future inflows is less than the initial cost. Also, if a future cash flow is itself negative (a cost or liability), its present value is also negative. In pension accounting, the present value of future benefit obligations can be very large and appears as a liability.

What discount rate should I use for personal financial decisions?

For personal decisions, the discount rate should reflect what you could reliably earn on the money elsewhere. If you have high-interest debt (credit cards at 20%+), use that rate, because paying it off is a guaranteed return at that rate. If you are comparing to investing in stocks, use your expected return (7 to 10%). For risk-free comparisons (comparing government payment streams), use the current Treasury rate. For most everyday personal finance decisions, 5 to 8% is a reasonable baseline.

How is present value used in bond pricing?

A bond's price is the sum of (1) the present value of all future coupon payments, treated as an ordinary annuity at the yield per period, plus (2) the present value of the face value (par) received at maturity. Price = PMT x [1 - (1 + r)^(-n)] / r + Face / (1 + r)^n. When the market yield rises above the coupon rate, this formula produces a price below par (discount bond). When the yield falls below the coupon rate, the price rises above par (premium bond).