What is Future Value (FV)?

Future Value is the value of a current asset at a future date, based on an assumed growth rate. FV = PV × (1 + r)^n for a lump sum, or FV = PMT × [(1+r)^n - 1] / r for an annuity. It answers: 'How much will my money grow to?'

What is Present Value (PV)?

Present Value is the current worth of a future sum of money, discounted at a specific interest rate. PV = FV / (1 + r)^n. It answers: 'How much is a future payment worth in today's dollars?' It's the inverse of Future Value.

An annuity is a series of equal periodic payments (PMT) made over a defined time period. Ordinary annuities have payments at the end of each period (mortgages, car loans). Annuities due have payments at the beginning (lease payments, insurance premiums). This calculator supports both payment timing options.

What is the difference between nominal and effective interest rates?

The nominal rate (I/Y) is the stated annual rate. The effective annual rate (EAR) accounts for compounding frequency: EAR = (1 + I/Y/C)^C - 1, where C is compounding periods per year. A 6% nominal rate compounded monthly has an EAR of 6.168%. This calculator supports different payment and compounding frequencies (P/Y and C/Y settings).

Finance Calculator

Time Value of Money · Solve FV · PV · PMT · I/Y · N · Equivalent of BA II Plus & HP 12C

Solving for: Future Value (FV)

How much will your money grow to?

N - Number of Periods ?

I/Y - Annual Interest Rate ?

PV - Present Value ?

PMT - Periodic Payment ?

FV - Future Value ?

Settings

P/Y - Payments/Year ?

C/Y - Compound/Year ?

PMT Timing

End of Period (Ordinary) Beginning (Due)

📊 Choose a solve mode (FV, PMT, I/Y, N, or PV), enter the known values, and the calculator will solve for the missing variable instantly.

Future Value (FV)

$9,455.36

10 periods · 6% · monthly payments

Sum of All Payments

$20,000

Total Interest

$9,455

Effective Annual Rate

6.168%

EAR after compounding

Periodic Rate

0.500%

per payment period

Total Cash Flow

$20,000

PV + all PMTs + FV

📊 Cash Flow Timeline

📋 Payment Schedule

Period	PV (Beg)	PMT	Interest	FV (End)

The Time Value of Money (TVM)

The Time Value of Money is the foundational concept of all finance: a dollar in hand today is worth more than a dollar promised in the future. This is true for three reasons:

Opportunity cost: A dollar today can be invested immediately and earn returns - a future dollar cannot.
Inflation: A future dollar will generally buy less than a dollar today due to rising prices.
Risk: Future cash flows are uncertain - a dollar promised in the future carries the risk of not materializing.

TVM is used to evaluate loans, bonds, mortgages, annuities, business investments, and retirement plans. This calculator replicates the 5-key TVM functionality of professional financial calculators like the Texas Instruments BA II Plus and HP 12C.

The Five TVM Variables

N: Total number of compounding periods (e.g., 360 for a 30-year monthly mortgage)
I/Y: Annual interest rate (the calculator adjusts for P/Y and C/Y automatically)
PV: Present Value - the value at time zero (positive = inflow, negative = outflow)
PMT: Periodic payment each period (negative if paying, positive if receiving)
FV: Future Value - the value at the end of all periods (0 for most loans)

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Sign convention: Use negative values for cash outflows (money you pay) and positive for inflows (money you receive). For a loan: PV is positive (you receive the loan), PMT is negative (you make payments). For savings: PV is negative (you deposit), FV is positive (you receive).

Core TVM Formulas

FV = PV(1+r)^N + PMT × [(1+r)^N - 1] / r × (1+r·T) r = periodic rate (I/Y ÷ C/Y adjusted for P/Y) · T = 0 (end) or 1 (beginning) Lump sum: FV = PV(1+r)^N Annuity FV: PMT × [(1+r)^N - 1] / r

PV = FV / (1+r)^N + PMT × [1 - (1+r)^(-N)] / r × (1+r·T) This is the present value formula, used to discount future cash flows back to today's dollars.

Ordinary Annuity vs. Annuity Due

The timing of periodic payments (PMT) changes the value of an annuity calculation significantly:

Ordinary Annuity (End of Period)

Payments occur at the end of each period. This is the most common type in finance:

Mortgage payments (due at end of month)
Car loan payments
Bond coupon payments
Most loan repayments

FV(ordinary) = PMT × [(1+r)^N - 1] / r PV(ordinary) = PMT × [1 - (1+r)^(-N)] / r

Annuity Due (Beginning of Period)

Payments occur at the beginning of each period. An annuity due is always worth more than an ordinary annuity at the same rate, because each payment compounds for one additional period:

Lease payments (first payment due immediately)
Insurance premiums
Rent (typically due at month's start)

FV(due) = FV(ordinary) × (1+r) An annuity due is always (1+r) times greater than an ordinary annuity.

Comparison: $1,000/month at 6%/yr for 10 years

Type	FV	PV
Ordinary Annuity (end)	$163,879	$90,073
Annuity Due (beginning)	$164,700	$90,524
Difference (1 extra period)	+$821	+$451

Nominal vs. Effective Interest Rates: P/Y and C/Y Settings

The nominal interest rate (I/Y) is the stated annual rate. The effective annual rate (EAR) is the actual annual yield after accounting for how frequently interest compounds within the year. Most financial contracts quote nominal rates, but EAR is what you actually earn or pay.

The EAR Formula

EAR = (1 + I/Y / C/Y)^C/Y - 1 For continuous compounding: EAR = e^(I/Y) - 1 Example: 6% nominal, monthly compounding → EAR = (1 + 0.06/12)^12 - 1 = 6.168%

Why P/Y and C/Y Matter

The P/Y setting (Payments per Year) defines how often you make payments - monthly, quarterly, annually. The C/Y setting (Compounding per Year) defines how often interest compounds. When P/Y ≠ C/Y, the calculator converts the nominal rate to a per-period effective rate before calculating:

Periodic Rate = (1 + I/Y / C/Y)^(C/Y / P/Y) - 1 This ensures accurate results when payment and compounding frequencies differ (e.g., monthly payments, semi-annual compounding on a Canadian mortgage).

Nominal Rate	C/Y	EAR	Real extra cost vs annual
6.00% / yr	Annually (1)	6.000%	baseline
6.00% / yr	Semi-annually (2)	6.090%	+$9/yr per $10,000
6.00% / yr	Quarterly (4)	6.136%	+$14/yr per $10,000
6.00% / yr	Monthly (12)	6.168%	+$17/yr per $10,000
6.00% / yr	Daily (365)	6.183%	+$18/yr per $10,000
6.00% / yr	Continuous	6.184%	+$18/yr per $10,000

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Canadian mortgages compound semi-annually (C/Y=2) but have monthly payments (P/Y=12). US mortgages compound monthly (C/Y=12, P/Y=12). Always set C/Y to match your loan agreement to get accurate results.

Net Present Value (NPV) & Internal Rate of Return (IRR)

NPV and IRR are extensions of TVM used in corporate finance and capital budgeting to evaluate whether an investment or project is worthwhile.

Net Present Value (NPV)

NPV discounts all future cash flows back to today's dollars using a required rate of return (discount rate), then subtracts the initial investment. A positive NPV means the investment creates value; a negative NPV destroys value.

NPV = -C₀ + C₁/(1+r) + C₂/(1+r)² + ... + Cₙ/(1+r)^n C₀ = initial investment · Cₙ = cash flow in period n · r = discount rate

Decision rule:

NPV > 0: Accept the investment (it creates value)
NPV = 0: Indifferent (exactly meets required return)
NPV < 0: Reject (destroys value)

Example: You invest $100,000 today to receive $30,000/year for 5 years at a 10% discount rate. NPV = -$100,000 + $30,000 × [1-(1.10)^-5]/0.10 = -$100,000 + $113,724 = +$13,724. The investment creates $13,724 of value above the required return.

Internal Rate of Return (IRR)

IRR is the discount rate that makes NPV exactly equal to zero - it's the investment's actual effective return rate. If IRR > required return (hurdle rate), the investment is acceptable.

0 = -C₀ + C₁/(1+IRR) + C₂/(1+IRR)² + ... + Cₙ/(1+IRR)^n IRR has no closed-form solution - it's found by iteration (Newton-Raphson method)

For the example above: IRR ≈ 15.2%. Since IRR (15.2%) > required return (10%), the investment is accepted.

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IRR limitation: With unconventional cash flows (multiple sign changes), multiple IRRs may exist, making the decision ambiguous. NPV is always the more reliable decision tool.

Real-World TVM Applications

Mortgage Payments (PV → PMT)

A $320,000 home loan at 6.57% for 30 years. Set N=360, I/Y=6.57, PV=320,000, FV=0, P/Y=12, C/Y=12. Solve for PMT = -$2,046.03/month. Total interest paid = $2,046 × 360 - $320,000 = $416,578.

Retirement Savings (PMT → FV)

Save $500/month for 30 years at 7% return. N=360, I/Y=7, PV=0, PMT=-500, P/Y=12. Solve for FV = $606,438 - built from $180,000 of contributions and $426,438 of compound growth.

Bond Valuation (FV, PMT, N → PV)

A $1,000 bond paying 5% semi-annual coupons, maturing in 10 years, when market rates are 6%. N=20, I/Y=6, PMT=25, FV=1000, P/Y=2, C/Y=2. Solve for PV = -$925.61. The bond trades at a discount because its coupon rate (5%) is below the market rate (6%).

Lease vs. Buy Analysis (NPV approach)

Compare buying equipment for $50,000 vs leasing at $1,200/month for 5 years. Use PV of lease payments at your cost of capital, then compare to outright purchase. TVM makes this comparison mathematically rigorous.

Perpetuity Value

A perpetuity pays a fixed amount forever. Its present value is simply:

PV(perpetuity) = PMT / r Example: $1,000/year at 5% discount rate → PV = $1,000 / 0.05 = $20,000

Growing Annuity

Cash flows that grow at rate g per period:

PV = PMT / (r - g) × [1 - ((1+g)/(1+r))^N] Gordon Growth Model: PV = D₁ / (r - g) for a growing perpetuity (e.g., stock valuation)

Discounted Cash Flow (DCF) & Business Valuation

Discounted Cash Flow analysis is the most rigorous framework for valuing businesses, real estate investments, and capital projects. It applies TVM principles to project all future cash flows, then discounts them to present value using an appropriate discount rate (typically the Weighted Average Cost of Capital, or WACC).

Step-by-Step DCF Valuation

Project free cash flows (FCF): Estimate how much cash the business generates each year after operating expenses and capital expenditures, but before debt payments. This requires revenue forecasts, margin assumptions, and capex plans.
Determine the discount rate (WACC): WACC = cost of equity × (E/V) + cost of debt × (D/V) × (1 - tax rate). Cost of equity is typically estimated using the Capital Asset Pricing Model (CAPM): r = risk-free rate + β × market risk premium.
Calculate the terminal value: Projects beyond the forecast horizon (usually 5-10 years) are captured in a terminal value. The Gordon Growth Model: TV = FCF(last year) × (1+g) / (WACC - g), where g is the long-run growth rate (typically 2-3%, near GDP growth).
Discount everything to present value: Sum the PV of each year's FCF plus the PV of the terminal value. This sum is the Enterprise Value of the business.
Subtract net debt: Enterprise Value - Net Debt = Equity Value. Divide by shares outstanding to get intrinsic value per share.

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DCF sensitivity: Small changes in the discount rate or growth assumptions produce dramatic changes in value. A 1% change in WACC or terminal growth rate can change a DCF valuation by 20-40%. Always run sensitivity analysis and treat DCF results as a range, not a precise number.

Weighted Average Cost of Capital (WACC)

WACC = (E/V) × Rₑ + (D/V) × Rₐ × (1 - Tₓ) E = equity value · D = debt value · V = E+D · Rₑ = cost of equity · Rₐ = pre-tax cost of debt · Tₓ = tax rate

CAPM: Capital Asset Pricing Model

Expected Return = Rƒ + β × (Rₘ - Rƒ) Rƒ = risk-free rate (US 10-yr Treasury) · β = systematic risk relative to market · (Rₘ-Rƒ) = equity risk premium (~5%) Example: Rƒ=4.5%, β=1.2, ERP=5% → Expected Return = 4.5% + 1.2 × 5% = 10.5%

Using This Calculator Like a BA II Plus

This calculator replicates the five-key TVM function of the Texas Instruments BA II Plus and HP 12C financial calculators, the industry standard for CFA exam candidates and finance professionals.

Key Differences from Consumer Calculators

Sign convention: Cash outflows are negative, inflows are positive. When taking a loan (receiving money): PV is positive. When making payments: PMT is negative.
P/Y and C/Y settings: These allow the calculator to handle cases where payment frequency differs from compounding frequency - critical for Canadian mortgages (semi-annual C/Y, monthly P/Y) and some corporate bonds.
N is total periods: N = 360 for a 30-year monthly mortgage. On the BA II Plus, you typically input N in periods, not years. Here you can input N directly.
FV for balloon payments: Use FV when a loan has a balloon payment at the end (not fully amortized). Most regular mortgages have FV = 0.

Common TVM Scenarios Quick Reference

Scenario	Solve for	Known inputs
Mortgage payment	PMT	N, I/Y, PV, FV=0
Loan balance after N payments	FV	N, I/Y, PV, PMT
Retirement savings goal	FV	N, I/Y, PV=0, PMT
Required monthly savings	PMT	N, I/Y, PV=0, FV
Bond price	PV	N, I/Y, PMT=coupon, FV=par
Bond yield (YTM)	I/Y	N, PV=price, PMT=coupon, FV=par
Payback period	N	I/Y, PV, PMT, FV
Lump sum growth	FV	N, I/Y, PV, PMT=0

Frequently Asked Questions

What is the time value of money?

TVM is the concept that a dollar today is worth more than a dollar in the future, because today's dollar can be invested and earn returns. It accounts for opportunity cost, inflation, and risk. Every financial calculation - loans, bonds, mortgages, retirement planning - is built on TVM principles.

What is the difference between PV and FV?

Present Value (PV) is the value today of a future sum, discounted at a given rate. Future Value (FV) is what an amount grows to over time at a given rate. They're inverses: FV = PV × (1+r)^N and PV = FV / (1+r)^N. If you know any four of the five TVM variables, this calculator solves for the fifth.

What does a negative PMT or PV mean?

Negative values represent cash outflows (money you pay). Positive values are inflows (money you receive). For a savings account: PV is negative (you deposit money) and FV is positive (you receive it later). For a loan: PV is positive (you receive the loan) and PMT is negative (you make monthly payments). Consistent sign convention is critical for correct results.

How do I calculate bond yield to maturity (YTM)?

Use the I/Y solve mode. Enter: N = periods to maturity, PV = current bond price (negative if you bought it), PMT = coupon payment each period (positive, as you receive it), FV = par/face value at maturity (positive). The calculator solves for I/Y, which is the yield to maturity. Multiply by C/Y if needed to get annual YTM.

What is net present value (NPV)?

NPV is the sum of all future cash flows discounted to present value, minus the initial investment. A positive NPV means an investment creates value above the required return; negative means it destroys value. It is the gold standard for capital budgeting decisions because it directly measures value creation in dollar terms.