Time Value of Money Calculator

Time Value of Money Cash Flow (optional)

The change in value from PV to FV can result from accrued interest being added to or deducted from the present value.

It can also result from adding or deducting an additional amount, plus the accrued interest, to or from the present value.

If there is an additional amount, it is known as an annuity. In this context, ‘annuity’ means a repeating cash flow (equal payments). If the amount is added to the PV, the cash flow is a credit annuity. If it is deducted from the PV, the cash flow is a debit annuity.

Having a cash flow is optional.

A loan is a debit annuity TVM calculation. The FV (or loan balance) should be less than the PV at the end of the cash flow term.

A 401(k) retirement account is a credit annuity TVM calculation. The FV (or account balance) should be greater than the PV at the end of the cash flow term.

TVM Calculator Notes

• If you select “add to PV” for cash flow type, the PV can always be 0. If you set another input to 0, the calculator will calculate the value of the second input. This allows you to solve for an input value and start with a 0 present value.
• If you select “deduct from PV” for cash flow type, the FV can always be 0. If you set another input to 0, the calculator will calculate the value of the second input. This allows you to solve for the input value that results in a 0 future value.
• The interest rate can be negative. A negative interest rate reverses the usual results.
• If you do not indicate a value to calculate by entering 0, the calculator will recalculate the PV when the cash flow is deducted from the PV.
• If there is nothing to calculate, the calculator will recalculate the FV when the cash flow is added to the PV.

Present Value of an Ordinary Annuity

Variable Definitions

R

Nominal annual interest rate.

i

Periodic interest rate.

f

Compounding frequency: the number of compounding periods per year.

n

Total number of periods.

PMT

Periodic cash flow amount (equal payments each period).

k

Period number of the cash flow, starting with 1.

For step-by-step guidance on solving the present value of an ordinary annuity equation, seecalculation steps explained, figure 2.

Present Value of an Annuity-Due

Variable Definitions

R

Nominal annual interest rate.

i

Periodic interest rate.

f

Compounding frequency: the number of compounding periods per year.

n

Total number of periods.

PMT

Periodic cash flow amount (equal payments each period).

k

Period number of the cash flow, starting with 1.

For step-by-step guidance on solving the present value of an annuity due equation, seecalculation steps explained, figure 4.

Interest Rate Equation

Variable Definitions

r

Periodic rate of return per period. For example, per year when cash flows are annual.

IRR

Nominal annualized rate of return, computed as IRR = r × f.

f

Frequency (the number of periods per year). For annual spacing, f = 1.

PMT

Cash flow at period index t. By convention, outflows are negative and inflows are positive. Values may differ across periods.

n

Total number of periods after t = 0. The summation from t = 0 to t = n includes both the initial cash flow at t = 0 and the final cash flow at t = n.

t

Period index. An integer with t = 0, 1, …, n, measured in equal time steps.

For step-by-step guidance on solving the interest rate equation, seesolution figure 2.

Cash Flow Amount Equation – Calculate the Periodic Cash Flow Amount

Variable Definitions

P

Payment amount.

L

Loan amount.

n

Number of periods (the loan term).

c

Monthly interest rate (nominal annual rate divided by 12).

For step-by-step guidance on solving the cash flow equation, seecalculation steps, figure 4.

Term Equation – Calculate the Number of Periods (N)

Variable Definitions

R

Nominal annual interest rate (quoted rate).

n

Number of compounding or payment periods per year.

i

Periodic interest rate.

A

Loan amount (principal).

P

Amount of each equal payment.

N

Total number of periods (loan term).

For step-by-step guidance on solving the term equation, seeterm equation, figure 2.

Future Value of an Ordinary Annuity Equation (with a starting amount)

For an ordinary annuity, the cash flows occur at the end of each period. To model this, set “First Contribution Date” to any date after “Start Date”. The calculator supports a stub (irregular-length) first period, although the equation itself does not.

Variable Definitions

R

Nominal annual interest rate.

f

Number of compounding periods per year.

i

Periodic interest rate.

PV

Present Value – the starting amount (may be 0).

PMT

Periodic cash flow amount. All cash flows are equal.

n

Total number of periods.

For step-by-step guidance on solving the future value of an ordinary annuity, seesteps to solve, figure 2.

Future Value of an Annuity Due Equation (with a starting amount)

For an annuity due, the cash flows occur at the beginning of each period. To model this, set “First Contribution Date” equal to “Start Date”.

Variable Definitions

R

Nominal annual interest rate.

f

Number of compounding periods per year.

i

Periodic interest rate.

PV

Present Value – the starting amount (may be 0).

PMT

Periodic cash flow amount. All cash flows are equal.

n

Total number of periods.

For step-by-step guidance on solving the payment equation, seesteps to solve, figure 4.