TVM Calculator (Time Value of Money Finance Calculator)

Of the five fields (number of periods N, annual rate I/Y, present value PV, payment PMT and future value FV), leave blank only the one you want to find and fill in the other four. The blank field is solved.

Periods N (number of payments)

Annual rate I/Y (%)

Present value PV ($)

Payment PMT ($)

Future value FV ($)

Payment timing

P/Y (payments per year)

C/Y (compounding per year)

Sign convention (same as a financial calculator): money that leaves you is negative, and money that comes to you is positive. For example, "borrow $20,000 and repay $2,000 a year" is PV = 20000 (in) and PMT = -2000 (out). Enter 0 in any field you do not use (a blank field tells the calculator to solve for it). This is a math calculation with a fixed rate and does not include fees, taxes or rate changes of real products.

Result and graph

Leave blank only the one field you want to find out of the five on the left and press "Calculate". The result and a graph will appear here.

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formulas and graphs
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Enter any four of the number of periods $N$, annual rate $I/Y$, present value $PV$, payment $PMT$ and future value $FV$, and the fifth is solved (the same five-key TVM design as financial calculators such as the BA II Plus and HP 12C)
The sign convention is also the same as a financial calculator: money that leaves you (deposits, savings, repayments) is negative, and money that comes to you (a loan you receive, payouts) is positive. Example: to save $2,000 a year, PMT = -2000
You can set the payment timing (end of period, as in most loans and savings plans, or beginning of period), P/Y (payments per year) and C/Y (compounding periods per year)
Along with the solved value, you also get the total of all payments and the total interest. When the number of periods is a whole number up to 20, a period-by-period table of interest and balance is shown too
Includes a balance graph. A plain-language explanation of the formulas and copy-and-paste formulas for Excel (FV, PV, PMT, RATE and NPER functions), Google Sheets and Python are all on this page

This page is a math calculation (the time value of money, TVM) that assumes the interest rate stays the same for the whole period. Real loans, savings and investments involve fees, taxes and changing rates, so check the exact amounts with your lender or bank. This page does not recommend any financial product. Use it for studying finance (CFP, CFA, business and engineering economics courses) and for understanding how loans and annuities work.

What is this calculation used for?

Studying finance and engineering economics (the six interest factors)

The six standard interest factors that students often memorize for engineering economics and finance courses (F/P, P/F, F/A, A/F, P/A and A/P) are all rearrangements of this one TVM equation. For example, F/P is just $(1+i)^N$, and F/A is $\dfrac{(1+i)^N-1}{i}$.
Checking them on this page with different numbers lets you understand the six as "one equation read in different ways" instead of six separate things to memorize, which also helps with harder problems.

Understanding a monthly mortgage payment

"Borrow $400,000 with a 30-year fixed-rate mortgage at 6.5%" is PV = +400,000, N = 360 (30 years × 12), monthly payments (P/Y = C/Y = 12) and FV = 0, and you solve for PMT. The answer is about $2,528.27 a month. The payments total about $910,178, so the total interest is about $510,178.
When you can see how the amount is built, you can check for yourself how much the loan amount, the rate and the term each matter. This is principal and interest only; a real monthly payment often also includes property taxes and insurance, plus fees, so check the exact amount with your lender.

Planning retirement withdrawals (how much you need to start with)

"I want to withdraw $40,000 a year for 25 years. If my savings earn 4% a year, how much do I need at the start?" is PMT = +40,000, N = 25 and FV = 0, and you solve for PV. The answer is about $624,883 (PV = −624,883, the money you put in at the start).
You receive $1,000,000 in total, so investment growth covers the difference of about $375,117. Seeing in numbers how much the needed amount changes when the rate changes is a big help in retirement planning (in real investing, returns are not fixed, and you can lose money).

Estimating how long it takes to reach a savings goal

"If I save $500 a month at 3% (compounded monthly), when will I reach $50,000?" is PMT = −500, FV = +50,000 and P/Y = C/Y = 12, and you solve for N. The answer is about 89.4 months (about 7.4 years).
At 0% it would take 100 months, so interest gets you there about 10 months sooner. Seeing how much the interest helps, in numbers, is what makes solving for N interesting.

Finding the interest in a car loan or installment plan

For a $30,000 car loan at a 4% APR for 60 months, set PV = +30,000, N = 60, P/Y = C/Y = 12 and FV = 0 and solve for PMT. The payment is about $552.50 a month and the total is about $33,150, so about $3,150 is interest.
In the other direction, if you are only told the monthly payment and the total, solving for I/Y reveals the real annual rate. This helps you compare car loans, leases, "buy now, pay later" plans and phone installment plans.

Formulas and graphs

The TVM equation (the basic time value of money formula, end-of-period payments)

Graph

Standard notation (the usual math form)

$PV$ $\times$ $(1+i)$ $N$ $+$ $PMT$ $\times$ $\dfrac{(1+i)^{N}-1}{i}$ $+$ $FV$ $=$ $0$

In words (symbols replaced with words)

① $PV$: present value $\times$ ② growth factor for one period ③ $N$: number of periods $+$ ④ payment each period $\times$ ⑤ future value annuity factor $+$ ⑥ $FV$: future value $=$ $0$

The formula in words

① Take the money now, the $PV$: present value

② and multiply it by the growth factor for one period $(1+i)$

③ once for each of the $N$: number of periods

④ Add the $PMT$: payment each period

⑤ times the future value annuity factor (how much a payment of 1 per period grows to with interest)

⑥ and the $FV$: future value . These three parts add up to 0: the money in and out balances exactly, including interest

Quick example

If you borrow $1,000 at 6% a year (PV = +1000, money in) and repay it all at once after 5 years with no payments in between (PMT = 0), the amount you repay (FV) is

present value (+1000) $\times$ growth factor (1.06) periods (5) $+$ $FV$: future value $=$ $0$

$1000 \times 1.06^{5} + FV = 0$

$FV = -1000 \times 1.06^{5} \approx -1338.23$

Key idea

The key to this equation is the sign convention. Money that leaves you is negative, and money that comes to you is positive. In the example, the minus sign in "FV = −1,338.23" says "you pay $1,338.23 after 5 years". In the same way, saving $2,000 every year is money that leaves you, so you enter PMT = −2000. This convention is shared by financial calculators such as the BA II Plus and HP 12C, and by Excel's FV, PV and PMT functions. $i$ is the interest rate per period (with yearly payments it is the annual rate itself; for 6%, $i = 0.06$). In the general form with $g$, payments at the beginning of the period multiply the $PMT$ term by $(1+i)$ one extra time, because money paid earlier earns one more period of interest.

Solving for the future value FV (end-of-period payments)

Graph

Standard notation (the usual math form)

$FV$ $=$ $-$ $($ $PV$ $\times$ $(1+i)$ $N$ $+$ $PMT$ $\times$ $\dfrac{(1+i)^{N}-1}{i}$ $)$

In words (symbols replaced with words)

⑥ $FV$: future value $=$ $-$ $($ ① $PV$: present value $\times$ ② growth factor for one period ③ $N$: number of periods $+$ ④ payment each period $\times$ ⑤ future value annuity factor $)$

The formula in words

① Take the $PV$: present value

② and multiply it by the growth factor for one period $(1+i)$

③ once for each of the $N$: number of periods

④ Add the $PMT$: payment each period

⑤ times the future value annuity factor and flip the sign of the total

⑥ and you get the $FV$: future value

Quick example

If you save $2,000 every year for 5 years at 3% (PV = 0, PMT = −2000, end of period), the amount you receive after 5 years (FV) is

$FV$: future value $=$ $-$ $($ payment (−2000) $\times$ annuity factor (about 5.31) $)$

$\dfrac{1.03^{5}-1}{0.03} \approx 5.3091$

$FV = -\left( 0 + (-2000) \times 5.3091 \right) \approx 10618.27$

Key idea

The money that leaves you, entered as a negative number ($2,000 saved every year), comes back as money you receive, a positive number (about $10,618). This flip in sign is the key to reading TVM answers. You put in $2,000 × 5 years = $10,000, so the difference of about $618 is interest. With payments at the beginning of each year, the $PMT$ term is multiplied by $(1+i)$ one extra time, so $FV \approx 10618.27 \times 1.03 \approx 10936.82$ dollars.

Solving for the number of periods N (logarithms, when PMT = 0)

Graph

Standard notation (the usual math form)

$N$ $=$ $\ln\!\left(-\dfrac{FV}{PV}\right)$ $\div$ $\ln(1+i)$

In words (symbols replaced with words)

③ $N$: number of periods $=$ ① log of the growth multiple $\div$ ② log of one period

The formula in words

① Find how many times the money grows (the growth multiple $-FV \div PV$), and take the log of the growth multiple $\ln(-FV \div PV)$

② divide it by the log of the growth factor for one period $\ln(1+i)$

③ and you get the $N$: number of periods

Quick example

If you deposit $1,000 at 8% a year (PV = −1000), the number of periods until it grows to $2,000 (FV = +2000, a multiple of 2) is

$N$: number of periods $=$ log of the growth multiple (2) $\div$ log of one period (1.08)

$-FV \div PV = -2000 \div (-1000) = 2$

$N = \ln 2 \div \ln 1.08 \approx 0.6931 \div 0.07696 \approx 9.006$

Key idea

$N$ sits in the exponent, so you solve for it with logarithms ($\ln$), the tool that brings an exponent down. Take the log of both sides of $(1+i)^N = -FV \div PV$ to get $N \ln(1+i) = \ln(-FV \div PV)$, which gives the formula above. When there are payments too ($PMT \neq 0$), set $K = PMT(1+i \cdot g) \div i$. Then the equation becomes $(PV + K)(1+i)^N = K - FV$, which you can also solve with logarithms (the calculator on this page uses this general form). If the growth multiple comes out as 0 or less (for example, you made a deposit but gave both amounts the same sign), there is no solution.

The special formula for a 0% rate (when i = 0)

Graph

Standard notation (the usual math form)

$PV$ $+$ $PMT$ $\times$ $N$ $+$ $FV$ $=$ $0$

In words (symbols replaced with words)

① $PV$: present value $+$ ② payment each period $\times$ ③ $N$: number of periods $+$ ④ $FV$: future value $=$ $0$

The formula in words

① Add the $PV$: present value

② the $PMT$: payment each period

③ taken once for each of the $N$: number of periods

④ and the $FV$: future value . The total is 0 (with no interest, it is just addition and subtraction)

Quick example

At 0% interest, if you first deposit $1,000 (PV = −1000) and save $100 every year for 12 years (PMT = −100), the amount you receive after 12 years (FV) is

present value (−1000) $+$ payment (−100) $\times$ periods (12) $+$ $FV$: future value $=$ $0$

$-1000 + (-100) \times 12 + FV = 0$

$FV = 1000 + 1200 = 2200$

Key idea

The TVM equation divides by $i$, so it cannot be used as is when the rate is 0% ($i = 0$). But with no interest, things are simple: all the money you put in just comes back, so it is plain addition. The calculator on this page switches to this formula automatically when $i = 0$.

TVM (the time value of money) is one equation that says money now, money each period and money in the future balance out, including interest. As long as you follow the sign convention (money out is negative, money in is positive), any of the five values (N, I/Y, PV, PMT, FV) can be solved from the other four. Only the annual rate I/Y cannot be isolated by rearranging, so it is found numerically (by bisection).

Symbols and terms

Symbols

$PV$	P V	Present value. The money that moves now. Positive if you borrow or receive it, negative if you deposit or invest it.
$FV$	F V	Future value. The money that moves at the end of the time period. Positive if you receive it at the end, negative if you pay it.
$PMT$	payment	The payment each period. The money that moves every period. Money that leaves you, such as savings or loan payments, is negative; money that comes in, such as rent you collect, is positive.
$N$	N	The number of periods (number of payments). With yearly payments it equals the number of years, but with monthly payments (P/Y = 12) you count months (10 years is $N = 120$).
$I/Y$	I over Y	Interest per year. The annual interest rate in %. The name comes straight from the key on a financial calculator.
$P/Y$	P over Y	Payments per year. How many payments $PMT$ there are in one year: 1 for yearly payments, 12 for monthly payments.
$C/Y$	C over Y	Compounding periods per year. How many times a year interest is added to the balance. Usually the same as $P/Y$; if it differs, the compounding is converted to the payment period.
$i$	lowercase i	The interest rate per period as a decimal. With yearly payments it is the annual rate itself (for 6%, $i = 0.06$); with monthly payments it is the rate per month.
$g$	g	A switch for the payment timing: $g = 1$ for the beginning of the period and $g = 0$ for the end. With beginning-of-period payments, the $PMT$ term is multiplied by $(1+i)$ one extra time.
$\ln$	natural log	The natural logarithm. A tool that finds how many times you multiply to reach a given multiple. It is used to solve for the number of periods $N$.

Terms

time value of money (TVM)	The idea that $10,000 today is worth more than $10,000 in the future, because money you have now can earn interest. It is the foundation of finance, and the equation on this page puts that idea into one formula.
present value	What future money is worth in today's money. For example, at 2% a year, $10,200 one year from now is worth the same as $10,000 today.
future value	How much money now, or regular savings, will be worth in the future, including interest.
discounting	Turning future money into its present value. You divide the future amount by $(1+i)^N$, so the value goes down. It is exactly the reverse of adding interest (compounding).
future value annuity factor	The multiple $\dfrac{(1+i)^N - 1}{i}$: how much a payment of $1 per period grows to with interest. In engineering economics it is the F/A factor, one of the six standard interest factors. You multiply it by the payment each period.
present value annuity factor	The multiple $\dfrac{1-(1+i)^{-N}}{i}$: what the right to receive $1 per period is worth today (the P/A factor). It is the present value version of the future value annuity factor, used for annuities and mortgages (the link between the loan amount and each payment).
cash flow	The flow of money in and out. TVM looks only at when, how much and in which direction money moves, and shows the direction with a sign (plus or minus).
sign convention	The rule that money leaving you (deposits, savings, repayments) is negative and money coming to you (a loan you receive, payouts) is positive. Financial calculators and Excel's financial functions share this rule, and it lets loans, savings and annuities all use the same single equation.
ordinary annuity (end mode)	Payments (PMT) made at the end of each period. Most loan payments and savings plans work this way, and it is the default on this page.
annuity due (begin mode)	Payments (PMT) made at the beginning of each period. The same payment starts earning interest one period earlier, so the $PMT$ term is multiplied by $(1+i)$ one extra time. Rent paid in advance and lease payments work this way.
financial calculator	A calculator with special TVM keys (N, I/Y, PV, PMT, FV). The Texas Instruments BA II Plus and the HP 12C are well known and are used in exams such as the CFA and CFP and in the workplace. This page is a web version of the same five-key design.
bisection method	A numerical method that finds an approximate answer by cutting the interval that contains the answer in half again and again, used when the answer cannot be isolated by rearranging. Solving for the annual rate I/Y requires solving an equation of degree N, which has no general formula, so this page uses bisection.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.

Positive and negative numbers (Grades 6–7)	Reading a negative number as an amount in the opposite direction (on this page, money that leaves you is negative) Being able to multiply negative numbers, as in $(-2000) \times 5 = -10000$
Percents (Grades 6–7)	Being able to write "6%" as the decimal 0.06 Knowing that "grows by 6%" is the same as "× 1.06"
Exponents (Grades 6–8)	Knowing that $1.06^{5}$ is "1.06 multiplied together 5 times"
Basics of compound interest ("Compound Interest Calculator" on this site)	Understanding compound interest, where the interest you earn also earns interest
Logarithms (Algebra 2 and precalculus, advanced)	Using logarithms to find how many times you multiply to reach a multiple (used when solving for N; if not, the calculator does it for you)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table for the future value FV (FV function)

Rate per period (6% = 0.06)	0.06
Number of periods N	10
Payment PMT	-2000
Present value PV	20000
Future value FV	=FV(B1,B2,B3,B4,0)

Table for the payment PMT (PMT function)

Rate per period (6% = 0.06)	0.06
Number of periods N	10
Present value PV	20000
Future value FV	0
Payment PMT	=PMT(B1,B2,B3,B4,0)

Table for the rate (RATE function)

Number of periods N	5
Payment PMT	0
Present value PV	-1000
Future value FV	1500
Rate per period	=RATE(B1,B2,B3,B4)

Table for the number of periods N (NPER function)

Rate per period (8% = 0.08)	0.08
Payment PMT	0
Present value PV	-1000
Future value FV	2000
Number of periods N	=NPER(B1,B2,B3,B4)

Table for the present value PV (PV function, beginning of period)

Rate per period (5% = 0.05)	0.05
Number of periods N	20
Payment PMT	-100
Future value FV	0
Present value PV	=PV(B1,B2,B3,B4,1)

Excel has built-in TVM functions (FV, PMT, RATE, NPER and PV), and they use the same sign convention as this page (money out is negative, money in is positive).
The first table shows -9,455.36 in B5, the second -2,717.36, the third about 0.0845 (an annual rate of 8.447%; format the cell as a percentage to read it easily), the fourth about 9.006, and the fifth 1,308.53.
The last argument of FV, PMT and PV is the payment timing: 0 or omitted = end of period, 1 = beginning of period. For monthly payments, enter the rate per month (annual rate ÷ 12) and N in months.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table for the future value FV (FV function)

Rate per period (6% = 0.06)	0.06
Number of periods N	10
Payment PMT	-2000
Present value PV	20000
Future value FV	=FV(B1,B2,B3,B4,0)

Table for the payment PMT (PMT function)

Rate per period (6% = 0.06)	0.06
Number of periods N	10
Present value PV	20000
Future value FV	0
Payment PMT	=PMT(B1,B2,B3,B4,0)

Table for the rate (RATE function)

Number of periods N	5
Payment PMT	0
Present value PV	-1000
Future value FV	1500
Rate per period	=RATE(B1,B2,B3,B4)

Table for the number of periods N (NPER function)

Rate per period (8% = 0.08)	0.08
Payment PMT	0
Present value PV	-1000
Future value FV	2000
Number of periods N	=NPER(B1,B2,B3,B4)

Table for the present value PV (PV function, beginning of period)

Rate per period (5% = 0.05)	0.05
Number of periods N	20
Payment PMT	-100
Future value FV	0
Present value PV	=PV(B1,B2,B3,B4,1)

Google Sheets uses the same function names (FV, PMT, RATE, NPER, PV) and the same sign convention as Excel. Copy the whole table, paste it into cell A1, and replace the inputs in column B with your own numbers.

How to calculate it in Python

import math

# Set the value you want to find to None (this example finds fv)
n = 10            # number of periods N (number of payments)
annual_rate = 6   # annual rate I/Y (%)
pv = 20000        # present value PV (money in = positive, money out = negative)
pmt = -2000       # payment PMT
fv = None         # future value FV
is_beginning = False  # True = beginning of period, False = end of period

g = 1 if is_beginning else 0
# Rate per period (yearly payments; for monthly payments divide by 12 and count n in months). Leave as None when solving for the rate
i = annual_rate / 100 if annual_rate is not None else None

def tvm_left_side(i_, n_, pv_, pmt_, fv_):
    """Left side of the TVM equation (the answer makes it 0)"""
    if abs(i_) < 1e-12:
        return pv_ + pmt_ * n_ + fv_
    growth = (1 + i_) ** n_
    return pv_ * growth + pmt_ * (1 + i_ * g) * (growth - 1) / i_ + fv_

if fv is None:
    fv = -(pv + pmt * n) if abs(i) < 1e-12 else -(pv * (1 + i) ** n + pmt * (1 + i * g) * ((1 + i) ** n - 1) / i)
    print(f"Future value FV = {fv:,.2f}")
elif pv is None:
    pv = -(fv + pmt * n) if abs(i) < 1e-12 else -(fv + pmt * (1 + i * g) * ((1 + i) ** n - 1) / i) / (1 + i) ** n
    print(f"Present value PV = {pv:,.2f}")
elif pmt is None:
    pmt = -(pv + fv) / n if abs(i) < 1e-12 else -(pv * (1 + i) ** n + fv) / ((1 + i * g) * ((1 + i) ** n - 1) / i)
    print(f"Payment PMT = {pmt:,.2f}")
elif n is None:
    if abs(i) < 1e-12:
        n = -(pv + fv) / pmt
    else:
        k = pmt * (1 + i * g) / i
        n = math.log((k - fv) / (pv + k)) / math.log(1 + i)
    print(f"Number of periods N = {n:.3f}")
else:
    # The annual rate cannot be isolated by rearranging, so search with bisection (some inputs have no solution)
    low, high = -0.9999, 10.0
    if tvm_left_side(low, n, pv, pmt, fv) * tvm_left_side(high, n, pv, pmt, fv) > 0:
        raise SystemExit("No solution in this range. Check the direction (sign) of the money")
    for _ in range(200):
        mid = (low + high) / 2
        if tvm_left_side(low, n, pv, pmt, fv) * tvm_left_side(mid, n, pv, pmt, fv) <= 0:
            high = mid
        else:
            low = mid
    i = (low + high) / 2
    print(f"Annual rate I/Y = {i * 100:.3f} %")

total_interest = abs(pv + pmt * n + fv)
print(f"Total interest = {total_interest:,.2f}")

Runs with the standard library only. Set the value you want to find to None, put numbers in the other four, and run it. The sign convention (money out is negative, money in is positive) is the same as the calculator on this page. Only the annual rate has no formula, so it is found by bisection (cutting the interval that contains the answer in half again and again).

How to write it in LaTeX and other math languages (copy and paste)

The TVM equation (the basic time value of money formula, end-of-period payments)

PV(1 + i)ᴺ + PMT × ((1 + i)ᴺ − 1)/i + FV = 0

PV(1+i)^{N} + PMT \cdot \dfrac{(1+i)^{N}-1}{i} + FV = 0

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>PV</mi>
    <msup>
      <mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>i</mi><mo>)</mo></mrow>
      <mi>N</mi>
    </msup>
    <mo>+</mo>
    <mi>PMT</mi>
    <mo>&#x22C5;</mo>
    <mfrac>
      <mrow>
        <msup>
          <mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>i</mi><mo>)</mo></mrow>
          <mi>N</mi>
        </msup>
        <mo>&#x2212;</mo>
        <mn>1</mn>
      </mrow>
      <mi>i</mi>
    </mfrac>
    <mo>+</mo>
    <mi>FV</mi>
    <mo>=</mo>
    <mn>0</mn>
  </mrow>
</math>

PV(1+i)^N + PMT((1+i)^N-1)/i + FV = 0

pv (1 + i)^n + pmt ((1 + i)^n - 1)/i + fv == 0

pv*(1 + i)^n + pmt*((1 + i)^n - 1)/i + fv = 0;

pv*(1 + i)^n + pmt*((1 + i)^n - 1)/i + fv == 0

PV(1+i)^N + PMT((1+i)^N-1)/i + FV = 0

Solving for the future value FV (end-of-period payments)

FV = −(PV(1 + i)ᴺ + PMT × ((1 + i)ᴺ − 1)/i)

FV = -\left( PV(1+i)^{N} + PMT \cdot \dfrac{(1+i)^{N}-1}{i} \right)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>FV</mi>
    <mo>=</mo>
    <mo>&#x2212;</mo>
    <mrow>
      <mo>(</mo>
      <mi>PV</mi>
      <msup>
        <mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>i</mi><mo>)</mo></mrow>
        <mi>N</mi>
      </msup>
      <mo>+</mo>
      <mi>PMT</mi>
      <mo>&#x22C5;</mo>
      <mfrac>
        <mrow>
          <msup>
            <mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>i</mi><mo>)</mo></mrow>
            <mi>N</mi>
          </msup>
          <mo>&#x2212;</mo>
          <mn>1</mn>
        </mrow>
        <mi>i</mi>
      </mfrac>
      <mo>)</mo>
    </mrow>
  </mrow>
</math>

FV = -(PV(1+i)^N + PMT((1+i)^N-1)/i)

fv = -(pv (1 + i)^n + pmt ((1 + i)^n - 1)/i)

fv := -(pv*(1 + i)^n + pmt*((1 + i)^n - 1)/i);

fv = -(pv*(1 + i)^n + pmt*((1 + i)^n - 1)/i);

FV = -(PV(1+i)^N + PMT((1+i)^N-1)/i)

Solving for the number of periods N (logarithms, when PMT = 0)

N = ln(−FV/PV) / ln(1 + i)

N = \dfrac{\ln\left(-\dfrac{FV}{PV}\right)}{\ln(1+i)}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>N</mi>
    <mo>=</mo>
    <mfrac>
      <mrow>
        <mi>ln</mi>
        <mrow>
          <mo>(</mo>
          <mo>&#x2212;</mo>
          <mfrac><mi>FV</mi><mi>PV</mi></mfrac>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mrow>
        <mi>ln</mi>
        <mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>i</mi><mo>)</mo></mrow>
      </mrow>
    </mfrac>
  </mrow>
</math>

N = ln(-FV/PV) / ln(1+i)

n = Log[-fv/pv]/Log[1 + i]

n := ln(-fv/pv)/ln(1 + i);

n = log(-fv/pv)/log(1 + i);

N = ln(-FV/PV)/ln(1+i)

The special formula for a 0% rate (when i = 0)

PV + PMT × N + FV = 0

PV + PMT \cdot N + FV = 0

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>PV</mi>
    <mo>+</mo>
    <mi>PMT</mi>
    <mo>&#x22C5;</mo>
    <mi>N</mi>
    <mo>+</mo>
    <mi>FV</mi>
    <mo>=</mo>
    <mn>0</mn>
  </mrow>
</math>

PV + PMT * N + FV = 0

pv + pmt n + fv == 0

pv + pmt*n + fv = 0;

pv + pmt*n + fv == 0

PV + PMT N + FV = 0

How to have ChatGPT do the calculation

You are a finance (time value of money, TVM) calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

Use the standard TVM equation PV(1+i)^N + PMT(1+i*g)*((1+i)^N - 1)/i + FV = 0 (g=1 for payments at the beginning of each period, g=0 for the end).
The sign convention is the same as a financial calculator: money that leaves you is negative, and money that comes to you is positive.

You borrow $20,000 at 6% a year (PV = +20000) and pay it off with equal payments at the end of each year for 10 years (FV = 0).
Find each of the following:
1. The yearly payment PMT
2. The total of all payments (PMT × 10)
3. The total interest (|PV + PMT×N + FV|)

Show the formulas you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula

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\(PV\)	P V	Present value. The money that moves now. Positive if you borrow or receive it, negative if you deposit or invest it.
\(FV\)	F V	Future value. The money that moves at the end of the time period. Positive if you receive it at the end, negative if you pay it.
\(PMT\)	payment	The payment each period. The money that moves every period. Money that leaves you, such as savings or loan payments, is negative; money that comes in, such as rent you collect, is positive.
\(N\)	N	The number of periods (number of payments). With yearly payments it equals the number of years, but with monthly payments (P/Y = 12) you count months (10 years is \(N = 120\)).
\(I/Y\)	I over Y	Interest per year. The annual interest rate in %. The name comes straight from the key on a financial calculator.
\(P/Y\)	P over Y	Payments per year. How many payments \(PMT\) there are in one year: 1 for yearly payments, 12 for monthly payments.
\(C/Y\)	C over Y	Compounding periods per year. How many times a year interest is added to the balance. Usually the same as \(P/Y\); if it differs, the compounding is converted to the payment period.
\(i\)	lowercase i	The interest rate per period as a decimal. With yearly payments it is the annual rate itself (for 6%, \(i = 0.06\)); with monthly payments it is the rate per month.
\(g\)	g	A switch for the payment timing: \(g = 1\) for the beginning of the period and \(g = 0\) for the end. With beginning-of-period payments, the \(PMT\) term is multiplied by \((1+i)\) one extra time.
\(\ln\)	natural log	The natural logarithm. A tool that finds how many times you multiply to reach a given multiple. It is used to solve for the number of periods \(N\).

TVM Calculator (Time Value of Money Finance Calculator)

Table of Contents

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What is this calculation used for?

Formulas and graphs

Symbols and terms

Symbols

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Good to know before you start

How to calculate it in Excel

How to calculate it in Google Sheets

How to calculate it in Python

How to write it in LaTeX and other math languages (copy and paste)

How to have ChatGPT do the calculation

How to Use

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