Use this calculator to solve for any of the five Time Value of Money (TVM) variables: Number of Periods (N), Interest Rate per Period (I/Y), Present Value (PV), Payment (PMT), or Future Value (FV). Enter four known values and leave one blank to solve for it.
Understanding the BA Plus Calculator and Time Value of Money (TVM)
The BA Plus Calculator, often referring to the Texas Instruments BA II Plus, is a powerful financial tool used to solve problems involving the Time Value of Money (TVM). TVM is a fundamental concept in finance, stating that a sum of money is worth more now than the same sum will be at a future date due to its potential earning capacity. This calculator helps you understand how interest rates, payment schedules, and time periods affect the value of money over time.
The Five Key TVM Variables:
N (Number of Periods): This represents the total number of compounding or payment periods over the life of the investment or loan. For example, a 30-year loan with monthly payments would have N = 30 * 12 = 360 periods.
I/Y (Interest Rate per Period): This is the interest rate applied during each period. It's crucial to ensure this rate matches the period length (e.g., if N is in months, I/Y should be the monthly interest rate). If an annual rate is given, you'll need to convert it (e.g., 6% annual rate compounded monthly is 0.5% per month). Enter it as a percentage (e.g., 0.5 for 0.5%).
PV (Present Value): This is the current value of a future sum of money or stream of cash flows. It represents the initial principal amount of a loan or the initial investment.
PMT (Payment): This is the amount of each regular payment made or received over the life of the investment or loan. Payments are assumed to be constant and occur at regular intervals.
FV (Future Value): This is the value of an asset or cash at a specified date in the future, equivalent in value to a specified sum today. It represents the ending balance of a loan or the accumulated value of an investment.
Cash Flow Sign Convention:
A critical aspect of using TVM calculators is understanding the cash flow sign convention. Generally:
Money received (inflow) is positive (+). For example, if you receive a loan, the PV would be positive. If you receive payments from an investment, PMT would be positive.
Money paid (outflow) is negative (-). For example, if you make a loan payment, the PMT would be negative. If you make an initial investment, the PV would be negative.
Consistency is key. If you enter PV as a positive loan amount, then your PMT (payments made) should be negative. If you enter PV as a negative initial investment, then FV (future value received) would typically be positive.
Annuity Types:
End (Ordinary Annuity): Payments occur at the end of each period. This is the most common assumption for loans and investments.
Begin (Annuity Due): Payments occur at the beginning of each period. This is common for rent payments or some lease agreements. Annuity due calculations typically result in a slightly higher future value or lower present value compared to an ordinary annuity, given the same inputs, because payments earn interest for one additional period.
Examples:
Example 1: Calculate Future Value (FV) of an Investment
You invest $1,000 today and plan to contribute an additional $100 at the end of each month for 5 years. The investment earns an annual interest rate of 6%, compounded monthly. What will be the future value of your investment?
N: 5 years * 12 months/year = 60 periods
I/Y: 6% annual / 12 months = 0.5% per period (enter as 0.5)
PV: -$1,000 (initial investment, an outflow)
PMT: -$100 (monthly contribution, an outflow)
FV: Leave blank (to be calculated)
Annuity Type: End
Expected Result: Approximately $7,112.48
Example 2: Calculate Present Value (PV) of a Loan
You want to take out a loan that requires monthly payments of $750 for 10 years. The annual interest rate is 4.5%, compounded monthly. What is the maximum loan amount (PV) you can afford?
N: 10 years * 12 months/year = 120 periods
I/Y: 4.5% annual / 12 months = 0.375% per period (enter as 0.375)
PV: Leave blank (to be calculated)
PMT: -$750 (monthly payment, an outflow)
FV: $0 (loan is fully paid off)
Annuity Type: End
Expected Result: Approximately $72,990.09
Example 3: Calculate Payment (PMT) for a Mortgage
You take out a $200,000 mortgage at an annual interest rate of 3.6%, compounded monthly, over 30 years. What will be your monthly payment?
N: 30 years * 12 months/year = 360 periods
I/Y: 3.6% annual / 12 months = 0.3% per period (enter as 0.3)
PV: $200,000 (loan received, an inflow)
PMT: Leave blank (to be calculated)
FV: $0 (loan is fully paid off)
Annuity Type: End
Expected Result: Approximately -$909.17 (a payment outflow)
Example 4: Calculate Number of Periods (N) for Savings Goal
You want to save $50,000. You currently have $5,000 and can contribute $200 at the end of each month. Your savings account earns an annual interest rate of 2.4%, compounded monthly. How many months will it take to reach your goal?
N: Leave blank (to be calculated)
I/Y: 2.4% annual / 12 months = 0.2% per period (enter as 0.2)
PV: -$5,000 (initial investment, an outflow)
PMT: -$200 (monthly contribution, an outflow)
FV: $50,000 (future goal, an inflow)
Annuity Type: End
Expected Result: Approximately 179.5 months
Example 5: Calculate Interest Rate (I/Y) for an Investment
You invested $10,000 five years ago. You also contributed $50 at the end of each month during that time. Today, your investment is worth $15,000. What was the annual interest rate (compounded monthly) you earned?
N: 5 years * 12 months/year = 60 periods
I/Y: Leave blank (to be calculated)
PV: -$10,000 (initial investment, an outflow)
PMT: -$50 (monthly contribution, an outflow)
FV: $15,000 (future value received, an inflow)
Annuity Type: End
Expected Result: Approximately 0.36% per month (which is about 4.32% annual)
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function getInputValue(id) {
var value = document.getElementById(id).value;
return value === "" ? null : parseFloat(value);
}
function tvmEquation(N, i, PV, PMT, FV, Type) {
if (i === 0) {
return PV + PMT * N + FV;
}
var term1 = PV * Math.pow(1 + i, N);
var term2 = PMT * ((Math.pow(1 + i, N) – 1) / i);
if (Type === 1) { // Annuity Due
term2 *= (1 + i);
}
return term1 + term2 + FV;
}
function solveFV(N, i, PV, PMT, Type) {
if (i === 0) {
return -PV – PMT * N;
}
var term1 = PV * Math.pow(1 + i, N);
var term2 = PMT * ((Math.pow(1 + i, N) – 1) / i);
if (Type === 1) { // Annuity Due
term2 *= (1 + i);
}
return -(term1 + term2);
}
function solvePV(N, i, PMT, FV, Type) {
if (i === 0) {
return -FV – PMT * N;
}
var term2 = PMT * ((Math.pow(1 + i, N) – 1) / i);
if (Type === 1) { // Annuity Due
term2 *= (1 + i);
}
return (-FV – term2) / Math.pow(1 + i, N);
}
function solvePMT(N, i, PV, FV, Type) {
if (i === 0) {
if (N === 0) return NaN; // Cannot solve if N is 0 and i is 0
return (-FV – PV) / N;
}
var numerator = -FV – PV * Math.pow(1 + i, N);
var denominator = ((Math.pow(1 + i, N) – 1) / i);
if (Type === 1) { // Annuity Due
denominator *= (1 + i);
}
if (denominator === 0) return NaN; // Avoid division by zero
return numerator / denominator;
}
function solveN(i, PV, PMT, FV, Type) {
if (i === 0) {
if (PMT === 0) {
if (PV + FV === 0) return 0; // Trivial case, or infinite solutions
return NaN; // Cannot solve if PMT is 0 and i is 0, and PV+FV != 0
}
return (-FV – PV) / PMT;
}
var x;
if (Type === 0) { // Ordinary Annuity
x = (PMT – FV * i) / (PV * i + PMT);
} else { // Annuity Due
x = (PMT * (1 + i) – FV * i) / (PV * i + PMT * (1 + i));
}
if (x <= 0) return NaN; // Logarithm of non-positive number
if (1 + i <= 0) return NaN; // Logarithm of non-positive base
return Math.log(x) / Math.log(1 + i);
}
function solveIY(N, PV, PMT, FV, Type) {
var bestI = 0;
var minAbsResult = Infinity;
var step = 0.0000001; // 0.00001%
var maxI = 5.0; // Up to 500% periodic rate
var minI = -0.99; // Down to -99% periodic rate
// Handle i = 0 case
if (Math.abs(tvmEquation(N, 0, PV, PMT, FV, Type)) < 0.0001) {
return 0;
}
// Search positive rates
for (var i = step; i <= maxI; i += step) {
var result = tvmEquation(N, i, PV, PMT, FV, Type);
if (Math.abs(result) < minAbsResult) {
minAbsResult = Math.abs(result);
bestI = i;
}
if (Math.abs(result) = minI; i -= step) {
var result = tvmEquation(N, i, PV, PMT, FV, Type);
if (Math.abs(result) < minAbsResult) {
minAbsResult = Math.abs(result);
bestI = i;
}
if (Math.abs(result) < 0.00001) { // Found a sufficiently close solution
return i;
}
}
return bestI; // Return the best found, even if not perfect
}
function calculateTVM() {
var N = getInputValue("N_periods");
var IY_percent = getInputValue("IY_rate");
var PV = getInputValue("PV_value");
var PMT = getInputValue("PMT_value");
var FV = getInputValue("FV_value");
var annuityType = document.getElementById("annuity_begin").checked ? 1 : 0;
var inputs = [N, IY_percent, PV, PMT, FV];
var blankCount = inputs.filter(function(val) { return val === null; }).length;
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = "";
if (blankCount !== 1) {
resultDiv.innerHTML = "Please leave exactly one field blank to solve for it.";
return;
}
// Convert IY_percent to decimal rate per period
var i_decimal = IY_percent !== null ? IY_percent / 100 : null;
var solvedValue;
var solvedFor = "";
if (N === null) {
if (i_decimal === null || PV === null || PMT === null || FV === null) {
resultDiv.innerHTML = "Error: Missing required inputs for N calculation.";
return;
}
solvedValue = solveN(i_decimal, PV, PMT, FV, annuityType);
solvedFor = "N (Number of Periods)";
} else if (IY_percent === null) {
if (N === null || PV === null || PMT === null || FV === null) {
resultDiv.innerHTML = "Error: Missing required inputs for I/Y calculation.";
return;
}
solvedValue = solveIY(N, PV, PMT, FV, annuityType);
solvedFor = "I/Y (Interest Rate per Period)";
// Convert back to percentage for display
solvedValue *= 100;
} else if (PV === null) {
if (N === null || i_decimal === null || PMT === null || FV === null) {
resultDiv.innerHTML = "Error: Missing required inputs for PV calculation.";
return;
}
solvedValue = solvePV(N, i_decimal, PMT, FV, annuityType);
solvedFor = "PV (Present Value)";
} else if (PMT === null) {
if (N === null || i_decimal === null || PV === null || FV === null) {
resultDiv.innerHTML = "Error: Missing required inputs for PMT calculation.";
return;
}
solvedValue = solvePMT(N, i_decimal, PV, FV, annuityType);
solvedFor = "PMT (Payment)";
} else if (FV === null) {
if (N === null || i_decimal === null || PV === null || PMT === null) {
resultDiv.innerHTML = "Error: Missing required inputs for FV calculation.";
return;
}
solvedValue = solveFV(N, i_decimal, PV, PMT, annuityType);
solvedFor = "FV (Future Value)";
}
if (isNaN(solvedValue) || !isFinite(solvedValue)) {
resultDiv.innerHTML = "Calculation Error: Please check your inputs. It might be impossible to solve with the given values (e.g., division by zero, logarithm of negative number).";
} else {
var formattedValue;
if (solvedFor.includes("I/Y")) {
formattedValue = solvedValue.toFixed(4) + "%"; // 4 decimal places for rate
} else if (solvedFor.includes("N")) {
formattedValue = solvedValue.toFixed(2); // 2 decimal places for periods
} else {
formattedValue = solvedValue.toLocaleString('en-US', { style: 'currency', currency: 'USD' }); // Currency format for PV, PMT, FV
}
resultDiv.innerHTML = "" + solvedFor + ": " + formattedValue;
}
}
function clearInputs() {
document.getElementById("N_periods").value = "";
document.getElementById("IY_rate").value = "";
document.getElementById("PV_value").value = "";
document.getElementById("PMT_value").value = "";
document.getElementById("FV_value").value = "";
document.getElementById("annuity_end").checked = true;
document.getElementById("result").innerHTML = "";
}