A semi-annual rate refers to an interest or yield calculation that compounds twice per year. This is the standard convention for the majority of bonds (including U.S. Treasuries) and is also used in specific jurisdictions, like Canada, for calculating fixed mortgage rates. The key distinction in this calculation is the difference between the Nominal Rate (the quoted annual percentage) and the Effective Annual Rate (EAR), which accounts for the intra-year compounding.
Calculation Logic
This calculator performs conversions based on the time value of money mechanics for semi-annual frequency ($m=2$). The logic follows these steps:
Periodic Rate Calculation: The annual nominal rate is divided by 2 to determine the rate applied every 6 months ($r_{periodic} = r_{nominal} / 2$).
Effective Annual Yield: Since the return from the first 6 months earns its own return in the second 6 months, the effective yield is higher than the nominal rate. Formula: $EAR = (1 + r_{periodic})^2 – 1$.
Total Periods: The duration in years is multiplied by 2 to get the total number of compounding events.
Real-World Example
If you purchase a bond with a 6% Nominal Annual Yield paid semi-annually:
Every 6 months, the periodic rate applied is 3% (6% / 2).
If you reinvest the coupon, the Effective Annual Rate becomes 6.09% ($1.03^2 – 1$).
On a $10,000 investment over 1 year, you would earn $609 instead of the simple $600 implied by the nominal rate.
This "yield pick-up" is critical for fixed-income investors comparing semi-annual bonds against annual-paying Certificates of Deposit (CDs) or monthly-compounding savings accounts.
function calculateSemiAnnualMath() {
var nominalInput = document.getElementById('nominalYieldInput').value;
var principalInput = document.getElementById('investmentPrincipal').value;
var yearsInput = document.getElementById('holdingPeriod').value;
var errorBox = document.getElementById('sarErrorDisplay');
var resultsBox = document.getElementById('sarResultsBox');
// Parse values
var nominalRate = parseFloat(nominalInput);
var principal = parseFloat(principalInput);
var years = parseFloat(yearsInput);
// Validation
if (isNaN(nominalRate) || isNaN(principal) || isNaN(years) || nominalRate < 0 || principal < 0 || years <= 0) {
errorBox.style.display = 'block';
resultsBox.style.display = 'none';
return;
}
errorBox.style.display = 'none';
// 1. Calculate Periodic Rate (Semi-Annual)
// r_periodic = r_nominal / 2
var periodicRateDecimal = (nominalRate / 100) / 2;
// 2. Calculate Effective Annual Rate (EAR)
// EAR = (1 + r_periodic)^2 – 1
var earDecimal = Math.pow((1 + periodicRateDecimal), 2) – 1;
// 3. Calculate Totals
var totalPeriods = years * 2;
// Future Value = P * (1 + r_periodic)^(2*t)
var futureValue = principal * Math.pow((1 + periodicRateDecimal), totalPeriods);
var totalReturn = futureValue – principal;
// Display Results
// Format percentages
document.getElementById('valPeriodicRate').innerHTML = (periodicRateDecimal * 100).toFixed(4) + '%';
document.getElementById('valEAR').innerHTML = (earDecimal * 100).toFixed(4) + '%';
document.getElementById('valTotalPeriods').innerHTML = totalPeriods.toFixed(1);
// Format currency
var currencyFormatter = new Intl.NumberFormat('en-US', {
style: 'currency',
currency: 'USD',
minimumFractionDigits: 2,
maximumFractionDigits: 2
});
document.getElementById('valFutureValue').innerHTML = currencyFormatter.format(futureValue);
document.getElementById('valTotalReturn').innerHTML = currencyFormatter.format(totalReturn);
resultsBox.style.display = 'block';
}