How to Calculate Par Rate

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Par Yield Calculator

Calculate the theoretical Par Rate from Zero-Coupon Spot Rates.

Enter the annual Zero-Coupon Spot Rates (%) for consecutive years to calculate the Par Rate for the final maturity year.

Calculated Par Rate

0.00%

How to Calculate Par Rate

The Par Rate (or Par Yield) is the coupon rate at which a bond's price equals its nominal face value (par value). In financial markets, specifically within fixed income analytics, calculating the par rate is essential for constructing the par yield curve from the theoretical zero-coupon spot rate curve.

This calculation determines what the coupon payment should be for a bond of a specific maturity so that it trades at exactly 100% of its face value today, assuming the current term structure of interest rates.

The Par Rate Formula

To calculate the Par Rate for a maturity of n years, we use the Discount Factors derived from the Spot Rates. The formula relates the sum of the present value of future coupon payments and the principal repayment to the current par price.

The mathematical formula is:

Par Rate = (1 – DFn) / Σ(DF1…DFn)

Where:

  • DFn is the Discount Factor for year n.
  • Σ(DF) is the sum of all Discount Factors from year 1 to year n.
  • Discount Factor (DFt) = 1 / (1 + rt)t, where rt is the spot rate for year t.

Example Calculation

Suppose you have the following Zero-Coupon Spot Rates:

  • Year 1: 3.0%
  • Year 2: 4.0%

Step 1: Calculate Discount Factors
DF1 = 1 / (1 + 0.03)1 = 0.9709
DF2 = 1 / (1 + 0.04)2 = 0.9246

Step 2: Apply Par Formula
Numerator = 1 – 0.9246 = 0.0754
Denominator = 0.9709 + 0.9246 = 1.8955
Par Rate = 0.0754 / 1.8955 = 0.03977 or 3.98%

Why is the Par Rate Important?

The Par Rate is crucial for valuing Interest Rate Swaps and determining the nominal yield of Treasury bonds. When banks or financial institutions quote a "yield curve," they are often referring to the Par Yield Curve. It represents the rate a high-quality issuer would pay on new bonds issued at par value today.

function calculateParRate() { // Retrieve inputs var s1 = document.getElementById('spotRate1').value; var s2 = document.getElementById('spotRate2').value; var s3 = document.getElementById('spotRate3').value; var s4 = document.getElementById('spotRate4').value; var s5 = document.getElementById('spotRate5').value; // Create an array of rates, filtering out empty ones to determine maturity var rates = []; if (s1 !== "") rates.push({ year: 1, rate: parseFloat(s1) }); if (s2 !== "") rates.push({ year: 2, rate: parseFloat(s2) }); if (s3 !== "") rates.push({ year: 3, rate: parseFloat(s3) }); if (s4 !== "") rates.push({ year: 4, rate: parseFloat(s4) }); if (s5 !== "") rates.push({ year: 5, rate: parseFloat(s5) }); // Validation if (rates.length === 0) { alert("Please enter at least the Year 1 Spot Rate."); return; } // Ensure consecutive years are entered (simple check) for (var i = 0; i < rates.length; i++) { if (rates[i].year !== i + 1) { alert("Please enter Spot Rates consecutively starting from Year 1 (do not skip years)."); return; } if (isNaN(rates[i].rate)) { alert("Please ensure all entered rates are valid numbers."); return; } } // Calculation Logic // Formula: C = (1 – df_n) / sum(df_1 to df_n) var discountFactorSum = 0; var finalDiscountFactor = 0; var discountFactors = []; for (var i = 0; i < rates.length; i++) { var r = rates[i].rate / 100; // Convert percentage to decimal var t = rates[i].year; // Calculate Discount Factor for this period: 1 / (1+r)^t var df = 1 / Math.pow((1 + r), t); discountFactors.push(df); discountFactorSum += df; // If this is the last year in our input, store it as the final DF if (i === rates.length – 1) { finalDiscountFactor = df; } } // Calculate Par Rate (Coupon) // Numerator: 1 – Discount Factor of the final maturity var numerator = 1 – finalDiscountFactor; // Denominator: Sum of all discount factors (annuity factor) var denominator = discountFactorSum; var parRateDecimal = numerator / denominator; var parRatePercent = parRateDecimal * 100; // Display Result var resultDiv = document.getElementById('resultContainer'); var resultValue = document.getElementById('parRateResult'); var summary = document.getElementById('calculationSummary'); resultDiv.style.display = 'block'; resultValue.innerHTML = parRatePercent.toFixed(3) + "%"; summary.innerHTML = "Calculated based on a " + rates.length + "-year maturity duration using the provided Spot Rate curve."; }

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