Calculate Spot Rate from Par Rate

Spot Rate Calculator from Par Rate

Understanding the relationship between par rates and spot rates is fundamental in fixed-income securities. A par rate is the coupon rate that makes a bond trade at its par value (face value). A spot rate, on the other hand, is the yield on a zero-coupon bond for a specific maturity. The spot rate curve (or zero-coupon yield curve) is a critical tool for pricing bonds and derivatives, as it reflects the market's expectations of future interest rates.

While bond coupons are paid periodically, a true discount bond (like a zero-coupon bond) pays its entire value at maturity. Therefore, the spot rate directly reflects the time value of money for a single cash flow at a specific future date. Calculating spot rates from observable par rates requires using a bootstrapping method, which iteratively determines each spot rate based on previously calculated shorter-term spot rates and the observed par yields of coupon-paying bonds.

This calculator simplifies the process by allowing you to input the par yields of coupon-paying bonds with different maturities. It will then derive the corresponding spot rates using the bootstrapping technique. This is particularly useful for financial analysts, portfolio managers, and investors looking to build a yield curve or value bonds more accurately.

function calculateSpotRates() { var maturities = []; var parYields = []; var spotRates = []; var resultHTML = ""; var m1 = parseFloat(document.getElementById("maturity1").value); var py1 = parseFloat(document.getElementById("parYield1").value) / 100; var m2 = parseFloat(document.getElementById("maturity2").value); var py2 = parseFloat(document.getElementById("parYield2").value) / 100; var m3 = parseFloat(document.getElementById("maturity3").value); var py3 = parseFloat(document.getElementById("parYield3").value) / 100; if (isNaN(m1) || isNaN(py1) || isNaN(m2) || isNaN(py2) || isNaN(m3) || isNaN(py3)) { document.getElementById("spotRateResult").innerHTML = "Please enter valid numbers for all inputs."; return; } if (m1 <= 0 || m2 <= 0 || m3 <= 0 || py1 < 0 || py2 < 0 || py3 = m2 || m2 >= m3) { document.getElementById("spotRateResult").innerHTML = "Maturities must be in increasing order."; return; } maturities.push(m1, m2, m3); parYields.push(py1, py2, py3); // Bootstrapping logic // Spot rate for maturity 1 var s1 = py1; spotRates.push(s1); // Spot rate for maturity 2 // (1 + s2 * m2) = (1 + py2 * m2) / (1 + s1 * m1) // 1 + s2 * m2 = (1 + py2 * m2) / (1 + s1 * m1) // s2 * m2 = (1 + py2 * m2) / (1 + s1 * m1) – 1 // s2 = [(1 + py2 * m2) / (1 + s1 * m1) – 1] / m2 if (1 + s1 * m1 === 0) { // Avoid division by zero document.getElementById("spotRateResult").innerHTML = "Calculation error for maturity 2. Check input values."; return; } var s2 = ((1 + py2 * m2) / (1 + s1 * m1) – 1) / m2; spotRates.push(s2); // Spot rate for maturity 3 // (1 + s3 * m3) = (1 + py3 * m3) / [(1 + s1 * m1) * (1 + s2 * (m2-m1))] 1, this calculation becomes iterative or requires more info. // Let's stick to the assumption that the first maturity implies the first spot rate. // For a generic case: (1 + s_m1 * m1) = 1 / (1 – py1 * m1) — This isn't right. // We'll assume the first maturity is 1 for simplicity of bootstrapping: if (m1 !== 1) { document.getElementById("spotRateResult").innerHTML = "For accurate bootstrapping, the first maturity (Maturity 1) should be 1 year."; return; } spotRates.push(py1); } // Maturity 2 var C2 = py2; // Annual coupon for year 2 bond var PV_CF1_M2 = C2 / Math.pow(1 + spotRates[0], maturities[0]); // Present value of coupon at time m1 var remaining_value_at_m2 = 1 – PV_CF1_M2; // Remaining value to be discounted at spot rate s2 var discount_factor_for_s2 = Math.pow(1 + s2, maturities[1]); // Placeholder, needs to be solved // Equation for s2: 1 = C2/(1+s1)^m1 + (C2+1)/(1+s2)^m2 // Rearranging: (1+s2)^m2 = (C2+1) / (1 – C2/(1+s1)^m1) var denominator_s2 = 1 – C2 / Math.pow(1 + spotRates[0], maturities[0]); if (denominator_s2 <= 0) { document.getElementById("spotRateResult").innerHTML = "Calculation error for Maturity 2. Denominator is zero or negative. Check inputs."; return; } var factor_s2 = Math.pow((C2 + 1) / denominator_s2, 1 / maturities[1]); var s2_calc = factor_s2 – 1; spotRates.push(s2_calc); // Maturity 3 var C3 = py3; // Annual coupon for year 3 bond var PV_CF1_M3 = C3 / Math.pow(1 + spotRates[0], maturities[0]); var PV_CF2_M3 = C3 / Math.pow(1 + spotRates[1], maturities[1]); var remaining_value_at_m3 = 1 – PV_CF1_M3 – PV_CF2_M3; // Equation for s3: 1 = C3/(1+s1)^m1 + C3/(1+s2)^m2 + (C3+1)/(1+s3)^m3 // Rearranging: (1+s3)^m3 = (C3+1) / (1 – C3/(1+s1)^m1 – C3/(1+s2)^m2) var denominator_s3 = 1 – C3 / Math.pow(1 + spotRates[0], maturities[0]) – C3 / Math.pow(1 + spotRates[1], maturities[1]); if (denominator_s3 <= 0) { document.getElementById("spotRateResult").innerHTML = "Calculation error for Maturity 3. Denominator is zero or negative. Check inputs."; return; } var factor_s3 = Math.pow((C3 + 1) / denominator_s3, 1 / maturities[2]); var s3_calc = factor_s3 – 1; spotRates.push(s3_calc); resultHTML += "

Calculated Spot Rates:

"; resultHTML += "
    "; resultHTML += "
  • Maturity " + maturities[0] + " Years: " + (spotRates[0] * 100).toFixed(4) + "%
  • "; resultHTML += "
  • Maturity " + maturities[1] + " Years: " + (spotRates[1] * 100).toFixed(4) + "%
  • "; resultHTML += "
  • Maturity " + maturities[2] + " Years: " + (spotRates[2] * 100).toFixed(4) + "%
  • "; resultHTML += "
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