How to Calculate Spot Rate from Par Rate

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In fixed-income mathematics, deriving the spot rate curve from the par rate curve is a fundamental process known as bootstrapping. While par rates represent the yield-to-maturity of a bond trading at its face value, spot rates represent the yield on a zero-coupon bond for a specific maturity period.

Spot Rate Bootstrapping Calculator

Enter the Annual Par Rates (as percentages) to derive the corresponding Spot Rates.

Understanding the Par Rate vs. Spot Rate

The Par Rate is the coupon rate for which the price of a bond equals its nominal (par) value. Because these bonds pay periodic coupons, their yield is influenced by cash flows at multiple points in time. In contrast, the Spot Rate (or Zero Rate) is the yield for a single cash flow occurring at a specific future date.

Financial analysts use spot rates to discount specific future cash flows more accurately than using a single average yield-to-maturity. To move from par rates to spot rates, we use the bootstrapping method, assuming that the first year's spot rate is identical to the first year's par rate (assuming annual coupons).

The Bootstrapping Formula

The general formula for a bond price $P$ based on spot rates $s_t$ is:

Price = [C / (1 + s1)^1] + [C / (1 + s2)^2] + … + [(C + Par) / (1 + sn)^n]

To calculate the spot rate for year 2 ($s_2$) when you have the 2-year par rate ($P_2$):

100 = [P2 / (1 + s1)^1] + [(100 + P2) / (1 + s2)^2]

Solving for $s_2$ requires algebraic isolation. As you move to year 3, you use the previously solved $s_1$ and $s_2$ to find $s_3$, and so on.

Step-by-Step Calculation Example

Imagine the following Par Yield Curve:

• Year 1: 2.0%
• Year 2: 3.0%

Step 1: For Year 1, the spot rate is equal to the par rate.
Spot Rate (s1) = 2.0%

Step 2: Solve for Year 2 Spot Rate (s2) using the Year 2 Par Rate (3.0%).
Assume a Par Value of 100. The coupon is 3.0 (3.0% of 100).
100 = [3 / (1 + 0.02)^1] + [103 / (1 + s2)^2]
100 = 2.9412 + [103 / (1 + s2)^2]
97.0588 = 103 / (1 + s2)^2
(1 + s2)^2 = 103 / 97.0588 = 1.0612
1 + s2 = sqrt(1.0612) = 1.03015
s2 = 3.015%

Frequently Asked Questions

Why is the spot rate usually higher than the par rate?
In an upward-sloping yield curve, the spot rate will be higher than the par rate for the same maturity because the spot rate must "compensate" for the fact that all value is back-loaded to the final maturity date, whereas par bonds pay earlier coupons at lower historical rates.

What is the spot rate curve used for?
It is used to price arbitrary sequences of cash flows, such as mortgage-backed securities, corporate bonds with unusual structures, or for valuing swaps and derivatives.

function calculateSpotRates() { var p1 = parseFloat(document.getElementById('par1').value) / 100; var p2 = parseFloat(document.getElementById('par2').value) / 100; var p3 = parseFloat(document.getElementById('par3').value) / 100; var p4 = parseFloat(document.getElementById('par4').value) / 100; if (isNaN(p1) || isNaN(p2) || isNaN(p3) || isNaN(p4)) { alert("Please enter valid numbers for all par rates."); return; } // Spot Rate 1: s1 = p1 var s1 = p1; // Spot Rate 2: 100 = (100*p2)/(1+s1) + (100 + 100*p2)/(1+s2)^2 // var C2 = 100 * p2 var c2 = 100 * p2; var den2 = 100 – (c2 / (1 + s1)); var s2 = Math.sqrt((100 + c2) / den2) – 1; // Spot Rate 3: 100 = (100*p3)/(1+s1) + (100*p3)/(1+s2)^2 + (100 + 100*p3)/(1+s3)^3 var c3 = 100 * p3; var den3 = 100 – (c3 / (1 + s1)) – (c3 / Math.pow(1 + s2, 2)); var s3 = Math.pow((100 + c3) / den3, 1/3) – 1; // Spot Rate 4: 100 = (100*p4)/(1+s1) + (100*p4)/(1+s2)^2 + (100*p4)/(1+s3)^3 + (100 + 100*p4)/(1+s4)^4 var c4 = 100 * p4; var den4 = 100 – (c4 / (1 + s1)) – (c4 / Math.pow(1 + s2, 2)) – (c4 / Math.pow(1 + s3, 3)); var s4 = Math.pow((100 + c4) / den4, 1/4) – 1; // Populate Results var rates = [ { year: 1, par: (p1 * 100).toFixed(3), spot: (s1 * 100).toFixed(3) }, { year: 2, par: (p2 * 100).toFixed(3), spot: (s2 * 100).toFixed(3) }, { year: 3, par: (p3 * 100).toFixed(3), spot: (s3 * 100).toFixed(3) }, { year: 4, par: (p4 * 100).toFixed(3), spot: (s4 * 100).toFixed(3) } ]; var resultHtml = ""; for (var i = 0; i < rates.length; i++) { resultHtml += "" + rates[i].year + "" + rates[i].par + "%" + rates[i].spot + "%"; } document.getElementById('resultBody').innerHTML = resultHtml; document.getElementById('resultsBox').style.display = 'block'; }