How to Calculate Spot Rate from Ytm

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Spot Rate Bootstrapping Calculator

Enter rates for years 1 to (n-1). Example for a 3-year bond: "4.5, 4.8"

Calculation Result:

The Spot Rate for Year is:

function calculateSpotRate() { var price = parseFloat(document.getElementById('bondPrice').value); var couponRate = parseFloat(document.getElementById('couponRate').value); var face = parseFloat(document.getElementById('faceValue').value); var n = parseInt(document.getElementById('yearsToMaturity').value); var priorRatesStr = document.getElementById('priorSpotRates').value; var resultArea = document.getElementById('resultArea'); var finalSpotRateSpan = document.getElementById('finalSpotRate'); var yearDisplay = document.getElementById('yearDisplay'); if (isNaN(price) || isNaN(couponRate) || isNaN(face) || isNaN(n)) { alert("Please fill in all bond details correctly."); return; } var couponAmount = (couponRate / 100) * face; var sumPV = 0; if (n > 1) { var ratesArray = priorRatesStr.split(',').map(function(item) { return parseFloat(item.trim()); }); if (ratesArray.length < (n – 1)) { alert("You must provide " + (n – 1) + " prior spot rate(s) for a " + n + " year bond."); return; } for (var i = 1; i < n; i++) { var r = ratesArray[i – 1] / 100; sumPV += couponAmount / Math.pow(1 + r, i); } } var remainingValue = price – sumPV; if (remainingValue <= 0) { alert("Invalid inputs: Present value of coupons exceeds bond price."); return; } var spotRate = Math.pow((couponAmount + face) / remainingValue, 1 / n) – 1; var spotRatePercentage = (spotRate * 100).toFixed(4); yearDisplay.innerText = n; finalSpotRateSpan.innerText = spotRatePercentage + "%"; resultArea.style.display = "block"; }

How to Calculate Spot Rate from YTM

In fixed-income analysis, the Yield to Maturity (YTM) is the internal rate of return of a bond, assuming all coupons are reinvested at that same rate. However, in the real world, interest rates differ for various maturities. This is where Spot Rates come in.

A spot rate is the yield for a zero-coupon bond for a specific maturity. Calculating spot rates from coupon-bearing bonds (using their YTM or market price) is a process known as Bootstrapping.

The Bootstrapping Formula

To find the spot rate ($s_n$) for year $n$, we use the current market price of the bond and the previously determined spot rates for years $1$ through $n-1$. The price of a bond is the sum of its discounted cash flows:

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

Where:

  • C: Annual Coupon Payment
  • F: Face Value (Par)
  • sₙ: Spot rate for year $n$

Step-by-Step Example

Suppose you have a 2-year bond with a face value of 100, a coupon rate of 5%, and a market price of 100. You already know the 1-year spot rate is 4.5%.

  1. Identify Cash Flows: Year 1 = 5 (coupon), Year 2 = 105 (coupon + face).
  2. Discount Year 1: PV of Year 1 coupon = 5 / (1 + 0.045)¹ = 4.7847.
  3. Solve for Year 2: 100 = 4.7847 + [105 / (1 + s₂)²].
  4. Isolate s₂: 95.2153 = 105 / (1 + s₂)² → (1 + s₂)² = 105 / 95.2153 = 1.10276.
  5. Calculate: 1 + s₂ = √1.10276 = 1.05012 → s₂ = 5.012%.

Why Spot Rates Matter

While YTM gives a "blended" average of the bond's return, the spot rate curve (or zero-coupon curve) is the true foundation for pricing any financial instrument. It allows investors to identify arbitrage opportunities and accurately value individual cash flows that occur at different points in time.

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