*Calculation assumes annual compounding and bootstrapping from a 1-year zero-coupon equivalent.
function calculateSpotRates() {
// 1. Get Input Values
var faceValue = parseFloat(document.getElementById('faceValue').value);
var couponRatePct = parseFloat(document.getElementById('couponRate').value);
var ytm1Pct = parseFloat(document.getElementById('ytm1').value);
var ytm2Pct = parseFloat(document.getElementById('ytm2').value);
// 2. Validation
if (isNaN(faceValue) || isNaN(couponRatePct) || isNaN(ytm1Pct) || isNaN(ytm2Pct)) {
alert("Please enter valid numeric values for all fields.");
return;
}
if (faceValue <= 0) {
alert("Face Value must be positive.");
return;
}
// 3. Convert Percentages to Decimals
var couponRate = couponRatePct / 100;
var ytm1 = ytm1Pct / 100;
var ytm2 = ytm2Pct / 100;
// 4. Calculate Cash Flows
var couponPayment = faceValue * couponRate;
// 5. Logic for Year 1 Spot Rate (S1)
// For a 1-period bond, Spot Rate = YTM
var s1 = ytm1;
// 6. Logic for Year 2 Spot Rate (S2) – Bootstrapping
// Step A: Calculate the Market Price of Bond 2 using its YTM
// Price = C / (1+YTM)^1 + (C+F) / (1+YTM)^2
var bond2Price = (couponPayment / Math.pow(1 + ytm2, 1)) + ((couponPayment + faceValue) / Math.pow(1 + ytm2, 2));
// Step B: Use S1 to find the Present Value of the first coupon
var pvCoupon1 = couponPayment / Math.pow(1 + s1, 1);
// Step C: Isolate the Year 2 Cash Flow PV
// Price = PV_Coupon1_using_S1 + PV_Final_using_S2
// PV_Final_using_S2 = Price – PV_Coupon1_using_S1
var pvFinalCashFlow = bond2Price – pvCoupon1;
// Step D: Solve for S2
// (C + F) / (1 + S2)^2 = pvFinalCashFlow
// (1 + S2)^2 = (C + F) / pvFinalCashFlow
// 1 + S2 = Sqrt((C + F) / pvFinalCashFlow)
if (pvFinalCashFlow <= 0) {
alert("Mathematical error: Implied price is invalid for spot rate calculation (Arbitrage violation). Check your inputs.");
return;
}
var s2 = Math.sqrt((couponPayment + faceValue) / pvFinalCashFlow) – 1;
// 7. Format Results
var s1Display = (s1 * 100).toFixed(4) + "%";
var s2Display = (s2 * 100).toFixed(4) + "%";
var priceDisplay = "$" + bond2Price.toFixed(2);
// 8. Display
document.getElementById('spot1Result').innerHTML = s1Display;
document.getElementById('price2Result').innerHTML = priceDisplay;
document.getElementById('spot2Result').innerHTML = s2Display;
document.getElementById('resultBox').style.display = "block";
}
How to Calculate Spot Rate from Yield to Maturity
Understanding the relationship between Yield to Maturity (YTM) and Spot Rates is fundamental to fixed-income arbitrage, bond pricing, and constructing the zero-coupon yield curve. While YTM assumes a flat yield curve for a single bond (discounting all cash flows at the same rate), spot rates represent the true interest rate for a specific cash flow at a specific moment in time.
This calculator utilizes the bootstrapping method to derive the theoretical Year 2 spot rate, given the YTMs of 1-year and 2-year bonds.
What is the Difference Between YTM and Spot Rate?
The Yield to Maturity (YTM) is an internal rate of return (IRR). It is a single average rate that sets the present value of a bond's future cash flows equal to its current market price. It assumes that you can reinvest all coupon payments at that same YTM rate.
The Spot Rate (or Zero-Coupon Rate) is the yield on a pure discount bond with no intermediate coupon payments. The spot rate curve (Term Structure of Interest Rates) allows for more accurate pricing because it discounts each specific cash flow (coupon or principal) at the rate specific to its timing.
The Bootstrapping Formula
Bootstrapping is a recursive process. We start with the shortest maturity liquid instrument (usually a T-bill) where the YTM equals the Spot Rate, and work forward in time.
General Bond Price Formula using Spot Rates:
Price = Σ [ CashFlowt / (1 + St)t ]
Step-by-Step Calculation Logic
To calculate the 2-year Spot Rate ($S_2$) from YTM, follow these steps:
Determine S1: For a 1-year bond, the Spot Rate ($S_1$) is equal to its YTM.
Calculate Price of Bond 2: Use the known Coupon Rate and YTM of the 2-year bond to calculate its market price.
Price = C / (1+YTM) + (C+Face) / (1+YTM)²
Discount First Coupon: Calculate the Present Value (PV) of the first year's coupon using $S_1$.
PV1 = C / (1 + S1)
Solve for S2: Subtract the PV of the first coupon from the total Price. The remaining value is the PV of the final cash flow discounted at $S_2$.
(C + Face) / (1 + S2)² = Price – PV1
Example Calculation
Let's calculate the spot rate manually using the following parameters:
Face Value: $1,000
Coupon Rate: 5% ($50 per year)
Year 1 YTM: 4.0%
Year 2 YTM: 5.0%
1. Find Spot Rate 1 (S₁)
Since the Year 1 bond matures in 1 period, $S_1 = 4.0\%$.
2. Find Price of Bond 2
We discount the cash flows of the 2-year bond using its YTM (5%):
Notice that the 2-year Spot Rate (5.025%) is slightly higher than the 2-year YTM (5.00%). This is typical for an upward-sloping yield curve; the YTM "averages down" the higher spot rate of the final principal payment with the lower spot rate of the earlier coupon.
Why is this important?
Investors use spot rates to identify arbitrage opportunities. If a bond's market price is lower than the sum of its cash flows discounted at the spot rates derived from other bonds, that bond is theoretically undervalued.