.fr-calculator-container {
max-width: 600px;
margin: 20px auto;
padding: 25px;
background-color: #f9f9f9;
border: 1px solid #e0e0e0;
border-radius: 8px;
font-family: Arial, sans-serif;
box-shadow: 0 2px 5px rgba(0,0,0,0.05);
}
.fr-calculator-container h2 {
text-align: center;
color: #333;
margin-bottom: 20px;
}
.fr-input-group {
margin-bottom: 15px;
}
.fr-input-group label {
display: block;
margin-bottom: 5px;
font-weight: bold;
color: #555;
}
.fr-input-group input {
width: 100%;
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
box-sizing: border-box;
font-size: 16px;
}
.fr-btn {
display: block;
width: 100%;
padding: 12px;
background-color: #2c3e50;
color: #fff;
border: none;
border-radius: 4px;
font-size: 18px;
cursor: pointer;
transition: background-color 0.3s;
}
.fr-btn:hover {
background-color: #34495e;
}
.fr-result {
margin-top: 20px;
padding: 15px;
background-color: #fff;
border: 1px solid #ddd;
border-radius: 4px;
text-align: center;
font-size: 18px;
font-weight: bold;
color: #27ae60;
min-height: 24px;
}
.fr-article {
max-width: 800px;
margin: 40px auto;
font-family: Arial, sans-serif;
line-height: 1.6;
color: #333;
}
.fr-article h2 {
color: #2c3e50;
margin-top: 30px;
border-bottom: 2px solid #eee;
padding-bottom: 10px;
}
.fr-article h3 {
color: #2c3e50;
margin-top: 25px;
}
.fr-article p {
margin-bottom: 15px;
}
.fr-article ul {
margin-bottom: 15px;
padding-left: 20px;
}
.fr-article li {
margin-bottom: 8px;
}
.fr-formula-box {
background-color: #f4f4f4;
padding: 15px;
border-left: 4px solid #2c3e50;
font-family: 'Courier New', monospace;
margin: 20px 0;
overflow-x: auto;
}
function calculateForwardRate() {
var r1Input = document.getElementById('spotRate1').value;
var t1Input = document.getElementById('time1').value;
var r2Input = document.getElementById('spotRate2').value;
var t2Input = document.getElementById('time2').value;
var resultDiv = document.getElementById('frResult');
// Validation
if (r1Input === "" || t1Input === "" || r2Input === "" || t2Input === "") {
resultDiv.style.color = "#c0392b";
resultDiv.innerHTML = "Please fill in all fields.";
return;
}
var r1 = parseFloat(r1Input);
var t1 = parseFloat(t1Input);
var r2 = parseFloat(r2Input);
var t2 = parseFloat(t2Input);
if (isNaN(r1) || isNaN(t1) || isNaN(r2) || isNaN(t2)) {
resultDiv.style.color = "#c0392b";
resultDiv.innerHTML = "Please enter valid numeric values.";
return;
}
if (t2 <= t1) {
resultDiv.style.color = "#c0392b";
resultDiv.innerHTML = "Longer period (Duration 2) must be greater than Shorter period (Duration 1).";
return;
}
// Calculation Logic
// Formula: f = [ (1 + r2)^t2 / (1 + r1)^t1 ] ^ (1 / (t2 – t1)) – 1
var decimalR1 = r1 / 100;
var decimalR2 = r2 / 100;
var numerator = Math.pow((1 + decimalR2), t2);
var denominator = Math.pow((1 + decimalR1), t1);
var timeDiff = t2 – t1;
var forwardRateDecimal = Math.pow((numerator / denominator), (1 / timeDiff)) – 1;
var forwardRatePercent = forwardRateDecimal * 100;
// Display Result
resultDiv.style.color = "#27ae60";
resultDiv.innerHTML = "Projected Forward Rate: " + forwardRatePercent.toFixed(4) + "%";
}
How to Calculate Forward Rates
In the world of finance and fixed-income investing, understanding how to calculate forward rates is essential for predicting future interest rate movements and identifying arbitrage opportunities. A forward rate represents the interest rate applicable to a financial transaction that will take place in the future, implied by current spot rates.
What is a Forward Rate?
A forward rate is an interest rate agreed upon today for a loan or investment that will commence at a specific future date. While a "spot rate" tells you the yield for a bond starting immediately, a forward rate tells you what the market expects interest rates to be between two future points in time.
For example, if you know the interest rate for a 1-year bond and a 2-year bond, you can calculate the "implied" rate for a 1-year bond that starts one year from now. This is the forward rate.
The Forward Rate Formula
The calculation relies on the "no-arbitrage" principle, which assumes that investing in a long-term bond should yield the same return as investing in a short-term bond and then reinvesting the proceeds at the forward rate.
Forward Rate (f) = [ (1 + r2)t2 / (1 + r1)t1 ] 1 / (t2 – t1) – 1
Where:
- r1 = Spot rate for the shorter time period.
- t1 = Duration of the shorter time period (in years).
- r2 = Spot rate for the longer time period.
- t2 = Duration of the longer time period (in years).
Step-by-Step Calculation Example
Let's calculate the 1-year forward rate starting 1 year from now. This is often denoted as the "1y1y" forward rate.
- 1-Year Spot Rate (r1): 3.00%
- 2-Year Spot Rate (r2): 4.50%
- Time 1 (t1): 1 Year
- Time 2 (t2): 2 Years
1. Convert Percentages to Decimals
r1 = 0.03
r2 = 0.045
2. Apply the Formula
We want to find the rate that bridges the gap between year 1 and year 2.
Numerator = (1 + 0.045)2 = 1.0452 = 1.092025
Denominator = (1 + 0.03)1 = 1.03
Ratio = 1.092025 / 1.03 = 1.060218
Exponent = 1 / (2 – 1) = 1
Forward Rate = 1.0602181 – 1 = 0.060218
3. Convert Back to Percentage
0.060218 * 100 = 6.0218%
This means the market implies that the interest rate for a 1-year loan starting next year will be approximately 6.02%.
Why are Forward Rates Important?
Investors and economists use forward rates for several critical functions:
- Predicting Market Expectations: They reveal the market's consensus on the future direction of interest rates.
- Hedging: Companies use forward rate agreements (FRAs) to lock in future borrowing costs and protect against rising rates.
- Bond Valuation: They help in determining if a bond is overvalued or undervalued relative to the yield curve.
Common Pitfalls
When learning how to calculate forward rates, ensure that your time periods ($t_1$ and $t_2$) are in the same unit (usually years). Also, remember that this calculation assumes annual compounding. If you are dealing with semi-annual bond yields (like US Treasuries), the formula requires adjustment to reflect semi-annual compounding periods.