function calculateRiskFreeRate() {
var raInput = document.getElementById("assetReturn").value;
var rmInput = document.getElementById("marketReturn").value;
var betaInput = document.getElementById("betaValue").value;
var resultDiv = document.getElementById("resultContainer");
var outputDiv = document.getElementById("rfOutput");
var explainDiv = document.getElementById("rfExplanation");
// Input Validation
if (raInput === "" || rmInput === "" || betaInput === "") {
resultDiv.style.display = "block";
outputDiv.innerHTML = "Invalid Input";
explainDiv.innerHTML = "Please enter values for all fields.";
return;
}
var ra = parseFloat(raInput);
var rm = parseFloat(rmInput);
var beta = parseFloat(betaInput);
if (isNaN(ra) || isNaN(rm) || isNaN(beta)) {
resultDiv.style.display = "block";
outputDiv.innerHTML = "Error";
explainDiv.innerHTML = "Please ensure all inputs are valid numbers.";
return;
}
// Singularity Check: Formula divides by (1 – Beta)
if (beta === 1) {
resultDiv.style.display = "block";
outputDiv.innerHTML = "Math Error";
explainDiv.innerHTML = "Calculation is undefined when Beta is exactly 1 (division by zero). Theoretically, if Beta is 1, Asset Return should equal Market Return.";
return;
}
// Logic: CAPM is E(Ra) = Rf + Beta * (E(Rm) – Rf)
// Rearranging for Rf:
// E(Ra) = Rf + Beta*E(Rm) – Beta*Rf
// E(Ra) – Beta*E(Rm) = Rf * (1 – Beta)
// Rf = (E(Ra) – Beta * E(Rm)) / (1 – Beta)
var numerator = ra – (beta * rm);
var denominator = 1 – beta;
var rf = numerator / denominator;
// Display Result
resultDiv.style.display = "block";
outputDiv.innerHTML = rf.toFixed(2) + "%";
// Contextual Explanation
if (rf < 0) {
explainDiv.innerHTML = "The result is negative. This might occur if a high-beta asset has a low return, or a low-beta asset has an excessively high return relative to the market.";
} else {
explainDiv.innerHTML = "Based on the provided Beta and Expected Returns, this is the implied Risk-Free Rate required to satisfy the CAPM equation.";
}
}
How to Calculate Risk-Free Rate With Beta
In financial modeling, the Risk-Free Rate is typically a known variable derived from government bonds (like the 10-year US Treasury yield). However, there are scenarios where you might want to reverse-engineer the Capital Asset Pricing Model (CAPM) to find the implied risk-free rate based on an asset's expected performance, its volatility (Beta), and the overall market return.
The Reverse CAPM Formula
To calculate the risk-free rate using Beta, we rearrange the standard CAPM equation. The standard equation calculates the Expected Return of an Asset ($E(R_a)$) as follows:
E(Ra) = Rf + β × (E(Rm) – Rf)
Where:
Rf = Risk-Free Rate
β = Beta of the asset
E(Rm) = Expected Return of the Market
To solve for the Risk-Free Rate ($R_f$), we perform the following algebraic steps:
In this scenario, the implied risk-free rate is 6%.
Why Calculate the Implied Risk-Free Rate?
Calculating the risk-free rate from Beta is often done for academic verification or arbitrage analysis. If the implied risk-free rate derived from a specific stock's pricing is significantly different from the actual yield on government bonds, it may indicate that:
The asset is mispriced (overvalued or undervalued).
The estimated Beta value is incorrect.
The market return expectations are not aligned with reality.
Understanding Beta Sensitivity
It is important to note that this calculation is highly sensitive to the value of Beta. As Beta approaches 1.0, the denominator $(1 – \beta)$ approaches zero, making the calculation extremely volatile. If Beta is exactly 1.0, the formula is mathematically undefined because, in CAPM theory, an asset with a Beta of 1.0 should have returns exactly equal to the market return, effectively cancelling out the risk-free rate term.