function calculateRiskFreeRate() {
var resultBox = document.getElementById('resultBox');
var resultValue = document.getElementById('resultValue');
var errorMsg = document.getElementById('errorMessage');
var Ri = parseFloat(document.getElementById('expectedReturn').value);
var Beta = parseFloat(document.getElementById('assetBeta').value);
var Rm = parseFloat(document.getElementById('marketReturn').value);
if (isNaN(Ri) || isNaN(Beta) || isNaN(Rm)) {
errorMsg.style.display = 'block';
errorMsg.innerText = "Please ensure all fields are filled with valid numbers.";
resultBox.style.display = 'none';
return;
}
if (Beta === 1) {
errorMsg.style.display = 'block';
errorMsg.innerText = "Beta cannot be exactly 1 for this calculation (division by zero). If Beta is 1, the Expected Return equals Market Return, and Risk-Free Rate cancels out.";
resultBox.style.display = 'none';
return;
}
errorMsg.style.display = 'none';
// CAPM Formula: Ri = Rf + Beta * (Rm – Rf)
// Rearranging for Rf:
// Ri = Rf * (1 – Beta) + Beta * Rm
// Ri – (Beta * Rm) = Rf * (1 – Beta)
// Rf = (Ri – (Beta * Rm)) / (1 – Beta)
var numerator = Ri – (Beta * Rm);
var denominator = 1 – Beta;
var rf = numerator / denominator;
resultValue.innerText = rf.toFixed(2) + '%';
resultBox.style.display = 'block';
}
How to Calculate Risk-Free Rate with Beta
The Risk-Free Rate is a foundational component of modern finance, typically represented by the yield on government treasury bonds. However, in financial modeling and academic exercises, you may need to solve for the "implied" risk-free rate based on an asset's expected performance, its risk profile (Beta), and the overall market performance. This reverse calculation is derived from the Capital Asset Pricing Model (CAPM).
The Mathematical Formula
The standard CAPM formula solves for the Expected Return ($E(R_i)$):
E(R_i) = R_f + β × (E(R_m) – R_f)
To calculate the Risk-Free Rate ($R_f$) using Beta, we rearrange the algebraic equation:
R_f = [E(R_i) – (β × E(R_m))] / (1 – β)
Where:
E(R_i): The Expected Return of the asset (Cost of Equity).
β (Beta): A measure of the asset's volatility relative to the market.
E(R_m): The Expected Return of the Market (typically S&P 500 or similar index).
Calculation Example
Imagine you are analyzing a tech stock. Analysts project the stock will return 12% next year ($E(R_i)$). The stock has a high volatility with a Beta of 1.5. The expected market return is currently 10%.
To find the implied risk-free rate used in this projection:
Calculate the weighted market return: $1.5 \times 10\% = 15\%$
Subtract this from the asset return: $12\% – 15\% = -3\%$
Calculate the beta divisor: $1 – 1.5 = -0.5$
Divide the result: $-3\% / -0.5 = 6\%$
The implied Risk-Free Rate in this scenario is 6.00%.
Why Perform This Calculation?
While the risk-free rate is usually a known input (obtained by looking at current Treasury yields), calculating the implied rate helps in:
Validation: Checking if a financial model's assumptions are consistent with current economic reality.
Arbitrage Analysis: Identifying discrepancies between an asset's priced return and the theoretical return dictated by its risk.
Academic Study: Understanding the mechanical relationship between volatility (Beta) and the baseline cost of money.