Risk-Free Rate (CAPM) Calculator
Calculated Risk-Free Rate:
Understanding the Risk-Free Rate in CAPM
The Capital Asset Pricing Model (CAPM) is a cornerstone of modern finance used to determine the theoretically required rate of return of an asset. While usually used to find the "Expected Return," we can algebraically rearrange the formula to isolate and solve for the Risk-Free Rate (Rf).
The Derivation Formula
The standard CAPM formula is:
E(Ri) = Rf + βi * [E(Rm) – Rf]
To solve for the Risk-Free Rate (Rf), we rearrange the equation as follows:
- Expand: E(Ri) = Rf + βi*E(Rm) – βi*Rf
- Factor Rf: E(Ri) = Rf(1 – βi) + βi*E(Rm)
- Isolate: Rf(1 – βi) = E(Ri) – βi*E(Rm)
- Final Formula: Rf = [E(Ri) – (βi * E(Rm))] / (1 – βi)
Key Definitions
- Expected Return of Asset (E(Ri)): The total profit anticipated from an investment, expressed as a percentage.
- Expected Return of Market (E(Rm)): The historical or projected return of a broad market index, such as the S&P 500.
- Beta (β): A measure of an asset's volatility in relation to the overall market. A beta of 1 moves exactly with the market.
- Risk-Free Rate (Rf): The theoretical return of an investment with zero risk, often represented by Government Treasury Bills.
Practical Example
Suppose you have an asset with an expected return of 9% and a Beta of 0.8. The general market is expected to return 11%. What is the implied risk-free rate?
Using the formula:
Rf = [9 – (0.8 * 11)] / (1 – 0.8)
Rf = [9 – 8.8] / 0.2
Rf = 0.2 / 0.2 = 1%
Important Note: This calculator encounters a mathematical singularity when Beta is exactly 1.0. In a perfect CAPM world, if β = 1, the asset return must equal the market return, and the risk-free rate becomes indeterminate through this specific rearrangement.