Deriving The CAPM Cost Of Equity

Why CAPM Adds a Risk Premium to the Risk-Free Return

An investor who supplies equity capital expects compensation for waiting and for bearing risk. In the Capital Asset Pricing Model (CAPM), the risk-free return provides the starting point. The additional return depends on how much an investment’s return moves with the market, not simply on how volatile that investment is by itself. This makes CAPM a model of the return required for systematic risk, the component of risk that cannot be eliminated by diversification.

For corporate finance, the CAPM estimate is commonly used as the cost of equity: the return shareholders require for bearing the risk of the company’s common equity. It is a model-based expected return, not a guarantee of the return that will actually be earned. This tutorial develops the equation and its inputs, aligned with the SOA VEE Accounting and Finance syllabus.

Beta measures an asset’s sensitivity to market movements, expressed relative to the market portfolio. A beta of 1 means that the asset has the market’s covariance-based exposure to market risk. A beta greater than 1 indicates greater systematic exposure than the market; a beta between 0 and 1 indicates less exposure. A negative beta indicates that the asset’s returns tend to move against the market, as measured by covariance. Beta is not the same as total volatility: an asset can have substantial asset-specific volatility without having correspondingly high beta.

Deriving the CAPM Required-Return Equation

The CAPM result follows from a particular equilibrium model. Investors consider a single investment period, agree on expected returns and covariances, can invest or borrow at the same risk-free rate, and face no taxes, transaction costs, or restrictions on buying and selling divisible assets. Investors choose portfolios using expected return and variance, and the market portfolio is the aggregate portfolio of risky assets. Under these assumptions, investors hold the same risky portfolio, with differences in their risk-free borrowing or lending.

To see the central relationship, consider an investor choosing exposures to risky assets while holding the remainder of wealth at the risk-free rate. Write \(x_i\) for the investor’s exposure to asset \(i\), and let \(R_p\) be the investor’s total portfolio return. A mean-variance investor with risk-aversion parameter \(A>0\) chooses exposures to maximize expected return minus a penalty for variance:

At an optimum, a small increase in exposure to asset \(i\) must not improve this objective. The change in expected return per unit of exposure is \(E[R_i]-r_f\); the change in portfolio variance is \(2\operatorname{Cov}(R_i,R_p)\). Setting the resulting first-order change in utility to zero gives:

In equilibrium, the investor’s risky holdings are proportional to the market portfolio. If the investor holds \(c\) units of the market portfolio’s risky exposure, then \(R_p\) consists of a risk-free component plus \(cR_M\). A risk-free component has zero covariance with a risky return, so \(\operatorname{Cov}(R_i,R_p)=c\operatorname{Cov}(R_i,R_M)\). Substitution gives \(E[R_i]-r_f=Ac\operatorname{Cov}(R_i,R_M)\). Applying the same relationship to the market portfolio itself yields \(E[R_M]-r_f=Ac\operatorname{Var}(R_M)\). Dividing the first equation by the second and rearranging produces the CAPM relationship:

The division assumes the market has positive variance and a nonzero expected excess return. In the model’s zero-market-risk-premium case, the first-order conditions instead imply zero expected excess return for the assets; that special case does not require division by the market risk premium. The general CAPM formula expresses the same economic idea: an asset’s expected return is the risk-free return plus compensation proportional to its systematic market exposure.

The beta definition also clarifies why the model focuses on covariance rather than total asset variance. An asset with returns that do not covary with the market has beta zero, even if its own returns are volatile. Within the CAPM, that asset receives no market-risk premium; its required return is \(r_f\). This conclusion depends on the model’s assumptions and is not a claim that every real-world risk is unimportant.

Applying the Equation

For calculation, keep the inputs on a consistent basis: the risk-free rate, market expected return, and beta must refer to compatible return periods and definitions. The market risk premium is a difference in returns, not the market expected return by itself. Once it is calculated, multiply that premium by beta and add the risk-free rate.

Interpretation, Limits, and a Common Trap

The CAPM equation is linear in beta. Holding \(r_f\) and the market risk premium fixed, an increase of 0.1 in beta changes the required return by \(0.1(E[R_M]-r_f)\). If the market risk premium is positive, a higher beta means a higher CAPM required return. A beta of 1 receives the market’s expected return; a beta of 0 receives the risk-free return; and a negative beta produces a return below the risk-free rate when the market premium is positive.

The equation depends on its model assumptions. Actual markets may have taxes, transaction costs, borrowing constraints, changing expectations, or risks that investors price in addition to market covariance. CAPM should therefore be understood as a structured estimate, not an identity that must describe every observed return. Its output is also an expected or required return, not a forecast that a particular realized return will occur.

A frequent calculation error is to use the expected market return where the market risk premium belongs. For example, with \(r_f=3.5\%\), \(E[R_M]=9.5\%\), and \(\beta_i=1.20\), the incorrect calculation is \(3.5\%+1.20(9.5\%)=14.9\%\). It applies beta to both the risk-free component and the market premium. The correction is to subtract the risk-free rate first: \(3.5\%+1.20(9.5\%-3.5\%)=10.7\%\). In CAPM, beta scales only the excess market return.