Estimating the cost of equity with the CAPM and the market model

Why does a valuation need a cost of equity?

A present value model needs a discount rate. For cash flows that belong only to shareholders, such as free cash flow to equity (FCFE), that rate is the cost of equity , also called the required return on equity. For free cash flow to the firm (FCFF), the cost of equity is one part of the weighted average cost of capital (WACC). The two names describe one number: what investors require is what the company’s equity costs.

Cash flow being discounted	Discount rate
Free cash flow to equity (FCFE)	The cost of equity
Free cash flow to the firm (FCFF)	The weighted average cost of capital (WACC), which blends the cost of debt and the cost of equity

The discounted cash flow module (Equities, Learning Module 6) assumed a discount rate. This module shows where it comes from, starting with the equity half. The discount rates lesson shows how the rate is used once you have it.

What does the CAPM say?

The capital asset pricing model (CAPM) says a stock’s expected return is the risk-free rate plus the stock’s beta times the expected market risk premium.

$$E[r_i] = r_f + \beta_i\left(E[r_m] - r_f\right)$$

There are three inputs: the risk-free rate $r_f$, the expected market risk premium (MRP) $E[r_m] - r_f$ (what the market is expected to earn over the risk-free rate), and the stock’s beta $\beta_i$. Plotted against beta, the equation is a straight line called the security market line (SML) : intercept $r_f$, slope the MRP. Read as a required return, it is the cost of equity. Quantitative Methods, Learning Module 8, derives it; here we use it.

One date, one country. The risk-free rate and the MRP reflect the economy, so they are the same for every company in it. Beta is the only thing that makes one company’s CAPM cost of equity differ from another’s. Across countries, the risk-free rate and the MRP differ too.

A high-beta stock has a beta above 1 and swings more than the market. A low-beta stock has a beta below 1 and swings less.

How do you get beta from correlation and volatility?

Beta is the covariance of the stock with the market divided by the market’s variance. Rewriting the covariance as correlation times the two standard deviations leaves a form that is easy to use.

$$\beta_i = \dfrac{\sigma_{i,m}}{\sigma_m^2} = \dfrac{\rho_{i,m}\,\sigma_i\,\sigma_m}{\sigma_m^2} = \rho_{i,m}\,\dfrac{\sigma_i}{\sigma_m}$$

Beta has two parts: how closely the stock moves with the market ($\rho$) and how volatile it is relative to the market ($\sigma_i \div \sigma_m$). A single stock is usually more volatile than the market, because it carries company-specific risk, so the volatility ratio is usually above 1. A correlation well below 1 can therefore still give a beta near 1. Beta equals the correlation only in the special case where the stock and the market are equally volatile.

The exam also runs it backwards: given the beta and the two volatilities, find the correlation.

$$\rho_{i,m} = \beta_i\,\dfrac{\sigma_m}{\sigma_i}$$

Pick what the question asks for, enter what it gives, and the rearranged formula and keystrokes follow.

Go deeper Why beta is covariance over variance

The market’s variance is the market’s covariance with itself, so the market’s own beta is exactly 1. A stock’s beta measures how many percentage points the stock tends to move for each point the market moves. Rewriting covariance as ρσ_iσ_m gives the form above, where the ratio of standard deviations converts “how closely” into “how far”.

What moves the cost of equity?

Every input changes over time, so the cost of equity, and with it the share price, can move even when expected cash flows do not. Hold the other inputs fixed and take Kestrel (beta 1.12):

Change	Effect on beta	Effect on the cost of equity
Higher correlation with the market	Up (ρ is in the numerator)	Up
Higher stock volatility σ_i	Up	Up
Higher market volatility σ_m	Down (σ_m is in the denominator)	Down
Higher risk-free rate, E[r_m] unchanged	None	Down if β is above 1, up if β is below 1

Exam tip If a question raises the risk-free rate and keeps the expected market return fixed, the premium shrinks. For a stock with beta above 1, the smaller premium outweighs the higher starting rate.

Go deeper The algebra behind the risk-free rate result

Rewrite the CAPM as $E[r_i] = (1 - \beta_i)\,r_f + \beta_i\,E[r_m]$. With $E[r_m]$ fixed, each point added to $r_f$ changes the cost of equity by $(1 - \beta_i)$ points: positive for a low-beta stock, zero at a beta of 1, negative for a high-beta stock.

How is beta estimated in practice?

From a regression of the stock’s past returns on an index’s returns. Analysts usually take beta from a data provider rather than compute ρσ_i/σ_m themselves, and providers may apply their own adjustments. The regression mechanics (slope, R-squared, standard error) are in Quantitative Methods, Learning Module 10.

The excess-return regression follows the CAPM directly. A short-term risk-free rate is subtracted from both the stock’s and the market’s returns, usually weekly or monthly over one to five years, and the slope is the historical beta.

$$r_{i,t} - r_{f,t} = \alpha_i + \beta_i\left(r_{m,t} - r_{f,t}\right) + \varepsilon_t$$

The market model is the same idea on raw returns. The two give similar betas. The market model is easier because it needs no risk-free return data, which is why data providers commonly use it.

$$r_{i,t} = \alpha_i + \beta_i\,r_{m,t} + \varepsilon_{i,t}$$

Beta reflects both the industry and the company. Businesses with steady demand and costs, such as mature consumer staples, tend to have betas below 1. Businesses whose revenue swings with volatile commodity prices tend to have betas well above 1. Market views of these risks change, so betas change too.

Drag the correlation and the two volatilities, and watch the fitted slope, the beta and the cost of equity move together.

What is adjusted beta?

A regression beta is called the raw beta . The adjusted beta is a weighted average of the raw beta and 1.0. Betas tend to drift toward 1.0 over time (mean reversion of beta ): high betas fall and low betas rise. A cost of equity discounts future cash flows, so a beta pulled toward 1.0 can be the better forecast.

$$\text{Adjusted beta} = w_1 \times \text{Raw beta} + w_2 \times 1.0, \quad w_2 = 1 - w_1$$

A widely used data provider (Bloomberg) uses $w_1$ = two-thirds (0.67) and $w_2$ = one-third (0.33).

A question can also show the raw and adjusted betas and ask for the weight.

Which choices change a beta estimate?

Three, and the analyst must choose and disclose them.

• The market portfolio. In theory the market portfolio holds every risky asset, traded or not. In practice the market portfolio proxy is a liquid public equity index chosen to fit the stock. To compare companies from different countries, use one broad index common to both. Beware a narrow local index in which the company is itself a big weight: the stock partly is the index, so the correlation and the beta come out high. A broad global index gives a different (often lower) beta for such a company.
• Noise. There are two gauges. R-squared (R²) is the share of the stock’s variation explained by the market (in a one-factor regression the correlation is √R²). The standard error of beta measures how precisely the slope is estimated. Low R² and a large standard error mean a noisy, imprecise beta.
• Horizon and frequency. The horizon and return frequency set how many observations you have. About 100 weekly returns over two years is a common default. Fewer observations (for example 60 monthly returns over five years) usually raise the standard error.

Estimating betastate the method you used

• Market index Know the idea broad and liquid; avoid a narrow index the stock dominates
• Regression form
  • Excess returns Know the idea subtract a short-term risk-free rate from both series
  • Market model Memorize raw returns, no risk-free data needed
• Noise Know the idea R-squared (R²) and the standard error of beta
• Horizon and frequency Know the idea about 100 weekly returns over 2 years is a common default
• Adjustment Memorize w₁ × raw + (1 − w₁) × 1.0

Exam tip Beta is an estimate, not a fact. Exam answers that call any one beta “the true beta” are wrong. The defensible answer names the trade-off.

Go deeper Why a narrow index pushes beta up

If one company makes up a large share of the index, its returns drive the index’s returns, so the correlation between the two is high almost by construction. A higher correlation raises beta even though nothing about the company’s real exposure to the wider economy has changed. A broad index dilutes the company to a small weight and measures its exposure to the economy.