EI 12 The Capital Asset Pricing Model, Market Model, and Other Factor-Based Equity Models

Valuing a company means discounting its expected future cash flows, and that requires a discount rate. Earlier equity work often simply assumed one. Here we build it, by deriving the return that a company’s equity investors require, which is the same thing as its cost of equity. That number is a central input to valuation: it is one component of the weighted average cost of capital (WACC) used to discount free cash flow to the firm (FCFF), and it is the discount rate applied directly to free cash flow to equity (FCFE).

The primary tool for this estimate is the capital asset pricing model (CAPM). It is a single-factor model: the expected return on a security or portfolio depends on the risk-free rate, the expected market risk premium (MRP), which is the excess of the expected market return over the risk-free rate, and the security’s beta.

This relationship is also called the security market line (SML) equation. The variable it solves for is the expected return on the stock, but that expected return is exactly what investors require to hold the stock, so it doubles as the required return on equity. Plotted against beta, the SML is a straight line that starts at the risk-free rate and rises with a slope equal to the market risk premium. A stock of average market risk sits at a beta of 1.0, where its required return equals the expected market return.

Beta itself measures how sensitive a stock is to market movements. Formally it is the covariance of the stock’s returns with the market divided by the variance of the market, and that ratio can be decomposed into a correlation and a ratio of standard deviations.

Here the covariance term is between security returns and market returns, the market variance is in the denominator of the first form, and the final form combines that correlation with the ratio of the stock’s return standard deviation to the market’s own. Because a single stock is usually more volatile than the whole market, that ratio of standard deviations typically exceeds 1.0, and the correlation then scales it down to arrive at beta.

Beta is the central measure in the CAPM. A stock with roughly average exposure to the market has a beta near 1.0. A low-beta stock, below 1.0, moves less than the index, while a high-beta stock, above 1.0, moves more. In practice analysts almost never plug covariances into the decomposed formula directly. Instead they estimate beta by regression, and reported figures from data providers often carry proprietary adjustments.

A common source is the Bloomberg BETA function. For Amazon.com, regressing two years of weekly stock returns on the S&P 500 produced a raw beta of 1.241, a correlation of 0.636, and a standard error on the beta of 0.15. The reported adjusted beta was 1.161. The adjusted figure is a weighted average of the raw regression beta and the market average of 1.0.

with the two weights summing to one. The adjustment reflects mean reversion: high betas tend to drift down over time and low betas tend to drift up, so pulling the estimate toward 1.0 can improve a forward-looking forecast built from historical data.

To estimate beta by regression, an analyst typically gathers one to five years of weekly or monthly returns for the stock and the market index. Subtracting the risk-free rate from each side gives excess returns, and the slope of the excess-return regression is the beta.

In everyday practice a simpler form, the market model, regresses the stock’s return directly on the market’s return without subtracting the risk-free rate.

The two forms give very similar betas, and the market model is easier because it does not require gathering realized risk-free returns. Whichever form is used, the resulting beta reflects both industry-level and company-specific characteristics. Firms with predictable demand and stable earnings, such as mature consumer staples companies, tend to show low betas. Firms exposed to volatile inputs tend to show high betas: Albemarle Corporation, a lithium producer supplying electric-vehicle battery makers, showed a market-model raw beta of about 1.605 and an adjusted beta of 1.403, consistent with the sharp swing in lithium prices during 2021 and 2022.

Choosing the market portfolio

In theory the market portfolio holds every risky asset, including non-traded assets such as human capital. In practice the proxy is a broad, liquid equity index, and the choice matters when a company is global. Taiwan Semiconductor Manufacturing (TSMC) trades mainly in Taipei, yet earns close to two-thirds of its revenue in the United States. Measured against the Taiwan capitalization-weighted index (TAIEX), its adjusted beta was 1.364, but TSMC is itself over 17 percent of that index, which lifts the correlation between the stock and the index to 88.4 percent. Switching the benchmark to the MSCI World Index lowered the adjusted beta to 1.281 over the same period. A narrowly defined local index can therefore distort a beta estimate.

Noise, horizon, and periodicity

Every beta is an estimate and carries noise. Two gauges are common. The R-squared of the market-model regression is the share of the stock’s variation explained by the market, and in a single-factor model the correlation is the square root of the R-squared; a low R-squared signals a noisier beta. More directly, a small standard error on the beta signals a more precise estimate. The analyst also chooses a horizon and a return frequency. A default of about 100 weekly observations over two years is common, but the choice is a judgment call. Re-estimating Amazon over five years of monthly returns gave a raw beta of 1.152, an adjusted beta of 1.101, and a higher standard error of 0.183 from only 59 observations, while a one-year daily regression produced a raw beta of 1.526 with a higher R-squared and a smaller standard error.

When a single-stock regression is too noisy, an analyst can build a beta from a set of comparable companies instead. Much of a lone stock’s return variation is diversifiable, so its regression beta carries a low R-squared and a wide standard error. A portfolio of similar firms averages away much of that idiosyncratic noise and correlates more tightly with the market.

The comparable approach also handles differences in financial leverage explicitly. Leverage raises equity risk, so two firms with identical business risk can show different equity betas simply because they carry different amounts of debt. The unlevered beta strips leverage out; it is the equity beta a firm would have with no debt, and it reflects pure asset risk. Because it cannot be observed directly, analysts average the observed levered betas of the comparable set and unlever that average.

In this expression the numerator is the average levered beta of the comparable firms, t is their average tax rate, and the ratio is their average debt to market value of equity. The estimate is sensitive to which firms are chosen.

The benefit of pooling shows up clearly when the same regression is run on a portfolio. For the five largest US airlines, individual betas ranged from 1.021 to 1.341, but the equally weighted portfolio had the lowest standard error and the highest correlation with the market.

The portfolio beta of 1.205 is also the average of the five individual betas, but its standard error of 0.118 is the smallest and its correlation of 0.542 is the highest. Notably, Southwest had both the lowest beta and the lowest debt-to-equity ratio, while American had the highest of each, showing that regression betas already embed leverage differences.

Once the group’s unlevered beta is in hand, the analyst releveres it using the target firm’s own leverage to obtain that firm’s equity beta.

The general ordering that results is captured by a simple inequality: the comparable set’s levered beta and the target firm’s levered beta both sit at or above the shared unlevered beta.

The size of these adjustments depends on how much leverage differs. When both the industry and the firm carry little debt, unlevering and relevering barely move the estimate; the effect is largest when leverage ratios diverge sharply. The industry route is especially useful for private companies with no traded shares and for firms that have recently changed their leverage, where a historical regression would not capture the new equity risk. One caveat applies: the unlevered beta is assumed to represent the asset risk of the target firm, so if the comparables and the firm differ in asset risk, the estimate is flawed.

The expected equity market risk premium is the extra return investors demand for holding risky equities rather than a risk-free asset. In practice analysts measure it as the gap between the expected return on a broad, well-diversified equity index and the yield on a government bond issued in a closely aligned country. Together with the government yield, the premium sets the required return for a stock of average risk, meaning a beta of 1.0. Premiums differ across countries and change over time, and even for developed markets their size is a frequent source of disagreement, because they are estimated and carry error.

A country risk premium for developing markets

Outside the most developed economies, analysts layer a country risk premium (CRP) onto the benchmark developed-market premium, and beta then multiplies the combined figure.

The country risk premium is the compensation for taking on the risk of a stock in that market. It equals the target country’s sovereign bond yield spread, quoted in a reserve currency such as the US dollar or the euro, scaled by the ratio of equity-market to bond-market return volatility.

Historical estimates

One family of methods takes the difference between the long-run average total return on a stock index and the long-run average total return on a government security from the same region. The approach assumes a long enough sample that the historical mean is an unbiased guide to future returns. Two methodological choices arise: whether to average arithmetically or geometrically, and which maturity of government security to treat as risk-free.

Two patterns are worth remembering. A geometric average is always smaller than the arithmetic average of the same data, because return volatility drags down compound growth. And because long-term government securities earn a maturity premium over short-term bills, a premium measured against long-term bonds is lower than one measured against bills. Since equities are fundamentally long-horizon claims, valuation generally favors a premium built on long-term government returns.

Forward-looking estimates

Because the premium is meant to reflect expected future returns, a forward-looking method is more natural. One approach uses the constant-growth dividend model on the whole index: setting the index level equal to its value and solving for the expected return gives the dividend yield plus the long-run dividend growth rate.

Subtracting the current government yield from that expected market return produces the implied market risk premium.

The dividend model can be extended to richer cash-flow measures, such as free cash flow to equity, or to multi-stage growth forecasts, but the logic is the same: current price and expected cash flows imply the return the market is pricing in.

Analysts must ultimately commit to a point estimate or a range for sensitivity analysis. Premiums in the most developed economies typically fall in the low-to-mid single digits, roughly 3 to 6 percent, while developing markets can reach high single or double digits. Time variation exposes a key contrast between the two approaches. A long rally lifts average historical returns, raising a historical premium and implying a higher discount rate, which paradoxically points toward lower valuations. A forward-looking premium built on current prices instead reads a rising market as investors accepting less compensation for risk, so it tends to fall as prices climb.

The CAPM recognizes a single source of systematic risk: variation in stock market returns. Investors are compensated only for the chance that the market underperforms low-risk government securities, and beta is the only risk metric. Empirical work has repeatedly shown that market risk is not the whole story, so researchers have searched for additional priced risks. Arbitrage pricing theory (APT) provides the general framework for a multi-factor model, allowing several sources of systematic risk rather than one.

Here each beta is the stock’s exposure to a factor and each expected factor value is that factor’s risk premium. Crucially, APT does not name the factors or fix their number. Each factor stands for a systematic risk that diversification cannot remove; the CAPM likewise assumes the representative investor already owns the fully diversified market portfolio, leaving only systematic risk. Unlike the CAPM, that systematic risk spans more than one factor, and the market return need not be one of them, though it often is. Identifying which factors matter is an empirical exercise.

Multifactor models in practice

Factors are built from company stock-market data, macroeconomic data, and company accounting data. By the early 2020s researchers had proposed and found positive results for hundreds of candidate factors, many of them likely the product of data mining or spurious correlation. Most practitioners therefore believe the field can be distilled to a small set that captures the relevant systematic risks. A widely used example is a four-factor model whose factors are excess market return, firm size (SMB, small minus big), firm value (HML, high minus low), and price momentum (UMD, up minus down).

The market factor matches the CAPM, but the other three each come from a zero-investment strategy. The size premium, SMB, is the return on going long small-capitalization stocks and short an equal amount of large-capitalization stocks. The value premium, HML, goes long stocks with high book-to-market ratios (value stocks, with low price-to-book) and short low book-to-market stocks (growth stocks). The momentum premium, UMD, goes long recent winners and short recent losers. Betas for these factors are estimated by regressing stock returns on the factor returns. Whereas the market beta averages 1.0, the other three factor betas average zero, so negative loadings are common. This four-factor design traces to the Fama and French three-factor model of 1993, to which Carhart added momentum in 1997.

The loadings reveal contrasting risk profiles. Apple, Nvidia, and MicroStrategy all carry above-average market betas, while Microsoft is near average. MicroStrategy has a large positive size loading, so it tends to do well when small stocks beat large ones; the other three load negatively on the size factor. All four have negative value loadings, marking them as growth stocks that perform better when high price-to-book names outperform. Nvidia and MicroStrategy also load negatively on momentum, so they tend to lag when the market rewards recent winners.

To turn factor betas into a cost of equity, each factor needs a risk premium. Using long US history, the four factors carry the following average monthly premiums, which annualize by multiplying by 12.

The direction of the gap is not fixed. A four-factor estimate can sit above or below the CAPM figure depending on the signs and sizes of a stock’s factor loadings; Nvidia happens to show a large overstatement by the CAPM. The market beta in a multifactor model also need not match the single-factor beta, and even when they coincide the required returns usually differ. These models feed not only valuation but also portfolio construction and performance evaluation, and the search for a reliable, compact set of factors remains an active area of research.