Market Risk, Firm-Specific Risk, Beta, and Alpha

Why can a stock outperform a short-term Treasury bill? The answer is not simply “because the stock is riskier.” A one-factor market model separates a security’s realized excess return into an intercept, a market-linked component, and one residual; that residual may contain company-specific surprises as well as omitted common influences. That separation helps us ask a better question: which risks did the investor actually bear, and which part of the outcome was merely a surprise?

This is an educational model, not a forecast or a recommendation to buy a security.

First, define the comparison

An excess return is a security’s return minus a reference risk-free rate over the same period and in the same currency:

$$r_{i,t}^{e}=R_{i,t}-R_{f,t}, \qquad r_{M,t}^{e}=R_{M,t}-R_{f,t}.$$

Here $i$ identifies a security, $M$ a broad market benchmark, and $t$ the observation period. If a stock returns 13% and the matched reference rate is 3%, the stock’s realized excess return is 10 percentage points. That arithmetic does not tell us whether the 10 points were expected, repeatable, or adequate compensation for risk.

Short-dated U.S. Treasury bills are often used as a practical nominal risk-free-rate proxy in U.S.-dollar examples. “Proxy” matters. Treasury securities carry the U.S. government’s full faith and credit, but a longer-term fixed-rate Treasury can lose market value when interest rates rise, especially if sold before maturity. Inflation can also erode the purchasing power of a nominal payoff. The original version of this note said the Treasury “cannot go bankrupt” and could simply “print money,” so its securities were unconditionally risk-free. That was an overstatement and is not needed for the model. The SEC’s Treasury-bond interest-rate bulletin explains the sale-price risk; Investor.gov’s bond overview also distinguishes interest-rate and inflation risks.

A one-factor way to sort realized returns

A simple market model writes the security’s realized excess return as

$$r_{i,t}^{e}=\alpha_i+\beta_i r_{M,t}^{e}+\varepsilon_{i,t}.$$

This is a statistical description using a specified market index and sample, not a law guaranteeing any future return.

• $\beta_i$ measures the security’s sensitivity to the benchmark’s excess return. A beta of 1.2 means that a one-percentage-point change in the benchmark’s excess return is associated with about a 1.2-percentage-point change in the model’s fitted security excess return, holding the intercept fixed. It does not mean the security’s total volatility must exceed the benchmark’s: firm-specific volatility also matters.
• $\alpha_i$ is the fitted intercept: the component the chosen one-factor benchmark does not explain when the market excess return is zero. It is a model-dependent parameter, not a special event that “kicks in” or a guaranteed stock-picking gain. An estimated alpha can change with the sample, benchmark, currency, return frequency, and omitted factors.
• $\varepsilon_{i,t}$ is the observation’s residual—the realized difference between the actual excess return and the fitted value $\alpha_i+\beta_i r_{M,t}^{e}$. A factory disruption or unexpected product success could contribute to a company-specific residual, although a one-factor residual can also contain omitted common factors, measurement error, and other model misspecification.

In an ordinary least-squares regression with an intercept, the fitted residuals average zero in that estimation sample and are uncorrelated with the fitted market regressor in that sample. That is a property of the fitting method, not a declaration that unpredictable events cannot have probabilities, or that tomorrow’s residual must be zero. More generally, treating $E[\varepsilon_i]=0$ and $\operatorname{Cov}(r_M^e,\varepsilon_i)=0$ as population statements requires assumptions about the model.

For a compact reference on beta as a regression measure, see OpenStax, Regression Applications in Finance. OpenStax’s CAPM discussion distinguishes market-related and diversifiable firm-specific risk. The CAPM’s expected-return equation is a different claim from this realized-return regression: it says $E[R_i]=R_f+\beta_i(E[R_M]-R_f)$ under the model’s assumptions, not that every realized return equals that expectation.

Work the 13% example without inventing a story

Suppose the matched risk-free rate is 3%, the market return is 10%, and the security return is 13%. The market’s excess return is 7% and the security’s is 10%. Imagine that a previously estimated one-factor model has $\beta_i=1.2$ and $\alpha_i=1.0\%$ for the same return period. Then the fitted security excess return is

$$\widehat{r_i^e}=1.0\%+1.2(7.0\%)=9.4\%.$$

The realized residual is $10.0\%-9.4\%=0.6\%$. The accounting check is

$$\underbrace{10.0\%}_{\text{realized excess return}}=\underbrace{1.0\%}_{\text{estimated intercept}}+\underbrace{8.4\%}_{\text{market-linked component}}+\underbrace{0.6\%}_{\text{realized residual}}.$$

Nothing about this one observation proves a persistent 1% alpha. We assumed the model estimates for the example; a real estimate would require dated return observations, a documented benchmark, and a fitting method. A sector-wide event, such as a change in laboratory-equipment demand, may affect several companies at once. If the broad index does not capture that common sector factor, the effect can appear in this single-factor model’s residual. It should not automatically be called security-specific alpha.

Beta is not the whole risk story

Beta can be estimated from the same-period covariance between security and market excess returns:

$$\beta_i=\frac{\operatorname{Cov}(r_i^e,r_M^e)}{\operatorname{Var}(r_M^e)}, \qquad \operatorname{Var}(r_M^e)>0.$$

The broad market exposure is often called systematic risk. The part not captured by that market benchmark is called residual or idiosyncratic risk in this model. A diversified portfolio can reduce many independent company-specific shocks, but diversification cannot remove broad market exposure. “Residual” is not a promise that all of it can be diversified away: residuals may correlate across holdings if the model leaves out common influences.

Because the intercept is constant within the fitted model, the exact variance identity is

$$\operatorname{Var}(r_i^e)=\beta_i^2\operatorname{Var}(r_M^e)+\operatorname{Var}(\varepsilon_i)+2\beta_i\operatorname{Cov}(r_M^e,\varepsilon_i).$$

Only when the market excess return and residual are uncorrelated does this simplify to

$$\sigma_i^2=\beta_i^2\sigma_M^2+\sigma_\varepsilon^2.$$

That missing covariance condition matters. The original note repeated the simplified formula without stating it. As a hypothetical unit check, if $\beta=1.2$, the standard deviation of market excess returns is 15%, and the residual standard deviation is 20%, assuming zero covariance, the security’s modeled standard deviation is

$$\sigma_i=\sqrt{(1.2\times15\%)^2+(20\%)^2}=\sqrt{724}\%\approx26.91\%.$$

Standard deviation is the square root of variance, a measure of dispersion in the same units as return. It is not itself a probability or a point where a Gaussian density has fallen to $1/e$ of its peak. For a normal density, one standard deviation from the mean places the density at $e^{-1/2}$ of its peak; $e^{-1}$ occurs at $\sqrt{2}$ standard deviations. The variance decomposition above does not require returns to be normally distributed.

What to take away

Beta tells us how strongly a return co-moves with a chosen market benchmark; it does not summarize every risk. Alpha is an estimated model intercept, not automatically manager skill. A residual is what the specified model did not explain in a particular period, not a mysterious source of guaranteed excess return. The practical habit is to state the return period, currency, reference rate, market index, and statistical assumptions before assigning a story to any one number.

This page revises an informal 2016 study note while preserving its original URL, publication date, series position, author, and archive identity. The sources above were checked on 2026-09-24. All worked numbers are hypothetical and are not current quotes or investment advice.