Variance Decomposition in Single-Factor Return Models

# Why additivity assumption holds in CAPM and factor models? (Screenshot of a textbook included)


# Why additivity assumption holds in CAPM and factor models? (Screenshot of a textbook included)












- All the excerpts are from the book investment, written by Bodie. At the bottom of this post, I attached pages of the the book that show a related part of my question.

## Question

1. Why the variance of term $E(r_{i}) = 0$?

> We can always decompose the return on any security into the sum of its expected return plus unanticipated components: $$r_{i} = E(r_{i}) + e_{i} $$ where the unexpected return, $e_{i}$ , has a mean of zero and a standard deviation of $\sigma_{i}$ that measures the uncertainty about the security return. When security returns can be well approximated by normal distributions that are correlated across securities, we say that they are joint normally distributed. This assumption alone implies that, at any time, security returns are driven by one or more common variables. When more than one variable drives normally distributed security returns, these returns are said to have a multivariate normal distribution. We begin with the simpler case where only one variable drives the joint normally distributed returns, resulting in a single-factor security market. Extension to the multivariate case is straightforward and is discussed in later chapters. If we suppose that there is one common factor, $m$, an unknown macroeconomic variable that affects all firms. Then we can decompose the sources of uncertainty into uncertainty about the economy as a whole, which is captured by m, and uncertainty about the firm in particular, which is captured by $e_{i}$ . In this case, we amend Equation written above to accommodate two sources of variation in return: $$r_{i} = E(r_{i}) + m + e_{i} $$ The macroeconomic factor, $m$, measures unanticipated macro surprises. As such, it has a mean of zero (over time, surprises will average out to zero), with standard deviation of $\sigma_{m}$. In contrast, $e_{i}$ measures only the firm-specific surprise. Notice that $m$ has no subscript because the same common factor affects all securities. Most important is the fact that $m$ and $e_{i}$ are uncorrelated, that is, because $e_{i}$ is firm-specific, it is independent of shocks to the common factor that affect the entire economy. The variance of $r_{i}$ thus arises from two uncorrelated sources, systematic and firm specific. Therefore, $$\sigma_{i}^2 = \sigma_{m}^2 + \sigma^2(e_{i})$$

- The third equation, which is about the variance works only when the term $E(r_{i})$ is zero. However, the book does not explicitly mention that this term's variance is zero.

2. Is the equation $r_{i} = E(r_{i}) + m + e_{i}$ the generalized form of the equation $R_{i}(t) = \alpha_{i} + \beta_{i}R_{M}(t) + e_{i}(t)$, which is mention in the excerpt below?

> Because rates of return on market indexes such as the S&P 500 can be observed, we have a considerable amount of past data with which to estimate systematic risk. We denote the market index by $M$, with excess return of $R_{M} = r_{M} - r_{f}$ , and standard deviation of $\sigma_{M}$. Because the index model is linear, we can estimate the sensitivity (or beta) coefficient of a security on the index using a single-variable linear regression. We regress the excess return of a security, $R_{i}= r_{i} - r_{f}$, on the excess return of the index, $R_{M}$ . To estimate the regression, we collect a historical sample of paired observations, $R_{i}( t )$ and $R_{M}( t )$, where t denotes the date of each pair of observations (e.g., the excess returns on the stock and the index in a particular month). The regression equation is $$R_{i}(t) = \alpha_{i} + \beta_{i}R_{M}(t) + e_{i}(t)$$

- If the former equation is just the more generalized version than the latter, what is the difference between $r_{i}$ and $R_{i}(t)$ in each equation?

#### Below are the excerpts of the pages where I have questions.

## Answer by Yoda And Friends (score 2, accepted)

https://quant.stackexchange.com/a/68026

To answer the first question recall that $$r_i = E[r_i] + \epsilon_i + m,$$ where $E[\epsilon_i] = E[m] = 0$ by assumption.

With this, we have: $$E[r_i] = E[E[r_i] + \epsilon_i + m] = E[r_i]$$ as one would expect. To find the variance: $$Var(r_i) = E[(r_i - E[r_i])^2] = E[(\epsilon_i + m)^2] = E[\epsilon_i^2 + 2m\epsilon + m^2].$$ From this we get $$Var(r_i) = E[\epsilon_i^2] + E[m^2] + 2E[\epsilon_im] = \sigma_{\epsilon}^2 + \sigma_m^2 + 2Cov(m, \epsilon_i),$$ where last equality follows since $E[\epsilon_i] = E[m] = 0$. If you assume $m$ and $\epsilon_i$ uncorrelated (or independent), you get the wanted result.

Regarding your second question: there you are observing the relationship between your stock-specific returns and the "market" returns. As the text-book shows, it is referring to its equation (8.5).