Volatility Decomposition of a Typical Firm

1. Introduction

This post reviews the analysis in Campbell, Lettau, Malkiel, and Xu (2001) who find that firm level volatility has been rising over the period from July 1962 to December 1997. I’ve posted the code I used here. What does this mean? The authors look at the day-to-day variations in the stock returns of all publicly listed firms on the NYSE, Amex, and NASDAQ exchanges in each month, t, during this sample. They then show how to decompose the variance of the daily returns each month for a typical firm (e.g., a firm selected randomly with probability proportional to its market cap) into a market-specific component, an industry-specific component, and a firm specific component. Campbell, Lettau, Malkiel, and Xu (2001) find that this firm-specific variance has been steadily rising as plotted in the figure below.

Annualized firm-specific variance.

Annualized variance within each month of daily firm returns relative to the firm’s industry’s value weighted return for the period from July 1962 to December 1997.

I begin by detailing how Campbell, Lettau, Malkiel, and Xu (2001) estimate their market-specific, industry-specific, and firm-specific (i.e., idiosyncratic) volatility components. A natural first approach would be to estimate I industry-level regressions:

(1)   \begin{align*} r_{i,t} &= \beta_{i,m} \cdot r_{m,t} + \tilde{\epsilon}_{i,t} \end{align*}

and J firm-level regressions:

(2)   \begin{align*} \begin{split} r_{j,t} &= \beta_{j,i} \cdot r_{i,t} + \tilde{\eta}_{j,t} \\ &= \beta_{j,i} \cdot \beta_{i,m} \cdot r_{m,t} + \beta_{j,i} \cdot \tilde{\epsilon}_{i,t} + \tilde{\eta}_{j,t} \end{split} \end{align*}

where r_{m,t} denotes the value-weighted excess return on the market, r_{i,t} denotes the value-weighted excess return on industry i, and r_{j,t} denotes the excess return on stock i. You could then just insert the realized \beta_{j,i} and \beta_{i,m} terms into expressions for the variation in r_{i,t} and r_{j,t} to get the desired result, no?

(3)   \begin{align*} \begin{split} \mathrm{Var}[r_{i,t}] &= \beta_{i,m}^2 \cdot \mathrm{Var}[r_{m,t}] + \mathrm{Var}[\tilde{\epsilon}_{i,t}] \\ \mathrm{Var}[r_{j,t}] &= \beta_{j,m}^2 \cdot \mathrm{Var}[r_{m,t}] + \beta_{j,i}^2 \cdot \mathrm{Var}[\tilde{\epsilon}_{i,t}] + \mathrm{Var}[\tilde{\eta}_{j,t}] \end{split} \end{align*}

The problem here is that \beta_{j,m} and \beta_{j,i} are hard to estimate and may well vary over time. Thus, a \beta-independent procedure is necessary. After describing this procedure in Section 2, I then replicate the variance time series used in Campbell, Lettau, Malkiel, and Xu (2001) in Section 3. Finally, in Section 4 I conclude by extending the sample period to December 2012 and discussing the interpretation of the results.

The analysis in Campbell, Lettau, Malkiel, and Xu (2001) ties in closely with numerous other findings in asset pricing, macroeconomics, and behavioral finance. In the empirical asset pricing literature, Ang, Hodrick, Xing, and Zhang (2006) find a puzzling result that firms with high idiosyncratic volatility “have abysmally low average returns.” e.g., stocks with the highest idiosyncratic volatility have -1.06{\scriptstyle \%/\mathrm{mo}} lower excess returns than stocks with the lowest idiosyncratic volatility. In a macroeconomic context, this analysis is directly supports the granular origins theory of Gabaix (2011) which proposes that the key source of aggregate macroeconomic fluctuations is idiosyncratic firm-specific shocks to large firms. Finally, in a behavioral finance setting, the fact that market and industry models explain so little of the variation in firm-level stock returns (somewhere between 20{\scriptstyle \%} and 30{\scriptstyle \%}) suggests a new kind of problem for traders: scarce attention. “What this information consumes is rather obvious,” writes Herbert Simon. “It consumes the attention of its recipients. Hence a wealth of information creates a poverty of attention, and a need to allocate that attention efficiently among the overabundance of information sources that might consume it.” As highlighted in Chinco (2012), it takes time to sift through all of the competing information about each and every firm and subset of firms.

2. Statistical Model

In this section I explain how Campbell, Lettau, Malkiel, and Xu (2001) compute the market-wide, industry-level, and idiosyncratic contributions to the variation in a firm’s daily returns. The challenge to doing this is that estimating the firm-specific \betas empirically is noisy and unreliable. To get around this challenge, Campbell, Lettau, Malkiel, and Xu (2001) decompose the daily return variance of a “typical” firm rather than every firm. e.g., they think about the determinants of the daily return variance of a firm selected at random each month from the market with probability proportional to its relative market capitalization.

To see why looking at a typical firm might be helpful here, define the relative market capitalization of an entire industry, w_{i,t}, and the relative market capitalization of a particular firm, w_{j,t}, in month t as follows:

(4)   \begin{align*} w_{i,t} &= \frac{\sum_{j \in J[i]} \mathrm{MCAP}_{j,t}}{\sum_{j \in J} \mathrm{MCAP}_{j,t}} \qquad \text{and} \qquad w_{j,t} = \frac{\mathrm{MCAP}_{j,t}}{\sum_{j \in J} \mathrm{MCAP}_{j,t}} \end{align*}

where \sum_{i \in I} w_{i,t} = 1 and \sum_{j \in J} w_{j,t} = 1. I use the notation that j denotes a particular firm in the market, and use J[i] \subset J to denote the subset of firms in industry i: J[i] = \{ j \in J \mid \mathrm{Industry}(j) = i \}. The key insight is that the value weighted sums of the industry-level and firm-specific \betas have to sum to unity:

(5)   \begin{align*} 1 &= \sum_{i \in I} w_{i,t} \cdot \beta_{i,m} \qquad \text{and} \qquad 1 = \sum_{j \in J[i]} \left( \frac{w_{j,t}}{w_{i,t}} \right) \cdot \beta_{j,i} \end{align*}

This observation follows mechanically from the fact that the market return is just the value-weighted sum of its industry constituents and the industry returns are just the value-weighted sums of their firm constituents:

(6)   \begin{align*} r_{m,t} &= \sum_{i \in I} w_{i,t} \cdot r_{i,t} \qquad \text{and} \qquad r_{i,t} = \sum_{j \in J[i]} \left( \frac{w_{j,t}}{w_{i,t}} \right) \cdot r_{j,t} \end{align*}

By sampling appropriately, you can eliminate the pesky \betas from Equation (3) by converting their weighted sums into 1s.

How would you go about decomposing the variance of a typical firm in practice? First, consider estimating the market-wide and industry-specific variance components in a \beta-independent fashion. Instead of running the regression in Equation (1), consider computing the difference between each industry’s value weighted return, r_{i,t}, and the value weighted return on the market, r_{m,t}:

(7)   \begin{align*} r_{i,t} &= r_{m,t} + \epsilon_{i,t} \end{align*}

where \epsilon_{i,t} now lacks the tilde and is connected to \tilde{\epsilon}_{i,t} via the relationship:

(8)   \begin{align*} \epsilon_{i,t} &= \tilde{\epsilon}_{i,t} + (\beta_{i,m} - 1) \cdot r_{m,t} \end{align*}

Since \epsilon_{i,t} is not a regression residual, it will not be orthogonal to the market return, \mathrm{Cov}[r_{m,t},\epsilon_{i,t}] \neq 0. Thus, when computing the the variance of the value weighted industry return, r_{i,t}, we get:

(9)   \begin{align*} \mathrm{Var}[r_{i,t}] &= \mathrm{Var}[r_{m,t}] + \mathrm{Var}[\epsilon_{i,t}] + 2 \cdot \mathrm{Cov}[r_{m,t},\epsilon_{i,t}] \\ &= \mathrm{Var}[r_{m,t}] + \mathrm{Var}[\epsilon_{i,t}] + 2 \cdot (\beta_{i,m} - 1) \cdot \mathrm{Var}[r_{m,t}] \end{align*}

Applying the sampling trick described above then allows us to remove the \betas by averaging over all industries:

(10)   \begin{align*} \sum_{i \in I} w_{i,t} \cdot \mathrm{Var}[r_{i,t}] &= \sum_{i \in I} w_{i,t} \cdot \left\{ (2 \cdot \beta_{i,m} - 1) \cdot \mathrm{Var}[r_{m,t}] + \mathrm{Var}[\epsilon_{i,t}] \right\} \\  &= \mathrm{Var}[r_{m,t}] + \sum_{i \in I} w_{i,t} \cdot \mathrm{Var}[\epsilon_{i,t}] \end{align*}

This result says that if you select an industry i \in I with probability \mathrm{Pr}[i] = w_{i,t}, then the expected variance of this industry’s daily returns in month t will consist of a market component, \mathrm{Var}[r_{m,t}], and an industry-specific component, \sum_{i \in I} w_{i,t} \cdot \mathrm{Var}[\epsilon_{i,t}]. When the market component is big:

(11)   \begin{align*}  \frac{\mathrm{Var}[r_{m,t}]}{\mathrm{Var}[r_{m,t}] + \sum_{i \in I} w_{i,t} \cdot \mathrm{Var}[\epsilon_{i,t}]} \end{align*}

then most of the variation in value weighted industry returns tends to come from broad market-wide shocks. Conversely if the industry-specific component is relatively large, then most fo the variation in value weighted industry returns tends to come from different industry-specific shocks which are only felt in their particular corner of the market.

Next consider estimating the firm-specific variance component in a \beta-independent way using the same procedure. Instead of running the regression in Equation (2), I compute the difference between each firm’s excess returns and the value weighted excess returns on its industry:

(12)   \begin{align*} r_{j,t} &= r_{i,t} + \eta_{j,t} \\ \eta_{j,t} &= \tilde{\eta}_{j,t} + (\beta_{j,i} - 1) \cdot r_{i,t} \end{align*}

Since \eta_{j,t} is not a regression residual, it will no longer be orthogonal to the value weighted industry return, \mathrm{Cov}[r_{i,t},\eta_{i,t}] \neq 0, and thus:

(13)   \begin{align*} \mathrm{Var}[r_{j,t}] &= \mathrm{Var}[r_{i,t}] + \mathrm{Var}[\eta_{j,t}] + 2 \cdot \mathrm{Cov}[r_{i,t},\eta_{i,t}] \\ &= \mathrm{Var}[r_{i,t}] + \mathrm{Var}[\eta_{i,t}] + 2 \cdot (\beta_{j,i} - 1) \cdot \mathrm{Var}[r_{i,t}] \end{align*}

However, the same sampling trick means that the expression for the value weighted average variance over all stocks within each industry will be \beta-independent:

(14)   \begin{align*} \sum_{j \in J[i]} \left( \frac{w_{j,t}}{w_{i,t}} \right) \cdot \mathrm{Var}[r_{j,t}] &= \sum_{j \in J[i]} \left( \frac{w_{j,t}}{w_{i,t}} \right) \cdot \left\{ (2 \cdot \beta_{j,i} - 1) \cdot \mathrm{Var}[r_{i,t}] + \mathrm{Var}[\eta_{j,t}] \right\} \\  &= \mathrm{Var}[r_{i,t}] + \sum_{j \in J[i]} \left( \frac{w_{j,t}}{w_{i,t}} \right) \cdot \mathrm{Var}[\eta_{j,t}] \end{align*}

The interpretation of this equation is similar to the interpretation of the market-to-industry decomposition above. Putting both of these pieces together then gives the full decomposition as follows:

(15)   \begin{align*} \sum_{j \in J} w_{j,t} \cdot \mathrm{Var}[r_{j,t}] &= \sum_{i \in I} \mathrm{Var}[r_{i,t}] + \sum_{j \in J} w_{j,t} \cdot \mathrm{Var}[\eta_{j,t}] \\ &= \mathrm{Var}[r_{m,t}] + \sum_{i \in I} w_{i,t} \cdot \mathrm{Var}[\epsilon_{i,t}] + \sum_{j \in J} w_{j,t} \cdot \mathrm{Var}[\eta_{j,t}] \\ &= \sigma_{\mathrm{Mkt},t}^2 + \sigma_{\mathrm{Ind},t}^2 + \sigma_{\mathrm{Firm},t}^2  \end{align*}

3. Trends in Volatility

I now replicate these 3 variance measures from Campbell, Lettau, Malkiel, and Xu (2001) using daily and monthly CRSP data on NYSE, AMEX, and NASDAQ stocks over the sample period form July 1962 to December 1997. I restrict the data to include only common stocks with share prices above \mathdollar 1. The riskless rate corresponds to the 30 day T-Bill rate. First, to compute the empirical analogue of the market variance component in Equation (15), I compute the mean and variance of daily excess returns on the value-weighted market:

(16)   \begin{align*} \hat{\mu}_{\mathrm{Mkt},t} &= \frac{1}{S} \cdot \sum_{s = 1}^S r_{m,t-s} \\ \hat{\sigma}_{\mathrm{Mkt},t}^2 &= \frac{1}{S} \cdot \sum_{s = 1}^S \left( r_{m,t-s} - \hat{\mu}_{\mathrm{Mkt},t} \right)^2 \end{align*}

where S denotes the number of days in month t. I plot the annualized market component of the variance of daily firm returns in the figure below.

Annualized market-wide variance.

Annualized variance within each month of the daily value weighted market return in excess of the 30 day T-bill rate for the period July 1962 to December 1997.

Next, I compute the industry-specific contribution to the variance of a typical firm’s daily excess returns using the Fama and French (1997) industry classification codes. There are 48 industries in this classification system, and I code all stocks without a specified industry as their own group leading to 49 total industries. Clicking on the image below gives a plot of the number of firms in each of these industries.

Industry size distribution.

Click to embiggen. Number of firms in each industry from July 1962 to December 2012.

The industry-specific contribution to the variance in daily excess returns of a typical firm is then given by:

(17)   \begin{align*} \hat{\sigma}_{\mathrm{Ind},t}^2 &= \sum_{i \in I} w_{i,t} \cdot \left\{ \frac{1}{S} \cdot \sum_{s = 1}^S \left( r_{i,t-s} - r_{m,t-s} \right)^2 \right\} \end{align*}

I plot the resulting time series in the figure below.

Annualized industry-specific variance.

Annualized variance within each month of daily value weighted industry returns in excess of the value weighted market return for the period from July 1962 to December 1997.

Finally, I compute the idiosyncratic contribution to the variance of the daily excess returns of a typical firm as follows:

(18)   \begin{align*} \hat{\sigma}_{\mathrm{Firm},t}^2 &= \sum_{j \in J} w_{j,t} \cdot \left\{ \frac{1}{S} \cdot \sum_{s = 1}^S \left( r_{j,t-s} - r_{i,t-s} \right)^2 \right\} \end{align*}

This is the time series I plotted in the introduction. Empirically, it seems that the overwhelming majority of the daily variation in firm-level excess returns comes from idiosyncratic shocks. One ways to quantify this statement is to look at the “model” fit each month:

(19)   \begin{align*} 1 - \mathrm{Err}_t &= \frac{\hat{\sigma}_{\mathrm{Mkt},t}^2 + \hat{\sigma}_{\mathrm{Ind},t}^2}{\hat{\sigma}_{\mathrm{Mkt},t}^2 + \hat{\sigma}_{\mathrm{Ind},t}^2 + \hat{\sigma}_{\mathrm{Firm},t}^2} \end{align*}

where the model corresponds to a market model with industry factors. The (1 - \mathrm{Err}_t) term captures the fraction of the daily variation in firm-level excess returns that is explained by model and industry factors in each month and is consistently below 0.30. i.e., more than 70{\scriptstyle \%} of the daily variation is explained by firm-specific shocks!

Model fit.

Fraction of the daily variation in firm-level excess returns that is explained by value-weighted market and industry factors in each month from July 1962 to December 1997.

4. Discussion

There are a couple of interesting take away facts from this analysis. First, the nature of the 3 variance time series dramatically changes after December 1997 when the original sample period in Campbell, Lettau, Malkiel, and Xu (2001) ends. Specifically, the post 1997 time period is dominated by a pair of volatility spikes which are not firm-specific: the dot-com boom and the financial crisis. When compared to these events, the slow run up in the firm-specific variance component looks relatively minor.

Updated sample.

Top Panel: Annualized variance within each month of the daily value weighted market return in excess of the 30 day T-bill rate for the period July 1962 to December 1997. Middle Panel: Annualized variance within each month of daily value weighted industry returns in excess of the value weighted market return for the period from July 1962 to December 2012. Bottom Panel: Annualized variance within each month of daily firm returns relative to the firm’s industry’s value weighted return for the period from July 1962 to December 2012.

Nevertheless, even with these large macroeconomic shocks, the majority of the variation in daily firm-level excess returns is driven by firm-specific information. e.g., even during this later period the model fit, (1 - \mathrm{Err}_t), scarcely crosses the 40{\scriptstyle \%} threshold in spite of all of the systemic risk in the market! What’s more, there is substantially cyclicality in the model fit at roughly the 5{\scriptstyle \mathrm{yr}} horizon. i.e., once every 5{\scriptstyle \mathrm{yr}} the predictive power of the value weighted market and industry factors grows and then shrinks by roughly 10{\scriptstyle \%} or around 1/2-to-1/3 of its baseline. One way to interpret this finding is that there should be 5{\scriptstyle \mathrm{yr}} year cycles in the profitability of technical analysis using market-wide factors.

Updated sample.

Fraction of the daily variation in firm-level excess returns that is explained by value-weighted market and industry factors in each month from July 1962 to December 2012.