One of my U of R projects has been research into dynamic capital structure, meaning how firms mix debt and equity. Firms make decisions about capital structure in the face of uncertainty. One of the sources of uncertainty they face comes in the form of the return on their investment. Firms transform investment into profits through their use of capital (raw materials, machines, and so forth), and the rate at which investment turns into profitability is called the firm’s productivity of capital.

For various reasons, the same amount of productivity can produce different amounts of profit in different periods. This can be due to some multiplier to the productivity function, a productivity shock, which can be either positive or negative, and examples exist for each (e.g. worker strikes, favorable weather, etc). The productivity shock process can be thought of as a process which evolves over time in a somewhat predictable manner.

In the academic finance literature, the productivity shock process is almost always thought of as a mean reverting process. This means that the process has some fundamental level and although at any given time may differ from that level, the firm expects that the shock will fluctuate around a known mean. My line of research breaks this assumption, and thinks of the firm as receiving a combination of mean-reverting shocks and permanent shocks. The difference between a mean-reverting (also called transitory) shock and a permanent shock has to do with the firm’s expectation about the future values of the shock. In a permanent shock, there is no expectation of mean reversion.

There are two main sections to this post. Section A bit more theory and math, and even a model contains a description of the model. Section How to solve talks about the solution and numerical methods used to find this example. Since this forum is not strictly academic, I am freely mixing methods, explanations, and code. I think that this may be useful, as often papers, even one that use complex solution methods, describe the entirety of these methods with a tiny appendix. By describing my solution methods in detail, it

• Helps me clarify my understanding
• Allows other people to find any errors in my logic or code
• Helps others in their own work

Capital structure refers to how firms utilize both debt and capital to fund investment. Firms work to maximize the sum of present and expected future value of equity value. They make investments, and can fund them by issuing corporate debt, or issuing equity. The decision to do this is complicated and is influenced by many factors including the current and expected future value of capital.

Assume that your profit is given by equation where is productivity, is capital stock, and is the curvature of the production function. The general style of these productions are called Cobb-Douglas production functions. I am interested in the dynamics of . is a stochastic process where at each time it takes a value . is typically assumed to follow an autoregressive process of type AR(1). An equation for an AR(1) process is:

where the 1 refers to the 1-period lag on the right side of the equation. is the autoregressive coefficient. AR(1) processes are by definition mean-reverting, meaning that over time, will approach some mean value.

Firms in reality receive shocks to their productivity from a variety of sources – changes in legislation, technology, relations with suppliers and labor, etc, and the nature of these shocks differ with regard to their expected permanence or transience. E.g., a new technology may be seen as a permanent shock while a supply chain concern or labor dispute may be more transient in nature. An AR(1) process is by definition transient when the autoregressive parameter is less than one, and thus using this process as part of a model of firm production assumes that all shocks are of the latter type. A permanent shock looks like:

After reading or skimming the above section, you may be thinking “that’s great. How do I solve it?”. That is the purpose of this section.

A very simple first question to ask is, along what dimension am I looking at leverage?

There are actually quite a few possibilities:

1. Take the average, pooled over all firms and time periods. However, it seems that there is more to say about the entire data set of corporate leverage ratios than can be expressed by one number.
2. At every distinct point in time, pool all firms. This would create a 1-dimensional time series. This is doing a little better.
3. For every period, compute means by industry. This creates a panel. However, doing this naievely creates a few problems (see the next section for help). At any given time, some firms enter the sample and some firms are leaving the sample. One way to mitigate this is take a slice of the data set known as a balanced panel, where the firms are chosen such that they exist for the entire window of time of interest. But, this introduces survivorship bias, because the sample is biased towards firms which endure all the economic woes of throughout the timeframe and do not leave the sample.
4. Look at leverage by leverage decile, year by year. This is an interesting way to see what the bottom, top, and middle leverage levels are doing. It is important to realize that this does not necessarily capture what the firms at each level are doing. This is because firms are free to change their leverage and thus their relative usage of leverage may change over time.
5. As a response to #4, At time period 1, sort leverage by decile and see which firms are in each decile. Then track these firms over time. This can be very interesting, taking caution that picking different starting time periods will most definitely change the groupings of firms.

There are plenty more.

I wrote some code to detect the largest balanced panels in the Compustat database. That code is coming tomorrow. I picked a balanced panel that started in the mid 1980′s for this analysis. I used the data.table package to hold the data, as it allows for keyed data tables that allow extremely fast, in-place subsetting.

For the first plot, the code was as follows.

Subset the data to get just the first time period. Using cut2 from Hmisc library, divide into deciles and label them A..J.

min.date = min(dt.t$date);
dt.tmin = subset(dt.t, date == min.date);
dt.tmin$initialDEC = with(dt.tmin, cut2(book_lev, g=10), labels=1:10);
dt.tmin$initialDEC = factor(dt.tmin$initialDEC, labels = LETTERS[1:10]);

initialDEC is a factor representing the initial decile. The merge does exactly what you think it should, meaning add a column called initialDEC to the the original data set, dt.t, merging on the column cusip.

## prepare to merge
setkey(dt.tmin, cusip);
setkey(dt.t, cusip);

## do the merge
dt.t = dt.t[dt.tmin]

Create a factor called timedec, which is the interaction of the initial decile (initialDEC) and each time period. There should be 10 * (max(dt.t$year) - min(dt.t$year) + 1) distinct values of this factor. Then, set this as they key, and take the means according to this group.

dt.t$timedec = paste(dt.t$date, "--", dt.t$initialDEC, sep="")
setkey(dt.t, timedec)
td.means = dt.t[,list(tdmean=mean(book_lev)), by=timedec]

This unwraps the data. td.means becomes a nice little data.table with 10 * (max(dt.t$year) - min(dt.t$year) + 1) rows.

timedate =  strsplit(as.vector(td.means$timedec), "--")
tddf = as.data.frame(matrix(unlist(timedate), ncol=2, byrow=TRUE))
colnames(tddf) = c("DATE", "iDEC");
td.means$date = tddf$DATE;
td.means$iDEC = tddf$iDEC;
td.means$year = extractyear(td.means$date)

Then, use ggplot2 to do the plotting:

p = ggplot(td.means, aes(x=year, y=tdmean, group=iDEC))
p = p + geom_line(aes(colour=iDEC))
p = p + opts(title = expression("Firm leverage at t=0 decile sort"))
p = p + scale_y_continuous('Book leverage')
print(p)

How to use R (and lots of cores) to find all possible balanced panels in a ragged data set.