Optimal Capital Structure

An ongoing research project

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Contents

1. Overview

2. Modelling Cap Structure with Flannery and Rangan

3. Validation

4. Conclusion and Next Steps

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1. Overview

Summary

• When looking into firms’ capital structures we deal with questions like: How do firms finance their operations? Do firms have leverage targets? How quickly do they approach these targets? What are the drivers of the targets? What are the impediments to achieving those targets?

• We are not the first to ask this questions, and the literature contains little consensus on the answers. We have reviewed the state of the art literature and have based our model in a well received paper by Flannery and Rangan from 2006.

• Flannery and Rangan, 2006 (F&R 2006) propose a regression approach to proof that firms partially adjust to target leverage ratios in each period. Among the most popular capital structure theories (trade-off,pecking order and market timing) this paper is based on the trade-off theory.

• If we ask ourselves why F&R 2006? We first may want to answer: Does the trade-off theory hold?
  • On the one hand our results suggest that profitable firms may lower their leverage level by choosing to hold on to their internal funds (retained earnings) to take advantage of future investment opportunities (This is in linel with Flannery and Rangan, 2006 but also with many others like Strebulaev, 2004 Hennessy and Whited, 2005; Huang and Ritter, 2009; Haron and Ibrahim, 2012…). This is indeed in line with the pecking order theory in which firms are expected to use retained earnigns to finance operatios.

  • On the other hand, our results also suggest that firms complete over half of their required leverage adjustment in less than 2 years. Such a rapid adjustment towards a firm-specific capital ratio suggests that pecking order or market timing does not dominate most firms’ debt ratio decisions.

  • Furthermore, the literature has shown other evidences that indirectly confirm the “trade-off theory”. For example, Leary and Roberts (2005) find that a typical firm changes the book value of its debt by more than 5% of book assets about once per year, and conclude that ‘‘Firms do indeed respond to equity issuances and equity price shocks by appropriately rebalancing their leverage over the next one to four years’’ (p. 32). This is line with a rapid adjustment speed because, if we define ‘‘appropriately rebalancing’’ as closing 90% of the initial leverage gap, ‘‘one to four years’’ corresponds to an adjustment speed that exceeds 40%.

• Now, it is important to take into account that while F&R 2006 aim to proof a capital strucuture theory, our goal is to predict next period’s leverage ratio for a given firm. Therefore, rather than restricting ourselves to the model proposed by F&R 2006, we use F&R 2006 as our baseline model and test model variations that might suit our usecase better.

Flannery and Rangan, 2006, Approach:

• The approach is based on the trade-off theory: A company chooses how much debt finance and how much equity finance to use by balancing the costs and benefits.

• They assume that there is a target leverage and use Market Debt Ratio as leverage metric:

\[ MDR_{i,t}=\frac{D_{i,t}}{D_{i,t}+MKTCap_{i,t}} \implies MDR_{i,t+1}^* = \beta \bar X_{i,t} \]

• Where \(MDR_{i,t+1}^*\) is the target debt nex period and \(\bar X_{i,t}\) is a vector of firm characteristics that expresses firms’ balance of costs and benefits.

• Now, under the trade-off hypothesis the \(\beta \neq 0\) and \(\Delta MDR_{i,t+1}^*\) is non-trivial as thereis friction created by adjustment costs that may prevent immediate adjustment. Again, as firm trades off its adjustment costs against the costs of operating with suboptimal leverage.

• Therfore firms permit a partial adjustment of the firms’ initial ratio toward target at the end of each period, which is the base of the model:

• Let the target leverage (I): \(MDR_{i,t+1}^* = \beta \bar X_{i,t}\)

• The change in \(MDR\) is defined as (II):

\[\Delta MDR_{i,t+1} = MDR_{i,t+1}-MDR_{i,t} = \lambda (MDR_{i,t+1}^*-MDR_{i,t})+\delta_{i,t+1}\]

• Plugging (I) into (II) we get a model for next period’s leverage:

\[MDR_{i,t+1} = (\lambda \beta) \bar X_{i,t} + (1-\lambda) MDR_{i,t}+\delta_{i,t+1}\]

• The rationale of the trade-off theory is included in \(\bar X_{i,t}\) which is a vector of:
  • EBIT/TA: A firm with higher earnings per asset dollar could prefer to operate with either lower or higher leverage. Lower leverage might occur as higher retained earnings mechanically reduce leverage, or if the firm limits leverage to protect the franchise producing these high earnings. Higher leverage might reflect the firm’s ability to meet debt payments out of its relatively high cash flow.
  • MB: Market to book ratio of assets. A higher MB is generally taken as a sign of more attractive future growth options, which a firm tends to protect by limiting its leverage.
  • DEP/TA: Depreciation as a proportion of total assets. Firms with more depreciation expenses have less need for the interest deductions provided by debt financing.
  • LnTA: Log of (real) total assets. Larger firms tend to operate with more leverage, perhaps because they are more transparent, have lower asset volatility, or have better access to public debt markets.
  • FA/TA: Fixed asset proportion. Firms operating with greater tangible assets have a higher debt capacity.
  • R&D/TA: Research and development expenses as a proportion of total assets. Firms with more intangible assets in the form of R&D expenses will prefer to have more equity. We are currently not including this.
• The research conducted by Flannery and Rangan, 2006 showed that the model above should be estimated with firm-specific fixed effects.

Model Variations

As part of the research, we play around with the components of \(\bar X_{i,t}\) by adding and removing variables in an attempt to come up with the best predictive model.

• F&R 2006: This is our baseline model based on Flannery and Rangan 2006.
• AR1: The main driver of the predictions in F&R 2006. Moreover, it is likely that we don’t have \(FA\), \(D&A\) and other information for some firms. $How would it look like to base our predictions on MDR information solely, letting the target debt ratio be the industry median MDR? We call AR1 to the special case of the baseline model where we only have MDR info.
• F&R 2006+Growth: It is likely that we will have forecasts for Sales, Earnings and Total Assets and we are able to compute expected growth variables from \(t\) to \(t+1\) that we can include to F&R 2006.
• AR1+Growth: AR1 plus growth variables, and EBIT/TA and lnTa as we’d have that info to compute the growth variables.
• Perfect Foresight: Using \(\bar X_{i,t+1}\) instead of \(\bar X_{i,t}\).

Data

We have conducted a detailed EDA on the data that can be found here.

Details about the sample construction:

• We have 14,491 US firms that report in USD in the database, from which we have 6,495 firms that have complete information to run the regression. The pooled panel for the regression has 64,024 observations.

• Our final database contains information from 1997 to 2020. We adjust total assets for inflation using the CPI index as of 2015.

• The Market Debt Ratio (our dependent variable) is constructed using time-matched market cap information from at least 7 weeks after the statement date. We use the first available end of month market cap after 7 weeks but no later than 2 months. As debt instruments, we use all the interest bearing debt items, both short-term and long-term.

• We have winsorized all the variables that play a role at the 99th percentile.

Main Stylized facts that we observe are the following:

• Summary statistics of the MDR and the other independent variables are in line with that reported in Table 1 of F&R 2006.

• There are some differences in the MDR ratios by size and sector, with median ratio of MDR being higher for larger firms and for firms in Utilities sector and Transportation sector and lower for firms in HiTech and Business Services industries.

• There is an upward trend in the actual MDR that we observe for 23% of the firms and a downward trend in the actual MDR that we observe for 15% of the firms.

• Literature shows mean reversion of MDR. Need to look into this.

• Over the entire sample period, the observed MDR series has a standard deviation of 25% (Similar to the 24.4% reported by Flannery and Rangan).

Validation and other tests

In order to diagnose if our predictions are good to be used, we have performed the following tests:

• Stylized facts on the aggregate are the same for the predicted MDRs.
• Actual vs. predicted visual inspection
• Performance metric: RMSE
• Residuals are not quite normal

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2. Modelling with Flannery and Rangan

The table below contains the specifications of the base model and the variations we present above.

## 
## ==============================================================
##                               Dependent variable:             
##                  ---------------------------------------------
##                                    MDR_lead                   
##                   F&R2006     AR1    F&R2006+Growth AR1+Growth
##                     (1)       (2)         (3)          (4)    
## --------------------------------------------------------------
## MDR              0.581***  0.604***     0.598***     0.608*** 
##                   (0.004)   (0.004)     (0.004)      (0.004)  
## EBIT_TA          -0.031***             -0.094***    -0.091*** 
##                   (0.003)               (0.004)      (0.003)  
## MB               -0.001***             -0.002***              
##                  (0.0002)               (0.0002)              
## DEP_TA            -0.016                 -0.012               
##                   (0.022)               (0.023)               
## FA_TA            0.050***               0.053***              
##                   (0.005)               (0.005)               
## lnTAcpi          0.018***               0.023***     0.027*** 
##                   (0.001)               (0.001)      (0.001)  
## IndMed           -0.114*** -0.087***   -0.064***    -0.053*** 
##                   (0.012)   (0.012)     (0.012)      (0.012)  
## Delta_EBIT_TA                          -0.098***    -0.097*** 
##                                         (0.003)      (0.003)  
## Delta_lnTAcpi                           0.049***     0.043*** 
##                                         (0.002)      (0.002)  
## Delta_lnSALEScpi                         -0.002       -0.001  
##                                         (0.001)      (0.001)  
## --------------------------------------------------------------
## Observations      63,385    63,385       57,529       57,529  
## R2                 0.351     0.339       0.368        0.365   
## Adjusted R2        0.285     0.272       0.296        0.293   
## ==============================================================
## Note:                              *p<0.1; **p<0.05; ***p<0.01

F&R 2006

• Lagged MDR’s coefficient, 0.58 implies that firms close on average 41.92 % of the gap between the current and the target debt ratio, yearly.

• Most of the lagged variables representing the target debt ratio carry significant coefficients with appropriate signs.
  • MB is negative as expected and significant, unlike in F&R 2006.
  • DEP_TA is negative as expected but not significant, unlike in F&R 2006.
  • FA_TA and lnTA have postive signs, magnitudes similar to F&R 2006, and are significant.
  • IndMed, is negative, which sounds counterintuitive to me as I would expect a firm in a given industry for which firms are more leveraged in median, to be more leveraged. It is also contradictory with F&R 2006 results.

• Model output for our example firm: MOODY’S CORP

A visual inspection of the output using F&R, 2006 of 87 other firms here.

• What do we see?
  • By model construction, we have that the distance between the predicted and past period’s actual is consistently 0.4192 as much as the distance between the target and past period’s actual.
  • As one could imagine from the regression summary, the predictions are largely driven by past MDR. The time series of the predicted looks like a shifted version of the actual.
  • Sometimes it even looks like the target debt ratio is driving the predictions further away from the actual (Eg last period for Moody’s). Would an AR1 do better in this case?

Model Variations

A visual inspection of the output using F&R, 2006 and the alternative models of 87 other firms here.

What model to pick?

• The regression output \(R^2\) (proportion of the variance that can be explained by the predictors): F&R2006+Growth> AR1+Growth>F&R2006>AR1. These show that the more variables to conform the target debt ratio, the better the predictions.

• We look at other model performance metrics: 1) We look at the correlation of the actual and predicted at the firm level (correlation squaed and r-squared should agree on the training data), and 2) RMSE represents the square root of the variance of the residuals. This means that if the residuals were normally distributed, roughly 68% of them would fall between \(+-RMSE\) around the mean residual, which in an ideal world, is zero.

  • On the overall sample: The metrics are consistent with the r-squared metrics.

• Mean cor and RMSE by size: We predict better for the larger firms. Which usually have more history. Perhaps the fe is better informed with more history. Or their capital structure is more stable. Or the single to noise ratio is better as we have more info.
• We investigate this further by controlling for the years of firm level historical information and running three F&R 2006 regressions: 1) Where all firms have exactly 20 years of history, 2) 10 years of history and 3) 5 years of history. We make sure that the firms are the same on the three regressions in an attempt to control for firm size and other features.

## 
## ==========================================
##                   Dependent variable:     
##              -----------------------------
##                        MDR_lead           
##                 20y       10y       5y    
##                 (1)       (2)       (3)   
## ------------------------------------------
## MDR          0.665***  0.582***  0.341*** 
##               (0.005)   (0.009)   (0.015) 
## EBIT_TA      -0.034*** -0.047*** -0.052***
##               (0.006)   (0.010)   (0.016) 
## MB            0.0001    -0.0001   -0.001  
##              (0.0004)   (0.001)   (0.001) 
## DEP_TA         0.003   0.232***   0.240** 
##               (0.036)   (0.065)   (0.110) 
## FA_TA        0.039***    0.015   -0.061***
##               (0.007)   (0.012)   (0.021) 
## lnTA         0.017***  0.027***  0.029*** 
##               (0.001)   (0.003)   (0.006) 
## IndMed       -0.132*** -0.055**  -0.074** 
##               (0.013)   (0.021)   (0.032) 
## ------------------------------------------
## Observations  26,300    13,150     6,575  
## R2             0.463     0.362     0.132  
## Adjusted R2    0.435     0.291    -0.086  
## ==========================================
## Note:          *p<0.1; **p<0.05; ***p<0.01

• We see that the R-squared is significantly better for model (1) than for model (3) and that that of model (2) is similar to the overall model’s.

• Now, the speed of adjustment decreases significantly with history length from around 66% for model (3) to 33% for model (1). I am not sure of what this means but wouldn’t expect the soa to be so susceptible to the depth of the firm level history.

• Should kick out of the sample firms with less than X years of history? How are fixed effects calibrated if there is little history?

• Back to what we wanted to find out: After controlling for length of firm level history, we still predict better for the larger firms, although the difference in performance is milder than before.

• Mean cor and RMSE by sector: We predict better for certain sectors. The sectors for which we predict (Utilities and Transportation) the best have also the largest firms on average versus the smallest (Agriculture and Business Services). Perhaps the effect is due to the size of the firms within the sectors. We could control for size and see if the differences persist. We report the rmse table here, the cor results can be found on the model diagnosis document here: model diagnosis document

• What if we had perfect foresight of X?

## 
## ================================================================================================================
##                                                         Dependent variable:                                     
##                    ---------------------------------------------------------------------------------------------
##                                                                 MDR                                             
##                    MDR+Perfect Foresight Perfect Foresight Perfect Foresight+Growth MDR+Perfect Foresight+Growth
##                             (1)                 (2)                  (3)                        (4)             
## ----------------------------------------------------------------------------------------------------------------
## MDR_lag                  0.541***                                                             0.556***          
##                           (0.004)                                                             (0.004)           
## EBIT_TA                  -0.079***           -0.066***            -0.108***                  -0.090***          
##                           (0.003)             (0.003)              (0.004)                    (0.003)           
## MB                       -0.007***           -0.007***            -0.009***                  -0.008***          
##                          (0.0002)            (0.0002)              (0.0003)                   (0.0002)          
## DEP_TA                   0.134***            0.491***              0.392***                   0.246***          
##                           (0.022)             (0.024)              (0.027)                    (0.023)           
## FA_TA                    0.103***            0.159***              0.167***                   0.097***          
##                           (0.005)             (0.005)              (0.006)                    (0.005)           
## lnTAcpi                  0.015***            0.012***              0.014***                   0.014***          
##                           (0.001)             (0.001)              (0.001)                    (0.001)           
## IndMed                   0.593***            0.757***              0.757***                   0.603***          
##                           (0.011)             (0.013)              (0.013)                    (0.011)           
## Delta_lnTAcpiL1                                                   -0.016***                   0.028***          
##                                                                    (0.002)                    (0.002)           
## Delta_EBIT_TAL1                                                    0.060***                   -0.006**          
##                                                                    (0.003)                    (0.003)           
## Delta_lnSALEScpiL1                                                -0.010***                   0.005***          
##                                                                    (0.002)                    (0.001)           
## ----------------------------------------------------------------------------------------------------------------
## Observations              57,529              63,385                57,529                     57,529           
## R2                         0.416               0.143                0.156                      0.421            
## Adjusted R2                0.350               0.056                0.060                      0.355            
## ================================================================================================================
## Note:                                                                                *p<0.1; **p<0.05; ***p<0.01

• The variance of the residuals is not constant…

tmp=regdtc[,err:=MDR_lead-`F&R2006`]
tmp[,sizebin:=.bincode(lnTAcpi,breaks=quantile(lnTAcpi,probs=seq(0,1,.2),na.rm=T),include.lowest = T)]
tmp[,.(mean=round(mean(err,na.rm=T),4),var=round(sd(err,na.rm=T)^2,4)*100),by=sizebin]

• Takeaways: See model diagnosis document for the mre detailed information.

  • There is heteroskedasticity. Perhaps we would be able to model the structure of the errors if we investigate further.
  • By looking at the \(R^2\) we see that a fair amount of variance remains unexplained. Are we able to capture more variation that we are not foreseeing?
  • By looking at model variations, we conclude that the more information, the better, although the improvements in \(R^2\) are not all that big.
  • Even when we include perfect foresight covariates to compute the target debt ratio, we are still missing out on a lot of variation. Is it unexplainable noise?
  • We predict better for the larger firms, partly because their fixed effects are more informed and partly because large firm dynamics seem to be easier to predict.
  • The length of the firm-level historical information does play a role in the quality of the predictions. Should we restrict our model to firms that have long enough history. Usually the larger firms have been around for longer…
  • We are not focusing on aggregate predictions as the aim is to build a firm-specific predictive model.
  • At a first glance, the results “makse sense” as most of the coefficients look good and through the visual inspection we are able to tell that some of the stylized facts mentioned above hold. However we formalize this in the next section.

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3. Validation

We run these tests with the predictions with the model F&R2006+Growth:

• Summary statistics of the actual MDR and the predicted MDR.

##     Variable N. Obs. Mean Median   SD   Min  Max  Q25  Q75
## 1:    Actual   63385 0.24   0.16 0.25  0.00 0.96 0.02 0.38
## 2: Predicted   63385 0.23   0.17 0.22 -0.15 1.17 0.08 0.35
## 3:    Target   63385 0.25   0.20 0.22  0.00 2.31 0.08 0.37

• SD actual is 25%. A good proxy for a similarly distributed panel should have similar SD. Predicitons and Target have 22%.

• Mean and Median of the actual MDR and the predicted MDR by Size:

##    sizebin MeanSize Mean_Act Mean_Pred Med_Act Med_Pred SD_Act SD_Pred
## 1:       1     1.72     0.22      0.24    0.10     0.16   0.27    0.25
## 2:       2     3.94     0.20      0.21    0.08     0.13   0.26    0.23
## 3:       3     5.54     0.20      0.21    0.10     0.15   0.25    0.21
## 4:       4     7.00     0.27      0.28    0.20     0.24   0.24    0.21
## 5:       5     9.04     0.30      0.32    0.26     0.29   0.22    0.19

• On average and median, our predicted MDR is above the actual MDR accross size quintiles.

• Larger companies have 36.36 % higer MDR than their smaller counterparts. We predict larger companies to have 33.33 %. On average.

• Firms in Utilities, Transportation and Communication are amongst the most leveraged with Utilities firms in utilities having on average 25% more debt than firms in Mining (4th most leveraged) and 172.41% more than firms in HiTech (the least leveraged). Our predictions yield 21.73 % and 158.86%, respectively. The figures are quite close, however I would perhaps expect more given that we are looking at the aggregate level and variance averages out. Plus results are largely driven by past period MDR and industry medians…

• There is an upward trend in the actual MDR that we observe for 22.9% of the firms. Our predictions show 39.36 and the targets 49.22%.

• There is a downward trend in the actual MDR that we observe for 15.81% of the firms.Our predictions show 15.21 and the targets 5.5%.

• These figures by size:

• Additional Testing to come:

  • Mean reversion in leverage shown in literature. Test.
  • What else could we look for?

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4.Conclusion and Next Steps

Conclusions

• We have tested Flannery and Rangan’s approach and some variations of their model in our data.

• The model they propose is inspired in the “trade-off” capital structure theory, which we view as appropiate given the literature as well as our results.

• In some cases, the model variations that we look at include “perfect foresight” variables. This is because the usecases in which we could use this research make the use of perfect foresight variables possible. For example, the Pro Forma Analizer starts the calculations with a change in sales from the previous period to the current. Another example: The CVaR engine has forecast for 30-year of Asset Value and Earnings.

• Coefficients make sense and are in line with what has been reported in the literature.

• The length in the firm-level history does play a roal in the speed of adjustment (coefficient of MDR on the regression table). I am not sure of how to interpret this result.

• performance metrics show that the more variables the better the model, so the conclusion at this point is to pick as many variables as possible.

• Conclusion about the model performance remains an open question, as we are unsure of whether the amount of variation that we capture is “good enough” or if there is still room for improvement. We expect to get some feedback about this.

• If we used existing literature as benchmark, we would conclude that our predictions are well enough to compete with the performance metric reported in various papers. However the state of the art literature is in general more theoretical and aims to show that a certain theory holds (in this case the trade-off theory). Our scope is different.

• Some of the validation tests show that the predictions perform quite well on the aggregate. However the goal is to predict firm-level target debt ratios.

Next steps

• Carry on with the testing?

• Apply the methodology to a real usecase (e.g. CVaR simulation) and see what we get.

• What else? Feedback is much appreciated.