Earnings mean reversion and dynamic optimal capital structure

Figures & data

Table 1. Definition of variables of the theoretical model.

$\mathrm{Eb}\left(x\right)$ = Equity before investment (equity in stage 1).

$\mathrm{Fb}\left(x\right)$ = Firm value before investment.

$\mathrm{Ub}\left(x\right)$ = Value of unlevered assets before investment.

$\mathrm{TBb}\left(x\right)$ = Tax benefits before investment.

$\mathrm{BCb}\left(x\right)$ = Bankruptcy costs before investment.

$\mathrm{Db}\left(x\right)$ = Debt before investment.

$R_0$ =  Coupon for $\mathrm{Db}\left(x\right)$.

$x_b$ = Bankruptcy threshold before investment.

$x_I$ = Investment trigger

$\mathrm{Ea}\left(x\right)$ = Equity after investment (equity in stage 2).

$\mathrm{Fa}\left(x\right)$ = Firm value after investment.

$\mathrm{Ua}\left(x\right)$ = Value of unlevered assets after Investment.

$\mathrm{TBa}\left(x\right)$ = Tax Benefits after Investment.

$\mathrm{BCa}\left(x\right)$ = Bankruptcy costs after Investment.

$D{a}_0\left(x\right)$ = Debt value of debt obtained at time zero after investment.

$D{a}_1\left(x\right)$ = Debt value of debt obtained at the investment trigger after investment.

$R_1$ =  Coupon for $D{a}_1\left(x\right)$.

$x_L$ = Bankruptcy threshold following investment (in stage 2).

$\tau$ = Corporate tax rate

$b$ = Proportional to unlevered assets bankruptcy costs

${\text{β}}_0$ = share of initial debt holders at bankruptcy in stage 2 under equal priority.

${\text{β}}_1=$ share of second debt holders at bankruptcy in stage 2 under equal priority.

I = Investment cost

RT = R0 + R1

Levb = $\mathrm{Db}\left(x\right)$ / $\mathrm{Fb}\left(x\right)$: Leverage ratio at t = 0

Crb = R0/ $\mathrm{Db}\left(x\right)$-r: Credit spread of initial debt at t = 0

Invb = $\text{I }\cdot \text{ J}\left(\text{x}\right)$ = Expected present value of investment costs

NBb(x) = $\mathrm{TBb}\left(x\right)$ – $\mathrm{BCb}\left(x\right)$: Net benefits of debt

At the investment trigger:

LevT = $\left(D{a}_0\right(x\left)\text{ + }D{a}_1\right(x)$) / $\mathrm{Fa}\left(x\right)$: Total leverage ratio at the investment trigger

ΔLev = LevT-Levb: Change in leverage relative to initial stage

Crb = R0/$D{a}_0\left(x\right)$-r: Credit spread of initial debt at the investment trigger

CrT = (R0 +R1)/ ($D{a}_0\left(x\right)+D{a}_1\left(x\right))$-r: Credit spread of total debt at the investment trigger

Table 2. Sensitivity analysis with respect to mean reversion speed (q).

Panel A: Optimal coupon and thresholds
Base case (θ = 1)
q	R0	R1	xI	vL	xb
0.10	0.52	0.55	1.341	−1.501	−1.191
0.15	0.65	0.45	1.238	−1.795	−1.209
0.20	0.78	0.45	1.149	−1.810	−1.134
0.25	0.88	0.47	1.077	−1.738	−1.053
0.30	0.96	0.49	1.021	−1.623	−0.958
Lower long-term profitability (θ = 0.75)
q	R0	R1	xI	vL	xb
0.10	0.36	0.54	1.473	−1.356	−1.094
0.15	0.39	0.4	1.409	−1.772	−1.186
0.20	0.46	0.37	1.342	−1.947	−1.169
0.25	0.52	0.39	1.276	−1.969	−1.137
0.30	0.57	0.42	1.217	−1.910	−1.094
Panel B: Firm value, equity, debt and leverage and credit spreads
Base case (θ = 1)
	Values at t = 0							Values at investment trigger T
q	Fb(x)	Db(x)	Eb(x)	Ub(x)	NBb(x)	Levb	Invb	Crb	LevT	ΔLev	CrT
0.1	20.556	7.997	8.518	26.720	1.692	0.389	7.856	0.005	0.485	0.096	0.0052
0.15	20.771	10.486	7.046	27.234	2.174	0.505	8.637	0.002	0.545	0.040	0.0020
0.2	21.166	12.787	6.114	27.683	2.707	0.604	9.224	0.001	0.628	0.024	0.0010
0.25	21.524	14.526	5.244	28.019	3.139	0.675	9.634	0.0006	0.698	0.023	0.0006
0.3	21.814	15.898	4.605	28.253	3.469	0.729	9.908	0.0004	0.752	0.023	0.0004
Lower long-term profitability (θ = 0.75)
	Values at t = 0							Values at investment trigger T
q	Fb(x)	Db(x)	Eb(x)	Ub(x)	NBb(x)	Levb	Invb	Crb	LevT	ΔLev	CrT
0.1	16.063	5.239	10.823	21.488	1.167	0.326	6.591	0.0087	0.431	0.104	0.0089
0.15	15.227	6.118	9.104	21.037	1.261	0.402	7.072	0.0037	0.44	0.038	0.0041
0.2	14.954	7.448	7.505	20.95	1.573	0.498	7.568	0.0018	0.499	0.001	0.0019
0.25	14.876	8.53	6.345	21.052	1.897	0.573	8.073	0.001	0.569	−0.004	0.0010
0.3	14.863	9.404	5.459	21.207	2.176	0.633	8.521	0.0006	0.633	0.001	0.0006

Notes: Initial earnings level x = 1, risk-free rate r = 0.06, tax rate τ = 0.15, proportional bankruptcy costs b = 0.5, growth option rate e = 2, investment cost I = 10. For the mean-reverting stochastic model parameters we use σ = 0.4, long-term mean θ = 1 or θ = 0.75 and vary mean reversion speed q. ΔLev stands for change in leverage and is calculated as Lev_T – Lev_b. Base case parameters highlighted in bold.

Table 3. Sensitivity analysis with respect to long-term profitability (θ).

Panel A: Optimal coupon and thresholds
θ	R0	R1	xI	vL	xb
0.6	0.31	0.53	1.571	−1.217	−0.974
0.75	0.36	0.54	1.473	−1.356	−1.094
1	0.52	0.55	1.341	−1.501	−1.191
1.25	0.73	0.58	1.237	−1.544	−1.234
1.5	0.99	0.61	1.156	−1.504	−1.210
1.75	1.29	0.63	1.093	−1.411	−1.132
2	1.64	0.62	1.047	−1.282	−0.976
2.25	2.03	0.57	1.017	−1.161	−0.764
Panel B: Firm value, equity, debt and leverage and credit spreads
	Values at t = 0							Values at investment trigger T
θ	Fb(x)	Db(x)	Eb(x)	Ub(x)	NBb(x)	Levb	Invb	Crb	LevT	ΔLev	CrT
0.6	13.635	4.246	9.387	18.243	0.97	0.311	5.578	0.013	0.405	0.094	0.0126
0.75	16.063	5.238	10.823	21.488	1.167	0.326	6.591	0.0087	0.430	0.104	0.009
1	20.556	7.997	12.555	26.72	1.692	0.389	7.856	0.005	0.485	0.096	0.0052
1.25	25.429	11.561	13.867	31.741	2.396	0.455	8.708	0.0031	0.548	0.093	0.0032
1.5	30.532	15.923	14.601	36.579	3.214	0.522	9.261	0.0022	0.611	0.089	0.0022
1.75	35.763	20.952	14.808	41.283	4.093	0.586	9.613	0.0016	0.667	0.081	0.0016
2	41.063	26.787	14.265	45.883	5.004	0.652	9.824	0.0012	0.716	0.064	0.0012
2.25	46.398	33.325	13.071	50.414	5.928	0.718	9.944	0.0009	0.754	0.036	0.0009

Notes: Initial earnings level x = 1, risk-free rate r = 0.06, tax rate τ = 0.15, proportional bankruptcy costs b = 0.5, growth option rate e = 2, investment cost I = 10. For the mean-reverting stochastic model parameters we use σ = 0.4, mean reversion speed q = 0.1 and vary long-term mean of earnings θ. ΔLev stands for change in leverage and is calculated as Lev_T – Lev_b. Base case parameters highlighted in bold.

Table 4. Sensitivity analysis with respect to current profitability level (x).

Panel A: Optimal coupon and thresholds
x	R0	R1	xI	vL	xb
−0.40	0.30	0.8	1.323	−1.481	−1.660
−0.25	0.32	0.8	1.324	−1.481	−1.616
0	0.37	0.7	1.327	−1.481	−1.508
0.25	0.41	0.7	1.330	−1.481	−1.422
0.50	0.44	0.6	1.332	−1.481	−1.358
0.75	0.48	0.6	1.336	−1.501	−1.274
1	0.52	0.6	1.341	−1.501	−1.191
1.25	0.59	0.5	1.352	−1.501	−1.049
1.30	0.62	0.45	1.357	−1.501	−0.989
1.35	0.68	0.4	1.369	−1.481	−0.873
Panel B: Firm value, equity, debt and leverage and credit spreads
	Values at t = 0							Values at investment trigger T
x	Fb(x)	Db(x)	Eb(x)	Ub(x)	NBb(x)	Levb	Invb	Crb	LevT	ΔLev	CrT
−0.4	10.391	4.583	5.807	13.44	1.176	0.441	4.225	0.0055	0.492	0.051	0.0054
−0.25	11.372	4.912	6.458	14.596	1.225	0.432	4.449	0.0051	0.491	0.059	0.0054
0	13.039	5.688	7.351	16.576	1.318	0.436	4.855	0.0051	0.491	0.055	0.0053
0.25	14.762	6.31	8.449	18.716	1.407	0.427	5.361	0.005	0.491	0.064	0.0054
0.5	16.563	6.78	9.780	21.069	1.493	0.409	6.000	0.0049	0.490	0.081	0.0053
0.75	18.476	7.393	11.079	23.688	1.589	0.4	6.801	0.0049	0.486	0.086	0.0052
1	20.556	7.997	12.555	26.720	1.692	0.389	7.856	0.005	0.485	0.096	0.0052
1.25	22.882	9.05	13.824	30.322	1.807	0.396	9.247	0.0052	0.484	0.088	0.0052
1.3	23.389	9.509	13.872	31.118	1.832	0.407	9.561	0.0052	0.483	0.077	0.0052
1.35	23.915	10.418	13.489	31.908	1.856	0.436	9.848	0.0053	0.485	0.05	0.0052

Notes: Risk-free rate r = 0.06, tax rate τ = 0.15, proportional bankruptcy costs b = 0.5, growth option rate e = 2, investment cost I = 10. For the mean-reverting stochastic model parameters we use σ = 0.4, mean reversion speed q = 0.1 and long-term mean of earnings θ = 1. ΔLev stands for change in leverage and is calculated as Lev_T – Lev_b. Base case parameters highlighted in bold.

Figure 1. The theoretical cross sectional relationship between leverage and return on asset.

Note: The figure uses numerical sensitivity results from the theoretical model in Table 4. The y-axis is Levb and the x-axis is a theoretical measure of return on assets taken as $x\left(1-\tau \right)/\mathrm{Ub}\left(x\right)$ which defines the after tax earnings divided by the value of unlevered assets Ub(x).

Table 5. Sensitivity analysis with respect to earnings volatility (σ).

Panel A: Optimal coupon and thresholds
σ	R0	R1	xI	vL	xb
0.24	0.72	0.46	1.015	−0.658	−0.415
0.30	0.61	0.5	1.135	−1.028	−0.772
0.40	0.52	0.55	1.341	−1.501	−1.191
0.50	0.46	0.64	1.541	−1.856	−1.563
Panel B: Firm value, equity, debt and leverage and credit spreads
	Values at t = 0							Values at investment trigger T
σ	Fb(x)	Db(x)	Eb(x)	Ub(x)	NBb(x)	Levb	Invb	Crb	LevT	ΔLev	CrT
0.24	20.822	11.644	9.175	28.187	2.475	0.559	9.84	0.0018	0.616	0.057	0.0018
0.3	20.58	9.711	10.867	27.373	2.08	0.472	8.873	0.0028	0.553	0.081	0.0028
0.4	20.556	7.997	12.555	26.72	1.692	0.389	7.856	0.005	0.485	0.096	0.0052
0.5	20.815	6.778	14.034	26.551	1.532	0.326	7.268	0.0079	0.45	0.124	0.0082

Notes: Initial earning level x = 1, risk-free rate r = 0.06, tax rate τ = 0.15, proportional bankruptcy costs b = 0.5, growth rate parameter e = 2, investment cost I = 10. For the mean-reverting stochastic model parameters we vary σ and use a mean reversion speed q = 0.1 and long-term mean of earnings θ = 1. ΔLev stands for change in leverage and is calculated as Lev_T – Lev_b. Base case parameters are highlighted in bold.

Table 6. Summary of directional effects on firm value, firm investment and default policies and leverage dynamics.

Parameter	Fb(x)	xI	vL	xb	Levb	LevT	ΔLev
Volatility (σ) ^1	U	+	–	–	–	–	+
Mean reversion speed (q)^2	U	–	U	U	+	+	–
Long-term profitability (θ)	+	-	U	U	+	+	-
Earnings level (x)	+	+	-	+	U	-	Inverted U
Growth expansion factor (e)	+	-	-	-	-	+	+
Capital investment cost (I)	-	+	+	+	+	-	-
Me-first priority rule (compared with equal priority)	+	+	-	-	-	Minor positive	+

Notes: The above table summarizes our sensitivity results. Note 1: the relation holds for ‘average’ long-term profitability: firm value is increasing when long-term profitability is low and decreasing when long-term profitability is high. All other effects of volatility remain unchanged. Note 2: When long-term profitability is low (high) firm value is decreasing (increasing).

Table 7. Sample of mean reverting and non-stationary firms.

Total non-financial and non-regulated firms	26,223
Number of firms with N = 40 consecutive earnings (oibdpq) excluding firms with growth >500% and no full parameter estimates of stochastic process	4,672
N. of firms classified as mean reverting	2,694
N. of firms classified as non-mean reverting	1,978
N. of mean reverting firms used in subsequent analysis with 1% quantile and 99% quantile for long-term mean and mean reversion speed removed and those with volatility higher than 5.	2,123
N. of non-mean reverting firms with volatility higher than 5 removed.	1,641
N. of unique mean reverting firms with complete observations on all financial variables (with 1% highest and lowest outliers removed)	1,572
Total number of firm-quarter year observation with full data for mean reverting firms	67,370
N. of unique non-mean reverting firms with complete observations on all variables (with 1% highest and lowest outliers removed)	1,300
Total number of firm-quarter year observation with full data for non-mean reverting firms	58,581

Figure 2. Example of a firm classified as mean reverting.

Notes: The figure shows an example of mean reverting firm with gvkey = 1609 with estimated parameters θ = 1.5, q = 0.23 and σe = 0.37. The figure show earnings (oibdpq) unadjusted for seasonality and scaled by the initial value of earnings of the firm for the whole periods of consecutive available data the firm.

Table 8. Descriptive statistics for sample of mean reverting and non-stationary firms.

Panel A: Mean reverting firms
	25% perc.	Median	Mean	75% perc.	Std
Market Leverage (gross)- Total	0.103	0.249	0.297	0.447	0.232
Profitability (ROA)	0.008	0.028	0.032	0.051	0.057
Size (Log of sales)	−3.055	−1.573	−1.542	−0.016	2.069
Market to book	0.644	0.922	1.174	1.386	0.896
Tangible assets	0.132	0.259	0.312	0.449	0.222
Herfindahl index	0.084	0.128	0.177	0.224	0.143
Cash ratio	0.019	0.060	0.111	0.150	0.137
Capex to assets (year-to-date)	0.009	0.021	0.033	0.043	0.036
Volatility (σ)	0.298	0.706	1.098	1.490	1.113
Long-Term profitability (θ)	0.407	0.989	1.426	2.097	2.223
Mean reversion speed (q)	0.347	0.566	0.751	0.928	0.594
Panel B: Non-stationary firms
	25% perc.	Median	Mean	75% perc.	Std
Market Leverage (gross)- Total	0.073	0.181	0.239	0.352	0.212
Profitability (ROA)	0.031	0.061	0.109	0.126	0.160
Size (Log of sales)	−2.015	−0.779	−0.846	0.313	1.811
Market to book	0.866	1.241	1.566	1.893	1.097
Tangible assets	0.121	0.256	0.316	0.471	0.236
Herfindahl index	0.080	0.119	0.159	0.199	0.127
Cash ratio	0.023	0.066	0.122	0.167	0.144
Capex to assets (year-to-date)	0.010	0.022	0.036	0.047	0.040
Volatility (σϵ)	0.361	0.712	1.168	1.587	1.154

Notes: The table provides descriptive statistics of all non-financial, non-regulated firms with at least 40 consecutive earnings (oibdpq) which are either mean reverting or non-stationary and have complete data on a set of variables needed for the analysis (see Table 5). Gross market leverage defined as long-term debt plus short-term debt divided long-term debt plus short-term debt plus the market value of equity, with market value of equity calculated using CRSP data (dlttq + dlcq)/ (dlttq + dlcq + price × shrout) while gross Profitability is defined as operating income after depreciation over total asset (oibdpq/atq). Volatility (σ) is the estimate of volatility as defined in eq. (29). Volatility (σ_ϵ) is defined as the estimated volatility of the standard error of eq. (26). Market-to-book is the market value of equity plus long-term debt divided by total assets ((equity + dlttq)/atq). Tangible assets are property plant and equipment over total assets (ppentq/atq). Herfindahl index a particular industry is defined as the sum of squared market shares for all firms in a two-digit SIC industry in a year-quarter. The market share of a firm is defined as sales of that firm in a year-quarter divided by the total sales in the industry of firm in that year-quarter. Cash ratio is cash and equivalents (cheq) scaled by total assets. Capex to assets is (capex) scaled by total assets. LT profitability (θ) is defined in eq. (28). Mean reversion speed is the estimated mean reversion speed as defined in eq. (27).

Table 9. Factors affecting leverage for mean reverting firms.

	Mean reverting	Non-stationary	All firms
	(1)	(2)	(3)
ROA	−0.493***	−0.123***	−0.159***
	(0.035)	(0.014)	(0.014)
Size	0.010***	0.006***	0.008***
	(0.001)	(0.001)	(0.000)
MB	−0.109***	−0.095***	−0.092***
	(0.002)	(0.001)	(0.001)
Tang.	0.168***	0.171***	0.166***
	(0.007)	(0.007)	(0.005)
Herf.	−0.036***	−0.034**	−0.024**
	(0.013)	(0.014)	(0.010)
Interaction of ROA with growth
MB × ROA	0.099***	0.053***	0.056***
	(0.019)	(0.005)	(0.005)
Revert vs Non-Revert
Revert			0.061***
			(0.010)
ROA × Revert			−0.327***
			(0.038)
MB × Revert			−0.021***
			(0.002)
ROA × MB × Revert			0.051**
			(0.02)
Quarter FE	Yes	Yes	Yes
Industry FE	Yes	Yes	Yes
R^2	0.328	0.391	0.350
Adj. R^2	0.326	0.389	0.348
Firm obs.	1,572	1,300	2,872
Total obs.	67,370	58,581	125,951
F-test (Prob)	<0.0001	<0.0001	<0.0001

Notes: The analysis is based on all non-financial, non-regulated firms with at least 40 consecutive earnings (oibdpq) (see Table 5). The dependent variable is gross market leverage defined as long-term debt plus short-term debt divided long-term debt plus short-term debt plus the market value of equity, with market value of equity calculated using CRSP data (dlttq + dlcq)/ (dlttq + dlcq + price × shrout). Profitability is defined as operating income after depreciation over total asset (oibdpq/atq). Size is the inflation adjusted log of sales (log(salesq/cpiind)). Market-to-book is the market value of equity plus long-term debt divided by total assets ((equity + dlttq)/atq). Tangible assets are property plant and equipment over total assets (ppentq/atq). Herfindahl index a particular industry is defined as the sum of squared market shares for all firms in a two-digit SIC industry in a year-quarter. The market share of a firm is defined as sales of that firm in a year-quarter divided by the total sales in the industry of firm in that year-quarter. The symbols *, **, and **** refer to estimates significantly different from zero at 10%, 5% and 1% confidence level, respectively. Heteroskedastic and autocorrelation robust standard errors clustered at firm level are provided in parenthesis.