# Debt-by-Price Ratio, End-of-Year Economic Growth, and Long-Term Prediction of Stock Returns

## Abstract

**:**

## 1. Introduction

## 2. The Technical Motivation

## 3. Data and the Prediction Model

#### 3.1. The Nonparametric Model

#### 3.2. Out-of-Sample Validation for Optimal Model and Bandwidth Selection

#### 3.3. Data

## 4. The Empirical Application

#### 4.1. The Relationship between Stock Returns and Government Debt

#### Univariate Regressions of Stock Returns on Price Ratios

#### 4.2. Third to Fourth Quarter Economic Growth

#### Univariate Regressions of Stock Returns on Economic Growth

#### 4.3. More Macroeconomic Variables

#### 4.4. The Multivariate Setting

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The United States annual debt-to-GDP ratio % growth (dark blue line), calculated using the annual gross federal debt as percent of gross domestic product [16]. The sample period is 1947–2020.

**Figure 2.**The most successful forecasting variables detected in this analysis (dark blue line). From top to bottom: debt-by-price ratio, third to fourth quarter GDP growth and term spread. Vertical lines (gray area) are the NBER business-cycles (recessions). The sample period is 1947–2020.

**Figure 3.**Risk-free benchmark. Relation between excess stock returns and debt-by-price ratio. Estimated nonparametric function $\widehat{m}$ (dark blue line), linear model (dotted red line), observations (green balls). The sample period is 1947–2020.

**Figure 4.**Risk-free benchmark. Relation between excess stock returns and third to fourth quarter GDP growth. Estimated nonparametric function $\widehat{m}$ (dark blue line), linear model (dotted red line), observations (green balls). The sample period is 1947–2020.

**Figure 5.**Risk-free benchmark. Relation between excess stock returns and term spread. Estimated nonparametric function $\widehat{m}$ (dark blue line), linear model (dotted red line), observations (green balls). The sample period is 1947–2020.

**Figure 6.**Risk-free rate benchmark. Relation between one year excess stock returns and the combination of third to fourth quarter GDP growth with the dividend-by-price ratio, earnings-by-price ratio, term spread and the debt-by-price ratio. Estimated nonparametric function $\widehat{m}$ (light blue surface), observations (green balls). The sample period is 1947–2020.

**Table 1.**Time-lagged predictive variables ${X}_{t-1}$ related to government debt and economic growth. Q/Q is the quarter-over-quarter growth rate from the third to the fourth quarter of a year, GDP is the nominal gross domestic product, IP the industrial production index, CON nominal personal consumption expenditures and DEB the total government debt.

Predictive Variables | ${\mathit{X}}_{\mathit{t}-1}$ |
---|---|

Growth in GDP (Q/Q) | $gd{p}_{t-1}=(GD{P}_{t-2}^{Q4}-GD{P}_{t-2}^{Q3})/GD{P}_{t-2}^{Q3}$ |

Growth in industrial production (Q/Q) | $i{p}_{t-1}=(I{P}_{t-2}^{Q4}-I{P}_{t-2}^{Q3})/I{P}_{t-2}^{Q3}$ |

Growth in personal consumption expenditures (Q/Q) | $co{n}_{t-1}=(CO{N}_{t-2}^{Q4}-CO{N}_{t-2}^{Q3})/CO{N}_{t-2}^{Q3}$ |

GDP-by-price ratio | $gp{r}_{t-1}=GD{P}_{t-1}/{P}_{t-1}$ |

Debt-by-price ratio | $dp{r}_{t-1}=DE{B}_{t-1}/{P}_{t-1}$ |

Debt-by-GDP ratio | $dg{r}_{t-1}=DE{B}_{t-1}/GD{P}_{t-1}$ |

**Table 2.**The table shows the summary statistics for the dependent and forecasting variables. ${Y}^{\left(R\right)}$ is the stock return in excess of risk free rate; gdp the third to fourth quarter GDP growth; ip the third to fourth quarter industrial production growth; con the third to fourth quarter consumption growth; gpr the GDP-by-price ratio; dpr the debt-by-price ratio and dgr the debt-by-GDP ratio. The sample period is 1947–2020.

${\mathit{Y}}^{\left(\mathit{R}\right)}$ | gdp | ip | con | gpr | dpr | dgr | |
---|---|---|---|---|---|---|---|

Mean | 6.49 | 1.47 | 0.82 | 1.46 | 1.36 | 0.76 | 59.5 |

Standard deviation | 15.41 | 1.20 | 2.01 | 0.99 | 0.52 | 0.3 | 22.14 |

**Table 3.**Predictive power for the one-year excess stock returns ${Y}_{t}^{\left(A\right)}$. The prediction problem is defined in (2). The predictive power (%) is measured by ${R}_{V}^{2}$ as defined in (3). The benchmarks ${B}^{\left(A\right)}$ considered are based on the short-term interest rate ($A\equiv R$), long-term interest rate ($A\equiv L$), earnings-by-price ratio ($A\equiv E$), and consumer price index ($A\equiv C$). The predictive variables ${X}_{t-1}$ used are given by dividend-by-price ratio ${d}_{t-1}$, earnings-by-price ratio ${e}_{t-1}$, debt-by-price ratio $dp{r}_{t-1}$ and GDP-by-price ratio $gp{r}_{t-1}$. The predictive power (%) for the one-year stock returns $ln{S}_{t}$ validated on the original scale without benchmark is measured by ${\tilde{R}}_{V}^{2}$ as defined in (5) in terms of back-transformed stock return prediction $\widehat{ln{S}_{t}}$ defined in (4). The sample period is 1947–2020.

Benchmark ${\mathit{B}}^{\mathit{A}}$ | ${\mathit{X}}_{\mathit{t}-1}$ | |||||||
---|---|---|---|---|---|---|---|---|

${\mathit{R}}_{\mathit{V}}^{\mathbf{2}}$ | ${\tilde{\mathit{R}}}_{\mathit{V}}^{\mathbf{2}}$ | |||||||

d | e | dpr | gpr | d | e | dpr | gpr | |

Short-term rate | 4.1 | 0.7 | 17.3 | 0.34 | −2.1 | −5.8 | 11.9 | −6.15 |

Long-term rate | 5.1 | 1.6 | 15.1 | 0.17 | 2.4 | −1.3 | 12.6 | −2.70 |

Earnings-by-price | 0.1 | −1.9 | 10.7 | 0.31 | 6.2 | 4.4 | 16.2 | 6.44 |

Inflation | 3.0 | −0.3 | 13.3 | 0.50 | −1.9 | −5.4 | 8.8 | 1.76 |

**Table 4.**Predictive power for the one-year excess stock returns ${Y}_{t}^{\left(A\right)}$. The prediction problem is defined in (2). The predictive power (%) is measured by ${R}_{V}^{2}$ as defined in (3). The benchmarks ${B}^{\left(A\right)}$ considered are based on the short-term interest rate ($A\equiv R$), long-term interest rate ($A\equiv L$), earnings-by-price ratio ($A\equiv E$), and consumer price index ($A\equiv C$). The predictive variables ${X}_{t-1}$ used on Panel A are given by the third to fourth quarter growth rates of GDP $gd{p}_{t-1}$, industrial production $i{p}_{t-1}$ and personal consumption expenditures $co{n}_{t-1}$. Panel B shows the fourth to fourth quarter (annual) growth rates of these corresponding variables. The predictive power (%) for the one-year stock returns $ln{S}_{t}$ validated on the original scale without benchmark is measured by ${\tilde{R}}_{V}^{2}$ as defined in (5) in terms of back-transformed stock return prediction $\widehat{ln{S}_{t}}$ defined in (4). The sample period is 1947–2020.

Benchmark ${\mathit{B}}^{\mathit{A}}$ | ${\mathit{X}}_{\mathit{t}-1}$ | |||||
---|---|---|---|---|---|---|

${\mathit{R}}_{\mathit{V}}^{\mathbf{2}}$ | ${\tilde{\mathit{R}}}_{\mathit{V}}^{\mathbf{2}}$ | |||||

gdp | ip | con | gdp | ip | con | |

Panel A: Third to fourth quarter | ||||||

growth rate | ||||||

Short-term rate | 15.4 | 15.7 | 17.3 | 9.9 | 10.2 | 11.9 |

Long-term rate | 14.5 | 15.2 | 16.2 | 12.1 | 12.8 | 13.8 |

Earnings-by-price | 15.7 | 14.9 | 15.2 | 20.9 | 20.1 | 20.5 |

Inflation | 17.2 | 13.3 | 14.6 | 13.0 | 8.9 | 10.2 |

Panel B: Fourth to fourth quarter | ||||||

growth rate | ||||||

Short-term rate | −0.4 | 1.0 | −5.9 | −6.9 | −5.4 | −12.8 |

Long-term rate | −1.2 | 0.9 | −5.9 | −4.1 | −1.9 | −8.9 |

Earnings-by-price | −0.3 | 0.8 | −5.9 | 5.9 | 6.9 | 0.6 |

Inflation | −0.2 | 3.0 | −6.0 | −5.3 | −2.0 | −11.4 |

**Table 5.**Predictive power for the one-year excess stock returns ${Y}_{t}^{\left(A\right)}$. The prediction problem is defined in (2). The predictive power (%) is measured by ${R}_{V}^{2}$ as defined in (3). The benchmarks ${B}^{\left(A\right)}$ considered are based on the short-term interest rate ($A\equiv R$), long-term interest rate ($A\equiv L$), earnings-by-price ratio ($A\equiv E$), and consumer price index ($A\equiv C$). The predictive variables ${X}_{t-1}$ used are given by the short-term interest rate ${r}_{t-1}$, long-term interest rate ${l}_{t-1}$, inflation ${\pi}_{t-1}$, term spread ${s}_{t-1}$, debt-by-GDP ratio $dg{r}_{t-1}$ and consumption–wealth ratio ${\widehat{cay}}_{t-1}$. The predictive power (%) for the one-year stock returns $ln{S}_{t}$ validated on the original scale without benchmark is measured by ${\tilde{R}}_{V}^{2}$ as defined in (5) in terms of back-transformed stock return prediction $\widehat{ln{S}_{t}}$ defined in (4). The sample period is 1947–2020.

Benchmark ${\mathit{B}}^{\mathit{A}}$ | ${\mathit{X}}_{\mathit{t}-1}$ | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{R}}_{\mathit{V}}^{\mathbf{2}}$ | ${\tilde{\mathit{R}}}_{\mathit{V}}^{\mathbf{2}}$ | |||||||||||

r | l | $\mathbf{\pi}$ | s | dgr | $\widehat{\mathbf{cay}}$ | r | l | $\mathbf{\pi}$ | s | dgr | $\widehat{\mathbf{cay}}$ | |

Short-term rate | 3.7 | 2.1 | −1.0 | 11.4 | 6.9 | 3.0 | −2.6 | −4.3 | −7.6 | 5.6 | 0.9 | −3.2 |

Long-term rate | 2.6 | 2.4 | −1.0 | 8.3 | 6.0 | 3.7 | −0.2 | −0.4 | −3.9 | 5.7 | 3.3 | 0.9 |

Earnings-by-price | −0.8 | −1.7 | −1.6 | 8.7 | 1.7 | 0.0 | 5.4 | 4.6 | 4.6 | 14.3 | 7.8 | 6.1 |

Inflation | 0.5 | 0.0 | 1.7 | 9.6 | 3.7 | 3.8 | −4.6 | −5.1 | −3.3 | 5.0 | −1.3 | −1.2 |

**Table 6.**Predictive power for the one-year excess stock returns ${Y}_{t}^{\left(A\right)}$ in a two-dimensional setting. The prediction problem is defined in (2). The predictive power (%) is measured by ${R}_{V}^{2}$ as defined in (3). The benchmarks ${B}^{\left(A\right)}$ considered are based on the short-term interest rate ($A\equiv R$), long-term interest rate ($A\equiv L$), earnings-by-price ratio ($A\equiv E$), and consumer price index ($A\equiv C$). The different pairwise combinations of predictive variables, ${X}_{t-1}$ are given by the combination of third to fourth quarter growth rates of GDP $gd{p}_{t-1}$, industrial production $i{p}_{t-1}$ and personal consumption expenditures $co{n}_{t}-1$ with dividend-by-price ratio ${d}_{t-1}$, earnings-by-price ratio ${e}_{t-1}$, term spread ${s}_{t-1}$ and debt-by-price ratio $dp{r}_{t-1}$. The predictive power (%) for the one-year stock returns $ln{S}_{t}$ validated on the original scale without benchmark is measured by ${\tilde{R}}_{V}^{2}$ as defined in (5) in terms of back-transformed stock return prediction $\widehat{ln{S}_{t}}$ defined in (4). The sample period is 1947–2020.

Benchmark ${\mathit{B}}^{\mathit{A}}$ | ${\mathit{X}}_{\mathit{t}-1}$ | |||||||
---|---|---|---|---|---|---|---|---|

${\mathit{R}}_{\mathit{V}}^{\mathbf{2}}$ | ${\tilde{\mathit{R}}}_{\mathit{V}}^{\mathbf{2}}$ | |||||||

(gdp, d) | (gdp, e) | (gdp, s) | (gdp, dpr) | (gdp, d) | (gdp, e) | (gdp, s) | (gdp, dpr) | |

Short-term rate | 25.3 | 23.0 | 23.4 | 28.6 | 20.4 | 18.0 | 18.4 | 23.9 |

Long-term rate | 24.8 | 22.9 | 20.6 | 25.9 | 22.7 | 20.6 | 18.3 | 23.8 |

Earnings-by-price | 18.7 | 17.5 | 18.3 | 23.8 | 23.7 | 22.6 | 23.3 | 28.5 |

Inflation | 25.2 | 23.9 | 22.1 | 26.2 | 21.3 | 20.0 | 18.1 | 22.4 |

(ip, d) | (ip, e) | (ip, s) | (ip, dpr) | (ip, d) | (ip, e) | (ip, s) | (ip, dpr) | |

Short-term rate | 16.3 | 13.9 | 20.7 | 25.4 | 10.9 | 8.3 | 15.6 | 20.5 |

Long-term rate | 17.2 | 13.6 | 18.0 | 23.2 | 14.8 | 11.2 | 15.6 | 21.0 |

Earnings-by-price | 13.6 | 11.6 | 17.6 | 19.9 | 18.9 | 17.1 | 22.7 | 24.8 |

Inflation | 13.8 | 11.3 | 17.6 | 19.9 | 9.4 | 6.8 | 13.3 | 15.8 |

(con, d) | (con, e) | (con, s) | (con, dpr) | (con, d) | (con, e) | (con, s) | (con, dpr) | |

Short-term rate | 24.3 | 22.4 | 19.3 | 25.9 | 19.3 | 17.3 | 14.1 | 21.1 |

Long-term rate | 24.1 | 22.4 | 16.5 | 23.7 | 21.9 | 20.2 | 14.1 | 21.6 |

Earnings-by-price | 16.7 | 15.5 | 15.8 | 18.2 | 21.9 | 20.7 | 21.0 | 23.2 |

Inflation | 19.3 | 17.2 | 15.4 | 19.6 | 15.2 | 13.0 | 11.1 | 15.5 |

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Mousavi, P.
Debt-by-Price Ratio, End-of-Year Economic Growth, and Long-Term Prediction of Stock Returns. *Mathematics* **2021**, *9*, 1550.
https://doi.org/10.3390/math9131550

**AMA Style**

Mousavi P.
Debt-by-Price Ratio, End-of-Year Economic Growth, and Long-Term Prediction of Stock Returns. *Mathematics*. 2021; 9(13):1550.
https://doi.org/10.3390/math9131550

**Chicago/Turabian Style**

Mousavi, Parastoo.
2021. "Debt-by-Price Ratio, End-of-Year Economic Growth, and Long-Term Prediction of Stock Returns" *Mathematics* 9, no. 13: 1550.
https://doi.org/10.3390/math9131550