# The Equity Curve and Its Relation to Future Stock Returns

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. The Term Structure of Expected Stock Returns Derived from Option Prices

#### 2.2. Hypotheses Derived from the Shape of the Equity Curve

**LEVEL**

**Hypothesis:**

**SLOPE**

**Hypothesis.**

**STEEPNESS**

**Hypothesis.**

#### 2.3. Limitations of the Approach

## 3. Empirical Results

#### 3.1. Data and Patterns of the Equity Curve

#### 3.2. Descriptive Analysis

#### 3.3. The Level of the Equity Term Structure and Subsequent Stock Returns

#### 3.4. The Slope of the Equity Term Structure and Subsequent Stock Returns

#### 3.5. Investment Strategies

#### 3.6. Alternative Definitions of the Slope of the Equity Curve

## 4. Conclusions

## Funding

## Conflicts of Interest

## Appendix A. Principal Component of Changes in the Equity Term Structure (Footnote 2)

Loadings | Explained Variance | Cumulative | |||||||
---|---|---|---|---|---|---|---|---|---|

1 m | 3 m | 6 m | 9 m | 1 y | 1.5 y | 2 y | |||

PC1 | 0.368 | 0.389 | 0.386 | 0.392 | 0.388 | 0.377 | 0.344 | 87% | 87% |

PC2 | 0.427 | 0.318 | 0.230 | 0.103 | −0.034 | −0.431 | −0.683 | 7% | 94% |

PC3 | −0.633 | −0.208 | 0.277 | 0.453 | 0.392 | −0.007 | −0.347 | 3% | 97% |

PC4 | 0.215 | −0.063 | −0.615 | −0.135 | 0.672 | 0.176 | −0.266 | 2% | 98% |

PC5 | 0.058 | −0.078 | −0.155 | 0.250 | 0.272 | −0.783 | 0.466 | 1% | 99% |

PC6 | 0.202 | −0.199 | −0.425 | 0.737 | −0.401 | 0.166 | −0.083 | 1% | 100% |

PC7 | −0.436 | 0.809 | −0.372 | 0.083 | −0.093 | −0.034 | 0.018 | 0% | 100% |

## Appendix B. Alternative Risk Aversion Coefficients (Footnote 3)

Panel A: Predictive Ability | |||||||||
---|---|---|---|---|---|---|---|---|---|

Slope | LOW Percentiles | HIGH Percentiles | LOW Minus HIGH | ||||||

5th | 10th | 25th | 25th | 10th | 5th | 5th | 10th | 25th | |

RA = 1 | 15.45% ** | 9.39% ** | 6.21% *** | −0.44% * | −0.05% | −4.64% ** | 20.08% *** | 9.43% *** | 6.65% *** |

RA = 2 | 12.38% ** | 5.72% ** | 4.57% *** | −6.00% *** | −8.46% *** | −11.17% ** | 23.55% *** | 14.18% *** | 10.58% *** |

RA = 5 | 11.38% ** | 6.59% ** | 2.45% ** | −5.53% *** | −7.74% *** | −8.99% ** | 20.36% *** | 14.33% *** | 7.97% *** |

RA = 10 | 14.53% ** | 8.91% ** | 0.60% * | −4.28% *** | −4.75% *** | −0.91% * | 15.44% *** | 13.66% *** | 4.87% ** |

Panel B: Investment Strategy | |||||||||

Slope | TIMING | STATIC | TC | OP after TC | c | c(t-stat) | |||

RA = 1 | 15.15% | 12.68% | 0.80% | 1.67% | 0.901 | 10.609 | |||

RA = 2 | 15.58% | 12.68% | 0.90% | 2.00% | 0.907 | 11.013 | |||

RA = 5 | 15.39% | 12.68% | 0.94% | 1.76% | 0.832 | 10.400 | |||

RA = 10 | 14.81% | 12.68% | 0.79% | 1.34% | 0.877 | 10.667 |

## References

- Ait-Sahalia, Yacine, and Andrew W. Lo. 2000. Nonparametric risk management and implied risk aversion. Journal of Econometrics 94: 9–51. [Google Scholar] [CrossRef] [Green Version]
- Black, Fischer, and Myron Scholes. 1973. The pricing of options and corporate liability. Journal of Political Economy 81: 637–54. [Google Scholar] [CrossRef] [Green Version]
- Bliss, Robert R., and Nikolaos Panigirtzoglou. 2004. Option implied risk aversion estimates. Journal of Finance 59: 407–46. [Google Scholar] [CrossRef]
- Brennan, Michael J. 1998. Stripping the S&P 500 index. Financial Analysts Journal 54: 12–22. [Google Scholar]
- Campbell, John Y., and Robert J. Shiller. 1988. Stock prices, earnings, and expected dividends. Journal of Finance 43: 661–76. [Google Scholar] [CrossRef]
- Cox, John C., and Stephen A. Ross. 1976a. A survey of some new results in financial option pricing theory. Journal of Finance 31: 383–402. [Google Scholar] [CrossRef]
- Cox, John C., and Stephen A. Ross. 1976b. The valuation of options for alternative stochastic processes. Journal of Financial Economics 3: 145–66. [Google Scholar] [CrossRef]
- Croce, Mariano M., Martin Lettau, and Sydney C. Ludvigson. 2015. Investor information, long-run risk, and the term structure of equity. Review of Financial Studies 28: 706–42. [Google Scholar] [CrossRef]
- Dimson, Elroy, Paul Marsh, and Mike Staunton. 2004. Irrational optimism. Financial Analyst Journal, 15–25. [Google Scholar] [CrossRef]
- Dumas, Bernanrd, Jeff Fleming, and Robert E. Whaley. 1998. Implied volatility functions: Empirical tests. Journal of Finance 53: 2059–106. [Google Scholar] [CrossRef] [Green Version]
- Estrella, Arturo, and Frederic S. Mishkin. 1998. Predicting U.S. recessions: Financial variables as leading indicators. Review of Economic Statistics 80: 45–61. [Google Scholar] [CrossRef]
- Estrella, Arturo, and Gikas Hardouvelis. 1991. The term structure as a predictor of real economic activity. Journal of Finance 46: 555–76. [Google Scholar] [CrossRef]
- Fama, Eugene F., and Robert R. Bliss. 1987. The information in long-maturity forward rates. American Economic Review 77: 680–92. [Google Scholar]
- Hasler, Michael, and Roberto Marfè. 2016. Disaster recovery and the term structure of dividend strips. Journal of Financial Economics 122: 116–34. [Google Scholar] [CrossRef]
- Jackwerth, Jens C. 2000. Recovering risk aversion from option prices and realized returns. Review of Financial Studies 13: 433–51. [Google Scholar] [CrossRef]
- Kang, Byung K., and Tong S. Kim. 2006. Option-implied risk preferences: An extension to wider classes of utility functions. Journal of Financial Markets 9: 180–98. [Google Scholar] [CrossRef]
- Kang, Byung K., Tong S. Kim, and Hyo S. Lee. 2014. Option-implied preference with model uncertainty. Journal of Futures Markets 34: 498–515. [Google Scholar] [CrossRef]
- Lettau, Martin, and Jessica A. Wachter. 2011. The term structures of equity and interest rates. Journal of Financial Economics 101: 90–113. [Google Scholar] [CrossRef] [Green Version]
- Litterman, Robert, and José Scheinkman. 1991. Common factors affecting bond returns. Journal of Fixed Income 1: 54–61. [Google Scholar] [CrossRef]
- Novosyolov, Arcady, and Daniel Satchkov. 2008. Global term structure modelling using principal component analysis. Journal of Asset Management 9: 49–60. [Google Scholar] [CrossRef]
- Rosenberg, Joshua V., and Robert F. Engle. 2002. Empirical pricing kernels. Journal of Financial Economics 64: 341–72. [Google Scholar] [CrossRef] [Green Version]
- Ross, Stephen. 2015. The recovery theorem. Journal of Finance 70: 615–48. [Google Scholar] [CrossRef] [Green Version]
- Rudebusch, Glenn D., and John C. Williams. 2009. Forecasting recessions: The puzzle of the enduring power of the yield curve. Journal of Business and Economic Statistics 27: 492–503. [Google Scholar] [CrossRef] [Green Version]
- Stoll, Hans R. 1969. The relation between put and call option prices. Journal of Finance 25: 801–24. [Google Scholar] [CrossRef]
- Treynor, Jack, and Kay Mazuy. 1966. Can mutual funds outguess the market? Harvard Business Review 44: 131–36. [Google Scholar]
- Van Binsbergen, Jules H., and Ralph S. Koijen. 2017. The term structure of returns: Facts and theory. Journal of Financial Economics 124: 1–21. [Google Scholar] [CrossRef] [Green Version]
- Van Binsbergen, Jules H., Hueskes Wouter, Ralph S. Koijen, and Evert B. Vrugt. 2013. Equity yields. Journal of Financial Economics 110: 503–19. [Google Scholar] [CrossRef]
- Vanguard. 2018. Available online: https://advisors.vanguard.com/web/cf/fas-investmentproducts/0968/overview (accessed on 18 April 2018).
- Welch, Ivo, and Amit Goyal. 2008. A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies 21: 1455–508. [Google Scholar] [CrossRef]

1 | Welch and Goyal (2008) analyse various ratios to forecast future stock returns using a predictive regression approach. They find that the predictive ability of many ratios is low. However, academics and practitioners alike use ratios to measure expected returns. |

2 | Results are shown in Appendix A. |

3 | Results are shown in Appendix B. |

**Figure 1.**The US equity curve for two days, a typical “normal” curve (July 2016) and a typical “crisis” curve (July 2010).

1 m | 3 m | 6 m | 9 m | 12 m | 18 m | 24 m | Slope 24 m–1 m | |
---|---|---|---|---|---|---|---|---|

Panel A: Recession periods | ||||||||

Median | 18.65% | 16.53% | 14.61% | 13.79% | 13.30% | 12.79% | 12.48% | −6.17% |

5th percentile | 11.52% | 11.64% | 11.23% | 10.27% | 9.45% | 8.14% | 7.39% | −4.12% |

95th percentile | 64.46% | 47.72% | 36.64% | 30.81% | 27.86% | 23.43% | 21.51% | −42.95% |

Standard dev. | 17.45% | 11.35% | 8.33% | 6.97% | 6.20% | 5.10% | 4.75% | |

Panel B: Non-recession periods | ||||||||

Median | 9.47% | 9.34% | 9.48% | 9.39% | 9.03% | 8.18% | 7.68% | −1.79% |

5th percentile | 4.49% | 4.87% | 5.42% | 5.83% | 5.97% | 6.04% | 5.85% | 1.36% |

95th percentile | 23.32% | 19.67% | 17.91% | 16.55% | 15.33% | 13.50% | 12.57% | −10.75% |

Standard dev. | 6.45% | 5.06% | 4.18% | 3.54% | 3.08% | 2.46% | 2.15% |

LOW Percentiles | HIGH Percentiles | HIGH Minus LOW | |||||||
---|---|---|---|---|---|---|---|---|---|

5th | 10th | 25th | 25th | 10th | 5th | 5th | 10th | 25th | |

Panel A: pth percentile of OIR | |||||||||

1 month | 4.54% | 5.08% | 6.70% | 16.12% | 21.90% | 27.94% | 23.40% | 16.82% | 9.42% |

1 quarter | 4.93% | 5.40% | 6.98% | 15.33% | 18.68% | 21.86% | 16.92% | 13.28% | 8.34% |

1 year | 6.00% | 6.30% | 7.48% | 12.76% | 14.86% | 16.33% | 10.33% | 8.56% | 5.27% |

Panel B: Median future return conditional on the pth percentile | |||||||||

1 month | 11.79% | 12.53% | 12.95% | 22.20% | 32.49% | 32.20% | 20.41% *** | 19.96% *** | 9.25% *** |

1 quarter | 17.35% | 16.92% | 15.56% | 22.32% | 29.66% | 26.58% | 9.23% *** | 12.74% *** | 6.76% ** |

1 year | 19.95% | 19.19% | 17.47% | 16.33% | 22.88% | 27.28% | 7.33% *** | 3.69% * | −1.14% |

Panel C: Mean future return conditional on the pth percentile | |||||||||

1 month | 8.78% | 8.25% | 6.51% | 14.72% | 25.38% | 23.27% | 14.49% *** | 17.13% *** | 8.21% *** |

1 quarter | 13.37% | 12.97% | 9.77% | 15.11% | 24.10% | 23.97% | 10.60% *** | 11.13% *** | 5.34% * |

1 year | 12.88% | 13.49% | 12.50% | 8.89% | 18.54% | 26.80% | 13.92% *** | 5.06% * | −3.60% |

LOW Percentiles | HIGH Percentiles | LOW Minus HIGH | |||||||
---|---|---|---|---|---|---|---|---|---|

5th | 10th | 25th | 25th | 10th | 5th | 5th | 10th | 25th | |

SLOPE | −16.26% | −10.98% | −5.99% | 0.53% | 1.82% | 2.16% | |||

Panel A: Median future return conditional on the pth percentile | |||||||||

1 month | 37.58% | 27.46% | 22.68% | 13.06% | 11.87% | 9.97% | 27.61% *** | 15.60% *** | 9.62% *** |

1 quarter | 30.15% | 26.50% | 22.66% | 16.42% | 16.02% | 13.78% | 16.38% *** | 10.48% *** | 6.24% ** |

1 year | 25.20% | 21.74% | 18.11% | 19.06% | 20.32% | 21.14% | 4.06% ** | 1.42% | −0.96% |

1 month–1 year | 12.38% ** | 5.72% ** | 4.57% *** | −6.00% *** | −8.46% *** | −11.17% ** | 23.55% *** | 14.18% *** | 10.58% *** |

Panel B: Mean future return conditional on the pth percentile | |||||||||

1 month | 29.59% | 19.50% | 14.20% | 5.64% | 6.79% | 4.82% | 24.77% *** | 12.71% *** | 8.56% *** |

1 quarter | 25.30% | 20.56% | 15.91% | 9.19% | 8.67% | 6.43% | 18.87% *** | 11.88% *** | 6.72% *** |

1 year | 24.42% | 17.85% | 11.47% | 12.42% | 14.00% | 14.40% | 10.02% ** | 3.85% * | −0.95% |

1 month–1 year | 5.17% ** | 1.65% * | 2.73% ** | −6.78% *** | −7.21% *** | −9.57% *** | 14.75% *** | 8.86% *** | 9.51% *** |

**Table 4.**Results from investment strategies (weights in stocks are displayed in the first six columns) which uses the slope of the equity curve to over-/under-weigh stocks.

LOW Percentiles | HIGH Percentiles | Ann. Performance | TM Regression | |||||||
---|---|---|---|---|---|---|---|---|---|---|

5th | 10th | 25th | 25th | 10th | 5th | TIMING | STATIC | c | c(t-stat) | |

SLOPE | −16.26% | −10.98% | −5.99% | 0.53% | 1.82% | 2.16% | ||||

(1) | 175% | 150% | 125% | 75% | 50% | 25% | 15.58% | 12.68% | 0.86 | 10.44 |

(2) | 175% | 150% | 125% | 100% | 100% | 100% | 15.18% | 12.68% | 0.80 | 10.81 |

(3) | 100% | 100% | 100% | 75% | 50% | 25% | 13.08% | 12.68% | 0.06 | 2.56 |

(4) | 175% | 100% | 100% | 100% | 100% | 100% | 14.71% | 12.68% | 0.95 | 11.88 |

(5) | 175% | 150% | 100% | 100% | 100% | 100% | 15.68% | 12.68% | 0.89 | 11.23 |

Reallocation Period | TIMING | STATIC | TC | OP after TC | c | c(t-stat) |
---|---|---|---|---|---|---|

1 | 15.58% | 12.68% | 0.90% | 2.00% | 0.907 | 11.013 |

2 | 15.38% | 12.69% | 0.45% | 2.25% | 0.871 | 12.094 |

3 | 14.64% | 12.68% | 0.28% | 1.68% | 1.163 | 13.712 |

4 | 14.65% | 12.69% | 0.21% | 1.75% | 0.687 | 8.764 |

5 | 14.25% | 12.70% | 0.17% | 1.38% | 0.923 | 11.996 |

6 | 14.69% | 12.68% | 0.14% | 1.87% | 0.697 | 7.257 |

7 | 14.69% | 12.68% | 0.13% | 1.88% | 0.656 | 6.527 |

8 | 14.78% | 12.68% | 0.11% | 1.99% | 0.759 | 7.860 |

9 | 14.78% | 12.69% | 0.09% | 2.00% | 0.703 | 6.027 |

10 | 14.37% | 12.77% | 0.09% | 1.51% | 0.611 | 6.205 |

Panel A: Predictive Ability | |||||||||
---|---|---|---|---|---|---|---|---|---|

SLOPE | LOW Percentiles | HIGH Percentiles | LOW Minus HIGH | ||||||

5th | 10th | 25th | 25th | 10th | 5th | 5th | 10th | 25th | |

2 y–1 m | 12.38% ** | 5.72% ** | 4.57% *** | −6.00% *** | −8.46% *** | −11.17% ** | 23.55% *** | 14.18% *** | 10.58% *** |

2 y–3 m | 14.53% *** | 9.75% *** | 2.40% ** | −4.96% *** | −7.79% *** | −9.37% *** | 23.90% *** | 17.54% *** | 7.36% *** |

2 y–6 m | 9.35% *** | 9.23% *** | 4.32% *** | −4.28% *** | −6.56% *** | −8.35% * | 17.70% *** | 15.79% *** | 8.60% *** |

2 y–1 y | 9.94% ** | 5.42% *** | 0.53% *** | −3.01% *** | −5.99% *** | −11.76% *** | 21.70% *** | 11.41% *** | 3.54% *** |

3 m–1 m | 10.60% * | 4.91% ** | 6.52% *** | −4.26% *** | −3.45% *** | 2.60% | 8.00% ** | 8.36% *** | 10.78% *** |

6 m–1 m | 16.53% * | 4.64% ** | 6.50% *** | −5.85% *** | −9.19% *** | −9.99% *** | 26.52% *** | 13.83% *** | 12.35% *** |

1 y–1 m | 14.63% ** | 4.10% ** | 4.28% *** | −6.71% *** | −8.57% ** | −10.53% *** | 25.16% *** | 12.68% *** | 10.99% *** |

Panel B: Investment Strategy | |||||||||

SLOPE | TIMING | STATIC | TC | OP after TC | c | c(t-stat) | |||

2 y–1 m | 15.58% | 12.68% | 0.90% | 2.00% | 0.907 | 11.013 | |||

2 y–3 m | 15.14% | 12.68% | 0.70% | 1.76% | 0.736 | 9.098 | |||

2 y–6 m | 14.70% | 12.68% | 0.62% | 1.39% | 0.661 | 8.082 | |||

2 y–1 y | 14.32% | 12.68% | 0.58% | 1.06% | 0.617 | 7.609 | |||

3 m–1 m | 16.37% | 12.68% | 1.31% | 2.38% | 0.957 | 10.929 | |||

6 m–1 m | 16.05% | 12.68% | 1.18% | 2.19% | 0.927 | 10.844 | |||

1 y–1 m | 15.44% | 12.68% | 1.01% | 1.75% | 0.925 | 11.120 |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Stotz, O.
The Equity Curve and Its Relation to Future Stock Returns. *J. Risk Financial Manag.* **2020**, *13*, 19.
https://doi.org/10.3390/jrfm13020019

**AMA Style**

Stotz O.
The Equity Curve and Its Relation to Future Stock Returns. *Journal of Risk and Financial Management*. 2020; 13(2):19.
https://doi.org/10.3390/jrfm13020019

**Chicago/Turabian Style**

Stotz, Olaf.
2020. "The Equity Curve and Its Relation to Future Stock Returns" *Journal of Risk and Financial Management* 13, no. 2: 19.
https://doi.org/10.3390/jrfm13020019