The bond-stock earnings yield differential model: additional applications and other models for stock market crash prediction

Abstract

In this second part, we discuss the predictive ability of the BSEYD model, applications of the BSEYD to the USA in 2007, Iceland in 2008, the Chinese stock market in 2009 and in 2015, and introduce other crash measures. These measures include the price-to-earnings ratio, Robert Shiller's Cyclically adjusted price-to-earnings ratio, Warren Buffet's ratio of the market value of all publicly traded stocks to the current level of the GNP and Sotheby's stock price.

Keywords:

• Stock market crashes
• BSEYD
• CAPE
• P/E ratio
• S&P 500
• Shanghai Stock ExchangeComposite Index

JEL Classification:

• G10
• G11
• G12
• G14
• G15

1. Testing the predictive ability of the BSEYD Model

Lleo and Ziemba (2015c) test the predictive ability of the BSEYD model statistically on a 51-year period starting on January 1, 1962, and ending on December 31, 2014 (12  846 daily data points). The methodology is based on a standard likelihood ratio test conducted on a ‘hit sequence’ derived from the time series of signals. This approach is analogous to the run tests that are used to determine whether a coin is fair. The methodology is complemented by a Monte Carlo study for small sample bias and by an analysis of the robustness of the measure. Here we highlight some of the key results.

Table 1 shows that the S&P 500 index experienced 22 crashes between January 1, 1962 and December 31, 2014, where a crash is defined as a decline of at least 10% from the previous peak in less than a year. The three largest crashes were, respectively, a 56.8% decline from peak to trough in 2007–2009, a 48.2% decline in 1973–1974 and a 33.5% decline in 1987. Overall, 5 crashes exceeded 20% and 13 exceeded 15%.

Table 1. The S&P 500 Index experienced 22 corrections between January 31, 1964 and December 31, 2014.

	Crash Identification Date	Peak Date	S&P Index at Peak	Trough date	S&P Level at trough	Peak-to-trough decline (%)	Peak-to-trough duration (in days)
1	1966-05-16	1966-02-09	94.06	1966-10-07	73.2	22.2%	240
2	1968-03-05	1967-09-25	97.59	1968-03-05	87.72	10.1%	162
3	1969-06-19	1968-11-29	108.37	1969-07-29	89.48	17.4%	242
4	1971-08-04	1971-04-28	104.77	1971-11-23	90.16	13.9%	209
5	1973-04-27	1973-01-11	120.24	1974-10-03	62.28	48.2%	630
6	1975-08-08	1975-07-15	95.61	1975-09-16	82.09	14.1%	63
7	1977-05-25	1976-09-21	107.83	1978-03-06	86.9	19.4%	531
8	1978-10-26	1978-09-12	106.99	1978-11-14	92.49	13.6%	63
9	1979-10-25	1979-10-05	111.27	1979-11-07	99.87	10.2%	33
10	1980-03-10	1980-02-13	118.44	1980-03-27	98.22	17.1%	43
11	1981-08-24	1980-11-28	140.52	1981-09-25	112.77	19.7%	301
12	1984-02-13	1983-10-10	172.65	1984-07-24	147.82	14.4%	288
13	1987-10-15	1987-08-25	336.77	1987-12-04	223.92	33.5%	101
14	1990-01-30	1989-10-09	359.8	1990-01-30	322.98	10.2%	113
15	1990-08-17	1990-07-16	368.95	1990-10-11	295.46	19.9%	87
16	1997-10-27	1997-10-07	983.12	1997-10-27	876.99	10.8%	20
17	1998-08-14	1998-07-17	1186.75	1998-08-31	957.28	19.3%	45
18	1999-09-29	1999-07-16	1418.78	1999-10-15	1247.41	12.1%	91
19	2000-04-14	2000-03-24	1527.46	2001-04-04	1103.25	27.8%	376
20	2007-11-26	2007-10-09	1565.15	2009-03-09	676.53	56.8%	517
21	2010-05-20	2010-04-23	1217.28	2010-07-02	1022.58	16.0%	70
22	2011-08-04	2011-04-29	1363.61	2011-10-03	1099.23	19.4%	157

What is the track record of the BSEYD model? Table 2 shows that the BSEYD model based on a normal distribution assumption went into the danger zone on 38 distinct occasions. The prediction proved correct in 29 cases, giving a 76% accuracy. The number of predictions is higher than the number of crashes because several distinct crash signals may precede a given crash. The results for the BSEYD model based on Cantelli's inequality, which accounts for fat tails, are similar with 28 correct predictions out of 39 signals, a 72% accuracy.

Table 2. Proportions of correct and incorrect predictions for each signal model.

Signal model	Total number of signals	Number of correct predictions	Proportion of correct predictions (%)
(1)	(2)	(3)	(4)
Standard	38	29	76.32%
Cantelli	39	28	71.79%

Table 3 shows the results of the statistical test. The maximum likelihood estimate of the signal in column 2 is equal to the historical accuracy of the models displayed in Table 2. The likelihood ratio Λ in column 3 is the ratio of the likelihood under the null hypothesis that the BSEYD predictions are purely random, to the likelihood using the estimated probability . The estimated test statistics, equal to , are asymptotically -distributed with one degree of freedom. The p-value is the probability of obtaining a test statistic that is higher than the one actually observed, assuming that the null hypothesis is true. The degree of significance and the p-value indicated in the table are both based on the -distribution with one degree of freedom. The critical values at the 95%, 99% and 99% levels are 3.8415, 6.6349 and 7.8794, respectively. Observe that the BSEYD measure based on a standard confidence interval is significant at the 99% confidence level, and the BSEYD model based on Cantelli's inequality is not far from the 99% mark. The null hypothesis that the BSEYD predictions are purely random can be rejected: the BSEYD has the ability to predict crashes over a one- to two-year horizon.

Table 3. Maximum likelihood estimate and likelihood ratio test results.

Signal model	ML estimate		Likelihood ratio Λ	Test statistics	p-value
Standard	76.32%	9.25E−10	0.0000	20.9073	0.00%
Cantelli	71.79%	8.40E−10	0.0003	16.3034	0.01%

*Significance at the 5% level. **Significance at the 1% level. ***Significance at the 0.5% level.

In addition to testing eight choices of parameters for the BSEYD model, Lleo and Ziemba (2015c) also tested the predictive ability of the logarithm of the BSEYD, the P/E ratio and the logarithm of the P/E ratio. They found that, historically, the original BSEYD model and its logarithmic version have shown higher predictive abilities than the P/E ratio. Among the models tested, the logarithm of the P/E ratio showed the least consistency and robustness.

2. The Iceland 2008 crash

In July 2006 Ziemba was in Iceland speaking at the international INFORMS Operations Research conference and became interested in the economic situation there. Let us first look at the long bond (10-year) versus earnings yield differentials for the USA, the UK, Japan, France and Germany as of July 12, 2006 as given in Table 4. Ziemba and Ziemba (2007) and Lleo and Ziemba (2012) discuss the Iceland bubble and subsequent crash.

Table 4. Long-bond (10 yr) versus earning yield differentials for major countries, July 12, 2006.

	S&P500	FTSE200	Nikkei225	CAC40	DAX30
Index	1259	5861	15249	4942	5638
(A) PE ratio	16.86	16.61	36.26	13.82	13.33
(B) Stock Return (1/A)	5.93%	6.02%	2.76%	7.24%	7.50%
(C) Bond Return (10 yr)	5.10%	4.67%	1.94%	4.10%	4.09%
(D) Crash Signal (C-B)	−0.83	−1.35	−0.82	−3.14	−3.41

Source: Ziemba and Ziemba (2007).

In Iceland, the 15 stocks in the index have weights of 26.5% to 1%, see Figure 1(a). The main point is that the three largest banks Kaupthing (26.5%), Lansbanki (13.0%) and Ghitnir (12.3%) were more than half the market capitalization. In addition, Actavis Banki had 9.9%, and FL Banki a further 6.7%. So the banks are well towardstwo-thirds of the index value. And index funds that track the market slightly over weight these banks to yield higher returns.

Figure 1. The Icelandic stock market: (a) Stock Market Index. Source: Glitnir (2006b), (b) The Iceland Equity Index and its growth in real terms from 1997 to 2006. Source: Glitnir (2006b).

Figure 1(b) shows the dramatic rise of the stock market particularly since 2004. It also shows how quickly drops occur. However, the notable sharp sell-offs have to a large extent been blips and there is a question whether these investments can continue to produce similar returns, and if not, whether that will prompt investors to seek other markets.

Table 5 provides two bond-stock measure calculations in 2006. The index measure was out of the danger zone. However, the non-financial sector was in the danger zones but its weight was not enough to pull down the whole market according to the script of the bond-stock crash measure. The low PE plus the lower long-term interest rates (about 9%) but substantially below the 14.25% projected to 16% short rates made the market look risky. So did the parabolic increase in the index in real terms from 1997 to 2006 shown in Figure 1(b). But there was no signal yet for a crash.

Table 5. Bond-stock measure calculations in Iceland in 2006.

index	16 nonfin	15 in index
(A) PE ratio		11.1
(B) Stock Return (1/A)	6.13%	9.01%
(C) Bond Return (5 yr)	11.00%	9.4%
(D) Crash Signal (C-B)	4.87%	0.39%

Figure 2 shows the bond-stock earnings yield crash measure by month. Finally, the question of whether or not the bond-stock earnings yield model predicted the crash is studied in Figures 2(a), (c) and (e) which use 95% one-sided confidence intervals using moving averages. These graphs show that the crash was predicted in 2007. For Kaupthing, the danger zone was penetrated on September 28, 2007, two months after the July 18 peak and less than a month before November 11 crash. Finally, for Glitnir, the signal was much earlier on October 10, 2006, some thirteen months before the crash. Finally, for Lansbanki, the danger signal was on February 13, 2007. We focus on the largest banks because they led the market into the collapse. The smallest stocks were in the danger zone in 2006 as discussed in Ziemba and Ziemba (2007) but not the large banks then.

Figure 2. Crash indicators, Iceland. (a) Glitnir, MA, (b) Glitnir, MA, (c) Kaupthing, MA, (d) Kaupthing, MA, (e) Lansbanki, MA and (f) Lansbanki, MA.

3. The US 2007–2009 crash

We investigate whether or not the bond-stock measure did predict the US 2007–2009 crash, specifically September 2008 to March 2009 period. There are numerous books concerning this period plus many articles and columns, even Ziemba has several in Wilmott. Starting in June 2007, Ziemba designed strategies and traded for an offshore BVI-based hedge fund for a group headed by a top trader. He had investments in Bear Stearns and in June 2007 asked for his money back. That took three months and gave him a strong signal of danger. As an astute trader, he hedged and studied carefully the market situation through technical indicators that he has developed. Ziemba remembers his words starting in the summer of 2007 ‘this is the big one’ …‘eventually the market will go to 666 on the S&P500’. In the fall of 2007, the S&P500 was about 1550. This was a rather bold call but a private one and it turned out to be very accurate. Rachel Ziemba was working in New York for Nouriel Roubini's company, Roubini Global Economics and he was predicting very boldly a serious financial meltdown starting in 2006 when the housing market was beginning its decline; see Figure 3 which gives the Case Shiller Home Price Index as of July 24, 2008. We see a sharp decline from 2005 to 2008. As of April 2011, he and other bears such as Yale Professor Robert Shiller were still pessimistic about the economy, real estate and financial markets. Dropping real estate has several depressive effects such as homeowners can no longer use house price gains to fund consumption and foreclosures. The March 2009 low was 660 and the subsequent rally has doubled the S&P500 to the 1330 area at the beginning of April 2011 then to the 1840 area in February 2014 (Figure 4). There is lots of discussion regarding whether or not this rally was low interest rate related to the Fed quantitative easing, or only game in town since real estate, bonds and cash looked unattractive. This is a case when the BSEYD signaled the rise in stock prices (see Figure 5).

Figure 3. Case Shiller Index as of July 29, 2008.

Figure 4. S&P500 and ten-year Treasury bond yields. Source: Robert Shiller data. (a) S&P500 price earnings ratios and (b) Treasury bond yield.

Figure 5. Crash indicator (moving average, 95% confidence): (a) Evolution of the S&P500 vs. crash indicator, (b) Crash indicator.

4. Predicting crashes on the Chinese Stock Market

4.1. The 2007 Shanghai stock index crash

Lleo and Ziemba (2012) analyzed the crash of the Shanghai Stock Exchange in early 2008. Figure 6 shows the rise of the Shanghai stock index from January 4, 2000, to February 25, 2014. The market bottomed at 1011 on July 11, 2005, then rose six-fold to peak at 6092 on October 16, 2007. Next, there was a crash of 11.98% from 5180 to 4560 on January 21–22, followed by another 7.19% fall from 4762 to 4419 on January 28, 2008.

Figure 6. The Shanghai Stock Exchange Composite Index, January 2000 to February 25, 2014.

Figures 7 and 8 show that the BSEYD model succeeded in predicting the crash. The signal goes into the danger zone and the market continues to rise, before crashing less than 12 months after the initial signal. Figure 7 uses a 95% confidence one-sided moving average interval using prior data out-of-sample. The danger signal occurs on November 12, 2006, some 11 months before the stock market peaked on October 16, 2007. Figure 8 uses a 99% one-sided confidence interval and gives the first danger signal on June 29, 2007, with the index at 3821. The market reached its peak less than 4 months later, on October 16, 2007.

Figure 7. BSEYD crash indicator (95% one-sided moving average confidence interval): Shanghai Stock Exchange Composite. First signal occurs on December 25, 2006. The market reaches its peak on October 16, 2007.

Figure 8. BSEYD crash indicator (99% one-sided moving average confidence interval): Shanghai Stock Exchange Composite.

4.2. Predicting equity market crashes in China

Lleo and Ziemba (2016) shows that the return distribution of the SHCOMP has fat tails, which indicates that extreme events are more likely to occur than a Normal distribution would predict. In fact, they counted 26 market movements with cumulative returns of 10% or more and 24 market movements with losses of 10% or more in the 25 years since the SHCOMP started trading. Table 6 presents the 18 downturns that occurred between December 19, 1990 and October 31, 2015. On average, a downturn lasted 199 days and caused a 35.1% decline in the value of the SHCOMP.

Table 6. The SHCOMP Index experienced 18 crashes between December 19, 1990 and October 31, 2015.

	Crash Identification Date	Peak Date	SHCOMP Index at Peak	SHCOMP Level at 20% Cutoff	Trough date	SHCOMP Level at trough	Peak-to-trough decline (%)	Peak-to-trough duration (in days)
1	1992-06-03	1992-05-25	1421.57	1137.26	1992-11-17	393.52	72.3%	176
2	1993-03-01	1993-02-15	1536.82	1229.46	1993-03-31	925.91	39.8%	44
3	1994-09-30	1994-09-13	1033.47	826.78	1995-02-07	532.49	48.5%	147
4	1995-10-04	1995-05-22	897.42	717.94	1996-01-22	516.46	42.5%	245
5	1996-12-17	1996-12-09	1247.66	998.13	1996-12-24	865.58	30.6%	15
6	1997-07-02	1997-05-12	1500.40	1200.32	1997-09-23	1041.97	30.6%	134
7	1998-08-17	1998-06-03	1420.00	1136.00	1998-08-17	1070.41	24.6%	75
8	1999-12-22	1999-06-29	1739.21	1391.36	1999-12-27	1345.35	22.6%	181
9	2001-09-17	2001-06-13	2242.42	1793.94	2002-01-22	1358.69	39.4%	223
10	2003-01-10	2002-07-08	1732.93	1386.35	2003-03-26	1384.86	20.1%	261
11	2004-06-25	2004-04-06	1777.52	1422.01	2004-09-13	1260.32	29.1%	160
12	2005-08-12	2004-09-23	1464.78	1171.82	2005-12-05	1079.19	26.3%	438
13	2007-11-27	2007-10-16	6092.06	4873.65	2008-11-04	1706.70	72.0%	385
14	2009-08-31	2009-08-04	3471.44	2777.15	2009-08-31	2667.75	23.2%	27
15	2010-08-18	2009-11-23	3338.66	2670.93	2010-09-20	2588.71	22.5%	301
16	2011-08-08	2010-11-08	3159.51	2527.61	2012-01-05	2148.45	32.0%	423
17	2012-12-03	2012-03-02	2460.69	1968.55	2012-12-03	1959.77	20.4%	276
18	2015-06-29	2015-06-12	5166.35	4133.08	2015-08-26	2927.29	43.3%	75

Table 7 displays the results for four measures, calculated with a confidence interval based on a normal distribution:

1. BSEYD0: BSEYD based on current earnings;

2. logBSEYD0: logarithm of the BSEYD0 measure;

3. BSEYD10: BSEYD using average earnings over the previous 10 years.

4. logBSEYD10: logarithm of the BSEYD10 measure.

Table 7. Maximum likelihood estimate and likelihood ratio test for the BSEYD0, PE0, BSEYD10 and CAPE10 and their logarithm.

Signal Model	Total number of signals	Number of correct predictions	ML Estimate		Likelihood ratio Λ	Test statistics	p-value
BSEYD0	4	3	75.00%	1.0547e−01	0.9840	0.0322	85.76%
logBSEYD0	6	3	50.00%	1.5625e−02	0.5590	1.1631	28.08%
BSEYD10	3	2	66.67%	1.4815e−01	0.9868	0.0265	87.07%
logBSEYD10	5	3	60.00%	3.4560e−02	0.8712	0.2758	59.95%

*Significant at the 5% level; **Significant at the 1% level; ***Significant at the 0.5% level.

The results for a confidence interval based on Cantelli's inequality are identical and have been omitted.

Because the BSEYD10 and logBSEYD10 require 10 years of earnings data, and the Bloomberg data series for 10-year government bonds only starts on October 31, 2006, we cannot use the full range of stock market data. The analysis in this section covers the period between October 31, 2006 and September 30, 2015. Over this period, the SHCOMP experienced six declines of more than 20% of its value.

The accuracy of the measures reaches a low of 50% for logBSEYD0 and a high of 75% for BSEYD0. By comparison, the uninformed prior probability that a day picked at random will precede a crash identification date by 252 days or less is . Overall, the BSEYD-based models do not perform as well as on the American market (Lleo and Ziemba 2015b). One possible explanation for this paradox is that the BSEYD-based measures tend to produce a signal earlier than the P/E ratio. For example, if we double the lead-time of the measures from 252 to 504 days, the accuracy of the logBSEYD10 model improves to 100% with all six crashes predicted. However, the same adjustment does not produce a similarly spectacular improvement for the BSEYD0 and logBSEYD0 models.

5. Some additional crash measures

5.1. The price-to-earnings ratio

Practitioners have used the price-to-earnings (P/E) ratio to gauge the relative valuation of stocks and stock markets since at least the 1930s (Graham and Dodd 1934, for example,discuss the use of the P/E ratio in securities analysis and valuation). In this section, we analyze the predictive ability of the P/E ratio calculated using current earnings.

Table 8 reports the evolution of the Price-Earnings (P/E) ratio over selected 20-year periods with high annual returns. In each period the P/E ends 1.6 to 4.7 times higher than it started. Furthermore, there is a 90% correlation between the annual returns and the ending P/E ratio.

Table 8. Evolution of the P/E Over Selected 20-Year Periods With High Annualized Returns.

Beginning Year	Ending Year	Annual Rate of Return	Beginning P/E	Ending P/E
1975	1994	9.6%	10.9	20.5
1977	1996	9.7%	11.5	25.9
1942	1961	9.9%	12.2	20.5
1983	2002	10.9%	7.3	25.9
1978	1997	11.9%	10.4	31.0
1981	2000	12.8%	8.8	41.7
1979	1998	12.9%	9.4	36.0
1982	2001	13.0%	8.5	32.1
1980	1999	14.0%	8.9	42.1

Note: Source: Bertocchi, Schwartz, and Ziemba (2015).

5.2. The cyclically adjusted price-to-earnings ratio

The drawback of the P/E ratio calculated using current earnings is that it might be might be overly sensitive to current economic and market conditions. Graham and Dodd (1934) warned against this risk and advocated the use of a P/E ratio based on average earnings over 10 years. In their landmark survey, Campbell and Shiller (1988) found that the of a regression of log returns on the S&P 500 over a 10-year period against the log of the price-earnings ratio computed using average earnings over the previous 10 and 30 years is significant (see Lleo and Ziemba 2015b, for a review of the literature and a discussion of the key results.). This led Shiller to suggest the use of a Cyclically Adjusted Price-to-Earnings ratio (CAPE), or a price-to-earnings ratio using 10-year average earnings, to forecast the evolution of the equity risk premium (see Shiller 2015).

Lleo and Ziemba (2015b) showed that a crash measure based on Shiller's CAPE produces statistically significant crash predictions, although the performance of this measure is slightly lower than the BSEYD's.

5.3. Warren Buffett's ratio of the market value of all publicly traded stocks to the current level of the GNP

In an article co-authored with Carol Loomis (Buffett and Loomis 2001), Warren Buffet discussed the ‘ market value of all publicly traded securities as a percentage of the country's business - that is, as a percentage of GNP:’

The ratio has certain limitations in telling you what you need to know. Still, it is probably the best single measure of where valuations stand at any given moment. And as you can see, nearly two years ago the ratio rose to an unprecedented level. That should have been a very strong warning signal.

Buffett and Loomis (2001) follow up on an earlier interview (Buffett and Loomis 1999), discussing the Dot.Com bubble and stock (over)valuation.

The idea behind the ratio of the market value of all publicly traded stocks to the current level of the GNP (MV/GNP) is to gauge the total market value of companies against the value of the goods and services that these companies produce. The market value of all publicly traded US securities reflects the capacity of US firms to generate revenue, and translate these revenues into stable earnings. The US GNP represents the market value of all the products and services produced by US citizens and companies regardless of where they are produced. By contrast, the US GDP is the market value of all the products and services produced in the USA, regardless of who produced it. To illustrate, the production of Apple computer equipments in China would be part of the US GNP but not GDP, while the cars produced in the USA by Toyota would count in the US GDP but not GNP. This argument justifies the use of the GNP in the ratio (see Figure 9).

Figure 9. Warren Buffett's market value of all publicly traded stocks to the current level of the GNP (MV/GNP).

Lleo and Ziemba (2015a) showed that Warren Buffett's market value of all publicly traded securities as a percentage of GNP (MV/GNP), and its parent the lorgarithm of the market value of all publicly traded securities as a percentage of GNP (lnMV/GNP), can be a statistically significant predictors of future market downturns. However, for these measures to have predictive value, we need to transform into a signal using time-varying confidence-based thresholds similar to what is done with the BSEYD, rather than the fixed thresholds suggested by Warren Buffett.

Using an arbitrary threshold fixed at 120%, the MV/GDP would have signalled at most 2 out of the 19 equity market corrections that occurred between 1970 Q4 and 2015 Q1. Using a time-varying threshold raises the number of correct prediction to 8, with a 70% accuracy.

5.4. Sotheby's stock price

The Sotheby's stock price has proved an accurate predictor of bubbles and crashes, forecasting the Japan (1989), US technology (2000) and US housing (2006) bubbles (Figure 10). Sotheby's is a high end auction house. The idea is that Sotheby's wealthy clients will tend to sell some of the items that they have accumulated in the recent boom years near the peak. Sotheby's auction volume and stock performance soars right at the top only to collapse once the market implodes.

Figure 10. Sotheby's share price (1989–2015).

In February 2016, Sotheby's stock price was around $23. To signal a bubble, the stock price would need to be above $60.

6. Conclusion

The BSEYD has a remarkable track record of predicting major stock market crashes around the world over the past 50 years. The BSEYD draws its success form from two sources: the information contained by stock prices and earnings in relation to prevailing government bonds yields, and the signal construction which includes a time-varying probabilistic threshold.

Disclosure statement

No potential conflict of interest was reported by the authors.