The bond-stock earnings yield model for stock market crash prediction: the basic idea and early applications

Abstract

We discuss the bond-stock earnings yield differential model (BSEYD) starting from when Ziemba first used it in Japan in 1988 through 2016 in various countries. The model has called many but not all crashes. Those have high interest rates in the most liquid long-term bonds relative to the trailing earnings-to-price ratio. In general, when the model is in the danger zone, almost always there will be a crash. The model called the 2000 and 2002 US crashes. A long horizon term study for the US, Canada, Japan, Germany, and the UK shows that being in the stock market when the bond-stock signal is not in the danger zone and in cash when it is in the danger zone provides a final wealth about double buy and hold for in these five countries.

Keywords:

• Stock market crashes
• BSEYD and Fed models
• Long-term investing

JEL Classification:

• G10
• G11
• G12
• G14
• G15

1. Introduction

This paper first discusses the bond-stock earnings yield differential model (BSEYD) crash prediction model that has worked well over time in the US, Japan and elsewhere. Other crash prediction models are discussed later in the paper and by Sornette and Zhou (2002), Sornette (2009), and Yan et al. (2012a,b). Jarrow et al. (2011) discuss when a bubble exists. Shiryaev et al. (2014, 2015) discuss a stopping rule model to exit and enter bubble type markets. The companion paper, Lleo and Ziemba (2016), discusses additional prediction applications.

In May 1988, Ziemba was invited by Yamaichi Securities to interview to be the first Yamaichi visiting professor of finance at the University of Tsukuba, a Japanese national university. Yamaichi wished to try to establish the study of finance, especially investments, in Japanese universities, which was not generally taught. They established a five-year program with five such visiting professors in succession. The teaching at the university (investments, security market anomalies, futures and options) was supplemented with a two-day a week consulting position in Tokyo some 60  km southwest of Tsukuba, with the research arm of Yamaichi Securities then the fourth largest securities firm in Japan and the sixth largest in the world. In his interview, he asked if he could study market imperfections (anomalies) and stock market crashes in two study groups with some of the young Yamaichi Research Institute employees who also came up to Tsukuba for my classes.

The proposal was accepted and each study group with about 10 eager young students in each group proceeded by me giving lectures on the US experience and they helped investigate the Japanese situation. They focused on the postwar period 1948–88 and much of what he learned appears in the book Invest Japan, Ziemba and Schwartz (1991) and the 1989–93 research papers of Ziemba and Schwartz and Stone and Ziemba (1993). Ziemba and his wife also wrote the book Power Japan (1992) that discussed the Japanese economy. Sandra had a pretty good idea right away that the Japanese policies that let to astronomically high land and stock prices and massive trade surpluses would lead to disaster and they would eventually lose most of the money that they received from selling cars, stereos and the like. We made a list of prestige buildings that the Japanese overpaid for in the 1987–89 era in Power Japan. Even at the height of their economic power in 1989 only 3% of Japanese assets were invested abroad.

The study groups started in August 1988 and ended a year later. Ziemba was asked to remain as a consultant for the fall of 1988 to complete a factor model discussed in Schwartz and Ziemba (2000) which was originally presented at a Berkeley Program in Finance meeting in Santa Barbara in September 1992. The factor model used anomaly ideas such as mean reversion, earnings surprise, momentum, price-earnings ratios, future earnings over price and value embedded in 30 variables to separate and rank stocks by their future mean performance from best to worst for all the stocks on the Tokyo Stock Exchange first section which was about 86% of the total capitalization. The model motivated by a similar model for the US by Jacobs and Levy (1988) was estimated yearly but updated monthly. The model performed well out of sample so was useful for hedge fund long-short trading as well as long-only investing. The hedge fund Buchanan Partners in London discovered the model which was discussed in Invest Japan when they bought the book and hired Ziemba to help them in their Japanese warrant trading which was largely long underpriced warrants and short overpriced stocks. Their trading was successful and the model, which was estimated using data during a stock market rise still worked when the decline came since variables such as earnings were the key drivers of the returns. An update of Japanese anomalies to 1994 appears in Comolli and Ziemba (2000).

2. The bond-stock equity return crash danger model

In the crash study group, Ziemba came up with a simple model in 1988 with only a single variable that being the difference between stock and bond rates of return.^† The idea was that stocks and bonds compete for investment dollars and, when interest rates are low, stocks are favored and when interest rates are high, bonds are favored. The main thing that he wished to focus on is that when the measure, the difference between these two rates, the long bond yield minus the earnings yield (the reciprocal of the price earnings ratio), was very large, then there was a high chance of a stock market crash. A crash was defined as a 10% fall in the index within one year from the start of the initial danger signal. The model explains the October 1987 crash. Indeed that application is how the idea evolved. Table 1 and figure  1 show this. The boxes indicate that there is extreme danger in the stock market because 30-year government bond yields are very much higher than usual stock market yields measured by the reciprocal of the previous year's reported price earnings ratio. These high interest rates invariably lead to a stock market crash. Here the danger indicator moved across a statistical 95% confidence line in April. The market ignored this signal but did eventually crash in October 1987. There was a similar signal in the US S&P500 around April 1999 and then a crash that began in August 2000 and a weak stock market in 2001/02 which is discussed below.

Figure 1. Bond and stock yield differential model for NSA, 1980–90.

Source: Ziemba and Schwartz (1991).

Table 1. S&P500 index, PE ratios, government bond yields and the yield premium over stocks, January 1984 to August 1988.

Returning to the story in Japan, in 1988–89, Ziemba asked one of the young colleagues in the crash study group, Sugheri Iishi, to check the accuracy of the bond-stock prediction model in Japan. They found that there were 20 10% plus crashes during the out of sample 40 years, 1949–89. Whenever this measure was in the danger zone (that is outside a 95% confidence band), there was a crash of 10% or more from the current level within one year. This was 12 out of 12, a splendid prediction record. Not all crashes had the measure in the danger zone but whenever it was there was a crash with no misses. Some eight crashes occurred for other reasons. Reinhart and Rogoff (2009) in their study of banking crises study some such circumstances that lead to stock market crashes of both interest rate- and non-interest rate-driven types. See also the classic book of Kindleberger and Aliber (2011) and Lleo and Ziemba (2015) who study hedge funds and bank trading disasters, how they occur and how they could be prevented.

So the measure was successful at predicting future crashes – but when and how deep there was no precise way to know. However, long-run mean reversion suggests that the longer the bull run is and the more over-priced the measure is, the longer and deeper the decline will probably be. Then one can use the measure as part of an econometric system to estimate future scenarios.

Each time the spread exceeded the 4.23 cutoff (which was higher than 95% confidence), there was a crash. The measure was way in the danger zone in late 1989 and the decline (the 21st crash) began on the first trading day of 1990 with the Nikkei stock average peaking at 38,916. See figure  1. It is too bad that Yamaichi's top management did not listen to Iishi when Ziemba sent him up to explain our results in Japanese; there was much greater danger in the market then they thought in 1989. By 1995, Yamaichi Securities was bankrupt and ceased to exist in 1999.

The model also indicated that the valuation was still high as of May 29, 1990, at 4.88. Not much later, the 22nd crash began. Interestingly, at the bottom of the 22nd crash on October 1, 1990, the NSA was at 20,222, which was almost exactly the mean. Meanwhile, the same calculation on May 29, 1990, for the S&P500 is shown in figure 2. Indeed, it was cheap, that is below the mean, since the September 1987 peak of 4.42. The May 29, 1990, value of 1.11 was, however, slightly above the mean level and the highest since the late fall of 1987.

Figure 2. Bond and stock yield differential model for the S&P500, 1980–1990.

Source: Ziemba and Schwartz (1991).

Japan has had weak stock and land markets for over twenty years, since the beginning of 1990. There are many factors for this that are political as well as economic. But the rising interest rates for eight full months until August 1990 is one of them. This extreme tightening of an over-levered economy was too much. Cheap and easily available money, which caused the big run-up in asset prices in the 1980s turned into expensive and unavailable money in the 1990s. This has parallels to the 2007–2009 US situation where easy available money, not necessarily cheap, turned into unavailable cheap money.

Despite the terrible earthquake and tsunami in Japan in March 2011, many analysts considered the stock market finally a buy because of very low valuations. They were proved right in 2013 with a large rally caused more by a lower yen than high earnings or low interest rates. With very low interest rates that are close to zero, the bond-stock earnings yield model is not in the danger zone. So the markets' attractiveness is from its PE ratio relative to other assets and markets. There are various ways that one can compute the upper and lower limits but experience has shown that with the various approaches, all of which use out of sample prior data, one usually has the same conclusion. In figure 2, the limits are simply the trailing mean plus or minus a standard deviation measure so the one-sided limit has 95% of the probability.^‡

Koivu et al. (2005) study the Fed model using a dynamic vector equilibrium correction model with data from 1980 to 2003 in the US, UK and Germany and show that the Fed model had predictive power in forecasting equity prices, earnings and bond yields. The model has been successful in predicting market turns, but in spite of its empirical success and simplicity, the model has been criticized. First, it does not consider the role played by time-varying risk premiums in the portfolio selection process while it does consider a risk-free government interest rate as the discount factor of future earnings. The inflation illusion (the possible impact of inflation expectations on the stock market) as suggested by Modigliani and Cohn (1979) is not taken into consideration. Second, the model assumes the comparability of earning price ratios, a real quantity, with a nominal, bond-induced, interest rate (Campbell and Vuolteenaho (2004), Asness (2000, 2003), and Ritter and Warr (2002) discuss these issues.) Consigli et al. (2009) propose a stochastic model of equity returns based on an extension of the model inclusive of a risk premium in which market corrections are endogenously generated by the bond-stock yield difference. The model accommodates both cases of prolonged yield deviations leading to a long series of small declines in the equity market and the case, peculiar of recent speculative bubbles, of a series of corrections over limited time periods. The inclusion of the yield differential as a key driver of the market correction process is tested and the model is validated with market data

Many of the critics focus on: (1) short-term predicability that we know is weak as does Giot and Petitjean (2008), (2) simply do not focus on the long-run value of the measure, or (3) dismiss it outright because of the real versus nominal versus real minor flaw as does Montier (2011). Consigli et al. (2009) use the model to estimate the current fair value of the S&P500. Of course, market and fair value can diverge for long periods. However, our concern is whether or not the model actually predicts stock market crashes, stock market rallies and good times to be in and out of stock markets. Berge et al. (2008) discuss the latter issue and found for five countries (the US, Germany, Canada, the UK and Japan) that the strategy stay in the market when it is not in the danger zone and move to cash otherwise provides about double the final wealth with less variance and a higher Sharpe ratio than a buy and hold strategy. There is some limited predictability of stock market increases but the evidence supports the good use of the model to predict crashes.

3. The 2000–2003 crash in the S&P500

The S&P500 was 470.42 at the end of January 1995. It was about 750 in late 1996 at the time of Alan Greenspan's famous speech on irrational exuberance in the US stock market. It peaked at 1527.46 on March 24, 2000, fell to 1356.56 on April 4, and then came close to this peak reaching 1520 on September 1, the Friday before Labor Day. The bond-stock crash model was in the danger zone virtually all of 1999 and it got deeper in the danger zone as the year progressed as the S&P500 rose from 1229.23 at the end of December 1998 to 1460.25 at the end of December 1999. The PE ratio was flat, increasing only from 32.34 to 33.29 while long bond yields rose from 5.47 to 6.69. The S&P500 fell to 1085 on September 7 prior to September 11, 2001.

Table 2 details this from January 1995 to December 1999. The spread reached three which was well in the 95% confidence danger band in April and rose to 3.69 in December 1999. The stage was set for a crash which did occur. Long-term mean reversion suggests that the 1996–2000 S&P500 values were too high relative to 1991–95 and a linear interpolation of the latter period gives a value close to that in 2002. Hong Kong was also in the danger zone.

Table 2. Bond and stock yield differential model for the S&P500, 1995–1999.

			a	b		b−c				a	b		b−c
Year	Month	S&P500 Index	PER	30-yr gov't bond	Return on stocks	Crash signal	Year	Month	S&P500 Index	PER	30-yr gov't bond	Return on stocks	Crash signal
1995	Jan	470.42	17.10	8.02	5.85	2.17	1997	Jul	954.29	23.67	6.78	4.22	2.56
	Feb	487.39	17.75	7.81	5.63	2.18		Aug	899.47	22.53	6.71	4.44	2.27
	Mar	500.71	16.42	7.68	6.09	1.59		Sep	947.28	23.29	6.70	4.29	2.41
	Apr	514.71	16.73	7.48	5.98	1.50		Oct	914.62	22.67	6.46	4.41	2.05
	May	533.40	16.39	7.29	6.10	1.19		Nov	955.40	23.45	6.27	4.26	2.01
	Jun	544.75	16.68	6.66	6.00	0.66		Dec	970.43	23.88	6.15	4.19	1.96
	Jul	562.06	17.23	6.90	5.80	1.10	1998	Jan	980.28	24.05	6.01	4.16	1.85
	Aug	561.88	16.20	7.00	6.17	0.83		Feb	1049.34	25.09	6.00	3.99	2.01
	Sep	584.41	16.88	6.74	5.92	0.82		Mar	1101.75	27.71	6.11	3.61	2.50
	Oct	581.50	16.92	6.55	5.91	0.64		Apr	1111.75	27.56	6.03	3.63	2.40
	Nov	605.37	17.29	6.36	5.78	0.58		May	1090.82	27.62	6.10	3.62	2.48
	Dec	615.93	17.47	6.25	5.72	0.53		Jun	1133.84	28.65	5.89	3.49	2.40
1996	Jan	636.02	18.09	6.18	5.53	0.65		Jul	1120.67	28.46	5.83	3.51	2.32
	Feb	640.43	18.86	6.46	5.30	1.16		Aug	97.28	27.42	5.74	3.65	2.09
	Mar	645.50	19.09	6.82	5.24	1.58		Sep	1017.01	26.10	5.47	3.83	1.64
	Apr	654.17	19.15	7.07	5.22	1.85		Oct	1098.67	27.41	5.42	3.65	1.77
	May	669.12	19.62	7.21	5.10	2.11		Nov	1163.63	31.15	5.54	3.21	2.33
	Jun	670.63	19.52	7.30	5.12	2.18		Dec	1229.23	32.34	5.47	3.09	2.38
	Jul	639.96	18.80	7.23	5.32	1.91	1999	Jan	1279.64	32.64	5.49	3.06	2.43
	Aug	651.99	19.08	7.17	5.24	1.93		Feb	1238.33	32.91	5.66	3.04	2.62
	Sep	687.31	19.65	7.26	5.09	2.17		Mar	1286.37	34.11	5.87	2.93	2.94
	Oct	705.27	20.08	6.95	4.98	1.97		Apr	1335.18	35.82	5.82	2.79	3.03
	Nov	757.02	20.92	6.79	4.78	2.01		May	1301.84	34.60	6.08	2.89	3.19
	Dec	740.74	20.86	6.73	4.79	1.94		Jun	1372.71	35.77	6.36	2.80	3.56
1997	Jan	786.16	21.46	6.95	4.66	2.29		Jul	1328.72	35.58	6.34	2.81	3.53
	Feb	790.82	20.51	6.85	4.88	1.97		Aug	1320.41	36.00	6.35	2.78	3.57
	Mar	757.12	20.45	7.11	4.89	2.22		Sep	1282.70	30.92	6.50	3.23	3.27
	Apr	801.34	20.69	7.23	4.83	2.40		Oct	1362.92	31.61	6.66	3.16	3.50
	May	848.28	21.25	7.08	4.71	2.37		Nov	1388.91	32.24	6.48	3.10	3.38
	Jun	885.14	22.09	6.93	4.53	2.40		Dec	1469.25	33.29	6.69	3.00	3.69

Source: Berge and Ziemba (2003)

The model for Japan was hard to interpret because there were high PE ratios but interest rates were close to zero so one had a close to 0–0 situation so the model did not apply to Japan in 1999. The model was not in the danger zone with return differences close to zero. We witnessed a dramatic fall in the S&P500 from its peak of 1527 in March 2000 to its September 17, 2000, low of 1085. Further declines occurred in 2001 and 2002. The lowest close to May 2003 was 768.63 on October 10, 2002. There was a lower close of 666 in March 2009, just before the big rally into 2016. This decline was similar to previous crashes.

4. Using the BSEYD model for long run investing

Berge and Ziemba (2003) and Berge et al. (2008) study this measure from 1970 and 1975 to 2000 in five major markets, namely the US, Germany, Canada, the UK and Japan.

They compare four strategies for each of the time periods based on the length of the sample from previous data (either 5 or 10 years) to determine the distribution type (historical or assumed normally distributed), the fractile for entry (70 %, 75 %, 80 % or 85%) and the fractile for exit (90 % or 95%), the confidence limits.

The 5-year data intervals were used for the 1980–2005 calculations and 10 years for the 1975–2005 calculations. The results vary slightly by strategy but the basic conclusions are the same. These results are summarized in table 3 for strategies 1 and 5 that have 5 and 10 years prior historical data, and 80% and 85% entry percentiles and 90% and 95% exit percentiles. All the results are in Berge et al. (2008) which also lists all the declines of 10%+ during this 20 year period.

Table 3. Evaluation of the performance of strategies 1 and 5 versus the market index in the US, Germany, Canada, UK and Japan from 1975 and 1980–2005.

		Overall Performance of the Strategies and the Stock Market
	Number of months	Mean log	Standard	Sharpe	Mean	Terminal
Strategy	in the index	return	deviation	ratio	excess return	value
US 1	319 (85,75%)	0.01194	0.03933	0.17814	0.00162	8,480.41
US Index (1975 2005)	372 (100%)	0.01032	0.04347	0.12401		4,649.75
US 5	273 (87,5%)	0.01254	0.03887	0.19773	0.00224	5,009.00
US Index (1980 2005)	312 (100%)	0.0103	0.04389	0.12405		2,490.17
G 1	302 (81.18%)	0.00881	0.0515	0.09157	0.00068	2,652.80
German Index (1975 2005)	372 (100%)	0.00813	0.05864	0.06886		2,061.27
G 5	279 (89.42%)	0.00963	0.05836	0.09405	0.00123	2,014.62
German Index (1980 2005)	312 (100%)	0.0084	0.06201	0.0687		1,373.26
C1	295 (79.3%)	0.00954	0.04243	0.08053	−0.0002	3,479.37
Canada Index (1975 2005)	372 (100%)	0.00974	0.04925	0.07337		3,743.49
C 5	273 (87.5%)	0.00937	0.04336	0.07839	0.00094	1,859.80
Canada Index (1980 2005)	312 (100%)	0.00843	0.04941	0.04977		1,387.23
UK 1	329 (88.44%)	0.01426	0.0502	0.14825	0.00102	20,163.11
UK Index (1975 2005)	372 (100%)	0.01324	0.05598	0.1147		13,787.40
UK 5	297 (95.19%)	0.01204	0.0442	0.12277	0.00101	4,275.28
UK Index (1980 2005)	312 (100%)	0.01103	0.04814	0.09179		3,121.47
J 1	222 (89.52%)	0.00612	0.05049	0.08427	0.00305	455.72
Japan Index (5/85 12/05)	248 (100%)	0.00307	0.05778	0.02085		213.88
J 5	172 (91.49%)	0.0012	0.0505	−0.00157	0.00168	125.35
Japan Index (5/90 12/05)	188 (100%)	−0.00048	0.05745	−0.03062		91.39

The initial wealth start at $100 (the US), €100 (Germany), C$100 (Canada), £100 (the UK), and 100¥ (Japan) and the terminal values are the gross performance using the strategy signals. The mean excess return is the average monthly excess return of the strategy over the stock market. In each country, the final wealth of the strategies exceeds (except for Canada) buy and hold for the stock market with some months in cash and a higher Sharpe ratio.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

† This difference model is a generalization of the ratio model, known as the Fed model. Koivu et al. (2005) study this model which seems to date from Fed minutes in 1996. See Yardeni (1997).

‡ Using a different index rather than the S&P500 has the same conclusion but slightly different results. Berge et al. (2008) used the MSCI index. The danger zone was entered in May 1987 and the correction occurred in October, four months later. During June, July and August, investors kept rebalancing their portfolios from the bond to the equity market (MSCI over the quarter) then the equity market fell 31.80% in the following quarter (September–November 1987) with the main decline in October.