Zhou Zhu Once in One Hundred Years Crisis SSRN-Id1462790

8/4/2019 Zhou Zhu Once in One Hundred Years Crisis SSRN-Id1462790

1/9Electronic copy available at: http://ssrn.com/abstract=1462790

Why Is the Recent Financial Crisis

a Once-in-a-century Event?

Guofu ZhouWashington University in St. Louis

and

Yingzi ZhuTsinghua University

First draft: June 2009

Current version: August 2009

We are grateful to Heber Farnsworth and Jack Strauss for valuable suggestions.

Corresponding Author: Guofu Zhou, Olin School of Business, Washington University, St. Louis, MO 63130.Phone: (314) 935-6384 and e-mail: [email protected]


2/9Electronic copy available at: http://ssrn.com/abstract=1462790

Why Is the Recent Financial Crisis

a Once-in-a-century Event?

In the recent financial crisis, the Dow Jones stock market index dropped about 54% from a high

of 14164.53 on October 9, 2007 to a low of 6547.05 on March 9, 2009. Alan Greenspan calls this

a once-in-a century crisis. While we do not know how he drew his conclusion, we show that the

probability of a stock market drop of 50% from its high within a century is about 90% based on the

popular random walk model of the stock prices. With a broad market index of the S&P500 and a

more sophisticated asset pricing model which captures more risks in the economy, the probability

rises to above 99%. The message of this paper is that a market drop of 50% or more is very likelyin long-run stock market investments, and the investors should be prepared for it.


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The Drawdown Probability

In the recent financial crisis, the Dow Jones stock market index dropped about 54% from a high

of 14164.53 on October 9, 2007 to a low of 6547.05 on March 9, 2009. Historically, financial crises,

including banking and currency crises, have occurred world-wide many times, and so there has

been a huge literature on them. Allen and Gale (2007) provide an excellent summary of various

crises and various theories of crisis. However, it is notoriously difficult to predict when the next

crisis will happen and what magnitude it will be. More theories and empirical studies are called

for. Ironically, if a crisis can be predicted, rational economic agents and governments would react

accordingly to avoid or prevent it, and the crisis would have not occurred in the first place.

Alan Greenspan calls the recent crisis a once-in-a century crisis.1 While we do not know how

he drew his conclusion, it will be of interest to examine what is the frequency of a crisis of such a

magnitude occurring if we take a simple model for the movement of stock prices. Theoretically, we

need to find the drawndown probability, i.e., the probability that the price will drop from a peak to

a new bottom in a model. The simplest and most popular model for the stock price is the random

walk model which says that stock prices move like the random walk of a drunken person. The

market participants have exploited all available information about the stock market, and so the

markets next move is unpredictable. However, a drunken person drifts to home and the random

walk theory of the stock market does permit the assets to appreciate at their long-term mean.

The model is the basic assumption underlying many finance theories, one of which is the famous

Black-Scholes option pricing theory. The option pricing theory is the theoretical foundation for the

growth of derivative products, which are at least partially responsible for the current crisis. The

random model is also well known to investors due perhaps to Malkiels (1981) lucid explanation in

his million-copy bestseller book. In analyzing long-term wealth accumulation, Jacquier, Kane, and

Marcus (2003), and Kan and Zhou (2009) also base their analysis on the random walk model.

Assuming the random walk model for the Dow Jones index, we calibrate the long-term mean

and volatility to the data, and find that the drawdown probability for the Dow to drop over 50%

from its high is about 90% in 100 years. The result is, however, sensitive to our estimates of the

long-term mean and volatility, which are 7.4% and 18.2% per annum, respectively.2 For example, if

1The Wall Street Journal, October 24, 2008.

2We use the ex-dividend long-term mean since the drawdown probabilities are close either including or excluding

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the mean is estimated at 6% or 9%, the drawdown probability will be about 97% or 83%, remaining

very high.

There are two important issues that are of interest for further examination. First, while the

Dow is the most popular index, it is not the most representative of the market. Hence, it will be

of interest to examine also the S&P 500 index, which is a broad market index and the benchmark

for institutional money management. Interestingly, the S&P 500 index made a high of 1565.15 on

the same day as the Dow, and got a low of 676.53 on the subsequent same date too. However,

the S&P dropped about 57% from its high. This is the likely result since it contains more smaller

stocks which are generally less resilient to crisis than larger stocks. The second issue is that

the random walk model may be too simplistic to account for enough risks in the economy. It

will be of interest to see how the drawdown probability changes in a more complex model. In

asset pricing, the long-run risks model of Bansal and Yaron (2004) is perhaps currently the most

viable model that successfully explains simultaneously the equity risk premium puzzle, the risk-

free rate puzzle, the high level of market volatility, and many other stylized facts about the stock

market, consumption and dividend-price ratio. This model is further extended by Zhou and Zhu

(2009) to better capture macroeconomic volatility than a single volatility model, and to reconcile

simultaneously the large negative market variance risk premium, differing predictability in excess

returns, consumption, dividends, and stock market volatility. Under the extended Bansal-Yaron

model, we find the drawdown probability is over 99% with an array of possible values for long-term

mean and volatility of the S&P 500 index.

Computations

The random walk model of a stock price assumes that

St+dt = St + t+dt, (1)

which says that the stock price tomorrow is todays price plus a random noise. This is a pure

random walk that has an unrealistic zero mean and can be theoretically negative. To account for

the long-term growth of the stock market, we can add a drift term. To avoid negative values, we

can assume the log price, rather than the price, follows the random walk model. Hence, we obtain

dividends, and since the published index does not contain dividends.

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a modified model,dStdt

= dt + dWt, (2)

where is the drift or long-term mean, capturing the average growth over time dt, is the long-

term mean volatility, and dWt is a noise with unit variance known as Brownian Motion. This

model is the simplest model of stock prices, known as a geometric random walk with drift when

the prices are taken as snap shots in discrete time, or as a Geometric Brownian Motion when the

prices are viewed as to move continuously. The model is fundamental in finance, and is the basis

underlying many theories.

We are interested in computing the drawdown probability, from a top to a bottom, condi-

tional on the drawndown to be bigger than a certain level h, say h = 50%, over time horizon T.

Mathematically, the maximum drawdown of St is defined as

MaxDD(T; , ) = 1 DS(T; , )1, (3)

where

DS(T; , ) = supt[0,T]

sups[0,t]

Ss/St

.

Hence, we need to evaluate

P(h) = Prob(MaxDD(T; , ) h), (4)

given that the stock price St follows the random walk model of Equation (2).

The probability can be computed analytically based on a formula due to Magdon-Ismail, Atiya,

Pratap, and Abu-Mostafa (2004). To do so, let {n} be the positive solutions of the eigenvalue

condition

tan n =2

hn.

Then,

P(h) = 24n=1

n sin n42n +

2h2 2he

h

2

1 e

22nT

2h2

2T

22

+ L, (5)

where L is a certain function of , , h and T, and is zero in the domain of parameters of our

interest. Mathematically, n increases with order n, and sin n changes sign in alternate terms. As

a result, the series expression for the drawndown probability, Equation (5), converges fast, with an

order of convergence O(1/n2).

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With daily data on the Dow Jones index from May 26, 1896 to June 19, 2008, we estimate the

long-term mean as the sample average return excluding dividends, and the estimated volatility is the

sample standard deviation. The results are = 7.4% and = 18.2% per annum, respectively. The

drawndown probability for the Dow to drop 50% from its high is then P(0.5) = 93.45%. To examine

the sensitivity of the probability to the estimates, Table 1 reports the drawndown probabilities for

values of and around the estimates. For a given value of = 18%, the probability changes

from 98.62% to 83.05%, as goes up from 5% to 9%. On the other hand, for a given value of

= 7%, the probability changes from 60.75% to 99.67%, as goes up from 15% to 21%. Hence,

the probability is sensitive to both estimates for and , and much more so to the estimate for .

In practice, it is well known that the volatility estimate is very accurate, but the mean is difficult

to estimate. Therefore, we need to focus more on the results when = 18%, which suggest thatthe drawndown probability is high, at least 83% at the most optimistic value of .

Table 1: Drawdown Probability

The table presents the probability for the market to drop over 50% from its high to a bottom over100 years horizon under various parameter estimates for the long-term mean and the long-termvolatility .

0.05 0.06 0.07 0.08 0.09

Random Walk Model

0.15 84.14 73.41 60.75 47.70 35.680.18 98.62 96.93 93.97 89.38 83.050.21 99.95 99.86 99.67 99.26 98.49

Long-run Risks Model

0.15 99.70 99.60 99.40 99.10 98.400.18 100.00 99.90 99.90 99.90 99.700.21 100.00 100.00 100.00 100.00 100.00

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In their extension of the Bansal-Yaron model, Zhou and Zhu (2009) consider the following

dynamic system for the consumption and dividend growths, dCt/Ct and dDt/Dt,

dCt

Ct= ( + Xt)dt +V

1tc + V

2t(1 c)dZ

1t

dXt = Xtdt + x

V1tx + V2t(1 x)dZ2t

dDtDt

= (d + Xt)dt + d

V1td + V2t(1 d)dBt + dc

V1tc + V2t(1 c)dZ1t

+ dx

V1tx + V2t(1 x)dZ2t + dv

V1tdw1t

dV1t = 1(V1 V1t)dt + 1

V1tdw1t

dV2t = 2(V2 V2t)dt + 2

V2tdw2t, 0 < 1 < 2,

(6)

where Xt captures the long-run growth prospects of the economy, V1t and V2t are long-run and

short-run volatilities, and all dZ1t, dZ2t, dBt, dw1t and dw2t are noises. The first equation says that

shocks in both long-run Xt and short-run dZ1t drive the consumption growth and asset prices. The

second equation is an autoregression model for the long-run risk. The third equation states that the

dividend process is also affected by both long-run and short-run risks as well as by its covariation

with the consumption. The last two equations allow market long-run and short-run volatilities to

evolve stochastically over time.

There are now diverse sources of risk in the system that determines the stock prices. As a

result, even if the total long-term mean and volatility are those given in Table 1, the drawndownprobabilities are much higher than before. Indeed, with one exception, all of them are greater than

99%. The intuition is that, after factoring more risks in the economy, the probability of a 50%

market drop is greater. In a one-hundred years time, such an event almost surely to occur.

Conclusion

Alan Greenspan calls the recent crisis a once-in-a century crisis, which can be interpreted to mean:

a) the crisis is rare; and b) it can occur at century frequency. This is related to Bogles (2008)

insightful Black Swan characterization of the recent financial crisis that it is rare, unpredictable,

extremely consequential, and appears predictable (finding the causes) after the fact.3 However,

the Black Swan school does not provide any indication of how frequently a Black Swan should be

observed.3Taleb (2007), the originator of the Black Swan concept, provides a detailed discussion of the Black Swans.

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What Alan Greenspan says (as we interpreted above), and what this article contributes based

on both simple and complex asset pricing models, is that such a rare event has a high probability

to occur in a century, though just a small probability in a given year. This is like the occurrence of

an catastrophe that can strike a person to death. At any given day or year, it is a small probability

event. However, over a hundred years, such an event is almost surely to occur. In terms of stock

market investments, the long-term investors should have prepared for a market drop of over 50%.

Do not panic (the implied volatility hit over 80% during a few months in late 2008 signaling great

fear of investors).4 Asset pricing theories that provide guidance to investing do allow for such

events. Instead, they should follow Bogles investment wisdom that for most of us, I think, with

a good asset allocation very diversified, very low cost this is not a time to flee the stock market

or, for that matter, the bond market. (see Bogle and Sullivan (2009)).

4As of the writing of this paper on July 21, the market is 8915.94, about 36% up from its recent low. It becomeshugely costly for those investors who sold at the panic and then stayed out.

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References

Allen, Franklin, and Douglas Gale. 2009. Understanding Financial Crises. New York: Oxford

University Press.

Bansal, Ravi, and Amir Yaron. 2004. Risks for the long run: a potential resolution of asset

pricing puzzles, Journal of Finance, vol. 59, no. 4 (August):14811509.

Bogle, John C. 2008. Black Monday and Black Swans, Financial Analysts Journal, vol. 64, no. 2

(March/April):3040.

Bogle, John C., and Rodney N. Sullivan. 2009. Markets in Crisis, Financial Analysts Journal,

vol. 65, no. 1 (January/February):1724.

Jacquier, Eric, Alex Kane, and Alan Marcus. 2003. Geometric Mean or Arithmetic Mean: A

Reconsideration. Financial Analysts Journal, vol. 59, no. 6 (November/December):4653.

Kan, Raymond, Guofu Zhou. 2009. What Will the Likely Range of My Wealth Be? Financial

Analysts Journal, vol. 65, no. 4 (July/August):6877.

Magdon-Ismail, Malik, Amir F. Atiya, Amrit Pratap, and Yaser S. Abu-Mostafa. 2004. On the

Maximum Drawdown of A Brownian Motion. Journal of Applied Probability, vol. 41, no. 1(March):147161.

Malkiel, B., 1981. A Random Walk Down Wall Street. Norton, New York.

Taleb, Nassim Nicholas. 2007. The Black Swan: The Impact of the Highly Improbable. New York:

Random House.

Zhou, Guofu, and Yingzi Zhu, 2009. A Long-run Risks Model with Long- and Short-run Volatil-

ities: Explaining Predictability and Volatility Risk Premium, Working paper, WashingtonUniversity in St. Louis.

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Zhou Zhu Once in One Hundred Years Crisis SSRN-Id1462790

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