Déjà vu around the world

“We seem to have a once-in-a-lifetime crisis every three or four years” --Leslie Rahl, founder of Capital Market Risk Advisors(1)

What started as a mortgage crisis in the US quickly spread to nearly every corner of the financial system when Lehman Brothers collapsed, Merrill Lynch sold itself to Bank of America, and AIG became strapped for cash--all in a single week. These and the events that followed shook investor confidence to the core. Stock markets around the world plummeted as exemplified by the FTSE 100 falling 65% from September 2008 to March 2009.

As the markets for many assets became illiquid, and credit dried up for almost everyone who needed it, the Bank of England, the US Federal Reserve, the US Treasury and their counterparts around the world took dramatic steps to restore liquidity to asset markets, stimulate lenders to make loans again, and shore up investor confidence in equity markets in an attempt to avoid a deep global recession. Political and fiscal policy leaders here in the colonies helped sell their $700 billion (£440 billion) bailout package last autumn as an extraordinary remedy for a "once-in-a-century event". This was echoed in November by Henry Paulson, the former US Secretary of the Treasury, who said the meltdown was a "once- or twice-in-a-100-years event" and former Federal Reserve chairman Alan Greenspan who characterised the crisis as a "once-in-a-century credit tsunami".

There's little doubt aspects of this crisis are unique and the economy is facing its hardest challenge since the Great Depression, but are severe economic crises the rare events Paulson, Greenspan, et al have suggested? A study of capital market history around the world suggests not, and perhaps nowhere more clearly than the UK. While Americans think of the greatest decline in stock market history as occurring during the 1930s, for British investors the worst decline was in the 1970s. After taking into account the impact of inflation and even after reinvesting all dividends, the British stock market fell almost 74% from April to November 1972 and took nearly a decade to recover to its previous level.(2)

If £1 was invested at the end of 1969 in the MSCI UK Gross Return index, though it would have grown to the equivalent of 5.6 times in purchasing power by the end of May 2009 (adjusted for inflation), the record is peppered with several long and severe declines.(3)

Looking at the prosperous island nation at the other side of Eurasia, the story is even more frightening. Over the same near-40 year period, the Japanese stock market is still in its second extended period of decline; and this one began nearly 20 years ago!

Furthermore, the capital market histories of the UK and Japan are not unique. If we take a look at the largest inflation-adjusted declines in eight industrialised countries (including the UK and Japan) over the past four decades, all of the largest markets suffered a major decline over the period, which clearly illustrates that level of stock risk is high indeed.

Modelling risk: The standard model
With large prolonged declines occurring with such frequency, you’d think the standard risk models investors use to make their asset-allocation decisions would assign a significant probability these events would occur. Think again. To see why, we need to look at how these models were formed.

To help make sense of the highly complex capital markets, financial economists in the 1960s and 1970s developed a set of mathematical models of the markets. The best known of these models are the Capital Asset Pricing Model (CAPM) of expected returns and the Black-Scholes Option Pricing Model. Their creators won the Nobel Prize in economics for their ground-breaking work. Each of these models is built on the assumption that the statistical distribution of market returns follows a 'normal', or bell-shaped, distribution.(4) And even though the historical data tells a different story, these models are firmly entrenched throughout the investment profession.

An alternative approach: Log-stable distributions
The distribution of monthly real total returns for the UK stock market from January 1970 to May 2009 shows that while in most months, the historical returns closely follow the curve, there are several months that have returns that fall far to the right or left of the lognormal curve. It is these outliers in the tails of the distributions that constitute both the opportunities and the risks of equity investing. This phenomenon is not unique to the UK market; rather, it is typical of equity markets throughout the world.

In the early 1960s, Benoit Mandelbrot, a mathematician teaching economics at the University of Chicago, was advising a doctoral student named Eugene Fama. Professor Mandelbrot had developed a statistical model for percentage changes in the price of cotton that had "fat tails". That is, the model assigned nontrivial probabilities to large percentage changes. In his doctorial dissertation, Fama applied Mandelbrot's model to stock prices and obtained promising results.(5) Until recently, however, the work of Mandelbrot and Fama had been largely ignored.(6)

In his dissertation, Fama assumed that the logarithm of stock returns followed a fat-tailed distribution called a "stable Paretian distribution," or stable distribution.(7) Hence, we refer to the resulting distribution of returns as a "log-stable distribution." If we lay the best-fitting log-stable distribution curve to the chart showing monthly total returns for the UK stock market from January 1970 to May 2009, though not perfect, we see that the log-stable model fits the historical distribution much closer than the log-normal both at the centre and the tails.

Risk measures
Our analysis of stock market drawdowns and return distributions strongly suggests that summarising risk with standard deviation omits much of the story. We expect to see modelling tools for advisers come to market in the near future that can account for large, prolonged drawdowns and fat tails.

One modelling approach that is currently used by some institutional money managers and risk analysts is to use fat-tailed models to develop measures of Value at Risk (VaR) and Expected Shortfall.(8) VaR describes the left tail in terms of how much capital can be lost over a given period of time. For example, a 5% VaR answers a question of the form: Having invested £10,000 there is a 5% chance of losing £'X' in 12 months. What is 'X'? Expected Shortfall is the expected loss of capital should VaR be breached and is therefore always greater than VaR.

VaR and Expected Shortfall depend on the investment horizon. Showing clients charts that depict this will help better communicate the risks of investing in various asset mixes over various time periods.

Conclusion
In every financial crisis, investors relearn the same message--there isn't a magic risk measure or model that can account for or predict every significant drop in the market. Economists and quantitative analysts have made incredible strides over the decades engineering new ways to explain the distribution of returns. These developments provide investors with valuable information to help them decide how to allocate their portfolios for any number of investing scenarios and mitigate risk. But they are not perfect.

As we've discussed, the record contains a much bumpier ride than many risk models would suggest. In addition to preparing clients’ portfolios for these occasional severe declines and taking other precautions, advisers would do well to keep reminding their clients of the risks they face as investors. Clients should be fully prepared to take on the 100-year floods they will surely face in the future.

This is an updated version of Paul Kaplan's original Déjà vu all over again article, which was first published June 4, 2009. An additional version of this updated article was published in InvestmentAdviser on October 5, 2009.

Footnotes:

1 As quoted by Christopher Wright, “Tail Tales,” CFA Institute Magazine, March/April 2007.

2 I obtained the historical monthly total returns and inflation from Morningstar EnCorr, an institutional asset-allocation software and data package.

3 We use a logarithmic scale for all growth of $1 charts.

4. For returns to follow a lognormal distribution means that logarithm one plus the return in decimal follows a normal distribution.

5. For an account of the work of Mandelbrot and Fama during this period, see Benoit Mandelbrot and Richard L. Hudson, The (Mis)Behavior of Markets, New York: Basic Books, 2004.

6. The idea of using fat-tailed distributions to model asset returns is starting to gain some traction. FinAnalytica was founded to provide investment analysis and portfolio construction software based on Mandelbrot and Fama’s work. Morningstar added distribution charts and forecasting models based on it to Morningstar EnCorr.

7. Strictly speaking, the assumption is that the logarithm of one plus the return in decimal form follows a stable Paretian distribution.

8. Expected Shortfall is also known as Conditional Value at Risk or CVaR.