Markets tanked yesterday after The Fed announced that it would begin “tapering” its policy of purchase of roughly $85 billion in bonds per month known as Quantitative Easing (“tapering” is in quotes because news outlets made such a big deal about Bernanke’s word choice). The U.S. stock market saw widespread declines; the NASDAQ lost over 2%, and the Dow realized similar declines.

At first, a 2% decrease in the NASDAQ (a traditionally tech-heavy market index that was started in 1971) doesn’t seem so bad. But if we employ some basic statistics we can see how unusual such a decrease is.

The first step is to get an idea for the basic shape of the distribution of NASDAQ returns. I used daily adjusted close price data from Yahoo! Finance over the range Feb. 5, 1971 to June 20, 2013. To compute returns, I took the Natural Logarithm of the quotient of the adjusted close in a given period divided by the adjusted close in the previous period. This formula yields the continuously compounded return, which allows us to more easily compute descriptive statistics. If we had taken the discrete return (i.e.( Price at time t – Price at time t-1)/Price at time t-1) then we can’t simply average the returns over a longer period for the periodic return. (note: the discrete return will **always **be greater than the continuously compounded return, but not by a lot). Below is a histogram of NASDAQ returns with an arrow indicating the return realized on June 20th after Bernanke’s speech.

The bins aren’t appropriately sized, but I was pressed for time. Anyway, we can see that the histogram of returns is approximately normally distributed. If you don’t believe me, a Chi-Square goodness-of-fit test yielded a P-value of virtually 0 and a Chi-Square test statistic of 10,687. The null hypothesis in this case is that the distribution of NASDAQ returns is NOT normally distributed. Clearly, we reject the null hypothesis with a very high degree of confidence in favor of the alternative hypothesis, which says that the NASDAQ returns fit the normal distribution.

Now that we know NASDAQ returns are normally distributed (which is equivalent to saying NASDAQ prices are log-normally distributed) we can easily compute the likelihood of realizing a return as low or lower than the -2.308% realized on June 20th. Given the mean daily return of 0.032899% with a standard deviation of 1.25792%, the probability is equal to:

Pr(X<-2.308%) = Pr(Z<-1.85) = 0.0313753 = 3.138%

In the second step, I just standardized the continuous random variable X = daily return. It is clear that a loss of this magnitude is relatively rare but not unheard of. In fact, it has occurred 400 times since the inception of the index.

In addition to stock market decreases, bond yields increased to their highest level since March of last year, with Treasury Notes pushing above 2.3%. Yields go up when the price of bonds go down.

It’s clear that investors aren’t thrilled about the Fed’s decision to stop pumping liquidity into the financial system. This could be because the Fed has decided to implement these changes on the basis of what many have called overly optimistic forecasts of economic conditions in the coming year(s). For example, the Fed has projected a 6.5-6.8% unemployment rate by the end of 2014, a figure many people (including the chief economist at Goldman Sachs) don’t believe.

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