### Do stock prices live in Mediocristan?: An Apple case study.

In his famous book Black Swan, Nassim Taleb introduced the concepts of Extremistan and Mediocristan (as two countries). He uses them as guides to define how predictable is the environment one’s studying. Mediocristan environments safely can use Gaussian distribution. In Extremistan environments, a Gaussian distribution is used at one’s peril. There are big fat tail distributions there.

In this case study, we looked at a specific stock, Apple (ticker: AAPL).

See the historical Apple price chart here:

We took the daily historical adjusted close prices, and then we calculated the daily %changes from it.

Let’s see how the distribution of the daily price %changes fit the Gaussian curve.

Since Apple IPO, we have 7000 days of data, which is 26 years.

The MATLAB code is not too difficult:

”

pChanges = closePrices(2:end) ./ closePrices(1:end-1) – 1;

aMean = mean(pChanges);

stDev = std(pChanges); % it uses the n-1 as a denominator

figure;

hold on; % plot 2 time series on each other

[nInBins, xout] = hist(pChanges, 600);

nInBins = nInBins ./ max(nInBins) .* 2.0 ; % convert the max to 2

bar(xout, nInBins);

x=-0.59:0.01:0.39;

y=gaussmf(x,[stDev aMean]); % generate Gaussian

plot(x,y)

”

The produced chart is here (you have to click it to see it properly in its full size).

What can we observe?

**The mean %change is 0.12%. The stDev is 3.02%.**

That looks quite a lot of standard deviation. It would mean that

– the price of AAPL changes more than 3% only 31% of the time (every 3rd day, ZScore = 1), or equivalently,

-the price of AAPL changes more than 6% 5% of the time. (every 20th day, ZScore = 2).

So, someone can argue that the stDev number: 3% shows that it is very volatile.

However, even this seemingly high volatility model cannot explain the AAPL real life price behaviour over the years.

Specifically, it cannot explain a -52% drop in a single day for example.

Let’s see some historical events in the Apple stock price:

1.

**Worst day: 2000-09-29: -52% single day loss.**

Apple had a grim earnings report on that day and it triggered many downgrades.

It was a brutal day for Apple:

”

Shares of the Cupertino, Calif.-based company** fell $27.75, or nearly 52 percent, to $25.75. Volume topped 132 million shares, more than 26 times the stock’s average daily volume of about 5 million shares.** Analysts at nearly a dozen financial institutions downgraded Apple and penned scathing reports on the company.

”

The generated Gaussian function (that fits to that mean and stDev) says that the probability of this is

P(-52% daily loss) = 2.4 * 10^(-65). (In a scientific notation it is: 2.4e-65).

It is a very, very small value.

**In average, this loss should occur every** 1/2.4*10^65 days. Let’s say, it realistically occurs every** 10^65 days.**

Just to illustrate how big value is this:

How many days old is the earth?

**Earth is 4.5B years old** that is 4,500,000,000 x 365 days = 4.5*10^9*365= 1.6*10^12 days.

So, **Earth is about 10^12 days old, and the event that Apple stock price drops -52% should occur every 10^65 days.**

**It shouldn’t have occurred in the lifetime of the Earth!**

Do you think there is a problem with the Gaussian mathematical model to describe financial data, or do you think the Gaussian function properly models real life events?

2.

**Second worst day: 1987-10-19, -25% single day loss.**

This was the famous Black Monday (1987) day when the Dow Jones dropped -22% on that single day. In itself, it was a Black Swan event.

The generated Gaussian function (that fits to that mean and stDev) says that the probability of this is

P(-25% daily loss) = 9.7 * 10^(-16). (in a scientific notation it is: 9.7e-16).

In average, this loss should occur every 1/9.7*10^16 days. Let’s say, it realistically occurs every 10^17 days.

(Again: Earth is 10^12 days old).

3.

**Best day: 1997-08-06, 33% single day gain.**

The event for the day was the following:

”

1997: Microsoft rescues one-time and future nemesis Apple with a $150 million investment that breathes new life into a struggling Silicon Alley icon

”

The generated Gaussian function (that fits to that mean and stDev) says that the probability of this is

P(33% daily gain) = 1.9 * 10^(-26). (in a scientific notation it is: 1.9e-26).

In average, this gain should occur every 1/1.9*10^26 days. Let’s say, it realistically occurs every 10^26 days.

(Again: Earth is 10^12 days old).

4.

**Second Best day: 1997-12-31, 24% single day gain.**

P(24% daily gain) = 2.7 * 10^(-14). (in a scientific notation it is: 2.7e-14).

**In average, this gain should occur every** 1/2.7*10^14 days. Let’s say, it realistically occurs **every 10^14 days.**

**(Again: Earth is 10^12 days old).**

In other words, **if Earth’s lifetime is 100x times bigger than it is, this event should occur only once, only on 1 day.**

Conclusion:

After this data, we contend that the price **time series of stocks doesn’t fit into the Gaussian model.**

Financial time series doesn’t belong to the world of Mediocristan. Unfortunately, the general mathematical models of risk that is used by banks, hedge funds or regulators are based on Gaussian distribution. We conclude that real life price series doesn’t work according to the mathematical model.

We would urge the investigation of other risk models: Levy-distribution, power laws or Mandelbrot’s fractals that we reckon would better fit real life data.

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