The Trump Rally, Really?

Today, The Dow Jones Industrial Average (DJIA) surpassed the 20,000 mark for the first time in history. At the time of the writing of this posting (12:31 PM on January 25), it is actually 20,058.29, so, I am not sure if it will close above 20,000 points, but, nevertheless, a lot of people are crediting this to Trump’s presidency, but I’m not so sure you can do that. First, the point must be made, that it is really the Obama economic policies that set the stage for this. On January 20, 2009, when Obama was sworn in, the Dow closed at 7949.089844 points. On November 8, 2016, when Trump won the election, the Dow closed at 18332.74023. So, during the Obama administration, the Dow increased by approximately 130.63%. I just wanted to make that point.

Now, the question that I wanted to investigate was would the Dow have closed past 20,000 points had Trump not been elected president. That is, assuming that the Obama administration policies and subsequent effects on the Dow were allowed to continue, would the Dow have surpassed 20,000 points.

For this, I looked at the DJIA data from January 20, 2009 (Obama’s first inauguration) to November 08, 2016 (Trump’s election). I specifically calculated the daily returns and discovered that they are approximately normally distributed using a kernel density method:

obamadowpdf

Importantly, one can calculate that the mean daily returns, \mu = 0.00045497596503813, while the volatility in daily returns, \sigma = 0.0100872666938282. Indeed, the volatility in daily returns for the DJIA was found to be relatively high during this period. Finally, the DJIA closed at 18332.74023 points on election night, November 08, 2016, which was 53 business days ago.

The daily dynamics of the DJIA can be modelled by the following stochastic differential equation:

S_{t} = S_{t-1} + \mu S_{t-1} dt + \sigma S_{t-1} dW,

where dW denotes a Wiener/Brownian motion process. Simulating this on computer, I ran 2,000,000 Monte Carlo simulations to simulate the DJIA closing price 53 business days from November 08, 2016, that is, January 25, 2017. The results of some of these simulations are shown below:

djiaclosingvaluesims

We concluded the following from our simulation. At the end of January 25, 2017, the DJIA was predicted to close at:

18778.51676 \pm 1380.42445

That is, the DJIA would be expected to close anywhere between 17398.0923062336 and 20158.94121. This range, albeit wide, is due to the high volatility of the daily returns in the DJIA, but, as you can see, it is perfectly feasible that the DJIA would have surpassed 20,000 points if Trump would not have been elected president.

Further, perhaps what is of more importance is the probability that the DJIA would surpass 20,000 points at any time during this 54-day period. We found the following:

probofexceeding

One sees that there is an almost 20% (more precisely, 18.53%) probability that the DJIA would close above 20,000 points on January 25, 2017 had Trump not been elected president. Since, by all accounts, the DJIA exceeding 20,000 points is considered to be an extremely rare/historic event, the fact that the probability is found to be almost 20% is actually quite significant, and shows, that it is quite likely that a Trump administration actually has little to do with the DJIA exceeding 20,000 points.

Although, this simulation was just for 53 working days from Nov 08, 2016, one can see that the probability of the DJIA exceeding 20,000 at closing day is monotonically increasing with every passing day. It is therefore quite feasible to conclude that Trump being president actually has little to do with the DJIA exceeding 20,000 points, rather, one can really attribute it to the day-to-day volatility of the DJIA!

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A Really Quick Derivation of The Cauchy-Riemann Equations

Here is a really quick derivation of the Cauchy-Riemann equations of complex analysis.

Consider a function of a complex variable, z, where z = x + iy, such that:

f(z) = u(z) + i v(z) = u(x+ iy) + i v(x+iy),

where u and v are real-valued functions.

An analytic function is one that is expressible as a power series in z.
That is,

f(z) = \sum_{n=0}^{\infty} a_{n} z^{n}, \quad a_{n} \in \mathbb{C}.

Then,

u(x+iy) + i v(x+iy) = \sum_{n=0}^{\infty} a_{n} (x+iy)^{n}.

We formally differentiate this equation as follows. First, differentiating with respect to x, we obtain

u_{x} + i v_{x} = \sum_{n=1}^{\infty} n a_{n} \left(x+iy\right)^{n-1}.

Differentiating with respect to y, we obtain

u_{y} + i v_{y} = i \sum_{n=1}^{\infty} n a_{n} \left(x + i y\right)^{n-1}.

Multiplying the latter equation by -i and equating to the first result, we obtain

-iu_{y} + v_{y} = \sum_{n=1}^{\infty} na_{n} \left(x+iy\right)^{n-1} = u_{x} + i v_{x}.

Comparing imaginary and real parts of these equations, we obtain

\boxed{u_{x} = v_{y}, \quad u_{y} = -v_{x}},

which are the famous Cauchy-Riemann equations.