The stock exchange market typically responds to socio-economic conditions. The active participation of key market players undoubtedly creates liquidity in the market. There are other factors that influence variations in the market besides the large transactions that key market makers make. For instance, markets react to news of social events, economic events, natural disasters, pandemics, etc. In certain conditions, the effects of those events drive price movement drastically. Systematic investors (investors who base their trade execution on a system that depends on quantitative models) need to prepare for future occurrences to preserve capital by learning from near-real-life conditions. If you inspect industries that involve considerable risk, for example, aerospace and defense, you will notice that the subjects learn through simulation. Simulation is a method that involves generating conditions that mimic the actual world so that subjects know how to act and react in a condition similar to preceding ones. In finance, we deal with large funds. They habitually take risk into account when experimenting and testing models.

In this chapter, we implement a Monte Carlo simulation to simulate considerable changes in the market without exposing ourselves to risk. Simulating the market helps identify patterns in preceding occurrences and forecasts future prices with reasonable certainty. If we can reconstruct the actual world, then we can understand the preceding market behavior and predict future market behavior. In the preceding chapters, we sufficiently covered models for sequential pattern identification and forecasting. We use the panda_montecarlo framework to perform the simulation. To install it in the Python environment, use pip install pandas-montecarlo.

When an investor trades a stock, they expect their speculation to yield returns over a certain period. Unexpected events occur, and they can influence the direction of prices. To build sustained profitability over the long run, investors can develop models to recognize the underlying pattern of preceding occurrences and to forecast future occurrences.

Let’s assume you are a prospective conservative investor with an initial investment capital of $5 million US dollars. After conducting a comprehensive study, you have singled out a set of stocks to invest in. You can use Monte Carlo simulation to further understand the risks of investing in those assets.

Understanding Value at Risk

The most convenient way to estimate risk is by applying the value at risk (VAR). It reveals the degree of financial risk we are exposing an investment portfolio to. It shows the minimum capital required to compensate for losses at a specified probability level. There are two primary ways of estimating VAR; we can either use the variance-covariance method or simulate it by applying Monte Carlo.

Estimate VAR by Applying the Variance-Covariance Method

The variance-covariance method makes strong assumptions about the structure of the data. The method assumes the underlying structure of the data is linear and normal. It is also sensitive to missing data, nonlinearity, and outliers. To estimate the standard VAR, we first find the returns, and then we create the covariance matrix and find the mean value and standard deviation of the investment portfolio. Thereafter, we estimate the inverse of the normal cumulative distribution and compute the VAR. Figure 7-1 shows the VAR of a portfolio alongside the simulated distributions.

Here we show you how to calculate the VAR by applying the variance/covariance calculation.

Assume:

• Investment capital is $5,000,000.

• Standard deviation from an annual trading calendar (252 days) is 9 percent.

• Using the z-score (1.65) at 95 percent confidence interval, the value at risk is as follows:

• $5,000,0000*1.645*.09 = $740 250

Listing 7-1 scrapes the stock prices for Amazon, Apple, Walgreens Boots Alliance, Northrop Grumman Corporation, Boeing Company, and Lockheed Martin Corporation (see Table 7-1).

from pandas_datareader import data

tickers = ['AMZN','AAPL','WBA',

           'NOC','BA','LMT']

start_date = '2010-01-01'

end_date = '2020-11-01'

df = data.get_data_yahoo(tickers, start_date, end_date)[['Adj Close']]

df.head()

Listing 7-1

Scraped Data

Table 7-1

Dataset

Attributes	Adj Close
Symbols	AMZN	AAPL	WBA	NOC	BA	LMT
Date
2010-01-04	133.899994	6.539882	28.798639	40.206833	43.441975	53.633926
2010-01-05	134.690002	6.551187	28.567017	40.277546	44.864773	54.192230
2010-01-06	132.250000	6.446983	28.350834	40.433144	46.225727	53.396633
2010-01-07	130.000000	6.435065	28.520689	40.850418	48.097031	51.931023
2010-01-08	133.520004	6.477847	28.559305	40.624100	47.633064	52.768509

Listing 7-2 specifies the investment weights of the portfolio.

weights = np.array([.25, .3, .15, .10, .24, .7])

Listing 7-2

Specify Investment Weights

From then on we specify the initial investment capital in the portfolio. See Listing 7-3.

initial_investment = 5000000

Listing 7-3

Specify Initial Investment

Listing 7-4 estimates the daily returns (see Table 7-2).

returns = df.pct_change()

returns.tail()

Listing 7-4

Estimate Daily Returns

Table 7-2

Daily Returns

Attributes	Adj Close
Symbols	AMZN	AAPL	WBA	NOC	BA	LMT
Date
2020-10-26	0.000824	0.000087	-0.021819	0.004572	-0.039018	-0.015441
2020-10-27	0.024724	0.013472	-0.032518	-0.024916	-0.034757	-0.016606
2020-10-28	-0.037595	-0.046312	-0.039167	-0.028002	-0.045736	-0.031923
2020-10-29	0.015249	0.037050	-0.030934	-0.004325	0.001013	0.004503
2020-10-30	-0.054456	-0.056018	0.015513	-0.008790	-0.026300	-0.006554

Table 7-2 highlights the first five rows of each stock’s daily return. Listing 7-5 estimates the joint variability between the stocks in the portfolio (see Table 7-3).

cov_matrix = returns.cov()

cov_matrix

Listing 7-5

Covariance Matrix

Table 7-3

Covariance Matrix

Attributes		Adj Close
	Symbols	AMZN	AAPL	WBA	NOC	BA	LMT
Adj Close	AMZN	0.000401	0.000159	0.000096	0.000092	0.000135	0.000081
	AAPL	0.000159	0.000318	0.000098	0.000098	0.000161	0.000093
	WBA	0.000096	0.000098	0.000303	0.000097	0.000131	0.000088
	NOC	0.000092	0.000098	0.000097	0.000204	0.000165	0.000150
	BA	0.000135	0.000161	0.000131	0.000165	0.000486	0.000162
	LMT	0.000081	0.000093	0.000088	0.000150	0.000162	0.000174

Listing 7-6 estimates the VAR. First, we specify the average daily returns, then we specify the confidence interval and cutoff value, and finally we obtain the mean and standard deviations. Subsequently, we find the inverse of the distribution.

conf_level1 = 0.05

avg_rets = returns.mean()

port_mean = avg_rets.dot(weights)

port_stdev = np.sqrt(weights.T.dot(cov_matrix).dot(weights))

mean_investment = (1+port_mean) * initial_investment

stdev_investment = initial_investment * port_stdev

cutoff1 = norm.ppf(conf_level1, mean_investment, stdev_investment)

var_1d1 = initial_investment - cutoff1

var_1d1

166330.5512926411

Listing 7-6

Estimate Value at Risk

At a 95 percent confidence interval, the investment portfolio of $5,000,000 will not exceed losses greater than 166,330.55 a day. Listing 7-7 prints the 10-day VAR.

var_arry = []

num_days = int(10)

for x in range(1, num_days+1):

    var_array.append(np.round(var_1d1 * np.sqrt(x),2))

    print(str(x) + " day VaR @ 95% confidence: " + str(np.round(var_1d1 * np.sqrt(x),2)))

1 day VaR @ 95% confidence: 166330.55

2 day VaR @ 95% confidence: 235226.92

3 day VaR @ 95% confidence: 288092.97

4 day VaR @ 95% confidence: 332661.1

5 day VaR @ 95% confidence: 371926.42

6 day VaR @ 95% confidence: 407424.98

7 day VaR @ 95% confidence: 440069.27

8 day VaR @ 95% confidence: 470453.84

9 day VaR @ 95% confidence: 498991.65

10 day VaR @ 95% confidence: 525983.39

Listing 7-7

Print 10-Day VAR

Listing 7-8 graphically represents the VAR over a 10-day period (see Figure 7-2).

plt.plot(var_array, color="navy")

plt.xlabel("No. of Days")

plt.ylabel("MaX Portfolio Loss (USD)")

plt.show()

Listing 7-8

10-Day VAR

Figure 7-2 shows that as we increase the number of days, the maximum investment portfolio increases too. In the next section, we explored Monte Carlo simulation.

Plot Simulations

Listing 7-11 plots the 10 simulations that the Monte Carlo simulation ran (see Figure 7-3).

mc.plot(title="")

Listing 7-11

Simulations

Figure 7-3 shows the simulation results. It highlights a dominant upward trend. Listing 7-12 tabulates raw simulations (see Table 7-5).

pd.DataFrame(mc.data).head()

Listing 7-12

Raw Simulations

Table 7-5

Raw Simulations

	Original	1	2	3	4	5	6	7	8	9
0	0.000000	-0.018168	-0.002659	-0.021098	-0.006357	0.018090	-0.010110	-0.006555	-0.004119	0.020557
1	0.017005	0.005521	-0.014646	0.000541	-0.009134	-0.000602	-0.006801	-0.003373	0.004342	-0.002257
2	-0.004800	-0.003900	-0.002494	0.027738	0.005924	-0.012834	-0.004095	-0.003315	0.000116	0.113851
3	0.015401	0.006688	0.001144	0.005586	-0.004924	0.011399	0.009817	-0.001273	0.006164	0.023355
4	0.001072	0.005617	0.010052	0.011748	0.007878	0.010080	-0.008666	-0.005126	0.036459	0.015438

Listing 7-13 returns the statistics of the simulation model (Table 7-6).

ev = pd.DataFrame(mc.stats, index=["s"])

ev

Listing 7-13

Monte Carlo Statistics

Table 7-6

Monte Carlo Statistics

	min	max	mean	median	std	maxdd	bust	goal
s	2.094128	2.094128	2.094128	2.094128	2.145155e-15	-0.164648	0.2	0.8

Table 7-6 highlights the dispersion. It also highlights the maximum drawdown (the maximum amount of loss from a peak).