Monte Carlo Simulation of Value at Risk in Python

If you recall the basics of the notebook where we provided an introduction on market risk measures and VAR, you will recall that parametric VAR simply assumes a distribution and uses the first two moments (mean and standard deviation) to compute the VAR; whereas for historical VAR, you use the actual historical data and use the specific datapoint (or interpolated values between 2 datapoints) for the confidence level.

VAR can also be computed via simulation. Which is a good way to provide a quick introduction to Monte Carlo simulation.

Simulated VAR at its core is quite simple. You basically take the moments (say mean and standard deviation if you assume a normal distribution), generate a simulated set of data with Monte Carlo simulation, and then get the required percentile. What this means is that we could also assume a non-normal distribution, say a t-distribution, and use that for simulation and to compute VAR.

First, let’s get market prices of AAPL from Quandl again, and compute the returns.

end = datetime.datetime.now()
start = end - datetime.timedelta(365)
AAPL = quandl.get('EOD/AAPL', start_date=start, end_date=end)

rets_1 = (AAPL['Close']/AAPL['Close'].shift(1))-1

We shall compute the mean and standard deviation of the AAPL returns first as we will use this later to perform Monte Carlo simulation.

mean = np.mean(rets_1)
std = np.std(rets_1)
Z_99 = stats.norm.ppf(1-0.99)
price = AAPL.iloc[-1]['Close']
print(mean, std, Z_99, price)

Out:
0.0016208298475378427 0.013753943856014762 -2.32634787404 220.79

Now, let’s compute the parametric and historical VAR numbers so we have a basis for comparison.

ParamVAR = price*Z_99*std
HistVAR = price*np.percentile(rets_1.dropna(), 1)

print('Parametric VAR is {0:.3f} and Historical VAR is {1:.3f}'
        .format(ParamVAR, HistVAR))

Out:
Parametric VAR is -7.064 and Historical VAR is -6.166

For Monte Carlo simulation, we simply apply a simulation using the assumptions of normality, and the mean and std computed above.

np.random.seed(42)
n_sims = 1000000
sim_returns = np.random.normal(mean, std, n_sims)
SimVAR = price*np.percentile(sim_returns, 1)
print('Simulated VAR is ', SimVAR)

Out:
Simulated VAR is  -6.7185294884

And that’s it! The full code can be found at the notebook here.

Monte Carlo Simulation of Value at Risk in Python

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