Simulating Deposit Account Openings

When desiging a set of simulations, you typically need to make 2 decisions.

  • The distribution that you will use to model the process being simulated
  • The statisitics (usually moments that you use to paraterize or fit the distribution )

Simulating the Number of Account Openings

Account openings over a period (say a month) is a random process whose value is a non-negative integer. That is, the number of openings in a month can be any of 0,1,2,3,4...

As such, this is classified as being a count process, which is typically fit using a Poisson Distribution.

The Poisson distribution has a single parameter, {% \lambda %} which determines the mean and variance of the distribution.
from scipy.stats import poisson # Set the expected average rate (mu / lambda) and the number of samples (periods) mu_value = 4.0 sample_size = 10 # Generate random numbers from the Poisson distribution random_numbers = poisson.rvs(mu=mu_value, size=sample_size) print(f"Nmber of accounts opened each period: {random_numbers}")

Simulating the Account Balance

The account balance for each newly opened account is also a random variable. Here, the variable can be any value greater than zero. (Although one could argue that there should be a reasonable bound on the upside).

A simple distribution with this property is the lognormal distribution.

To specify the lognormal distribution, you need to specify its mean and variance.
import numpy as np from scipy.stats import lognorm # 1. Define the parameters of the underlying normal distribution mu = 0.0 # Mean of the log-transformed data sigma = 0.5 # Standard deviation of the log-transformed data # 2. Map parameters to SciPy's notation shape_parameter = sigma # 's' in scipy scale_parameter = np.exp(mu) # 'scale' in scipy loc_parameter = 0 # Keep at 0 unless shifting the distribution # 3. Simulate random variables using .rvs() sample_size = 10000 simulated_data = lognorm.rvs( s=shape_parameter, loc=loc_parameter, scale=scale_parameter, size=sample_size, random_state=42 # Seed for reproducibility ) # Verification: Print the empirical mean vs theoretical expectation print(f"Empirical Mean: {np.mean(simulated_data):.4f}") print(f"Theoretical Mean: {np.exp(mu + (sigma**2)/2):.4f}")

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