Learn Risk Simulation: Value Ranges and Probabilistic Outcomes | Monte Carlo Simulation for Uncertainty and Risk

Understanding and managing risk is a central challenge in many fields, from finance to engineering. In simulation modeling, risk simulation involves exploring how uncertain variables can lead to a range of possible outcomes. By modeling value ranges and quantifying the likelihood of extreme events, you can gain valuable insights into the potential risks of a system or investment. Rather than relying on single-point estimates, risk simulation empowers you to make decisions under uncertainty by considering the full distribution of possible results.


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import numpy as np

# Simulate daily returns: assume 0.1% mean return, 2% standard deviation, 10,000 days
np.random.seed(42)
mean_return = 0.001
std_dev = 0.02
num_days = 10000
returns = np.random.normal(mean_return, std_dev, num_days)

# Calculate probability of a daily loss exceeding 3%
loss_threshold = -0.03
extreme_losses = returns < loss_threshold
prob_extreme_loss = np.mean(extreme_losses)

print(f"Simulated probability of a daily loss exceeding 3%: {prob_extreme_loss:.4f}")

The simulation above generates a large sample of daily returns and calculates how often losses greater than 3% occur. The resulting probability quantifies the risk of rare, extreme losses—sometimes called "tail risk." When interpreting such probabilities, it's important to recognize that even low-probability events can have significant impacts, especially in fields like finance, engineering safety, or project management. By understanding the likelihood of these rare events, you can make more informed decisions about risk management strategies, such as setting aside reserves, adjusting exposure, or implementing safeguards to reduce the impact of adverse outcomes.

Task

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You will implement a function random_walk_risk that estimates the probability that a random walk drops below a given threshold.

A random walk of length n_steps starts at 0. At each step, you add a normally distributed increment with mean 0 and standard deviation 1.

Your task:

Simulate the random walk n_simulations times.
For each simulation, record the minimum value reached.
Estimate the probability that the minimum value is below threshold.
Return this probability as a float.

Use only NumPy for the computations.

Solution

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Section 2. Chapter 3

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              123456789101112131415
            
import numpy as np

# Simulate daily returns: assume 0.1% mean return, 2% standard deviation, 10,000 days
np.random.seed(42)
mean_return = 0.001
std_dev = 0.02
num_days = 10000
returns = np.random.normal(mean_return, std_dev, num_days)

# Calculate probability of a daily loss exceeding 3%
loss_threshold = -0.03
extreme_losses = returns < loss_threshold
prob_extreme_loss = np.mean(extreme_losses)

print(f"Simulated probability of a daily loss exceeding 3%: {prob_extreme_loss:.4f}")