Вивчайте Measuring Portfolio Risk and Return | Risk Analysis and Portfolio Optimization

Understanding how to measure the risk and return of a portfolio is a cornerstone of modern portfolio theory. When you invest in multiple assets, you need to know not just what each asset might earn, but also how their returns interact. The expected return of a portfolio tells you the average outcome you might anticipate, while risk—often measured as standard deviation—captures the uncertainty or variability around that average. Diversification, which means spreading investments across various assets, plays a crucial role in managing risk. By combining assets that do not move exactly together, you can often reduce the overall risk of the portfolio without necessarily sacrificing return.


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import numpy as np

# Expected returns for two assets
returns = np.array([0.08, 0.12])  # 8% and 12% expected returns
weights = np.array([0.6, 0.4])    # 60% in asset 1, 40% in asset 2

# Covariance matrix between the two assets
cov_matrix = np.array([
    [0.04, 0.018],   # variances and covariance
    [0.018, 0.09]
])

# Portfolio expected return
portfolio_return = np.dot(weights, returns)

# Portfolio standard deviation (risk)
portfolio_variance = np.dot(weights.T, np.dot(cov_matrix, weights))
portfolio_std = np.sqrt(portfolio_variance)

print("Expected Portfolio Return:", round(portfolio_return, 4))
print("Portfolio Standard Deviation (Risk):", round(portfolio_std, 4))

To calculate a portfolio's expected return, you multiply each asset's expected return by its portfolio weight and sum the results. For risk, the calculation is more involved: you must account for both the individual variances of each asset and the way the assets' returns move together, which is captured by their covariances. The formula for portfolio variance (risk) is:

Portfolio Expected Return: the sum of (weight of asset × expected return of asset) for all assets;
Portfolio Variance: the sum of (weight_i × weight_j × covariance between asset i and asset j) for all combinations of assets.

In the two-asset example above, the weights and expected returns are used to find the average return, while the covariance matrix is used to calculate how the assets’ returns interact, leading to the overall portfolio risk.


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import pandas as pd

# Expected returns for four assets
returns = pd.Series([0.07, 0.09, 0.13, 0.11], index=["A", "B", "C", "D"])
weights = pd.Series([0.25, 0.25, 0.30, 0.20], index=["A", "B", "C", "D"])

# Covariance matrix for four assets
cov_matrix = pd.DataFrame([
    [0.025, 0.010, 0.012, 0.009],
    [0.010, 0.040, 0.014, 0.011],
    [0.012, 0.014, 0.060, 0.016],
    [0.009, 0.011, 0.016, 0.035]
], columns=["A", "B", "C", "D"], index=["A", "B", "C", "D"])

# Portfolio expected return
portfolio_return = (weights * returns).sum()

# Portfolio variance and standard deviation
portfolio_variance = np.dot(weights, np.dot(cov_matrix, weights))
portfolio_std = np.sqrt(portfolio_variance)

print("Expected Portfolio Return:", round(portfolio_return, 4))
print("Portfolio Standard Deviation (Risk):", round(portfolio_std, 4))

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Секція 3. Розділ 1

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Suggested prompts:

Can you explain how diversification reduces risk in a portfolio?

What does the covariance matrix represent in these calculations?

How would the results change if the asset weights were different?

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import numpy as np

# Expected returns for two assets
returns = np.array([0.08, 0.12])  # 8% and 12% expected returns
weights = np.array([0.6, 0.4])    # 60% in asset 1, 40% in asset 2

# Covariance matrix between the two assets
cov_matrix = np.array([
    [0.04, 0.018],   # variances and covariance
    [0.018, 0.09]
])

# Portfolio expected return
portfolio_return = np.dot(weights, returns)

# Portfolio standard deviation (risk)
portfolio_variance = np.dot(weights.T, np.dot(cov_matrix, weights))
portfolio_std = np.sqrt(portfolio_variance)

print("Expected Portfolio Return:", round(portfolio_return, 4))
print("Portfolio Standard Deviation (Risk):", round(portfolio_std, 4))

Portfolio Expected Return: the sum of (weight of asset × expected return of asset) for all assets;
Portfolio Variance: the sum of (weight_i × weight_j × covariance between asset i and asset j) for all combinations of assets.


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import pandas as pd

# Expected returns for four assets
returns = pd.Series([0.07, 0.09, 0.13, 0.11], index=["A", "B", "C", "D"])
weights = pd.Series([0.25, 0.25, 0.30, 0.20], index=["A", "B", "C", "D"])

# Covariance matrix for four assets
cov_matrix = pd.DataFrame([
    [0.025, 0.010, 0.012, 0.009],
    [0.010, 0.040, 0.014, 0.011],
    [0.012, 0.014, 0.060, 0.016],
    [0.009, 0.011, 0.016, 0.035]
], columns=["A", "B", "C", "D"], index=["A", "B", "C", "D"])

# Portfolio expected return
portfolio_return = (weights * returns).sum()

# Portfolio variance and standard deviation
portfolio_variance = np.dot(weights, np.dot(cov_matrix, weights))
portfolio_std = np.sqrt(portfolio_variance)

print("Expected Portfolio Return:", round(portfolio_return, 4))
print("Portfolio Standard Deviation (Risk):", round(portfolio_std, 4))

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Дякуємо за ваш відгук!

Секція 3. Розділ 1

Measuring Portfolio Risk and Return

1. What is the formula for calculating the expected return of a portfolio?

2. How does diversification affect portfolio risk?

3. Why is standard deviation used as a measure of risk in finance?

Measuring Portfolio Risk and Return

1. What is the formula for calculating the expected return of a portfolio?

2. How does diversification affect portfolio risk?

3. Why is standard deviation used as a measure of risk in finance?