To calculate the variance of returns, follow these steps:

Formula for Variance:

$\text{Variance} = \sum_{i=1}^n P_i (R_i - \bar{R})^2$ where:

• $P_i$ is the probability of the state of the economy.
• $R_i$ is the rate of return in that state.
• $\bar{R}$ is the expected return.

Step 1: Calculate the Expected Return ($\bar{R}$)

$\bar{R} = \sum_{i=1}^n P_i \cdot R_i$ Substituting the values: $\bar{R} = (0.33 \cdot -10.10\%) + (0.36 \cdot 11.60\%) + (0.31 \cdot 22.60\%)$

Step 2: Calculate the Variance

1. Compute $(R_i - \bar{R})^2$ for each state.
2. Multiply by the probability $P_i$ for each state.
3. Sum the results.

Let me calculate this step-by-step. Given data import pandas as pd

Probabilities and returns

data = { "State": ["Recession", "Normal", "Boom"], "Probability": [0.33, 0.36, 0.31], "Rate_of_Return": [-10.10, 11.60, 22.60], # percentages }

Create a DataFrame

df = pd.DataFrame(data)

Calculate expected return

df["Weighted_Return"] = df["Probability"] * df["Rate_of_Return"] expected_return = df["Weighted_Return"].sum()

Calculate variance

df["Deviation"] = df["Rate_of_Return"] - expected_return df["Squared_Deviation"] = df["Deviation"] ** 2 df["Weighted_Squared_Deviation"] = df["Probability"] * df["Squared_Deviation"] variance = df["Weighted_Squared_Deviation"].sum()

expected_return, variance### Results:

1. Expected Return ($\bar{R}$): 7.849%
2. Variance: 178.8337

Explanation:

The variance was calculated using the probability-weighted squared deviations of the returns from the expected return. Let me know if you'd like me to explain the steps further or clarify the results.

Questions for Exploration:

1. How does the variance affect investment decisions?
2. What is the standard deviation for the given returns?
3. Can we visualize the deviation of each state's return from the expected return?
4. How would a change in probabilities affect the variance?
5. What are real-world examples where such variance computations are critical?

Tip:

The square root of variance gives the standard deviation, which is often easier to interpret in terms of investment risk.