Consider the following stock price and no. of shares information: Stock No. of shares outstanding Stock prices at the end of 2022 (Rs) Stock prices at the end of 2023 (Rs) : : : : A 2,000 400 380 B…

a. Price-Weighted Index

The price-weighted index is calculated by summing the stock prices and dividing by a divisor. The divisor is typically the sum of the initial prices of the stocks in the index.

Step 1: Calculate the Price-Weighted Index for 2022

The formula for the price-weighted index is:

$$PW{I}_{2022}=\frac{\sum {\text{Stock Prices}}_{2022}}{\text{Number of Stocks}}$$

Since the divisor is not provided, we assume it is the sum of the initial prices (2022 prices) divided by the number of stocks (3). However, in practice, the divisor is often fixed to avoid manipulation. For simplicity, we will use the sum of the stock prices directly as the index value (since the divisor cancels out in percentage change calculations).

Sum of stock prices in 2022: $$400+150+700=1250$$

Thus, the Price-Weighted Index for 2022 is 1250.

Step 2: Calculate the Price-Weighted Index for 2023

Sum of stock prices in 2023: $$380+165+720=1265$$

Thus, the Price-Weighted Index for 2023 is 1265.

Step 3: Calculate the Percentage Change

The percentage change is calculated as:

$$\text{Percentage Change}=\left(\frac{PW{I}_{2023}-PW{I}_{2022}}{PW{I}_{2022}}\right)\times 100$$

Substituting the values: $$\text{Percentage Change}=\left(\frac{1265-1250}{1250}\right)\times 100=\left(\frac{15}{1250}\right)\times 100=1.2\%$$

Final Answer: The Price-Weighted Index for 2022 is 1250, the Price-Weighted Index for 2023 is 1265, and the percentage change is 1.2%.

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b. Market Value Weighted Index (Market Capitalization Weighted Index)

The market value weighted index is calculated by summing the total market value of all stocks and dividing by a divisor. The divisor is typically the sum of the initial market values (2022 market values).

Step 1: Calculate the Market Value for 2022

The market value for each stock is calculated as: $$\text{Market Value}=\text{No. of Shares Outstanding}\times \text{Stock Price}$$

For 2022:

• Stock A: $2000\times 400=800,000$
• Stock B: $1000\times 150=150,000$
• Stock C: $5000\times 700=3,500,000$

Sum of market values in 2022: $$800,000+150,000+3,500,000=4,450,000$$

Thus, the Market Value Weighted Index for 2022 is 4,450,000.

Step 2: Calculate the Market Value for 2023

For 2023:

• Stock A: $2000\times 380=760,000$
• Stock B: $1000\times 165=165,000$
• Stock C: $5000\times 720=3,600,000$

Sum of market values in 2023: $$760,000+165,000+3,600,000=4,525,000$$

Thus, the Market Value Weighted Index for 2023 is 4,525,000.

Step 3: Calculate the Percentage Change

The percentage change is calculated as:

$$\text{Percentage Change}=\left(\frac{4,525,000-4,450,000}{4,450,000}\right)\times 100=\left(\frac{75,000}{4,450,000}\right)\times 100=1.69\%$$

Final Answer: The Market Value Weighted Index for 2022 is 4,450,000, the Market Value Weighted Index for 2023 is 4,525,000, and the percentage change is 1.69%.

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c. Equally-Weighted Index

The equally-weighted index assigns equal weight to each stock in the index.

Step 1: Calculate the Equally-Weighted Index for 2022

The formula for the equally-weighted index is:

$$EW{I}_{2022}=\frac{\sum {\text{Stock Prices}}_{2022}}{\text{Number of Stocks}}=\frac{400+150+700}{3}=\frac{1250}{3}=416.67$$

Thus, the Equally-Weighted Index for 2022 is 416.67.

Step 2: Calculate the Equally-Weighted Index for 2023

For 2023: $$EW{I}_{2023}=\frac{380+165+720}{3}=\frac{1265}{3}=421.67$$

Thus, the Equally-Weighted Index for 2023 is 421.67.

Step 3: Calculate the Percentage Change

The percentage change is calculated as:

$$\text{Percentage Change}=\left(\frac{421.67-416.67}{416.67}\right)\times 100=\left(\frac{5}{416.67}\right)\times 100=1.20\%$$

Final Answer: The Equally-Weighted Index for 2022 is 416.67, the Equally-Weighted Index for 2023 is 421.67, and the percentage change is 1.20%.

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d. Difference in the Results for Three Stock Indexes

The differences in the results of the three stock indexes can be summarized as follows:

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Figure: Comparison of Index Values

Figure: Percentage Change in Indexes

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Python Program to Compute Indexes

# Import necessary libraries
import pandas as pd

# Data provided
data = {
    'Stock': ['A', 'B', 'C'],
    'No_of_shares': [2000, 1000, 5000],
    'Price_2022': [400, 150, 700],
    'Price_2023': [380, 165, 720]
}

# Create a DataFrame
df = pd.DataFrame(data)

# Calculate Price-Weighted Index
price_weighted_2022 = df['Price_2022'].sum()
price_weighted_2023 = df['Price_2023'].sum()
percentage_change_pw = ((price_weighted_2023 - price_weighted_2022) / price_weighted_2022) * 100

# Calculate Market Value Weighted Index
df['Market_Value_2022'] = df['No_of_shares'] * df['Price_2022']
df['Market_Value_2023'] = df['No_of_shares'] * df['Price_2023']
market_value_weighted_2022 = df['Market_Value_2022'].sum()
market_value_weighted_2023 = df['Market_Value_2023'].sum()
percentage_change_mvw = ((market_value_weighted_2023 - market_value_weighted_2022) / market_value_weighted_2022) * 100

# Calculate Equally-Weighted Index
equally_weighted_2022 = df['Price_2022'].mean() * len(df['Price_2022'])
equally_weighted_2023 = df['Price_2023'].mean() * len(df['Price_2023'])
percentage_change_ew = ((equally_weighted_2023 - equally_weighted_2022) / equally_weighted_2022) * 100

# Print results
print("Price-Weighted Index:")
print(f"2022: {price_weighted_2022}, 2023: {price_weighted_2023}, Percentage Change: {percentage_change_pw:.2f}%")

print("\nMarket Value Weighted Index:")
print(f"2022: {market_value_weighted_2022}, 2023: {market_value_weighted_2023}, Percentage Change: {percentage_change_mvw:.2f}%")

print("\nEqually-Weighted Index:")
print(f"2022: {equally_weighted_2022:.2f}, 2023: {equally_weighted_2023:.2f}, Percentage Change: {percentage_change_ew:.2f}%")

How it works: This Python script uses the pandas library to create a DataFrame from the given data. It calculates the price-weighted, market value-weighted, and equally-weighted indexes for 2022 and 2023, then computes the percentage change for each index type. The script outputs the index values and their respective percentage changes.