To solve the given problem, we will go through each part step by step. For price-weighted indices, the index is calculated as the average of the stock prices, divided by a divisor. The divisor is adjusted when there is a stock split, as it impacts the stock prices but not the overall value.

a. Divisor for the price-weighted index in year 2

When stock splits occur, the divisor must be adjusted. In this case, there is a stock split for stock B at time t = 2. Specifically, it is a 4-for-1 stock split (since the shares outstanding increase from 300 to 1200, which is 4 times the original).

We will compute the divisor for time t = 2:

1. The total price before the split (at t = 1) is calculated as: $\text{Total price at t=1} = P_A + P_B + P_C = 85 + 70 + 67 = 222$

2. Adjust stock B’s price for the split at t = 2. After the 4-for-1 split, B’s price becomes: $P_B' = \frac{35}{4} = 8.75$

3. The total price after the split at t = 2 is: $\text{Total price at t=2} = 90 + 8.75 + 70 = 168.75$

4. Since the total value should remain unchanged after the split, the divisor is adjusted as: $\text{Divisor} = \frac{\text{Total price at t=2 without split}}{\text{Total price after split}} = \frac{222}{168.75} \approx 1.315$

So, the divisor for the price-weighted index in year 2 is approximately 1.315.

b. Price-weighted index at t = 1

The price-weighted index is the sum of the prices at time t = 1 divided by the divisor (which is 1.0 before any adjustments):

$\text{Index at t = 1} = \frac{85 + 70 + 67}{1} = \frac{222}{1} = 74$

The price-weighted index at t = 1 is 74.

c. Price-weighted index at t = 2

At t = 2, using the adjusted divisor (calculated in part a) and the stock prices at time t = 2:

$\text{Index at t = 2} = \frac{90 + 8.75 + 70}{1.315} = \frac{168.75}{1.315} \approx 128.35$

Thus, the price-weighted index at t = 2 is 128.35.

Summary:

• Divisor at t = 2: 1.315
• Price-weighted index at t = 1: 74
• Price-weighted index at t = 2: 128.35

Would you like any further clarification on this problem or any of the calculations?

Here are 5 questions for further exploration:

1. How does a stock split impact the divisor of a price-weighted index?
2. Why do we adjust the divisor in a price-weighted index after stock splits?
3. How would the index value change if stock B had not split at time t = 2?
4. What are the differences between a price-weighted and a market capitalization-weighted index?
5. How does the number of shares outstanding affect the calculation of a price-weighted index?

Tip: Always double-check stock splits when calculating a price-weighted index to ensure accurate divisor adjustments!