Final answer:
To find the new portfolio beta after the stock exchange, subtract the beta contribution of the sold coal mining stock (0.05) from the initial portfolio beta (1.12), then add the beta contribution of the bought mineral rights company stock (0.12). The new portfolio beta will be 1.19.
Step-by-step explanation:
To calculate the new beta of the portfolio after selling the coal mining stock and purchasing the mineral rights company stock, you can use the portfolio beta formula. The initial portfolio beta is 1.12 for a $100,000 portfolio ($5,000 in each of 20 stocks). When you sell the coal mining stock with a beta of 1.00 and purchase the mineral rights company stock with a beta of 2.4, you are exchanging one stock beta for another while keeping the total investment the same.
The new portfolio beta can be calculated as follows:
- Total Portfolio Beta = Initial Portfolio Beta - Contribution of Sold Stock + Contribution of Bought Stock
- Contribution of Sold Stock = Weight of Sold Stock * Beta of Sold Stock
- Contribution of Bought Stock = Weight of Bought Stock * Beta of Bought Stock
The weight of each stock being 1/20 since you have 20 stocks of equal value ($5,000 each in a $100,000 portfolio).
So, the Contribution of Sold Stock = $5,000/$100,000 * 1.00 = 0.05 * 1.00 = 0.05
The Contribution of Bought Stock = $5,000/$100,000 * 2.4 = 0.05 * 2.4 = 0.12
Therefore, the new portfolio beta = 1.12 - 0.05 + 0.12 = 1.19.