You hold a diversified $100,000 portfolio consisting of 20 stocks with $5,000 invested in each. The portfolio's beta is 1.12. You plan to sell a stock with b = 0.90 and use the proceeds to buy a new stock with b = 1.80. What will the portfolio's new beta be?
To find the portfolio's new beta after selling a stock with a beta of 0.90 and buying a stock with a beta of 1.80, you need to calculate the weighted average of the betas.
Step 1: Calculate the total value of the portfolio before the transaction.
Since you have $5,000 invested in each of the 20 stocks, the total value of the portfolio is $5,000 x 20 = $100,000.
Step 2: Calculate the total beta of the portfolio before the transaction.
The portfolio's beta is already given as 1.12.
Step 3: Determine the value of the stock you are selling.
Since you are selling one stock with a value of $5,000, the total value of the stock being sold is $5,000.
Step 4: Determine the beta of the stock you are selling.
The beta of the stock being sold is given as 0.90.
Step 5: Determine the value of the stock you are buying.
Since you are using the proceeds from selling the stock to buy the new stock, the value of the stock being bought is also $5,000.
Step 6: Determine the beta of the new stock you are buying.
The beta of the new stock is given as 1.80.
Step 7: Calculate the new total value of the portfolio after the transaction.
Since you sell one stock and buy another with the same value, the total value of the portfolio remains $100,000.
Step 8: Calculate the new total beta of the portfolio.
To calculate the new beta, you need to subtract the beta of the stock being sold and add the beta of the new stock, both scaled by their respective weights.
The formula to calculate the weighted average beta is:
New Beta = (Total Beta Before - Beta of Stock Being Sold + Beta of New Stock) / Total Value of Portfolio
In this case, the formula would be:
New Beta = (1.12 x $100,000 - 0.90 x $5,000 + 1.80 x $5,000) / $100,000
Simplifying this equation gives:
New Beta = (112,000 - 4,500 + 9,000) / 100,000
New Beta = 116,500 / 100,000
New Beta = 1.165
Therefore, the portfolio's new beta after selling a stock with a beta of 0.90 and buying a stock with a beta of 1.80 will be approximately 1.165.