Currently you own a portfolio comprised of the following three securities. How much of the riskiest security should you sell and replace with risk-free securities if you want your portfolio beta to equal 90 percent of the market beta?
Stock Value/Beta
A 16,400/1.06
B 20,500/1.32
C 18,200/0.98
I don't know how to set this problem up. To find the portfolio Beta, you have to muliply by the weights. But if take money from B to put in T-Bills, the weights change. Is this just a trial and error question?
To solve this question, we need to calculate the weight of each security in the portfolio and then adjust the weights accordingly to achieve the desired portfolio beta.
Let's start by calculating the initial weights of the securities in the portfolio:
Total Portfolio Value = Value of A + Value of B + Value of C
Total Portfolio Value = $16,400 + $20,500 + $18,200
Total Portfolio Value = $55,100
Weight of Security A = Value of A / Total Portfolio Value
Weight of Security A = $16,400 / $55,100
Weight of Security A ≈ 0.2971 (rounded to four decimal places)
Weight of Security B = Value of B / Total Portfolio Value
Weight of Security B = $20,500 / $55,100
Weight of Security B ≈ 0.3719 (rounded to four decimal places)
Weight of Security C = Value of C / Total Portfolio Value
Weight of Security C = $18,200 / $55,100
Weight of Security C ≈ 0.3310 (rounded to four decimal places)
Now, let's assume that we sell a portion of Security B and replace it with risk-free securities (such as T-Bills). Let's call the weight we sell x (where 0 ≤ x ≤ 1).
After selling a portion of Security B, the adjusted weights become:
Weight of Security A = 0.2971 (unchanged)
Weight of Security B = 0.3719 - x (decreased by x)
Weight of Security C = 0.3310 (unchanged)
Now, we want the portfolio beta to equal 90 percent of the market beta. Let's denote the market beta as β_market and the desired portfolio beta as β_desired.
β_desired = 0.9 * β_market
Using the formula for portfolio beta:
β_portfolio = (Weight of A * β_A) + (Weight of B * β_B) + (Weight of C * β_C)
Given the beta values for each security:
β_A = 1.06, β_B = 1.32, β_C = 0.98
Substituting these values into the formula, we have:
0.9 * β_market = (0.2971 * 1.06) + ((0.3719 - x) * 1.32) + (0.3310 * 0.98)
Now, you can solve this equation for x. Start by simplifying the right-hand side of the equation:
0.9 * β_market = (0.314926 + (0.491208 - 1.32x) + 0.32338)
Combine like terms:
0.9 * β_market = 1.129514 - 1.32x
Rearrange the equation to isolate x:
1.32x = 1.129514 - 0.9 * β_market
x = (1.129514 - 0.9 * β_market) / 1.32
Now you can substitute the market beta (β_market) given in the problem, and calculate the value of x.