You hold a diversified portfolio consisting of a $5,000 investment in
each of 20 different common stocks. The portfolio beta is equal to
1.12. You have decided to sell a lead mining stock (b = 1.0) at $5,000
net and use the proceeds to buy a like amount of a steel company stock
(b = 2.0). What is the new beta of the portfolio?

Hi trae!!

Beta of portfolio = sum of weighted betas

For 20 stocks, each having a market value of $5000, the portfolio is
worth $100,000 and each stock has the same weight of 1/20 = 0.05 = 5%

Beta of portfolio = SUM(Xi*Bi) =               for  i = 1 to 20
                  = SUM(0.05*Bi) =  
                  = 0.05 * SUM(Bi)  


Assume that the sold stock is the one with subindex 1, then B1 = 1 and:

Beta of portfolio = 0.05*B1 + SUM(0.05*Bi) =     i = 2 to 20
                  = 0.05*1.0 + SUM(0.05*Bi) =
                  = 0.05 + SUM(0.05*Bi) = 1.12
Then:
SUM(0.05*Bi) = 1.12 - 0.05 = 1.07           i = 2 to 20

Finally we have that:
Beta of portfolio = 0.05*B1 + SUM(0.05*Bi) =     i = 2 to 20
                  = 0.05*B1 + 1.07

If we replace the first stock in the portfolio with one of the same
value ($5000) but with  beta B = 2.0, then:

New Beta of portfolio = 0.05*B1 + SUM(0.05*Bi) =     i = 2 to 20
                      = 0.05*2.0 + 1.07 =
                      = 0.10 +1.07 =
                      = 1.17

So the so beta of new portfolio is 1.17.

I hope that this helps.

Regards.
livioflores-ga

There's another easier way to calculate your new beta in this
question. This works because all your stocks were worth $5000 each and
you sold away a stock, also worth $5000.

Original total beta = 1.12 X 20
                    = 22.4
Subtracting the beta of 1.0 (sold away), and adding the beta of 2.0 (new stock), 
new total beta = 22.4 - 1.0 + 2.0
               = 23.4
Therefore, average beta for new portfolia
               = 23.4 / 20
               = 1.17

Do remember that this method only works when you have all the stocks of equal value.