(a) Since APPL and MSFT account for the entire large cap technology component of S&P 500 and their proportion of index is based on their weights, this large cap technology sector is 5% (3% + 2%) or 0.05 of the total portfolio.
(b) The beta of the large cap technology sector can be determined as follows.
Portfolio beta = ∑W_{i}β_{i}
In the current case, the portfolio is large cap technology sector where AAPL occupies 3/5 weight and MSFT occupies 2/5 weight of the portfolio. With the given betas,
Portfolio beta = (3/5)*1.04 + (2/5)*1.05 = 0.624 + 0.42 = 1.044
The beta of this large cap technology sector is 1.044.
(c) The beta of the market portfolio is generally 1. Since the S&P 500 is the market portfolio, its beta is 1.
(d) The weight or fraction of the remaining stocks in S&P is 1 - 5% = 95%. With the formula in (b), the combined beta of these two stocks is
Beta = 0.03*1.04 + 0.02*1.05 = 0.0522
Portfolio beta = 1
Hence beta of remaining stocks = 1 - 0.0522 = 0.9478
The beta of the remaining stocks in the S&P is 0.9478.