@David Harper CFA FRM will be able to answer your question regarding the formula that is being used, but I want to clarify that you are referring to the following question, which is located on Page 17 of the Bodie Study Notes under the End of Chapter Q&A section. It is always helpful if you include the full question and answer in your forum post, especially during this very busy time before the exam. This not only helps you to receive an answer more quickly, but also allows other members to use the search function to find your thread if they have the same questions.
End of Chapter Question 8(b)
Assume that security returns are generated by the single-index model, R(i) = α(i) + β(i)*R(M) + e(i), where Ri is the excess return for security i and RM is the market’s excess return. The risk-free rate is 2%. Suppose also that there are three securities A, B, and C, characterized by the following data:
b) Now assume that there are an infinite number of assets with return characteristics identical to those of A, B, and C, respectively. If one forms a well-diversified portfolio of type A securities, what will be the mean and variance of the portfolio’s excess returns? What about portfolios composed only of type B or C stocks?
If there are an infinite number of assets with identical characteristics, then a well-diversified portfolio of each type will have only systematic risk since the non-systematic risk will approach zero with large n:
If you have ONLY Type A securities in your well diversified (this means that the idiosyncratic term σ^2(e) equals zero) portfolio then your portfolio variance equals:
σ^2(Type A Securities) = beta^2*σ^2(m) which yields >>> 0.8^2*0.2^2 = 0.0256
σ^2(Type B Securities) = beta^2*σ^2(m) which yields >>>1.0^2*0.2^2 = 0.04
and so forth....
The key information to remember:
If we have an n-equally weighted portfolio (in our case e.g. Type A Securities consisting only ofA securities, or Type B securities consisting only of B securities) each with weight (1/n) the portfolio variance equals: σ^2(p) = 1/n*σ^2(i) + (n-1)/n*σ^2(i)
1. σ^2(i) = average variance of all assets in the portfolio 2. cov = average covariance of all pairings in the portfolio cov(a1,a2,a3,a4,a5....)
The equally weighted portfolio variance equals the sum of two components:
1. unsystematic risk (idiosyncratic risk) which is the variance term, σ^2(i)
2. systematic risk which is the covariance term, cov
Both components are affected by portfolio size. When the idiosyncratic term gets large (towards N, or infinity), the idiosyncratic risk gets to zero (because 1/n approaches zero the more N, no. of securities, you have in the portfolio) as you diversify away all the idiosyncratic risk of the individual Type A securities in the portfolio.
The term (n-1)/n*cov gets closer to the average covariance as n gets large because (n-1)/n approaches 1.