In this problem, we will go through the steps of managing an active portfolio. We will provide a set of instructions on how to construct the portfolio.

With a passive strategy, we believe that we cannot identify which assets are better than others, after accounting for their risk as measured by $.$ With an active strategy, we believe that we are able to identify assets that have an abnormal return, that is$,$ an $.$

When we want to add a stock to any actively managed portfolio, there is a trade$-$off. On the one hand, we love to get the extra return from the $,$ that is$,$ the abnormal return of the stock for which we do not have to take market risk. On the other hand, we add firm$-$specific risk to the portfolio by exposing ourselves to individual companies.

This trade$-$off is measured through the information ratio. A stock with a large information ratio has a large $$ or small firm$-$specific risk, or both. Using this line of thinking, there is a clear rule of deciding the weights in an active portfolio.

In the following, we are using the following notation: The market portfolio has the index $M,$ the existing active portfolio has the index $A,$ and the stock you want to add to this portfolio has the index $S.$

You are given the following information about the excess return and standard deviation of the market portfolio:

$E[R_M]=0\mathrm{.}09,{}_M=0\mathrm{.}12.$

You are given the following information about the existing active portfolio, which was estimated using a single$-$factor model:

${}_A=0\mathrm{.}044,{}_A=1\mathrm{.}25,{}^2(e_A)=0\mathrm{.}04$

The stock we wish to add has the parameters:

${}_S=0\mathrm{.}12,{}_S=1\mathrm{.}05,{}^2(e_S)=0\mathrm{.}2$

a$)$ Find the systematic and non$-$systematic components of the variance of Stock S$.$

Using the rules for building an active portfolio, we find that we should build a new active portfolio $N$ which consists of $\frac{5}{8}$ of A and $\frac{3}{8}$ of $S.$

b$)$ Find the $,$ and variance from the firm$-$specific risk of the new active portfolio $N.$ That means, find ${}_N,{}_N$ and ${}^2(e_N).$