Assume an investor’s universe consists of three stocks, Stock 1, 2 and 3. The return of each stock is denoted as &#11990

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Assume an investor’s universe consists of three stocks, Stock 1,
2 and 3. The return of each stock is denoted as 𝑟𝑖 where 𝑖 ∈ 1, 2,
3. The weight of each stock in the market portfolio is denoted as
𝑤𝑖 . The standard deviation of each stock is 𝜎𝑖 and lastly, the
covariance between two stocks is given by 𝜎𝑖,𝑗 . Let 𝒘 be a 3 x 1
matrix of weights and Σ be a 3 x 3 variance-covariance matrix.
a) Show that the variance of the market portfolio 𝜎𝑀 2 = 𝒘′𝚺𝒘 is
given by the expression below. 𝜎𝑀 2 = 𝑤1 2𝜎1 2 + 𝑤2 2𝜎2 2 + 𝑤3 2𝜎3
2 + 2(𝑤1𝑤2𝜎1,2 + 𝑤1𝑤3𝜎1,3 + 𝑤2𝑤3𝜎2,3)
b) Confirm that 𝑟𝑀 = 𝒘′𝒓 = Σ𝑤𝑖𝑟𝑖 = 𝑤1𝑟1 + 𝑤2𝑟2 + 𝑤3𝑟3. Also note
that the covariance between the return of asset 𝑖 and the market
(which consists of these three assets) is given by 𝐶𝑜𝑣(𝑟𝑖 , 𝑟𝑀) =
𝜎𝑖,𝑀 = 𝐶𝑜𝑣(𝑟𝑖 , 𝑤1𝑟1 + 𝑤2𝑟2 + 𝑤3𝑟3) Using the above show that the
market variance 𝜎𝑀 2 = Σ𝑤𝑖𝜎𝑖,𝑀
c) What is the relationship between 𝜎𝑖,𝑀 and 𝜎𝑀 2 ? Can we think
of the ratio 𝜎𝑖,𝑀 𝜎𝑀 2 as the contribution of a stock to the risk
of the market portfolio?