Question

An equally weighted portfolio consists of 22 assets which all have a standard deviation of 0.154. The average covariance between the assets is 0.046. Compute the standard deviation of this portfolio. Please enter your answer as a percentage to three decimal places (i.e. 12.345% rather than 0.12345 -- the percent sign is optional).

Answer 1

No. of stocks in the portfolio = 22

weights of all the stocks in the portfolio are equal (w), therefore,

w₁ = w₂ = w₃ =..............w₂₂ = 1/22

σ₁ = σ₂ = σ₃ = ........=σ₂₂ = 0.154

standard deviation of the stocks is σ = 0.154

average covariance between the assets = Cov(i,j) = 0.046

The formula to calculate variance for n stocks is given by:

σ_p² = ΣΣw_i*w_j*Cov(i,j)

where Cov(i,j) is the pairwise Covariance between stock i and stock j

Now, for a portfolio of n equal-weightage stocks with equal standard deviation (average standard deviation) and the same pairwise correlation (avg correlation) between different pairs.

There will be total of n variance terms and nC2 covariance or correlation terms terms (pairs)

We can calculate the variance of the portfolio of n stocks with equal weights using the below formula:

σ_p² = n*σ²*w²+2*[n*[(n-1)/2]*w²*Cov(i,j)= 22*0.154²*(1/22)² + [22*21*(1/22)²*0.046] = 0.001078+0.0439090909090909 = 0.0449870909090909

Therefore, estimated standard deviation = (0.0449870909090909)^1/2 = 0.212101605154442

Answer -> The estimated standard deviation of the portfolio is 21.2101605154442%