Question

Q4. Suppose you develop a mutual fund that includes 500 NASDAQ stocks, all with equal weights in the fund's portfolio. The average return standard deviation of the stocks is 44 percent, and the average pairwise correlation among the stocks is 0.30. What is your estimate of the standard deviation of the fund's portfolio?

Answer 1

No. of stocks in the portfolio = 500

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

w₁ = w₂ = w₃ =..............w₅₀₀ = 1/500

σ₁ = σ₂ = σ₃ = ........=σ₅₀₀ = 0.44

average standard deviation of the stocks is σ = 0.44

and pairwise correlation among the stocks ρ = 0.30

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

σ_p² = ΣΣw_i*w_j*σ_i*σ_j*ρ_i,j

where ρ_i,j is the pairwise correlation 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]*ρ*w²*σ²= 500*0.44²*(1/500)² + [500*499*0.30*(1/500)²*0.44²] = 0.0003872+0.05796384 = 0.05835104

Therefore, estimated standard deviation = (0.05835104)^1/2 = 0.24155959927107

Answer -> The estimated standard deviatin of the fund's portfolio is 24.156%