Question

An equally weighted portfolio consists of 25 assets which all have a standard deviation of 0.219. The average covariance between the assets is 0.082. 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

For a n asset portfolio, var = W1^2*Var1 + W2^2*Var2 + ..... + WN^2*VarN + 2*W1*W2*Cov(1,2) + ...... (all the possible covariance combinations)

For the first part, W1^2*Var1 + W2^2*Var2 + .... WN^2*VarN

W1=W2=....=WN = 100/25 = 4% (since it is an equally weighted portfolio)

Also, Var1=Var2=.....=VarN = 0.219^2 = 0.04796 (given in question, SD is same for all)

So the first part becomes = 25 * (0.04^2*0.04796) = 0.19184% (since all the terms are same, we can compute one and just multiply it by 25)

For the second part, the covariance terms,

since there are 25 assets and we can have cov between 2 assets at a time, 25C2 = 300 such terms will occur

2*W1*W2*Cov(1,2) --> here only the Cov part will differ for each asset. So, this can be written as,

2*W^2*[cov(1,2) + cov(1,3)+ ....+ cov(24,25)]

= 2* 0.04^2 * (0.082*25) = 0.6560% [Note: Sum of Cov was needed, Avg was given; so SUM = AVG*N]

So, portfolio variance = adding the two parts = 0.19184% + 0.6560% = 0.8478%

SD of portfolio = SQRT(var) = 9.208%