Question

Assume N securities. The expected returns on all the securities are equal to 0.01 and the variances of their returns are all equal to 0.01. The covariances of the returns between two securities are all equal to 0.005.

Answer 1

Number of securities in a portfolio = N

Expected returns on all the securities E_i = 0.01

Variances of their returns =0.01

i. Covariances of the returns between two securities = 0.005

Expected return of the portfolio is E(R) = w1R1 + w2Rq + ...+ wn Rn

E(R) =
`( (1)/(N)*0.01 + (1)/(N)*0.01 + .... + (1)/(N)*0.01 ) * N`

E(R) = 0.01

Expected return of N asset portfolio E(R) = 0.01

Variance of N asset portfolio
`\sigma ^2 = ( \sum_(k=1)^(n)(r_k- E(r))^2)/(n-1)`

where,

k is the specific return of the asset,

E(r) is the expected return.

ii. As N gets large, the portfolio's diversified risk rapidly decreases.

iii. The portfolio is well-diversified if the assets in a well-diversified portfolio exhibit a negative correlation or less correlation to each other.