Question

A) Consider a portfolio of cancer research projects, and assume for simplicity that each project's financial return has a standard deviation of 400%. If the correlation coefficient between each pair of projects is 0.5, what is the standard deviation of an equal-weighted portfolio of: (Note: Your answer should be a number in percentage form. Do not enter '%'.)

10 projects? ____% 100 projects ___% 1000 projects ____%

B) Now assume the projects are statistically independent and therefore uncorrelated. What is the standard deviation of an equal-weighted portfolio of: (Note: Your answer should be a number in percentage form. Do not enter '%'.)

10 projects? ____% 100 projects ___% 1000 projects ____%

Answer 1

Solution:

A)

Portfolio risk of many related assets = Sqrt { (sigma^2 / n) + ( (n-1) / n ) * Corr * sigma^2 }

Where sigma is the std deviation = 400% or 4 , Corr is the correlation between the assets =.5 , n is the number of assets

Sigma for a portfolio of 10 projects = Sqrt ( (4^2 / 10 ) + (9/10) * .5 * 4^2) = Sqrt { (16/10) + (9/10)*.5*16 } = Sqrt (1.6 + 7.2) = Sqrt (8.8 ) = 2.966

Sigma for a portfolio of 100 projects = Sqrt ( (4^2 / 100 ) + (99/1000) * .5 * 4^2) = Sqrt { (16/100) + (99/100)*.5*16 } = Sqrt (.16 + 7.92) = Sqrt (8.08) = 2.842

Sigma for a portfolio of 1000 projects = Sqrt ( (4^2 / 1000 ) + (999/1000) * .5 * 4^2) = Sqrt { (16/1000) + (999/1000)*.5*16 } = Sqrt (.016 + 7.992) = Sqrt (8.152 ) = 2.855

B) If the projects are uncorelated the Corr = 0 and formula for the portfolio risk becomes

Portfolio risk = Sqrt(Sigma^2 / N)

Sigma for a portfolio of 10 projects = Sqrt ( 4^2 / 10 ) = Sqrt (16/10) = Sqrt (1.6 ) = 1.264

Sigma for a portfolio of 100 projects = Sqrt ( 4^2 / 100 ) = Sqrt (16/100) = Sqrt (.16 ) = .4

Sigma for a portfolio of 1000 projects = Sqrt ( 4^2 / 1000 ) = Sqrt (16/1000) = Sqrt (.016 ) = .1264