Question

Assume that the average variance of return for an individual security is 50 and that the average covariance is 10. What is the expected variance of an equally weighted port- folio of 5, 10, 20, 50, and 100 securities?

Answer 1

The average variance of return for an individual security = 50

The average covariance = 10

Expected variance of an equally weighted portfolio of 5; where n = 5

Expected variance = (1/n)* (average variance) + {(n^2 –n)/n^2} *(average covariance)

= (1/5) * 50 + {(5^2 -5)/5^2}*10

= 10 + {20/25} * 10

= 18

Expected variance of an equally weighted portfolio of 5 is 18

Expected variance of an equally weighted portfolio of 10; where n = 10

Expected variance = (1/n)* (average variance) + {(n^2 –n)/n^2} *(average covariance)

= (1/10) * 50 + {(10^2 -10)/10^2}*10

= 5 + {90/100} * 10

= 14

Expected variance of an equally weighted portfolio of 10 is 14

Expected variance of an equally weighted portfolio of 20; where n = 20

Expected variance = (1/n)* (average variance) + {(n^2 –n)/n^2} *(average covariance)

= (1/20) * 50 + {(20^2 -20)/20^2}*10

= 2.5 + {380/400} * 10

= 12

Expected variance of an equally weighted portfolio of 20 is 12

Expected variance of an equally weighted portfolio of 50; where n = 50

Expected variance = (1/n)* (average variance) + {(n^2 –n)/n^2} *(average covariance)

= (1/50) * 50 + {(50^2 -50)/50^2}*10

= 1 + {2450/2500} * 10

= 10.8

Expected variance of an equally weighted portfolio of 50 is 10.8

Expected variance of an equally weighted portfolio of 100; where n = 100

Expected variance = (1/n)* (average variance) + {(n^2 –n)/n^2} *(average covariance)

= (1/100) * 50 + {(100^2 -100)/100^2}*10

= 0.5 + {9900/10000} * 10

= 10.40

Expected variance of an equally weighted portfolio of 100 is 10.40