Question

Part 1: 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 portfolio of 5,10,20,50, and 100 securities?

Part 2: In part 1, how many securities need to be held before the risk of a portfolio is only 10% more than minimum?

Answer 1

Part 1

expected variance of an equally weighted portfolio = (1/no. of securities)*average variance + [(no. of securities - 1)/no. of securities]*average co-variance

expected variance of an equally weighted portfolio of 5 securities = (1/5)*50 + [(5 - 1)/5]*10 = 0.2*50 + 0.8*10 = 10 + 8 = 18

expected variance of an equally weighted portfolio of 10 securities = (1/10)*50 + [(10 - 1)/10]*10 = 0.1*50 + 0.9*10 = 5 + 9 = 14

expected variance of an equally weighted portfolio of 20 securities = (1/20)*50 + [(20 - 1)/20]*10 = 0.05*50 + 0.95*10 = 2.5 + 9.5 = 12

expected variance of an equally weighted portfolio of 50 securities = (1/50)*50 + [(50 - 1)/50]*10 = 0.02*50 + 0.98*10 = 1 + 9.8 = 10.8

expected variance of an equally weighted portfolio of 100 securities = (1/100)*50 + [(100 - 1)/100]*10 = 0.01*50 + 0.99*10 = 0.5 + 9.9 = 10.4

Part 2

Risk of portfolio 10% more than minimum = average co-variance*(1+10%) = 10*(1+0.10) = 10*1.10 = 11

11 = (1/no. of securities)*average variance + [(no. of securities - 1)/no. of securities]*average co-variance

11 = (1/no. of securities)*50 + [(no. of securities - 1)/no. of securities]*10

11*no. of securities = 50 + (no. of securities - 1)*10

11*no. of securities = 50 + 10*no. of securities - 10

11*no. of securities - 10*no. of securities = 50 - 10

1*no. of securities = 40

no. of securities = 40/1 = 40