Question

Suppose you have invested 25% of your portfolio in four different stocks. The mean and standard deviation of the annual return on each stock are shown in the table below. The correlations between the annual returns on the four stocks are also shown in this file.

Distributions of returns (assumed normal)
	Mean	Stdev
Stock 1	15%	20%
Stock 2	10%	12%
Stock 3	25%	40%
Stock 4	16%	20%
Correlation matrix
	Stock 1	Stock 2	Stock 3	Stock 4
Stock 1	1.00
Stock 2	0.80	1.00
Stock 3	0.70	0.75	1.00
Stock 4	0.60	0.55	0.65	1.00

a. What is the probability that your portfolio's annual return will exceed 20%?
  %

b. What is the probability that your portfolio will lose money during the year?
  %

Answer 1

Distributions of returns (assumed normal)
	Mean	Stdev	Variance
Stock 1	0.15	0.2	0.04
Stock 2	0.1	0.12	0.0144
Stock 3	0.25	0.4	0.16
Stock 4	0.16	0.2	0.04	Weight square

expected return

=0.25*(0.15+0.1+0.25+0.16) = 0.165

Mean = 16.5%

standard deviation = σ²(port) = ΣΣw(i)w(j)σ(i)σ(j)ρ(i,j)

Std dev = 20.2%

a) What is the probability that your portfolio's annual return will exceed 20%?

Given that the standard deviation is 20.2%,

P(X>20) = 1-P((20-16.5)/20.2) = 1-P(0.173) = 1-0.5687 = 0.43122 = 43.12%

b) What is the probability that your portfolio will lose money during the year?

P(X<0) = P((0-16.5)/20.2<0) = 20.70%