Question

Assume second quarter investment returns for a particular 2011 portfolio can be described by the Normal model N(0.067, 0.028).

   a. What is the probability that returns are between 1% and 3.5%?

   b. At least what percentage of returns is needed to attain the top 5% of all possible returns?

Answer 1

Solution :

Given that ,

mean = = 0.067

standard deviation = = 0.028

a) P(0.01 < x < 0.035) = P[(0.01 - 0.067)/ 0.028) < (x - ) /  < (0.035 - 0.067) / 0.028 ) ]

= P(-2.04 < z < -1.14)

= P(z < -1.14) - P(z < -2.04)

Using z table,

= 0.1271 - 0.0207

= 0.1064

b) Using standard normal table,

P(Z > z) = 5%

= 1 - P(Z < z) = 0.05  

= P(Z < z) = 1 - 0.05

= P(Z < z ) = 0.95

= P(Z < 1.645 ) = 0.95  

z = 1.645

Using z-score formula,

x = z * +

x = 1.645 * 0.028 + 0.067

x = 0.113

x = 11.3%