Question

A Portfolio consists of seven investment products. The expected return of each investment, in million GPB, is normally distributed as follows: Investment I ~ N(70, 16); Investment II ~ N(40, 25); Investment III ~ N(60, 9); Investment IV ~ N(20, 4); Investment V ~ N(20, 16); Investment VI ~ N(30, 9); Investment VI ~ N(10, 4); The returns from the seven investments are independent.

Find the distribution of the total Portfolio return. Report the mean, the variance and the standard deviation. (40%)

If the total return exceeds 260 million GPB, a bonus will be given. What is the probability that this bonus will be given? (30%)

If the total return is less than 230 million GPB, the client will look for other firms to handle his money in the future. What is the probability that the firm will keep this customer? (30%)

Answer 1

Sum of Independent normal distributions is a normal distribution with expected return and variance as follows

Portfolio returns =Sum of returns = 70+40+60+20+20+30+10 = 250

Portfolio Variance = Sun of Variances = 16+25+9+4+16+9+4 = 83

Portfolio Standard Deviation = ?83 = 9.11

Standardizing Prob (X > 260) = P((260-250)/(9.11)) = P(Z>1.097) = (1 -P(Z<1.097))

= 1 - 0.8640 = 0.136 = 13.6%

Probability of Bonus to be given is 13.6%

Standardizing probability

P(X< (230-250)/(9.11)) = P(z < -2.195) = 0.141 = 14.1%

Probability that client will look for other firms = 14.1%

Probability that firm will keep this customer = 1-14.1% = 85.9%