Question

Suffolk Investment Club has two funds: the fixed income fund and the variable income fund. Let X (Y) denote the annual return of the fixed income fund (the variable income fund). X fits a Normal distribution with mean 7% and standard deviation 2%, Y fits a Normal distribution with mean 13% and standard deviation 8%. Correlation between X and Y is -0.4. An ISOM 201 student has invested 30% of his money in the fixed income fund and the remaining 70% of his money in the variable income fund.

What is the probability that the annual return of the student's investment portfolio will be between 10% and 15%? [up to two decimals]

Answer 1

Let W be the annual return for of the student's investment portfolio.

W = 0.3X + 0.7Y.

E[W] = E[0.3X + 0.7Y] = 0.3E[X] + 0.7E[Y] = 0.3 * 7 + 0.7 * 13 = 11.2%

Var[W] = Var[0.3X + 0.7Y] = 0.3² Var[X] + 0.7² Var[Y] + 2 * 0.3 * 0.7 * Cov[X, Y]

0.3² Var[X] + 0.7² Var[Y] + 2 * 0.3 * 0.7 * Corr[X, Y] * SD[X] * SD[Y]

= 0.09 * 2² + 0.49 * 8² + 0.42 * -0.4 * 2 * 8

= 0.36 + 31.36 - 2.688

= 29.032

SD[W] = = 5.388%

Thus, W ~ N(11.2 , 5.388)

Probability that the annual return of the student's investment portfolio will be between 10% and 15%

= P(10 < W < 15)

= P(W < 15) - P(W < 10)

= P[Z < (15 - 11.2)/5.388] - P[Z < (10 - 11.2)/5.388]

= P[Z < 0.7053] - P[Z < -0.2227]

= 0.7597 - 0.4119

= 0.3478