A broker has calculated the expected values of two different financial instruments X and Y. Suppose that E (x) = $100, E (y) = $90 SD (x) = 90$ and SD (y) = $8. Find each of the following.

a. E (X + 10) and SD (X + 10)

b. E (5Y) and SD (5Y)

c) E (X + Y) and SD (X + Y)

d) What assumption must you make in part c?

Expectation is linear, meaning

E (a X + b Y) = E (a X) + E (b Y)

= a E (X) + b E (Y)

If X = 1 and Y = 0, we see that the expectation of a constant, E (a), is equal to the constant, a.

Use this property to compute the expectations:

E (X + 10) = E (X) + E (10) = $110

E (5Y) = 5 E (Y) = $450

E (X + Y) = E (X) + E (Y) = $190

Variance has a similar property:

V (a X + b Y) = V (a X) + V (b Y) + Cov (X, Y)

= a^2 V (X) + b^2 V (Y) + Cov (X, Y)

where "Cov" denotes covariance, defined by

E[ (X - E (X)) (Y - E (Y)) ] = E (X Y) - E (X) E (Y)

Without knowing the expectation of X Y, we can't determine the covariance and thus variance of the expression a X + b Y.

However, if X and Y are independent, then E (X Y) = E (X) E (Y), which makes the covariance vanish, so that

V (a X + b Y) = a^2 V (X) + b^2 V (Y)

and this is the assumption we have to make to find the standard deviations (which is the square root of the variance).

Also, variance is defined as

V (X) = E[ (X - E (X)) ^2] = E (X^2) - E (X) ^2

and it follows from this that, if X is a constant, say a, then

V (a) = E (a^2) - E (a) ^2 = a^2 - a^2 = 0

Use this property, and the assumption of independence, to compute the variances, and hence the standard deviations:

V (X + 10) = V (X) = => SD (X + 10) = SD (X) = $90

V (5Y) = 5^2 V (Y) = 25 V (Y) = => SD (5Y) = 5 SD (Y) = $40

V (X + Y) = V (X) + V (Y) = => SD (X + Y) = √[SD (X) ^2 + SD (Y) ^2] = √8164 ≈ $90.35