2017 October 27,

Math 265: Day 20

Carleton College, Prof. Joshua R. Davis

Due at the start of class on Day 22

Complete these problems. Write them up carefully, in the order assigned, for handing in with the rest of your homework.

This first set of exercises concerns arbitrary continuous random variables X and Y. You will prove that E[X Y] ≤ sqrt(E[X²] E[Y²]). I recommend that you follow these steps.

• Explain why E[(X - λ Y)²] ≥ 0 for any number λ.
• Explain how E[(X - λ Y)²] is a degree-2 polynomial in λ.
• If a degree-2 polynomial is always non-negative, then what does that say about its discriminant (the thing under the square root in the quadratic formula)?
• Finish the proof.

These next exercises are related to the exercises before and after them.

• For arbitrary continuous random variables X and Y, show that Cov(X, Y) ≤ SD[X] SD[Y].
• What does this inequality say about Corr(X, Y)?

In our investment application from class, we computed E[R] and V[R], where R = w₁ R₁ + w₂ R₂ + ... + w_n R_n. For simplicity we assumed that R₁, R₂, ..., R_n are uncorrelated.

• What happens to V[R] if we do not assume that the R_i are uncorrelated? Does V[R] increase or decrease? Or does it depend on the details of the assets in question?
• Building on our example with n = 3, illustrate your answer to the preceding exercise with specific numbers.