Suppose the expected returns on assets $A$ and $B$ are $7\%$ and $5\%$ and their volatilities ($\sigma_{i}$) are $40\%$ and $20\%$. What are the expected return and risk of an equally weighed portfolio of $A$ and $B$ if the correlation is $0.5$?

Write your answer as two numeric values separated by a space. Write percentages out, i.e., $7\%$ should be entered as $0.07$.

What is the Lagrangian for the optimization problem

$\begin{array}{rl} \mbox{minimize:} & \mbox{s1^2 * w1^2 + 2 * rho * s1 * s2 * w1 * w2 + s2^2 * w2^2} \\ \mbox{subject to:} & \mbox{w1 + w2 - 1 = 0} \\ & \mbox{mu1 * w1 + mu2 * w2 - mu = 0} \end{array}$

Use $\mbox{l1}$ and $\mbox{l2}$ for the multipliers.

Use Lagrange's method to solve the following optimization problem.

$\begin{array}{rl} \mbox{minimize:} & x + y \\ \mbox{subject to:} & x^{2} + y^{2} - 1 = 0 \end{array}$

The answer must be expressed exactly.

What is the Taylor polynomial of order 3 for the function $f(x) = log(1 + x)$ around the point $a = 0$?

Find $k$ such that the function

$\displaystyle f(x) = \frac{x^{5} - 2x^{2} + x -1}{x^{2} + 2x + 5} = O(x^{k}) \quad$as $\quad x \rightarrow \infty$

What is the Taylor polynomial of order 2 for the function

$f(x, y) = e^{x + y}$

around the point $a = (0, 0)$?

What is the radius of convergence of the Taylor series expansion of the function $f(x) = e^{x}$?