MTH 4500: Introductory Financial Mathematics
# Midterm 1 (Practice 1)

**Problem 1**
**Problem 2**
**Problem 3**
**Problem 4**
**Problem 5**
**Problem 6**
Assume that there are \(n\) risky securities.
It is known that among all portfolios whose expected return is \(3\), the minimal risk has the portfolio \(V_1\). Also, among all portfolios with expected return \(10\), the minimal risk has the portfolio \(V_2\). Assume that the covariance matrix of the returns \(K_{V_1}\) and \(K_{V_2}\) of the portfolios is
\[C=\left[\begin{array}{cc}2&1\newline 1&1\end{array}\right].\] Determine the maximal possible return for a portfolio whose risk is \(3\).

Find the rate of continuous compounding equivalent to daily compounding of \(90\%\), if we assume that a year has \(365\) days.

The three year bond has the face value USD 100, and pays coupons annually according to the following schedule: In the first year, the bond pays a coupon of USD \(5\), and in the second year it pays USD \(10\). At the maturity, the owner of the bond receives only the face value and no additional coupon payments are issued. Assume that the continuously compounding rate is \(9\%\). Find the price of this bond.

Compute the risk measured by the standard deviations \(\sigma_{K}\) for the risky security whose return depends on the market scenario:

\[ \begin{array}{|c|c|c|}\hline \mbox{Scenario} & \mbox{Probability}& \mbox{Return }K \newline \hline \omega_1& 0.3& 2\%\newline \hline \omega_2&0.7 & 22\%\newline \hline\end{array} \]

Three risky securities have expected returns \(\mu_1=10\%\), \(\mu_2=20\%\), \(\mu_3=9\%\), standard deviations \(\sigma_1=0.15\), \(\sigma_2=0.25\), \(\sigma_3=0.25\), and correlations \(\rho_{12}=0.4\), \(\rho_{23}=0.1\), and \(\rho_{31}=-0.3\).

**(a)**Compute the expected return \(\mu_V\) and the risk \(\sigma_V\) of the portfolio with weights \(w_1=-17\%\), \(w_2=35\%\), \(w_3=82\%\).**(b)**Find the weights of the portfolio of the minimal risk that can be constructed using the given three securities. What is the expected return and risk of this minimal risk portfolio?

Suppose that there are several risky securities and that their efficient frontier is given by \(\sigma=\sqrt{0.29-4\mu+20\mu^2}\).

**(a)**What is the expected return and risk of the minimal variance portfolio?**(b)**Add a risk-less asset with return \(R=7\%\). Find the expected return and risk of the market portfolio.