MTH 4500: Introductory Financial Mathematics
# Midterm 1 (Practice 2)

**Problem 1**
**Problem 2**
**Problem 3**
**Problem 4**
**Problem 5**
**Problem 6**
Given \(n\) risky securities, assume that their expected returns are \(m=[\begin{array}{cccc}\mu_1&\mu_2&\dots& \mu_n\end{array}]\) and that the covariance matrix between their returns is \(C\). For each vector of weights \(w=[\begin{array}{c}w_1&w_2&\dots&w_n\end{array}]\) denote by \(\mu(w)\) the return and by \(\sigma(w)\) the risk of the portfolio whose weights are \(w_1\), \(\dots\), \(w_n\). Find the maximum of the function \(F(w)=\frac{\mu(w)}{\left(\sigma(w)\right)^2}\). We are assuming that all securities are risky (i.e. that all variances are non-zero) and that their covariance matrix is invertible.

**(a)**Assume that the discrete interest is compounded annually and that the rate is \( 30\% \). What is the value of the investment of USD \( 100 \) that is withdrawn after one year?**(b)**Assume that the discrete interest is compounded annually and that the rate is \(r=10\%\). What is the value of the investment of USD \(100\) that is withdrawn after the period of \(3\) years?

The three year bond has the face value USD 270, and pays coupons annually in the amount of USD 54. The last coupon is paid at maturity. The interest rate is \(50\%\) and the interest is simple and compounded annually. Determine the price of the bond.

Assume that there are two risky securities whose returns \( K_1 \) and \( K_2 \) depend on the market scenario in the following way: \[ \begin{array}{|c|c|c|c|}\hline \mbox{Scenario} & \mbox{Probability}& \mbox{Return }K_1 &\mbox{Return }K_2 \newline \hline \omega_1& 0.8& 20\% & 10\%\newline \hline \omega_2&0.2 & -80\% & 60\%\newline \hline\end{array} \]

**(a)**Determine the expected returns and the risks of the two given securities.**(b)**Determine the covariance matrix between the returns of the given two securities.

Assume that there are four risky securities whose expected returns and covariance matrix are given by \[ m=\left[\begin{array}{cccc} 20\%&30\%&40\%&20\%\end{array}\right]\quad\quad\quad \mbox{and}\quad\quad\quad C=\left[\begin{array}{cccc}2 & 3&0&0\newline 3&5&0&0\newline 0&0&1&2\newline 0&0&2&5\end{array}\right].\] Assume that the rate of risk free return is \( R=30\% \). Determine the weights of the market portfolio.

Suppose that there are four risky securities \(S_1\), \(S_2\), \(S_3\), \(S_4\) and suppose that among all portfolios whose expected return is \(5\) the minimal risk has the portfolio with the weights \((10\%, 20\%, 30\%, 40\%)\). Suppose that among all portfolios with expected return \(10\) the minimal risk has the portfolio with weights \((15\%, 25\%, 35\%, 25\%)\). Among all portfolios with expected return \(8\), find the one with the minimal risk.