MTH 4500: Introductory Financial Mathematics
# Midterm 1 (Practice 3)

**Problem 1**
**Problem 2**
**Problem 3**
**Problem 4**
**Problem 5**
**Problem 6**

The US dollar (USD) is a home currency for dealer \( A \), while euro is the home currency for dealer \( B \). The dealers \( A \) and \( B \) use the following rates for currency exchange: \[ \begin{array}{|c|l|l|}\hline \mbox{dealer }A & \mbox{buy}&\mbox{sell}\newline \hline \mbox{EUR } 1& \mbox{USD }4& \mbox{USD } 7\newline \hline \end{array} \quad\quad \begin{array}{|c|l|l|}\hline \mbox{dealer }B & \mbox{buy}&\mbox{sell}\newline \hline \mbox{USD } 1& \mbox{EUR }2& \mbox{EUR }9\newline \hline \end{array}. \] Prove that there is an opportunity for risk-free profit. You are allowed to assume that each currency can be borrowed without interest.

It is known that the interest is compounded continuously and that an investment of \( P \) dollars today will result in \( Q \) dollars in one year. What is the annual rate of continuously compounded interest in terms of \( P \) and \( Q \)?

Consider a bond with face value \( F \) that matures in \( 21 \) years and that pays coupons in the amount \( C \) at the end of years \( 2 \) and \( 5 \). What is the formula for the price of this bond if the continuous compounding rate is \( r \)?

Assume that there are two securities \( S_1 \) and \( S_2 \) whose initial prices at time \( 0 \) are \( S_1(0)=60 \) and \( S_2(0)=100 \). The prices at time \( 1 \) depend on the market scenario in the following way: \[ \begin{array}{|c|c|c|c|}\hline \mbox{Scenario} & \mbox{Probability}& S_1(1) & S_2(1) \newline \hline \omega_1& 0.3& 66 & 170\newline \hline \omega_2&0.7 & 42 & 70\newline \hline\end{array} \]

**(a)**Determine the covariance matrix between the returns of the given two securities.**(b)**Determine the portfolio of the minimal variance that can be constructed using these two securities.

Assume that there are three risky securities each of which has the risk equal to \( 1 \). Assume that the covariance between each two of the given securities is non-negative. If the risk of the portfolio with weights \( \overrightarrow w=\left[\begin{array}{ccc} 20\% & 60\% & 20\%\end{array}\right] \) is equal to \( \frac{\sqrt{44}}{10} \), determine the covariance matrix between the returns of the given three securities.

Assume that the expected returns of \( n \) given securities are \( m=\left[\begin{array}{cccc}\mu_1&\mu_2&\dots&\mu_n\end{array}\right] \). Let \( C \) be the covariance matrix between the returns and assume that the sum of all components of the vector \( mC^{-1} \) is equal to \( 18 \). For each vector of weights \( w=[\begin{array}{cccc}w_1&w_2&\dots&w_n\end{array}] \) denote by \( \mu(w) \) and \( \sigma(w) \) the expected return and the risk of the portfolio whose weights are \( w_1 \), \( \dots \), \( w_n \). Find the vector of weights \( w \) that minimizes the function \( F(w)=\sigma^2(w) + 15\mu(w) \). We are assuming that all securities are risky (i.e. that all variances are non-zero) and that their covariance matrix is invertible.